BOOKSELLERS. * Web- ?tb. i I YOKOHAMA ENGINEERING LIBRARY A HANDBOOK OF ELECTRICAL TESTING BY H. R. KEMPE, TECHNICAL OFFICER, POSTAL TELEGRAPHS ; MEMBER OP THE COUNCIL OF THE SOCIETY OP TELEGRAPH-ENGINEERS AKD ELECTRICIANS ; ASSOCIATE MEMBER OP THE INSTITUTION OF CIVIL ENGINEERS. FOURTH EDITION. (ADOPTED BY THE POSTAL TELEGRAPH DEPARTMENT.) E. & F. N. SPON, 125, STEAND, LONDON. NEW YOEK : 35, MUEKAY STEEET. 1887. GIFT OF c LIBRARY "" ::' : .. : NOTE. Is the present Edition I have not only taken advantage, as far as possible, of the many friendly suggestions which have been made to me (especially by Dr. A. Muirhead) for the improve- ment of the original work, but have added a considerable amount of new matter, besides thoroughly revising the old. I have more particularly to thank Messrs. Elliott Brothers, Messrs. Latimer Clark, Muirhead and Co., Mr. B. Pell (of Messrs. Johnson and Phillips's), and Mr. P. Jolin, for the illustrations of apparatus which have been added. I am also greatly indebted to Messrs. W. T. Glover and Co. for per- mission to insert their Table (III.) of the Weights, Resist- ances, &c., of Pure Copper Wire, and to Mr. Herbert Taylor for the Table (IX.) of Data of recent Submarine Cables. H. R. K. ENGINEER-IN-CHIEF'S OFFICE, ELECTRICIAN'S DEPARTMENT, GENERAL POST OFFICE. London, July 1887. 869173 CONTENTS. CHAPTER I. PAG* SIMPLE TESTING 1 CHAPTER II. RESISTANCE COILS 10 CHAPTER III. GALVANOMETERS 18 CHAPTER IV. SHUNTS 67 CHAPTER V. MEASUREMENT OF GALVANOMETER RESISTANCE 79 CHAPTER YI. MEASUREMENT OP THE INTERNAL RESISTANCE OF BATTERIES .. .. 113 CHAPTER VII. MEASUREMENT OF THE ELECTROMOTIVE FORCE OF BATTERIES 137 VI CONTENTS. CHAPTER VIII. PAGE THE WHEATSTONE BBIDGE 188 CHAPTER IX. LOCALISATION OP FAULTS 242 CHAPTER X. KEYS, SWITCHES, CONDENSERS, AND BATTERIES 270 CHAPTER XL MEASUREMENT OF POTENTIALS 284 CHAPTER XII. MEASUREMENT OF CURRENT STRENGTH 301 CHAPTER XIII. MEASUREMENT OF ELECTROSTATIC CAPACITY 325 CHAPTER XIV. THE THOMSON QUADRANT ELECTROMETER 348 CHAPTER XV. MEASUREMENT OF HIGH RESISTANCES 364 CHAPTER XVI. MEASUREMENT OF RESISTANCES BY POTENTIALS CONTENTS. Vll CHAPTER XVII. PAGE LOCALISATION OF FAULTS BY FALL OF POTENTIALS 386 CHAPTER XVIII. TESTS DURING THE LAYING OF A CABLE .. CHAPTER XIX. JOINT-TESTING 402 CHAPTER XX. SPECIFIC MEASUREMENTS 408 CHAPTER XXI. CORRECTIONS FOB TEMPERATURE .. .. 414 CHAPTER XXII. LOCALISATION OF FAULTS OF HIGH RESISTANCE 428 CHAPTER XXIH. LOCALISATION OF A DISCONNECTION FAULT IN A CABLE 439 CHAPTER XXIV. A METHOD OF LOCALISING EARTH FAULTS IN CABLES 447 CHAPTER XXV. GALVANOMETER RESISTANCE 457 Viii CONTENTS. CHAPTEE XXVI. PAGB SPECIFICATION TOR MANUFACTURE OF CABLE. SYSTEM OF TESTING CABLE DURING MANUFACTURE 461 CHAPTER XXVII. MISCELLANEOUS 490 TABLES. L NATURAL TANGENTS 526 II. BESISTANCE OF A KNOT-POUND OF COPPER WIRE OF VARIOUS CONDUCTIYITIES, AT 75 FAHR 528 HI. THE RELATIVE DIMENSIONS, LENGTHS, RESISTANCES (AT 60 FAHR.), AND WEIGHTS, OF PURE COPPER WIRE 528 IV. COEFFICIENTS FOR CORRECTING THE OBSERVED KESISTANCE OF PURE COPPER WIRE AT ANT TEMPERATURE TO 75 FAHR., OR AT 75 TO ANY TEMPERATURE 529 V. COEFFICIENTS FOR CORRECTING THE OBSERVED KESISTANCE OF ORDINARY COPPER WIRE AT ANY TEMPERATURE TO 75 FAHR., OR AT 75 TO ANY TEMPERATURE 530 VI. COEFFICIENTS FOR CORRECTING THE OBSERVED KESISTANCE OF " SlLVERTOWN " GUTTA PERCHA AT ANY TEMPERATURE TO 75 FAHR 531 VII. COEFFICIENTS FOR CORRECTING THE OBSERVED RESISTANCE OF " WlLLOUGHBY SMITH'S " GUTTA PERCHA AT ANY TEMPERATURE TO 75 FAHR 532 VIII. MULTIPLYING POWER OF SHUNTS EMPLOYED WITH A GALVANOMETER OF 6000 OHMS RESISTANCE 533 IX. MULTIPLYING POWER OF SHUNTS EMPLOYED WITH A GALVANOMETER OF 10,000 OHMS RESISTANCE 534 X. STANDARD WIRE GAUGE 535 XI. ELECTRICAL AND MECHANICAL DATA OF RECENT SUBMARINE CABLES . 536 INDEX 537 A HANDBOOK OF CHAPTER I. SIMPLE TESTING. 1. IN order to be able to make measurements of any kind, it necessary to have certain standard units with which to make comparisons. For example, in the case of length, or weight, we have as standards the foot and the pound. Some of the units are dependent upon two of the other units ; the unit of " work," for example, is the foot-pound, or the work done in raising a pound 1 foot high. Now in electrical measurements we require units of a like character. Those with which we have to deal chiefly are electromotive force, the unit of which is called the volt ; resistance, the unit of which is the ohm ; also we have the unit of current, which FIG. 1. is dependent upon the two foregoing units, and which is called the ampere. 2. If the two poles of a battery be joined by a conductor a current will flow, and the strength of this current will vary directly as the electromotive force of the battery, and inversely as the total resistance in the circuit. This relation is known as " Ohm's law." If the electromotive force is expressed in volts and the resistance in ohms, then the resulting current will be in amperes. 3. Suppose now a battery of a resistance r and electromotive force E, a galvanometer of a resistance G, and a wire of a resistance K, be joined up in circuit, as shown by Fig. 1. By B 2 HANDBOOK OF ELECTRICAL TESTING. the foregoing law, the strength of current C, which will flow out of the battery and through the galvanometer, will be c- E -R+r+G The current, in flowing through the galvanometer, produces a deflection of its aeedle t .which deflection will remain constant providedjtbe elect^n>Qfciy6> force of the battery and also the resistances remain constant. < If now K be a wire whose re- sist a tf^e" "fet xgqtiiTG l , o fiixd, aiid which we can replace by another wire the value of whose resistance can be varied at pleasure, then by adjusting this latter so that the deflection of the galvanometer needle becomes the same as it was before the change of resistances was made, this resistance gives the value of our unknown resistance E. This method of testing, known as the substitution method, although exceedingly simple, is a very good and accurate one if a little ordinary care be taken in making it. Its correctness is only limited by the sensibility of the galvanometer to small changes of strength in the current affecting it, and by the accuracy with which the variable resistance can be adjusted. It should be mentioned, however, that for reasons which will become obvious when the subject of testing is gone further into, the resistance of the battery and galvanometer used in making a test of the kind should be small compared with the resistance being measured. 4. Next, suppose the galvanometer to have its scale so graduated that the number of divisions on it will, by the de- flection of the needle, accurately represent the comparative strength (C) of currents which may pass through it. Let the battery, galvanometer, and resistance be joined up as at first, then, as before, = E+7TG ; or ' E = (R +r + G )' Now remove E, and insert any other known resistance p, in its place. Calling the new strength of current, d, then But we have seen that E = C (E + r -f G), therefore SIMPLE TESTING. 3 that is K = ^(p + r + G)-(r + G). [1] Now, as we have supposed the deflections of the galvanometer needle to be directly proportional to the strengths of current which produce them, we may, instead of C and C x , write in our formula the deflections of the galvanometer needle which those strengths produce. Calling, then, a the deflection obtained with the strength C, and a x that with the strength Ci, .our formula [1] becomes R = ^ (p + r + G) - (r + G). [2] In order to find R, it is necessary to know G, which is usually marked on the galvanometer by the manufacturer, r also must be known, but as it is difficult to determine its value accurately, it is best to use a battery whose resistance is very small in comparison with the other resistances in the circuit, and which may consequently be neglected ; in this case we may write our formula R = ^ (p + G) - G. [3] Having then obtained a with R and a x with p, we can find the value of R. For example. With a galvanometer whose resistance was 100 ohms, and a battery whose resistance could be neglected, we obtained with a resistance R a deflection of 20 divisions (a), and with a re- sistance of 200 ohms (p) a deflection of 30 divisions (a,). What was the unknown resistance R ? 30 R = - (200 + 100) - 100 = 350 ohms. A\J 5. Next, suppose it is required to find the resistance of a galvanometer. From equation [3], by multiplying up, we find that R a = p a x + G aj G a, by arranging the quantities G c^ G a = R a p B 2 4 HANDBOOK OF ELECTKICAL TESTING. or G (aj a) = R a p a 1? therefore G = Ea-p,. a x a If, then, with a known resistance B, we obtain a deflection of a divisions, and with a known resistance p we obtain a deflection of ttj divisions, we can determine G. l For example. With a galvanometer (G) and a battery whose resistance could be neglected, we obtained with a resistance of 350 ohms (R) a deflection of 20 divisions (a), and with a resistance of 200 ohms (p) a deflection of 30 divisions (04). What was the resistance of the galvanometer ? 350 x 20 - 200 X 30 G = ~ 30-20 - 6. Lastly, when the resistance of our battery is considerable, and it is required to find its value, from equation [2] by multi- plying up, we find R a = p a! + *" i + G a, r a G a, by arranging the quantities r > 3, ,, 12 ,, j 4, O The resistance of our wires would then be No. 1, 20000 5 = 4000 ohms. 2, 20000 6 = 3333 3, 20000 12 = 1666 4, 20000 3 = 6666 These results are the total insulation resistances of the wires, which may be of various lengths. To get comparative results, it is necessary to obtain the insulation resistance of some unit length of each wire, such as a mile. Now, it will be readily seen that the greater the length of the wire the greater will be the leakage, and consequently the less will be the insulation resistance, or, in other words, this resistance will vary inversely as the length of the wire. To obtain, then, the insulation resistance, or insulation as it is simply called, all we have to do is to multiply the total insulation by the length of wire. Thus, for example, if No. 1 wire was 100 miles long, its insulation per mile would be 4000 X 100 = 400,000 ohms. It is usual to fix a standard insulation per mile, and if the result is below that standard, the line is considered faulty. 200,000 ohms per mile is the standard adopted by the Postal Telegraph Department. 10. The rule of multiplying the total insulation by the mileage of the wire to get the insulation per mile is not strictly correct, more especially for long lines, as it assumes that the leakage is the same at every point along the line. This, how- SIMPLE TESTING. ever, is clearly not the case, as a little of tlie current leaking out at one point leaves a smaller quantity to leak out at the next. In fact, we really measure the last portion of the line with a weaker battery than we do the first. The true law is, however, somewhat complex, and will be considered hereafter.* 11. We have hitherto considered the galvanometer deflections to be directly proportional to the currents producing them, but in no galvanometer is this the case if the deflections are measured in degrees ; in such a case they are proportional to some function of those degees, such as the tangent. Thus, if we were reading oif the scale of degrees on a tangent galvanometer, that is to say a galvanometer in which the strengths of currents are directly proportional to the tangents of the angle of deflection which those currents produce, we should have to find the tangents of those degrees of deflection before multiplying and dividing. For example. If with a tangent galvanometer we obtained with our standard resistance of 1000 ohms a deflection of 20, and with the unknown resistance (E) a deflection of 15, we should have | R tan 20 x 1000 = '364 x 1000 tan 15 -268 t 1358 ohms. When measuring the insulation resistance of a line of tele- graph, having taken the constant, we should join up our instru- ments and line, as shown by Fig. 2. In making a measurement FIG. 2. Earth. ari of this kind, it is usual to have the positive pole of the battery to earth, so that a negative (zinc) current flows out to the line, as a zinc current will show best any defective insulation in the wire, a positive current having the effect, to a certain extent, of sealing a fault up, more especially if the defect is in any under- ground work which may be in the circuit. * See Appendix. 8 HANDBOOK OF ELECTRICAL TESTING. The foregoing method of measurement is, as a rule, sufficiently accurate for all practical purposes. Greater accuracy may, how- ever, "be obtained with but little extra trouble by allowing for the resistance of our battery and galvanometer in the following manner : Instead of multiplying the constant deflection by the 1000 ohms standard resistance, multiply it by 1000 plus the resist- ance of the galvanometer and battery, and having divided the result by the deflection obtained with the line wire in circuit, subtract from the result the resistance of the galvanometer and battery. For example. With a standard resistance of 1000 ohms, a tangent galvano- meter of a resistance of 50 ohms, and a battery of a resistance of 100 ohms, we obtained a deflection of 30, and with the line wire in circuit a deflection of 10. What was the exact insulation resistance of the line ? Insulation! resistance j tan 30 (1000 + 50 + 100) tan 10 577 X 1150 176 - (50 -f 100) - 150 = 3760 ohms. When a large number of wires have to be measured for insulation daily, it is very convenient to have a table con- structed on the following plan : EARTH READINGS. Constant Readings through 1000 ohms. 1 2 3 4 20 20852 10423 6945-0 5205-0 21 21992 10993 7324-6 5489-5 22 23146 11570 7709-3 5777-9 23 24318 12155 8099-5 6070-2 24 25507 12750 8495-5 6367-1 In this table the first vertical column represents the deflec- tions in degrees obtained with a tangent galvanometer through a standard resistance of 1000 ohms, and the top row of degrees SIMPLE TESTIN 7 G. 9 are the deflections obtained with the line wire in circuit. The numbers at the points of intersection of a vertical with a hori- zontal column give the resistances corresponding to those deflections, these resistances being calculated from the formula Insulation ") tan constant reading x 1000 resistance j " tan earth reading Thus the constant deflection, or reading, with the 1000 ohms standard resistance being 22, and the deflection with the line wire (the earth reading) being 2, the resistance required is seen at a glance to be 11,570 ohms. Before proceeding to the more intricate systems of measure- ment, we will consider some of the instruments which would be used in making measurements such as we have described. 10 HANDBOOK OF ELECTRICAL TESTING. CHAPTEK II. RESISTANCE COILS. 12. THE essential points of a good set of resistance coils are, that they should not vary their resistance appreciably through change of temperature, and that they should be accurately adjusted to the standard units, which adjustment ought to be such that not only should each individual coil test according to its marked value, but the total value of all the coils together should be equal to the numerical sum of their marked values. It will be frequently found in imperfectly adjusted coils that although each individual coil may test, as far as can be seen, correctly, yet when tested altogether their total value will be one or two units more or less than the sum of their individual values ; because although an error of a fraction of a unit may not be perceptible in testing each coil individually, yet the accumulated error may be comparatively large. The wire of the coils is either of platinum-silver alloy or of German silver ; the former material has the advantage that its resistance changes but very slightly by variation of temperature ; this variation not amounting to more than *031 per cent, per degree centigrade. Platinum-silver is, however, rather ex- pensive, and consequently, where the highest possible accuracy is not of great importance, German silver, whose percentage of resistance variation per degree centigrade is -044, is used. Recently a new metallic compound called platinoid, which is a combination of tungsten, copper, nickel, and zinc, has been dis- covered by Mr. F. W. Martino. This alloy, besides being very inexpensive, has a lower co-efficient of resistance variation by change of temperature than even platinum-silver, this percentage being as low as *021 per degree centigrade; it is therefore likely to come into extensive use. The wire is usually insulated by two coverings of silk, and is wound double on ebonite bobbins, the object of the double winding being to eliminate the extra current which would be induced in the coils if the wire were wound on single. By double winding, the current flows in two opposite directions on the bobbin, the portion in one direction eliminating the in- ductive effect of the portion in the other direction. When KESISTANCE COILS. 11 wound, the bobbins are saturated in hot paraffin wax, which thoroughly preserves their insulation, and prevents the silk covering from becoming damp, which would have the effect of partially short circuiting the coils and thereby reducing their resistance. The small resistances are made of thick wire, the higher ones of thin wire, to economise space. When bulk and weight are of no consequence, it is better to have all the coils made of thick wire, more especially if high battery power is used in testing, as there is less liability of the coils to become heated by the passage of a current though them. The individual resistances of a set of coils are generally of such values that, by properly combining them, any resistance from 1 to 10,000 can be obtained. One arrangement in general use has coils of the following values : 1, 2, 2, 5, 10, 10, 20, 50, 100, 100, 200, 500, 1000, 1000, 2000, 5000 ohms. These numbers enable any resistance from 1 to 10,000 to be obtained, using a minimum number of coils without fractional values. With these numbers, however, it is a matter of some little difficulty to see at once what coils it is necessary to put into circuit in order to obtain a particular resistance ; and as it is often necessary to be quick in changing the resistances, the following numbers are frequently used : 1, 2, 3, 4, 10, 20, 30, 40, 100, 200, 300, 400, 1000, 2000, 3000, 4000, which enables any particular resistance, that is required to be inserted, to be seen almost at a glance. The way in which the different coils are put in circuit is shown by Fig. 3. The ends of the several resistances, e, c, c, are connected between the brass blocks, 6, 6, 6, . Any of the FIG. 3. coils can then be cut out of the circuit between the first and last blocks, by inserting plugs, p, as shown, which short-circuit the coils between them ; thus, if all the plugs were inserted, there would be no resistance in circuit, and if all the plugs were out, all the coils would be in circuit. 12 HANDBOOK OF ELECTBICAL TESTING. 13. There are various ways of arranging the coils in sets ; one of the most common is that shown in outline by Fig. 4, and in general view by Fig. 5. This form is much used in sub- marine cable testing. The brass blocks, here shown in plan, are screwed down to a plate of ebonite which forms the top of the box in which the coils are enclosed. The ebonite bobbins are fixed to the lower surface of the ebonite top, the ends of the wires being fixed to the screws which secure the brass blocks. The holes shown in the middle of the brass blocks are con- venient for holding the plugs that are not in use. FIG. 4. FIG. 5. It will be seen that six terminals, A, B, C, D, E, F, are provided : when we only require to put a resistance in circuit, the two terminals D and E would be used. The use of the other ter- minals and of the movable brass strap S, will be explained hereafter. 14. In using a set of resistance coils, one or two precautions are necessary. First of all, it should be seen that the brass shanks of the plugs are clean and bright, as the insertion of a dirty plug will not entirely short-circuit the coil it is intended to cut out. It is a good plan, before commencing to test, to give the plugs a scrape with a piece of glass or emery paper, taking care to rub RESISTANCE COILS. 13 off any grains of grit which may remain sticking to them after this has been done. When a plug is inserted, it should not be simply pushed into the hole, but a twisting motion should be given it in doing so, that good contact may be ensured ; too much force should not be used, as the ebonite tops may thereby be twisted off in extracting the plugs. Care also should be taken that the neighbouring plugs are not loosened by the fingers catching them during the operation of shifting a plug. Before commencing work it is as well to give all the plugs a twist in the holes, so as to ensure that none of them are left loose. On no account must the plugs be greased to prevent their sticking, and their brass shanks should be touched as little as possible with the fingers. 15. For taking the insulation resistance of a line in the manner described in the last chapter, such an elaborate set of coils is not of course wanted. A single coil of a resistance of 1000 ohms in a box with two terminals, to which the ends of the coils are attached, is all that is required. 16. One of the most useful sets of coils for general purposes is represented in outline by Fig. 6, and in general view by Fig 7. The general arrange- ment of resistances, it will be seen, is the same as that shown by Fig. 4. Two keys, however, are provided (drawn u in Fig. 6 in elevation, for dis- tinctness). The contact point of the right-hand key is con- nected, as shown by the dotted line, with the middle brass block of the upper set of re- sistances, the terminal B' at the end of the key corresponding in fact, when the key is pressed down, with the terminal B shown in Fig. 4. In like manner the terminal A' corresponds with the terminal A. In the place of the movable strap of brass between A and D (Fig. 4), a plug marked INF. (infinity) is provided, which answers the same purpose ; an infinity plug is also placed at the first bend of the coils on the right hand of the figure. When we require simply to insert a resistance in a circuit, we should use the terminals A' and E, the left-hand key being pressed down when the deflection of the galvanometer needle is to be noted. The current can thus be conveniently cut off or FIG. 6. ,3 * 10 "40 30 20 | E B" 14: HANDBOOK OF ELECTRICAL TESTING. put on when required, by releasing or depressing the key. Care should be taken that the two infinity plugs are firmly in their FIG. 7. places, to ensure their making good contact. For the same purpose the key contacts should be occasionally touched with emery paper or a fine file. * Another set of coils, known as the " Dial " pattern, is repre- sented in general view by Fig. 8 ; these will be again referred FIG. 8. to hereafter (Chapter VIII). In this pattern (as will be seen from the Fig.) ten brass blocks are arranged radially around a central circular block. One disadvantage of the arrangement is that it is difficult to clean the surface of the ebonite on which the brass blocks are mounted ; in a somewhat similar pattern this disadvantage is got over by substituting a rectangular bar RESISTANCE COILS. 15 for the central circular block, and arranging five of the brass blocks in a row on one side and five on the other side of the same. By this arrangement a piece of rag can easily be passed between the blocks and the central bar, and the surface of the ebonite on which the blocks and bar are mounted be readily cleaned. SLIDE EESISTANCE COILS. 17. Fig. 9 shows the principle of this method of arranging ^Resistance Coils. The coils, which are generally all of equal value, are connected between brass blocks, as in Fig. 3, but instead of plugs being FIG. 9. inserted between the blocks to cut the various coils out of circuit, a sliding piece, B, is provided which can be moved along a rod with which it is in connection. The slider has a spring fixed to it which presses against the brass blocks ; it is evident, then, that any required resistance can be inserted between A and B, that is between A and a terminal fixed to the end of the rod, by simply sliding the piece B along the rod. The object of arranging the coils in this manner is more par- ticularly to enable the ratio of A B to B C to be varied, whilst the sum of the two, that is to say the whole length, A C, remains constant ; this is sometimes required to be done. These coils are sometimes set in a circle instead of a straight line, the contact-piece B being a spring forming a radius of the circle. This is a very compact and useful arrangement. 18. For some tests a long straight wire of German silver or other metallic compound is employed in the place of the resistance coils. It is important that this wire should be made of a perfectly uniform alloy, and should be of the same diameter throughout, so that its resistance may be directly proportional to its length ; thus, if the slider were at the middle point of the wire, the resistance on each side should be exactly the same. If, as is sometimes the case, it is required to use a long wire of this kind, it would be inconvenient to have it straight ; in such a case, therefore, the wire is wound spirally on a cylinder of ebonite or other insulating material, the two ends being con- nected to the metal axes, these latter being in connection with terminals. The sliding contact-piece is moved along parallel with the axes of the cylinder by a screw which gears with the 16 HANDBOOK OF ELECTRICAL TESTING. cylinder, and which is therefore revolved by the handle which turns the latter ; the contact of the slider with the wire is made when required by pressing the former with the finger. The arrangement, in fact, is a modified form of Jacobi's Rheostat. The Thomson- Jolin Eheostat. 19. This apparatus, which is shown by Fig. 10, has been recently devised by Sir William Thomson and Mr. P. Jolin, of Bristol, and is a valuable modification of the original rheostat of Wheatstone, the apparatus being entirely free from the defects which characterised the latter instrument. FIG. 10. In the new rheostat the wire is guided between the cylinders, so as to be laid on them spirally, by means of a travelling /nut on a long screw. The screw is turned by the handle, and carries a toothed wheel which gears into two other toothed wheels ; one of the latter turns one of the cylinders, and the other a loose shaft carrying the other cylinder ; a spring fixed to this shaft acts on the last-named cylinder which surrounds it on the principle of the main spring of a watch. By this arrangement the wire is kept tightly stretched and the barrels can be turned backwards or forwards without the wire becoming slack. The guiding nut is also arranged to stop the motion of the screw shaft at each end of the range, and so prevent the possibility of over-winding ; it also carries an index, which moves along a graduated scale and indicates the number of turns of wire on the insulating cylinder. The conducting cylinder and the wire are both of " platinoid," RESISTANCE COILS. 17 a metallic alloy which has been recently introduced, and which has properties which make it specially suitable for the purpose. It has very high electric resistance, very small temperature variation of resistance (as has previously been pointed out on p. 10), and it remains with its surface almost, if not altogether, untarnished in the air. On account of the last-name 1 property, the contact between the wire and the conduct ing-cylinder, is as perfect as can be desired; and continuity of action, which was a great difficulty in the old Wheatstone instrument, is (according to Sir William Thomson) absolutely complete. 20. It is evident that a much finer adjustment of resistance can be obtained by the slide wire than by the slide resistance coils, but inasmuch as the length of the wire and the smallness of its diameter must be limited, it does not admit of very large variations of resistance being obtained. By combining, how- ever, a slide-wire resistance with plug resistance coils, this difficulty can be got over, though in tests which we shall describe it is preferable to use the slide coils. 21. Slide resistance coils, though very convenient, are not absolutely n-cessary for varying the ratio of the resistances in the manner described ; for it is evident that A B and B C could be two sets of resistance coils in which, to adopt the slide resistance principle, the resistances would have to be increased in one set and diminished in the other, or vice versa, care being taken that the same resistance is added in one set as is taken out in the other. 18 HANDBOOK OF ELECTEICAL TESTING. CHAPTEK III. GALVANOMETEKS. 22. FOR the class of tests in which it is required, by adjusting certain resistances, either to bring the needle to zero, or to the same deflection in making two measurements, as described on pages 1 and 2, a galvanometer having its scale graduated to degrees would be sufficient. It should be provided with an astatic pair of needles suspended by a cocoon fibre, the end of the latter being attached to a piece of metal connected to a screw by the twisting of which the FIG. 11. needles can be lowered down on to the coil, so that there would be no danger of the fibre being fractured when the instrument has to be moved about. Such an instrument is shown by Fig. 11. When the galvanometer is to be used it should be placed on a firm table, and the screw connected to the fibre turned until the needles swing clear of the coil. The instrument should then be placed in such a position that the top needle stands as nearly as possible over the zero points. It should next be carefully levelled by means of the levelling screws attached to its base, until the metal axis which connects the two needles together is exactly in the centre of the hole in the scale-card through which it passes. The adjustment of the needles to zero is much facilitated in the instrument by making the coil movable about the centre of the scale-card by means of a simple handle attached direct to the coil. The final touch can thus be given without shaking the needles, which would render exact adjustment difficult. In some galvanometers there is a scale graduated to degrees attached to the coil, so that the angle through which it is turned can be seen if required. This scale is employed when using the instrument as a Sine galvanometer. GALVANOMETEBS. 19 THE SINE GALVANOMETER. 23. We before stated that the strengths of currents producing certain deflections are not directly proportional to those deflec- tions, but to some function of them, such as the tangent. In measuring strengths of currents by means of a sine galvanometer we proceed as follows : The needle is first adjusted to zero. The current whose strength is to be measured is then allowed to flow, and a deflection of the needle produced. The coil is now turned round ; this causes the needle to diverge still more with respect to the stand of the instrument, but the angle which it makes with the coil becomes less the farther the latter is turned, and finally a point is reached at which the needle is again parallel to the coil that is, its ends are again over the zero points on the scale-card. The reason of this is, that the deflective action of the coil on the needle is always the same, provided the current strength does not vary, but the farther the needle moves from the magnetic meridian, the greater becomes its tendency to return to that meridian, and finally when the needle becomes parallel to the coil, the deflective force of the latter exactly balances the reactive force of the earth's magnetism. The strength of the current which produces the deflection of the needle will then be directly proportional to the sine of the angle through which the coil has been turned. The sine galvanometer, though but rarely used, is a very accurate instrument, in so far that its results are entirely inde- pendent of the shape of the coil, size of the needle, &c. The only precaution necessary is to see that when the needle is at zero at starting it is brought back exactly to zero. Indeed it is not absolutely necessary that the starting point be zero the law of the sines holds good if the needle be at, say, 5 when com- mencing, but in this case, by the turning of the coil, the needle must be brought back to 5, and not to zero. THE TANGENT GALVANOMETER. 24. The tangent galvanometer, which is perhaps the most useful and convenient instrument for general purposes, consists essentially of coils of wire wound in a deep groove in the cir- cumference of a circular ring, a magnetic needle being placed at the centre of the latter over a graduated circle. The length of this needle must be small compared with the diameter of the coils so as to ensure, as far as possible, the magnetic influence of the current on the needle being the same at whatever angle the- o 2 20 HANDBOOK OF ELECTRICAL TESTING. needle may be with respect to the coil. Theoretically to effect this result, the magnet should be a mere point, but this is of course impossible, and practically it is sufficient for the coil to be eight or ten times as large in diameter as the length of the needle. Upon the influence of the coil on the needle being the same, whatever angle the needle takes up with respect to it, depends the truth of the proposition, that the strengths of cur- rents circulating in the coil are directly proportional to the tangents of the angles of deflection of the needles. For a 6 or 7 inch ring, a needle about three-quarters of an inch in length is a con- venient size, and gives sufficiently accurate results for all practical purposes. Tlie needle must be so placed that its central point is at the axis of the coils and also in the same plane with them. 25. The principle of the instrument is as follows : Let n s be the needle in its normal position, i. e. the position where it is parallel to the magnetic meridian and also parallel to the ring or coils. Let ^ s 1 be the position the needle takes up when deflected by FIG. 12. the action of the coils. Draw c d at right angles to w x Sj making c n equal to ^ d ; draw a c and d a, each at right angles to c d ; also draw w t a parallel to n s and w x ! at right angles to n s. Now the position which the needle takes up is due to the fact that the deflective notion of the coils and the directive force of the earth's magnetism when resolved at right angles to the needle are equal and oppo- site in effect. The first of these forces / 2 , acts at right angles to n s, and the second, /! acts parallel to n s ; then if a j and a l w, represent the forces /i and f. 2 respec- tively, c Wj and d n r will represent the resolved forces at right angles to n t j, which forces are equal since equilibrium is produced ; let their value be/. Now since a Wj is parallel to n o, and a c parallel to n { o, the an^le &-%=&' is equal to the angle o ; * also since Wj a l is perpendicular to n o, and M! d is perpendicular to n^ o the angle a 1 n^ d is equal to the angle a. We consequently have f =. / 2 cos a, and, f = f\ sin a, therefore / 2 cos a = /i sin a, * * Euclid,' book i., prop. 34. GALVANOMETERS. 21 or o 1 cos a but/! (the directive force of the earth's magnetism) is constant, therefore f 2 (the deflective force of the coils) is proportional to tan a, that is to say, the current strength, C, in the ring or coils is proportional to tan a, * or C = tan a X a constant. 26. Fig. 13 shows a form of tangent galvanometer which is used by the Postal Telegraph Department.f The mag- netic needle (which is j of an inch long) has a long pointer of gilt copper, about five inches long, fixed at right angles to it; when the needle is parallel to the coil, each end of this pointer is over the zero of a graduated scale. One of these scales is divided to true degrees, and the other to numbers pro- portional to the tangent of those degrees, so that if we read off two deflections from the degrees scale, the other extremity of the pointer will indicate, approximately, numbers proportional to the tangents of those two degrees of deflection. Now as the strengths of currents producing certain deflections are proportional to the tangents of the degrees of those deflec- tions, if we read off from the degrees scale we must, as we have explained in Chapter I. ( 11), reduce the degrees to tangents, from a table of tangents, J before working out a formula which has reference to the strengths of currents. If, however, we read * Professor J. P. Joule and Professor Jack point out in vol. vi. pp. 135, 147, and 151, of the ' Proceedings of the Manchester Literary and Philo- sophical Society,' that if the needle be of a considerable length, then if a be the angle of deflection, I the magnetic length of the needle (generally about | of the actual length), and d the magnetic diameter of the coil, the correction to be supplied to the tangent of the angle of deflection is \ (4 tan 2 o - 1) sin 2 a, which correction is additive at great deflections, and subtractive at small ones. At a certain deflection this correction vanishes, that is to say we have or < 4 tan 2 a - 1 = 0, tan o = J = tan 26 30'. The exact arrangement of this instrument is described in Chapter VII. Table I. 22 HANDBOOK OF ELECTRICAL TESTING. FIG. 13. off from the tangent scale, no reduction is necessary, and the numbers can be at once inserted in the formula. GALVANOMETERS. 23 To avoid parallax error, in consequence of the needle being elevated above the scale, a piece of looking-glass is fixed close to the tangent scale, so that when we look at the end of the needle and see that the reflected image of the pointer coincides with the pointer itself, we know that we are looking at the end of the pointer perpendicularly with the scale. As the instrument is generally only provided with a looking- glass near the tangent scale, it is necessary when reading off from the degrees scale to run the eye along the pointer to the looking-glass end and see whether the reflected image corre- sponds with the pointer at that end ; if it does, we may be sure that when we look at the degrees scale we do so correctly. 27. Before using the galvanometer it should be seen that the pointer has not become bent, but stands at right angles to the magnet, and that when suspended it turns freely. On no account should the magnet suspension be oiled, as quite the opposite effect to what is intended will be produced by so doing. Care should be taken that the scale is in its proper position, so that when the two ends of the pointer are over the zero points, the pointer stands at right angles to the coils. The correct setting of the position of the scale with reference to the coil is a mechanical adjustment which when once properly effected is not likely to alter ; it is, however, as well to verify its correctness by means of a set square before the instrument is brought into general use. The pointer attached to the magnetic needle is very subject to accident, and is often badly adjusted. The correctness of the setting can be verified by placing the galvanometer so that the pointer stands at zero, and then send- ing a current through the coil first in one direction and then in the other. The deflections on either side of zero in this case should be equal ; if they are not, the position of the pointer relative to the needle should be readjusted until the required equality of deflections on either side of zero is obtained. Care should be taken when making this adjustment to place the instrument clear of any unequally distributed masses of iron, otherwise unequal deflections may be obtained although the pointer and magnet are correctly set. Angle of Maximum Sensitiveness. 28. In using the tangent galvanometer it is always as well to avoid obtaining either very high or very low deflections. The reason of this is, that any small change in the strength of a current traversing the galvanometer will produce the greatest effect on the needle when the latter stands at some deflection 24: HANDBOOK OF ELECTRICAL TESTING. which is neither very high or very low. The galvanometer is, in fact, most sensitive when the needle points, under the influence of a current, at that deflection. Thus, for example, suppose we had a current which produced a deflection of 5, and this current was increased say by i^th, then the deflection would be increased to 5 30', because tan 5 : tan 5 30' : : 1 : 1 T V . Next suppose the needle stood at 80, and the current was, as before, increased by T ^th, then the deflection would be increased to 80 54', for tan 80 : tan 80 54' : : 1 : 1 T V . Lastly, let us suppose the needle stood at 43, then by the increase in the current the deflection would have changed to 45 43', for tan 43 : tan 45 43' : : 1 : 1 T V. In the first case, then, when the deflection was low, the increase was 5 30' -5= 30'; in the second case, when the deflection was high, 80 54' - 80 = 54'; and in the third case, when the deflection was of a medium, value, 45 43' - 43 = 2 43'. What, then, is the deflection at which this increase is greatest ? The point to be determined is, what deflection is increased most by any small alteration in the current producing that deflection ? If C be a current giving a deflection of a, and d a current a little stronger, say, which increases this deflection to (oj -f- 8), we have to find what value given to c^ , makes 8 as large as possible when C and C 4 are very nearly and ultimately equal. We have C : d :: tan ai : tin (a? + 8), therefore tan(a l +8) = ^tan ai . [A] Now we have to make 8 a maximum, supposing that the foregoing equation holds good. 4incL yali<-e of T\ tf * ^-^V v.. g^-^TM^X - >t^ C - c *^. GALVANOMETERS. 25 Since 8 is to be a maximum, tan 8 must also be a maximum. Now .._. tan a, -f- tan 8 C t tan (ttj + v) = 5~i 5~ = TT tan a x , therefore tan ai -f tan 8 = ~ tan ai (1 - tan o^ tan 8), therefore tan 8 ( 1 -f - tan 2 af) = tan ai (~ - l\ \ C / \U / therefore 0, C /C \ V ^^ J We have then to find what value of tan aj makes this fraction a maximum, and this we shall do by finding what value makes the denominator of the fraction a minimum. Now tan a, = tan ttl ^ C and this will be a minimum when that is, when 1 = tan 2 a,", or, tan a, = ; but as Cj and C are ultimately equal, becomes equal to 1, therefore tan ai = VT = 1 = tan 45. 29. We see then that in order to make the tangent galvano- meter as sensitive as possible we should obtain the deflection of its needle as near 45 as possible ; 45 is in fact the angle of maximum sensitiveness. Every galvanometer has an angle of maximum sensitiveness, 26 HANDBOOK OF ELECTRICAL TESTING. although it is not the same in all. The angle can, however, be found experimentally (see ' Calibration of Galvanometers,' p. 46), and should be marked on the instrument for future reference. 30. If we require to adjust two currents in two different measurements so that they should be equal in both cases, it is evident that the needle of the galvanometer employed to measure them should in each case show the same deflection. In making the two measurements, we take the deflection obtained by one current as the standard, and then in making the second measurement we adjust the current until the same deflection is obtained. Now the accuracy with which this. current can be adjusted depends upon the sensitiveness of the galvanometer to a change in the strength of the current, and we have seen that this sensitiveness is at a maximum when the deflection is 45. If, therefore, we employ a tangent galvano- meter for such a test as that just mentioned, we should endea- vour in both measurements to bring the needle to 45. 31. In what way can the property of the galvanometer be taken advantage of when comparing two deflections ? We must in such a case endeavour to obtain both deflections as near 45 as possible. To do this we should have to get one deflection on one side, and the other deflection on the other side, of 45. But then the question arises, should we get the deflections at an equal distance on either side, or one closer to the 45 than the other, and if so, should the higher or the lower deflection be the closer of the two ? Now a little consideration will make it clear that if the two deflections in question are taken either near or 90, they will be much closer together than if they were taken near 45, for the reason that the tangents of high or low deflections differ more widely from one another than do the tangents of medium deflections. But we have shown that when deflections are high or low, any increase or decrease in the strength of the current producing those deflections has less effect than when the de- flections are of a medium value. It is therefore evident that it is most advantageous to get the deflections as wide apart as possible. Let then tan represent the stronger, and tan < the weaker current, and let one current be times as strong as the other. We then have to find what values of and < make a maximum, supposing that tan = n tan rz rv? -. 7~yZ *rvv -sK;', ry '3 iuvf'/ GALVANOMETERS. 27 If in the last investigation we substitute < for 8, for a^, Q and n for , we can see that in order to get the required result C we must make and, since tan = n tan 0, tan0 =~= = V w" V n If one current strength is to be twice as great as the other, then n = 2 ; consequently, tan0 = V~2 = 1-41421 = tan 54 44' = tan 54J, and tan = = 70711 = tan 35 16 f = tan 35j. V 2 These then are the deflections that theoretically it is best to obtain in making a test with a tangent galvanometer in which one current is to be twice as strong as the other. But practi- cally we may make the deflections 55 and 35^, as these are more convenient to adjust to, and tan 55 is, within 1', exactly double tan 35^. If we examine the theoretical deflections 54 44' and 35 16' it will be seen that 54 44' - 45 = 9 44', and 45 - 35 16' = 9 44', or in other words, the angular deflections on either side of 45 are in this case the same. Let us then see whether they are so when n has any value other than 2. The angular deflection between 45 and will be - 45, that between 45 and <, 45 - <, 28 HANDBOOK OF ELECTEICAL TESTING. but we know, since tan = An and tan = -, that that is +*m fl tan >n <'-*)= tan that is tan (45 - 0) = tan (0 - 45), or 45 - = - 45 ; showing that these angular deflections are the same whatever be the value of n. This is a very useful fact, as it shows that when we are making a test in which two deflections are involved whose relative values are unknown, we should so adjust the resist- ances, &c., that the deflections are obtained, as near as possible, at equal distances on either side of 45. To sum up, then, we have Best Conditions for using the Tangent Galvanometer. 32. When a test is made in which only one deflection is con- cerned, then that deflection should be as near 45 as possible. If there are two deflections to be dealt with, then these should be as nearly as possible at equal distances on either side of 45. If one of these deflections is to be double the other, then 55 and are the most convenient to employ. 33. Although it is usual to take the readings on the tangent galvanometer, starting with the pointer at the ordinary zero, i. e. with the needle parallel to the plane of the ring or coils, yet it is not absolutely necessary that this arrangement should be adopted ; the instrument can be used when the needle in its normal position makes an angle with the plane of the ring. Under the latter conditions, however, the current strength will not be in direct proportion to the tangent of the angle of deflection. GALVANOMETERS. 29 FIG. 14. Let the dotted line A B, Fig. 14, represent the plane of the coils, and let n s be the needle in its normal position, i. e. in the plane of the magnetic meridian ; also let n v s : be the position which the needle takes up under the influence of the current. Let /3 be the angle which the needle makes normally with the coils, and let a -|- ft be the angle throiipjh which the needle turns when deflected to the position % s t . Draw c rtlat right angles to % s^ making c n^ qqual to ^ d ; draw c a and d ! each at right angles to c d ; also draw w x a parallel to n s, and ^ a at right angles to AB. Now since a ^ is parallel to n o, and a c parallel to n^ o, the angle c a n-^ is equal to the angle a -f /5 ; also since n { a,^ is perpendicular to A o, and j d is perpendicular to ^ o, the angle a : r^ d is equal to the angle a. We consequently have therefore or / = / 3 cos a, and / = /j sin (a + / 3 cos a = / sin (a -}- /?) , ' sin (a + /8) * /3 ^""SH 5 " ~- _ ,. ain a cos ff -f- sin j3 cos a ( ~ cos a [A] = /j (tan a cos y3 + sin =/! cos /3 (tan a -f tan ^) . so that cos /3 being a constant quantity, the strength of a current is directly proportional to (tan a -f- tan j3) which is the reading on the tangent scale ( 26) if the figures on the latter * If the angle j8 had been on the right instead of the left hand side (as in Fig. 14) of the coils AB, the angle o still being the angle Aon^ then we should have had _sin(a-)8 ) /3 cos a 30 HANDBOOK OF ELECTRICAL TESTING. are re-arranged so that the zero is at the division at which the needle points in its normal position. Fig. 15 shows a scale so re-arranged, the new figures being additional to the old ones ; such a scale has been adopted in the tangent instruments used for testing purposes in the Postal Telegraph Department. FIG. 15. Apartffrom the fact that the adoption of the foregoing " skew " method of using the tangent galvanometer, gives an increased range to the instrument, a considerable increase of sensitiveness in the case of high deflections is also obtained by it, i. e. a current which would move the needle of the instrument through a given angle from the old zero, will move it through a much larger angle from the new or " skew " zero. This, however, is only the case if the first angular deflection in question (the one from the old zero) exceeds a certain value, if it is less than this value, then the deflection for a given current will be less from the skew than from the old zero. Let be the angular deflection obtained with a given current when the needle is deflected from the ordinary zero, then but if the needle had been at the skew zero, then with the same current we should have had sm(a+/3) J J 1~ , o COS a GALVANOMETERS. 31 therefore , , o sin (a + /3) tan = - =-- L cos a Suppose we have /2 = 60, and suppose the current to be of such a strength as to turn the needle through an angle of 120, then in this case a = 60, and we consequently have _ sin 120 an 4 " cos 60 ; but sin 120 = sin (180 - 60) = sin 60, therefore, or f = 60; that is to say, the angle through which the needle would have been turned when the zero was 60 to one side of zero, would be twice what it would be if it had been deflected from the ordinary zero. The relative values of the deflections, with a given current, from the ordinary and from the skew zero, approach nearer to an equality in proportion as the deflections become smaller; at a certain point they become equal, and then the relative values become reversed, i. e. for the same current the deflection from the skew zero becomes less than the deflection from the ordinary zero. Let us determine at what point the deflections from the two zeros become the same. We have in this case COS a COS a therefore sin (a + /5) = sin a ; if now the angle a is negative, that is to say, if the angular deflection from the skew zero is less than the angle /3, then we have sin (/3 - a) = sin a, or j8 - a = o, that is F = 2a, or 32 HANDBOOK OF ELECTRICAL TESTING. that is to say, whatever be the angle /3 (the angular distance of the skew from the ordinary zero) then a current sufficient to /3 move the needle a distance of ~ from the ordinary zero would 2i move the needle the same distance from the skew zero. If the Reflection from the old zero be less than ^ , then the deflection from the skew zero will be less still, so that there is no advantage in the j8 use of the skew zero unless the deflections exceed ^~ 2 From what has been proved, it is obvious that the greater we make /3 the greater will be the deflection obtained with a given current, but there is a practical limit to increasing /?, for the larger we make the latter the more does the deflective action of the coil tend to act in a direction parallel but opposite to the earth's magnetism, the consequence being that the resultant of the two forces is a comparatively small quantity, and the friction of the pivot, &o., prevents the needle from settling down to the true angle representing the force of the current. Under such conditions large errors in the readings may result. Were it not for this fact the instrument would increase in actual sensitive- ness up to the point at which ft = 90, at which point the needle would not move unless acted upon by a current exceeding in deflective force the intensity of the earth's magnetism ; when the current exceeded this value the needle would swing com- pletely round through an angle of 180. 34. "What is the angle of maximum sensitiveness in the case of a tangent galvanometer with a skew zero ? Keferring to page 24, it is obvious, since the current strength is in proportion to tan a -}- tan /3, that equation (A) on the page referred to becomes Q tan ( ai + 8) + tan /5 = ^1 (tan a\ + tan /3) , O or tan ( ai + 8) = ^-(tan a, + tan j8) - tan /5. O Now we have to make 8 a maximum, supposing that the fore- going equation holds good. Since 8 is to be a maximum, tan 8 must also be a maximum. Now _ tan ai + tan 8 d tan (<*!-{- 6 ) = 5- T5 = ^j- (tan c^ 1 tan ttj tan 8^ C v + tan ) - tan /2, GALVANOMETERS. 33 therefore tan ^ + tan 3 = f 1 (tan a, + tan /3) - tan [1 - tana^tanS ], therefore tan 8 f"l + ^ tan ai (tan d + tan /3) - tan a/ tan therefore tan 8 = 1 - tan a! tan /3 + ~ tan ai (tan a* + tan j8) - 1 - tan a, tan ff . C, tan^ + tan/f +o" tana '- We have then to find what value of tan c^ makes this fraction a maximum, and this we shall do by finding what value makes the denominator of the fraction a minimum. Let tan af = a, C tan (3 = 6, and -^ = K, then we have to determine what value of a makes 1 - ab T7T + Ka a -j- o a minimum. Now I l-a& and this will be a minimum when 34 HANDBOOK OF ELECTRICAL TESTING, that is, when in but as G! and C are ultimately equal, p^- 1 , that is K, becomes LI equal to 1, therefore, a + b = v T+~F, therefore a 2 + 6 2 + 2a& = 1 -f 6 2 , therefore 2 a & = 1 - a 2 , therefore that is _ 00 _ 1 - tan 2 a x c 2 tan ai or cot (90 - /3) = cot 2 fl^ , therefore 90 - j8 = 2 a, or Since /3 cannot be greater than 90, or less than (unless it has a negative value), we see that c^ must lie between the ordinary zero and 45 from it. In the case of a galvanometer where = 60, we have n O _ AKO 60 _ 1 KO 0-1 = 4rO - 1O , that is (60 + 15), or 75, from the skew zero. 35. In order that a tangent galvanometer when used in the ordinary way may give accurate results, it is obviously necessary that the magnetic needle, or rather the magnetic axis of the same, be strictly parallel to the magnetic plane of the coils, that is to say, the angle /3 must be equal to nothing. When the latter is the case, the angular deflection for a given current should be the same to whichever side of zero the needle is deflected. If it is found that these deflections are different, we can deter- mine from the two results what is the magnitude of the angle GALVANOMETEKS. 35 j8 (Fig. 14, page 29). Keferring to this figure, let 0^ be the angular movement of the needle from its position of rest, then or O /j O 00 . a =0! - (3 ; therefore from equation (A) (page 29), we can see that in this case /3=/ 'cO S Sl - = If the same current is now sent in the reverse direction, and the angular movement of the needle from its position of rest is 2 , we have sin 02 /3 " Jl cos (0 2 + py therefore sin 0! sin 2 therefore sin 0! sin 2 cos 0! cos /3 -f sin ^ sin cos 2 cos /3 - sin 2 sin fi ' : therefore cot 0! cos /3 + sin fi = cot 2 cos /3 - sin /8, therefore cot 0! + tan ^ = cot 2 - tan 0, that is 2 tan = cot 2 -cot Of, or To make the instrument read correctly the graduated dial plate would have to be turned round through the angle /3, in the direction in which the needle moved when the largest of the two deflections was obtained ; the zero point will then be correctly set, and the tangent of the angle of deflection taken from this zero will represent directly the current strength. When the needle is provided with a pointer, the simplest method of making the correction is to bend the pointer as ex- plained in 27 (page 23), until equal readings are obtained, with the same current, on both sides of zero. D 2 36 HANDBOOK OF ELECTRICAL TESTING. 36. A form of tangent galvanometer, which is in very general use for lecture and educational purposes, is shown by Fig. 16. This instrument is known as Gaugain's galvanometer, though actually, it is a modification by Helmholtz of the original FIG. 16. instrument of Gaugain. It was pointed out by the latter, that if the magnetic needle were suspended, not at the centre of the coil, but at a point on the axis at a distance from the centre equal to half the radius of the coil, then the chief error due to the magnetic needle not being infinitely short, disappears. Helmholtz improved upon this arrangement by placing a second coil, similar to the first, at an equal distance on the other side of the magnet ; by this means, the error due to the centre of the magnetic needle not being truely at the point indicated by Gaugain, is got rid of. In order that the ratio between the diameter of the coil to its distance from the centre of the magnet may be preserved with reference to every turn of which the coil is composed, these turns should be wound on a conical surface as in the instrument shown by Fig. 16. It is pointed out by Clerk Maxwell, however,* that such a method of winding * ' Electricity and Magnetism, by J. Clerk Maxwell,' vol. ii. p. 318. GALVANOMETERS. 37 is quite unnecessary, as the conditions may be satisfied by coils of a rectangular section, which can be constructed with far creater accuracy than coils wound on an obtuse cone. OBACH'S GALVANOMETER. 37. In this galvanometer, which is shown by Fig. 17, the ring instead of being fixed as in the ordinary tangent instru- ment is movable about an horizontal axis ; by this means the FIG. 17. deflective action of the ring on the needle can be reduced from the full effect (when the ring is in the usual vertical position) down to zero (when the ring is in an horizontal position), so that the instrument has a very wide range, a range which in practice is 100 times as great as that of an ordinary tangent 38 HANDBOOK OF ELECTRICAL TESTING. FIG. 18. galvanometer, thus enabling either weak or very powerful currents to be measured. The effect of setting the ring at an angle to the vertical position is as follows : In Fig. 18, let a b be the vertical position of the ring, and j 6j, the latter when inclined at an angle \j/. Draw ! c at right angles to a^ bj, and a l d at right angles to a 6, then the angle c a 1 d equals the angle ^. Now if c a 1? that is / 2 , represents the mag- netic force of the ring when the latter is traversed by a current, this force being at right angles to the ring, then a l d, that is / 3 , will be the resolved force at right angles to the vertical a I. We have then = sec h or / 2 = / 3 sec \f/ ; that is to say, the magnetic force of the ring is equal to its deflective force on the needle multiplied by the secant of the angle at which the ring is set. But the magnetic force of the ring is in direct proportion to the current strength, and the resolved deflective force is in direct proportion to the tangent of the angle of deflection (a) of the needle of the instrument. Hence the strengths of currents circulating in the ring are directly proportional to the tangents of the angles of deflection of the needle multiplied by the respective secants of the angles of inclination of the ring ; or we may say C = tan a x sec* \j/ x a constant. It must be obvious that there are several ways in which the instrument can be used. In the first place it can be made use of as an ordinary tangent galvanometer, the ring being set at such an angle as would cause the deflections obtained to be brought as nearly as possible in the neighbourhood of 45 (the angle of maximum sensitiveness); the current strengths in this case would of course be directly proportional to tne tangents of the angles of deflection. Again, the ring could be moved so that the same deflection of the needle is obtained with each current being measured; in this case, the current strengths will of course be directly pro- portional to the secants of the angles at which the ring had to be set in the different cases. Inasmuch as the adjustment of GALVANOMETERS. 39 the position of the ring is dependent upon the observation of the movement of needle, it is best to arrange that the latter shall point as nearly as possible at the angle of maximum sensitiveness, i. e. at 45. The " equality " method of using the instrument consists in znoving the ring until it is found that the angular deflection of the needle, and the angle through which the coil has been turned are the same ; in this case we get C = tan \j/ x sec i]/ x a constant. As only a single angle has to be dealt with for a particular measurement, the products of tangents and secants can be calcu- lated beforehand and embodied in a table. In the ordinary tangent galvanometer, the deflective action of the ring acts in the same plane as that in which the needle turns ; but in the Obach instrument, the deflective force, being at an angle with this plane, tends to make the needle dip when the ring is inclined. In order to avoid this tendency, the FIG. 19. arrangement shown by Fig. 19 is adopted. The needle nsis fixed near to the upper end of a thin vertical axle a &, the lower end of the latter being provided with a cylindrical brass weight w. This weight offers but little additional momentum to the whole system round the vertical axis, whilst the movement 40 HANDBOOK OF ELECTRICAL TESTING. round the horizontal axis is completely prevented. The alu- minium pointer p q, is situated in the same plane as the scale ; the ends are flattened and provided with a fine slit, which serves as an index for reading the deflections ; the bottom of the box in which the needle turns being blackened, the reading can be taken without parallax, and therefore very accurately. The magnetic needle n s, has a biconical shape, which entirely pre- vents the shifting of the magnetic axis from its original position, as was sometimes found to be the case with the old broad needles. Adjustments are provided by which the cocoon fibre /, serving to suspend the needle, can be raised or lowered, as well as accurately centred. In order to damp the oscillations of the needle, a shallow, cylindrical box, about 8 centimetres in diameter, and 1J centi- metres deep is provided; this box has two radial partitions which can be slid in or out; the axle of the needle passing through the centre of this box, carries a light and closely fitting vane. By sliding the partitions more or less into the box, various degrees of damping can be obtained ; and if they are right in, the motion is practically dead beat. The scale over which the needle turns is provided with degree and also with tangent divisions. The scale fixed to the ring enables the inclination of the latter to be read to T Vth of a degree ; this scale is also engraved with secant divisions, so as to avoid the necessity of reducing the degrees to secants by means of a table. In order to enable the " constant " of the instrument, i.e. the deflection due to a given current, to be made the same at any place when the instrument is being used, an auxiliary magnet (seen in the figure) is placed at the side of the instrument ; this magnet can be turned round an horizontal axis passing through its neutral point and the centre of the needle, and is at right angles to the diameter on which the ring is turned. This magnet does not affect the zero position, and moreover, if placed exactly vertical with its magnetic axis, it does not alter the original constant, which then only depends upon the horizontal terrestrial component, more or less modified by the surroundings ; but if it is dipped, the horizontal force acting on the needle is either augmented or diminished, according to the direction in which the magnet is turned, and to the amount of dip given. The ring of the instrument, it should be mentioned, is of gun- metal, and serves for the purpose of measuring strong currents, whilst tine wire wound in a groove in the ring enables weaker currents also to be measured. The relative values of the deflective actions of the ring and of the fine wire upon it, are GALVANOMETERS. 41 so adjusted that a current of 1 ampere through the ring gives exactly the same deflection as an electromotive force of 1 volt at the terminals of the fine wire. Since we have C = tan a x sec \[/ x a constant we can easily see if any particular instrument is properly made and the scales correctly graduated ; for if we pass a constant current through the ring, and set the latter at different in- clinations, then the products of the secants of the angles of inclination of the ring and the tangents of the corresponding angles of deflection obtained, should be the same in every case. METHOD OF READING GALVANOMETER DEFLECTIONS. 38. The reading of galvanometer deflections requires con- siderable method, in order that accurate results may be obtained in making measurements. Let A and B (Fig. 19 A) be two contiguous division marks on the galvanometer scale. Now, by observation, we can always determine without difficulty whether the pointer lies exactly FIG. 19A. 1 BA -r 4 BA -r -T -r Deflection = A. Deflection = A f. Deflection = A . Deflection = Af. over A or over B, or whether it lies exactly midway between the two ; and further, if it does not occupy either of these exact positions, we can judge without difficulty whether it lies nearest to A or to B. This is equivalent to saying that we can be certain of the magnitude of the deflection within a quarter of a degree. Thus, supposing the pointer stood between A and B, but nearer to A than to B, then we should call the deflection "AJ," and supposing the deflection was actually very nearly 42 HANDBOOK OF ELECTEICAL TESTING. equal to A, then AJ would be a quarter of a division, or degree, too much. ; if, on the other hand, the deflection was very nearly equal to AJ, then AJ would be a quarter of a division, or degree, too little. In one case the error would be a plus one, and in the other a minus one, but in either case its maximum value would be J only. We have, in fact, the rule that if A be the smaller of two contiguous deflections A and B, then when the pointer is exactly over A, the deflection should be called " A " ; if nearer to A than to B, then it should be called "A"; if exactly midway between A and B, it should be called "AJ"; and lastly, if the pointer is nearer to B than to A, then the deflection should be called "A}" ; thus, for example, if A and B (Fig. 19A) were the 57 and 58 division marks respectively on the scale ; then in case 1 the deflection would be taken as 57, in case 2 the deflection would be taken as 57J ; and again, in cases 3 and 4 the deflections would be taken as 57j and 57f re- spectively. By keeping to these instructions, then, we can be sure of the magnitude of a deflection within of a division or degree. 39. If we are making a measurement with a tangent galvano- meter and we read from the degrees scale, and if we have two deflections to deal with, one of which is to be a proportional part of the other (Usually one-half), then after the first deflec- tion has been observed it has to be reduced to a tangent,* and then the latter being divided, say, by two, the corresponding deflection is ascertained from the tangent table; the resistances, &c., are then adjusted till the required second deflection is as nearly as possible obtained. If we find that the halved tangent does not exactly correspond to a deflection in the table, then we must take, say, the nearest deflection below the exact value, and then take care to adjust so that the deflection of the pointer is a little above that angle. Thus suppose the first deflection to be 58, then the tangent of 58 is 1-6003, and 1 '^ Q 3 = '8001 ; a now the nearest number below this in the table is * 7954, which is the tangent of 38 J ; in adjusting the deflection, therefore, we should take care that we get it rather more than 38 J. Degree of Accuracy attainable in reading Galvanometer Deflections. 40. If the galvanometer scale be so graduated that the number of divisions of deflection directly represent the pro- portionate strengths of the currents producing those deflections, * Table I. GALVANOMETERS. 4<* then an error of, say ^th of a division in d divisions will represent a percentage error, y, in the strength of the current represented by d, which is given by the proportion y : JL : : 100 : d, or 1 per cent. [A] d If, however, the instrument be a tangent galvanometer and the deflection be read from the degrees scale, then an error of 10 in d will not represent an error of m X o per cent., for in this case we must have the proportion 7o : tan d - tan d :: 100 : tan d, or 1 O _ (tan d- tan - 1>00 per cent [B] d J For example. If the deflection d were 46 divisions, then J of a division error (^) would be an error, y, of i X 100 y = = 54 per cent. in the current strength represented by the deflection d ; but if the deflection were 46, then J error would be an error, y , of in the current strength. 41. In cases where we have two deflections to deal with, one of which, or the tangent of one of which, has to be -^th (usually J) of the other, then after we have ascertained, as accurately as we can judge, the magnitude of the first deflection d, the latter (or the tangent of the latter) is divided by n, and then the resist- ances, &c., in the circuit of the galvanometer are adjusted until the deflection - ( or the deflection corresponding to - - J is obtained as accurately as possible. Now in adjusting to this- 44 HANDBOOK OF ELECTRICAL TESTING. latter deflection we are liable to make a plus or minus error of -th of a division or degree as in the first case, and as - n ( d \ i I or tan J m ay itself contain an error due to d being ~th of a division or degree wrong in the first instance, the new deflection may be more than ~th of a division or degree out. What then is the " total possible percentage of error which may exist in the second deflection " ? Now the absolute error which may be made in the two deflec- tions must be the same in both cases, viz. ^ , but the percentage value of the latter will be directly proportional to the value of the deflections ; thus a J division error in 50 divisions is a J per cent. error, but a J division error in 25 divisions is a 1 per cent, error ; in fact, if y be the percentage error (corresponding to the absolute error ~) in d divisions, then n y will be the percentage error (corre- sponding to the absolute error J.) in - divisions. Now if d con- tains a percentage error y, then must also contain a percentage error y ; consequently if we make a percentage error of n y in - 71 when d already contains a percentage error y, then - must contain a total percentage error, P, of or snce _ ; X 100 7= ~TT we get r-i^cn..). [0] * Strictly speaking this is not absolutely correct, for it assumes that the second percentage should be calculated on - , whereas it ought to be calculated 71 d+1 on - - ; but as i is small compared with d, the consequent error is small also. GALVANOMETERS. 45 (1) For example. If d and ^ were 58 divisions and J division, respectively, and further, if the deflection d had to be halved, that is, if n = 2, then we should get r = i-^-^5 x 3 = 1-3 per cent. 58 If we have to deal with degrees of deflection instead of divisions, then in the case of a tangent galvanometer we should have tan < - 1) 100 + (^L- 1) / V tan d, / tan /tan d-~ tan 100 = \ tan d tan df where tan ! = . (2) For example. If cf, ^, and w, were 58, J, and 2, respectively, then we should have tandi = - '8001 (= tan 38 J), therefore It may he pointed out that this last example shows the possible percentage of error which may occur when making a halved current test with the tangent galvanometer under the best possible conditions. Practically, therefore, we may say that under no possible conditions could the deflection error in a halved current test be regarded as being less than If per cent. As will be seen when we come to consider such tests, other sources of error are met with which still further reduce the degree of accuracy with which the tests can be made. 42. Although in formulae [B] and [D] the function of the deflections has been taken as the tangent, yet the formulae apply equally well in cases where the current strengths are propor- tional to any other function of the deflections. 46 HANDBOOK OF ELECTRICAL TESTING. CALIBRATION OF GALVANOMETERS.* 43. The deviations in degrees of the needle of a galvano- meter which is not of the tangent form are not generally pro- portional to any simple function of those degrees, yet it is easy to determine the relative values of the deflections in terms of the currents which would produce them, that is, to calibrate the scale. In order to do this, it is simply necessary to join up in circuit with the galvanometer, a battery, a set of resistance coils, and also a galvanometer, the values of whose deflections are known (a tangent galvanometer, for example). This being done, and the galvanometers being set so that their needles are at zero, we insert sufficient resistance in the circuit to reduce the deflection in one of the instruments to 1, and then by means of a " shunt " (Chapter IV.) we also reduce the deflection of the needle of the second galvanometer to 1. We now reduce the resistance in the circuit step by step so as to produce deflections of 1, 2, 3, 4, &c., from the needle of the galvano- meter whose scale is required to be calibrated. As each deflec- tion is obtained we observe and note the corresponding deflection on the tangent instrument. When the whole range of the scale (or as much of it as is considered necessary) of the instru- ment under calibration has been gone through, we can construct a table for use with it by writing down opposite the various degrees of deflection the tangents of the deflections which were obtained on the tangent instrument and which corresponded to the deflections in question. The table so constructed would be used precisely in the same way as would the table of tangents in the ease of a tangent galvanometer, the use including, it may be remarked, the determination of the percentage value of an error in a deflection. It may also be remarked that the angle of maximum sensitiveness would be the deflection which was obtained when the needle of the tangent instrument pointed to 45. THE THOMSON GALVANOMETER. 44. The accuracy with which measurements can be made depends chiefly upon the sensitiveness of the galvanometer employed in making those measurements. The Thomson re- flecting galvanometer supplies this requisite sensitiveness, and is the instrument which is almost invariably employed when great accuracy is required, and also when very high resistances have to be measured. * See also p. 76, 73. GALVANOMETERS. 47 Description. 45. The principle of the instrument is that of employing a very light and small magnetic needle, delicately suspended within a large coil of wire, and of magnifying its movements by means of a long index hand of light. This index hand is obtained by throwing a beam of light on a small mirror fixed to the suspended magnetic needle, the ray being reflected back on to a graduated scale. This scale being placed about 3 feet distant from the mirror, it is obvious that a very small angular movement of the mirror will cause the spot of light reflected on the scale to move a considerable distance across it. The needle being very small, and being placed in the centre of a large coil, the tangents of its deflections are approxi- mately directly proportional to the strength of the currents & FIG. 20. producing them. _D In Fig. 20, let L be a lamp which throws a beam upon the mirror m, which has turned through a small angle, and Q reflected the beam on the scale at D. Let d be the distance through which the beam has moved on the scale from the zero point at L, and let I be the distance between the scale and the mirror. Now the angle through which the beam of light turns will be twice the angle through which the mirror turns ; this is clear if we suppose the mirror to have turned through 45, when the reflected beam will be at 90, or at right angles to the incident beam. If, then, we call a the angle through which the beam of light turns, will be the angle through 2t which the mirror will have turned. Let -^ and - 2 be the two 2> 2i angles through which the mirror has been turned by two currents, of strengths Cx and 2 respectively, then d. ""*-.. tan x tan a 2 V 1 -{- tan 2 being positive, as the angles are less than 90. y >-^ 48 HANDBOOK OF ELECTRICAL TESTING. Z being the distance of the scale from the mirror, let di and d. 2 be the distances traversed on the scale by the beam of light, then d l d. 2 tan ! = y> tan o 2 = therefore when d^ and d 2 do not differ- largely , then we may take but when this is not so a small error is observable. For instance, suppose di = 50, and d 2 = 300. According to the last formula this would show that one current is just six times as strong as the other, but by the correct formula, taking Z = 1500 divisions (which would be about its value), we find that G! : C 2 : : 300 (Vl500 2 + 50* - 1500) : 50 (Vl500 2 + 300 2 - 1500), that is C, : C 2 : : 250 : 1485, or OjiCa :: 50 :297; so that when extreme accuracy is required we cannot take the strengths of currents as being exactly proportional to the number of divisions of deflection on the scale. The galvanometer, as usually constructed, consists essentially of a very small magnetic needle, about three-eighths of an inch long, fixed to the back of a small circular mirror, whose diameter is about equal to the length of the magnet. This mirror, which is sometimes a plano-convex lens, of about six feet focus, is suspended from its circumference by a cocoon fibre devoid of torsion, the magnetic needle being at right angles to the fibre. The mirror is placed in the axis of a large coil of wire, which completely surrounds it, so that the needle is always under the influence of the coil at whatever angle it is deflected to. A beam of light from a lamp placed behind a screen, about three feet distant from the coil, falls on the mirror, and is FIG. 21, Front Elevation. J real size. 50 HANDBOOK OF ELECTRICAL TESTING. reflected back on to a graduated scale placed just above the point where the beam emerges from the lamp. The scale is, as we have before said, straight, and is usually graduated to 360 divisions on either side of the zero point. It is not absolutely necessary that the working zero be the middle or zero point of the scale ; it is a very common practice to adjust the instrument so that the reflected beam of light normally falls near the end of the scale ; by this adjustment an extreme range of 360 X 2, or 720 divisions can be obtained. 46. The Thomson galvanometer is made in a variety of forms : Fig. 21 gives a front, and Fig. 21 A a side elevation (with glass shade, &c., removed) of one very common pattern. FIG. 2lA. Side Elevation. (Shade removed.') I real size. It consists of a base formed of a round plate of ebonite, provided with three levelling screws ; two spirit-levels, at right angles to one another, are fixed on the top of this plate, so that the whole instrument can be accurately levelled : sometimes one circular level only is provided, but the double level is much the best arrangement. From the base rise two brass columns, between which a brass plate is fixed, rounded off at the top and bottom. Against the faces of this plate are fixed the coils (c, c, c, c) of the instrument. GALVANOMETERS. 51 The brass plate lias shallow countersinks on its surface for the faces of the coils to fit into, so that they can be fitted in their correct places without trouble or danger of shifting. Bound brass plates press against the outer surfaces of the coils by means of screws, and keep them firmly in their places. There are two round holes in the brass plates coinciding with the centre holes in the coils. The coils themselves, which are four in number, are wound on bobbins of thin insulating material, the wire being heaped up towards the cheek of the bobbin which bears against the brass plate. This heaping up is done in accordance with a law of Sir William Thomson, so as to obtain, as far as possible, a maximum effect out of a minimum quantity of wire. The edges of the coils are covered with shellac, so as to protect the wire from injury. Within the holes in the brass plate are placed two little magnets, n s and s n* formed of watch-spring highly magnetised ; they are connected together by a piece of aluminium wire, so as to form an astatic pair of needles. A small groove is cut in the brass plate, between the upper and the lower hole, for the aluminium wire to hang freely in. An aluminium fan is fixed at right angles to the lower needle; this fan acts as a damper, and tends to check the oscillations of the needles and to bring them to rest quickly, In front of the top needle is fixed the mirror. It is suspended by a fibre attached at its upper end to a small stud which can be raised or lowered when required ; when this stud is pressed down as far as it will go the needles rest on the coils, and the tension being taken off the fibre, there is no danger of breaking the latter by moving the instrument. One end of each coil is connected to one of the four terminals in front of the base of the instrument, the other ends being connected to one another through the medium of the small terminals placed midway on either side of the coils. The connections are so made, that when the two middle terminals on the base of the instrument are joined together the whole four coils are in the circuit of the two outer terminals, so that they all four act on the magnetic needles. 47. As it is often convenient to be able to couple up the four coils in different ways so as to vary their total resistance, in the instruments manufactured by the Indiarubber, Gutta- percha and Telegraph Works Company the ends of all the four * In the more recent instruments it is usual to have several small magnets placed one above the other at a short distance apart, in the place of a single magnet. E 2 52 HANDBOOK OF ELECTRICAL TESTING. coils are connected to terminals in a manner designed by Messrs. March Webb and E. K. Gray, and shown by Fig. 22. This figure represents the base of one of these instruments. Lines are engraved on the ebonite base to show the routes followed FIG. 22, by the various coils. Arrows also are engraved alongside the lines to show the directions in which the currents must flow in order that all the coils may tend to turn the galvanometer needle in the same direction. There are five possible ways of coupling up all the coils together, so as in each case to produce a different resistance. The following will show the various methods : I. To obtain total resistance of all the coils in series, connect terminals 2 and 3, 4 and 5, 6 and 1. GALVANOMETERS. 53 II. To obtain f resistance, connect terminals 2 and 3, 2 and 5, 7 and 6, 7 and 4 III. To obtain J resistance, connect terminals 2 and 3, 4 and 8, 1 and 5, 6 and 7. IV. To obtain T 3 ^ resistance, connect terminals 2 and 8, 1 and 3, 4 and 5, 6 and 7. V. To obtain -^ resistance, connect terminals 1 and 3, 3 and 5, 5 and 7, 6 and 8, 4 and 6, 2 and 4. In each case the leading wires from the galvanometer must be connected to terminals 1 and 8. Keferring again to Fig. 21 ; over the coils a glass shade is placed, from the middle of the top of which a brass rod rises. A short piece of brass tube slides over this rod, with a weak steel magnet, slightly curved, fixed at right angles to it. This FIG. 23. FIG. 24. magnet can be slid up or down the rod, or twisted round, as occasion may require. For fine adjustments a tangent screw is provided, -which turns the brass rod round, and with it the magnet. Figs. 23 and 24 show modified forms of the instrument, 54: HANDBOOK OF ELECTRICAL TESTING. which, however, in general arrangement are similar to the pattern which has been described. In the more recent galvanometers manufactured by Messrs. Elliott Brothers, the brass plates, which, in the older instru- ments secured the coils in their places, are hinged to the frame, whilst the coils themselves are permanently fixed to the plates ; by this arrangement the magnetic needles, with their mirror, fibre-suspension, &c., attachments, can be got at, if required, with the greatest facility. Altogether this improvement is one of the most convenient that has been made. About 5000 or 6000 ohms is usually the total resistance of the coils of these galvanometers. Fig. 25 shows a portable reflecting galvanometer which is very useful, especially for travelling purposes ; the three legs are hinged at their junction with the lower part of the coil frame, so that they can be folded together, and thus made to occupy but little space. Owing to the instrument being pro- vided with but two coils (one in front of, and the other behind, the needle) its sensitiveness is not quite so great as that of the larger instruments with four coils, but for general purposes it is an excellent piece of apparatus. FIG. 25. FIG. 26. 48. We have said that the mirror is sometimes made of a plano-convex lens. This is done so as to obtain a sharp image of the spot of light on the scale. The width of the spot can be regulated by means of a brass slider fixed over the hole in the screen, through which the beam emerges from the lamp. A much better arrangement than the spot of light is now provided with most instruments. The hole through which the light emerges is made round, about the size of a sixpence, with GALVANOMETERS. 55 a piece of fine platinum wire stretched vertically across its diameter. A lens is placed a little distance in front of this hole, between the scale and galvanometer, so that a round spot of light, with a thin black line across it, is reflected on the scale. This enables readings to be made with great ease, as the figures on the scale can be very distinctly seen. (The mirror in this arrangement may be a plane one.) When the spot of light only is used, it is necessary to partially illuminate the scale with a second lamp. The general appearance of a back view of the scale frame with the lamp placed in position, is shown by Fig. 26. Jacob's Transparent Scale. 49. The position of the ordinary form of scale for the Thomson's galvanometer is to a certain extent inconvenient, especially to near-sighted persons. Mr. F. Jacob has completely remedied this inconvenience by the arrangement shown in front view and cross section by Fig. 27. In this fig. B is a wooden scale-board with a longitudinal slot, as shown at ; P is the paper scale, cut FIG. 27. \ // M- 35 26 /// I,,,,!,,.,! 1 1 1 ?7 38 39 4O ! i 1 i ! i I ) Jf \ so that all the division lines reach the lower edge ; A is a slip of plane glass with its lower half finely ground from one end of the slip to the other, on the side towards C : the scale is so placed that the lower end of the division lines just touches the ground part of the glass slip. The image of the slit with a fine wire stretched across it is focussed in the ordinary manner on the ground part of the glass, and will of course be clearly seen by the observer on the opposite side of the scale ; as the line and printed divisions are in the same plane, there is no parallax ; and a great increase in accuracy of reading the position of the hair line is obtained, owing to the greater ease of observing that two lines coincide when end on to one another, than when super- imposed ; and further, from the circumstance that the room need not be darkened. The lamp and its slit is placed on one side and reflects the beam of light on to the galvanometer by a mirror or total reflection priism, and by means of two long plane 56 HANDBOOK OF ELECTKICAL TESTING. mirrors the actual distance between the galvanometer and scale is reduced, so as to have everything close to the observer's hand. The scale adopted is divided into half millimetres, and it is per- fectly easy to read to a quarter of a division, and with a hand magnifying-glass still further. This arrangement has been adopted in the testing-rooms of Messrs. Siemens Brothers and Co., at Woolwich, and gives great satisfaction. 50. In the testing-rooms at the Silver town Telegraph Works, the scales employed are of large dimensions, being about 5 feet long, and are set at a distance of several feet from the galvano- meter. By this arrangement a greatly magnified image of tho round spot of light with the black line across it is obtained, and the divisions on the scale being of correspondingly large dimensions, the readings can be made with great facility, and with very little fatigue to the eye. The only objection to the arrangement is the space which it necessarily occupies, but as it is not often that many instruments require to be set up in the same room, this need hardly be taken into account. To set up the Galvanometer. 51. It is essential, before proceeding to set up the instrument for use, to see that the ebonite base is thoroughly dry and clean, so that there may be no leakage from the wires to interfere with the tests taken. Indeed, it is as well to place the galvanometer and the other apparatus to be used on a large sheet of gutta- percha or ebonite, more especially if the room in which the tests are to be made is at all damp. Sometimes little ebonite cups are provided for the levelling screws of the instrument to stand in, which answers the purpose of insulating very thoroughly. The instrument should be set up on a very firm table in a basement storey. It is almost useless to test with it in an upper room, as the least vibration sends the spot of light dancing and vibrating to and fro. At all cable works the instrument is placed on a solid brick table built on the earth so that no vibration can possibly affect it. A suitable table being 'chosen, set the galvanometer in any convenient position, and adjust the levelling screws until the bubbles of the level or levels show the instrument to be perfectly level. Now remove the glass shade, and gently raise the stud at the top of the coils by squeezing the tips of the fingers between the head of the stud and the top of the brass plate in which it runs. If the stud is raised by a direct pull, there is almost a certainty GALVANOMETERS. 57 of its coming up with a jerk and breaking the fibre. On no account must the stud be twisted round, except to get rid of any torsion which may exist in the fibre when it has been replaced after becoming broken. The stud being raised sufficiently high to allow the mirror to swing clear of the coils, replace the glass shade, screw the brass rod with the magnet, on to its top, and set the magnet about half-way up the rod, the poles being placed so as to assist in keeping the magnetic needles north and south. The scale lamp being lighted, place it in position on the scale stand, the edge of the wick being turned towards the brass slider which regulates the width of the beam of light. Having opened the slider to its full extent, the scale and lamp should be placed about 3 feet from the galvanometer, so that it stands parallel with the faces of the coils and so that a line drawn at right angles to the scale from the lamp-hole will pass through the centre of the galvanometer. The reflected beam of light should then fall fairly on the scale. If too high, this may be remedied by propping up the scale, and if too low, by screwing up the levelling screws of the galvanometer. Should the light be too high on the scale, it will be found an easier matter to prop up the scale than to lower the galvanometer by means of the levelling screws. The spot of light should now be set at the zero point on the scale by turning the regulating magnet by means of the screw; the spot should next be focussed, by advancing or retreating the lamp and scale, until a sharply defined image is obtained on the scale. The width of the slit may then be diminished, by means of the brass slide, until a thin line of light only is obtained on the scale. If the round spot of light with the line across it is used, the focussing must be made so that the black line is sharply defined. The position of the scale and galvanometer being once ob- tained, their positions on the table may be marked for future occasions, or, at least, the exact distance of the scale from the galvanometer noted, so that it can be placed right without trouble. The instrument being now ready for use, if it is not required to be sensitive, place the regulating magnet low down ; if, on the contrary, it is required to be sensitive, place it high up. 52. To obtain the maximum sensitiveness : Kaise the magnet to the top of the bar, and then turn it half round, so that its poles change places. The magnet will now be opposing the earth's magnetism, and consequently will tend to turn the magnetic needles round. If the magnet is at the top of the rod, the 58 HANDBOOK OF ELECTKICAL TESTING. effect of the magnetism of the earth on the magnetic needles will be more powerful than the magnetism of the regulating- magnet, and the needles will tend to keep north and south ; but by placing the regulating magnet lower down, a point is reached where the earth's magnetism is just counteracted. Under these conditions the needles will stand indifferently in any position. By placing the regulating magnet about an inch higher than the position which gives this exact counteraction, the magnetism of the earth will be just sufficient to keep the magnets north and south, and consequently the spot of light at the zero on the scale, and at the same time leaves the magnets free to be moved by a very slight force. It will be noticed with the regulating magnet in this position, that in order to get the spot of light at the zero point, the magnet must be turned in the opposite direction to that in which it is required that the needles should move. It is not advisable to adjust the instrument too sensitively, because it is difficult then to keep the spot exactly at zero, as any slight external action may throw it a degree or two out. 53. The presence of iron near the instrument is not prejudicial to its correct working, so long as the metal remains stationary. The experimenter should, however, remove any keys or knives he may have about him, as they very much affect the galvanometer if he moves about much. These precautions may seem too minute, but as the very object of the Thomson galvanometer is to enable measurements to be made with accuracy, all likely causes of disturbance should be avoided. 54. If the fibre of the instrument by any chance gets broken, the top front plate must be unscrewed, when the coil which it secures can be removed, and the mirror and its appendages got at. Care should be taken, when replacing the fibre, that only a fine thread from the cocoon silk is used, or the sensitiveness of the instrument will be much diminished. The operation requires care, and must be done in a room free from draughts. When the ends of the fibre are passed through their respective holes, and tied, a small drop of shellac varnish may be dropped on them, which will prevent their becoming loose. It is as well to let the needles remain suspended for a time, so that the fibre may become stretched to its normal length before being used. The suspending stud should always be pressed down before removing the instruments. 55. A resistance box, containing three shunts, is provided with the galvanometer, of the values ^th, -ggth, and ^^th of the resistance of its coils, which values, as we shall show in the GALVANOMETERS. 59 next chapter, enable us to reduce the sensitiveness of the galva- nometer to its T Vth, T i^th, and T ^Vo^h part respectively. Fig. 28 shows a form of this shunt. By inserting a plug into one or other of the holes, the required shunt is inserted. The numbers are sometimes marked as T Vth, T ^th, and j-^^th, instead of ^th, ^th, and -^^th, thereby indicating that FIG. 28. FIG. 29. ferealsize the particular shunt reduces the deflections of the needle to that particular fraction, but they have really just the same adjustment in both cases. The shunts are sometimes enclosed in a round brass box, as shown by Fig. 29, which is perhaps a more portable and elegant form than the foregoing. The two broad strips of copper shown in Fig. 28 are used for the purpose of connecting the box with the galvanometer. The blank plug-hole is for the purpose of short-circuiting, which should always be done when the instrument is not actually in use. THOMSON'S DEAD-BEAT GALVANOMETER. 56. Great inconvenience and loss of time in testing often arise from the needle of the galvanometer not settling down at once to the angle of deflection it should take up when under the influence of a constant current, but oscillating to and fro several times before it finally comes to rest, and again acting in the same way when the current is taken off and the needle returns 60 HANDBOOK OF ELECTRICAL TESTING. towards the zero point. The object of the dead-beat galvanometer is to avoid these inconvenient oscillations. Fig. 30 shows the arrangement invented by Sir William Thomson for efiecting this object. A is a brass tube, whose end a a, which is screwed, is closed by a piece of glass. B is a short piece of tube, which is screwed, and whose end b I is similarly closed by a piece of glass. C is a third short piece of tube, into which the ends of A and B screw. The length of this tube is such that when the whole arrangement is united together there is a very small space between the ends a a and b b ; a small air-tight cell in fact is formed. Hanging midway inside C is a mirror m, with a magnetic needle fixed to it, as in the ordinary Thomson galvanometer. This mirror very nearly fits inside the tube, there being only just room for it to swing freely ; it is suspended by a very short fibre. The space between a a and 6 &, although very small, is just sufficient to enable the mirror to turn through an angle large enough to give a good deflection of the spot of light on the scale. The complete arrangement is inserted in the centre of a single galvanometer coil, so that the mirror occupies the same position that it does in the ordinary galvanometer. Owing to the air inside the cell being'so closely confined, the violent movement of the mirror is checked when it is acted upon by a current passing through the coils, and the consequence is that the mirror, instead of overshooting the mark and then recoiling, turns with a gradually decreasing velocity towards GALVANOMETERS. 61 its final deflection and ceases to move when the latter is reached. The same thing takes place when the current is cut off; in this case the spot of light moves back to zero and ceases to move at that point. The suspension fibre being very short, the mirror cannot turn so freely as the one in the ordinary galvanometer ; its sensitive- ness is therefore not quite so great, but it is sufficiently so for most purposes for which the latter would be used. The fibre is very easily replaced when broken. One end being attached to the mirror, the other is passed through a small hole in the side of C, and is then drawn sufficiently tight to suspend the mirror inside the tube so that it does not touch the sides, a drop of shellac is then applied to the hole, which closes it and fixes the fibre. In some cases the cell is filled with paraffin oil, which still further tends to check the move- ment of the mirror. THE D'ARSONVAL DEPREZ DEAD-BEAT GALVANOMETER. 57. The main peculiarity of this instrument lies in the fact that, whereas in almost all galvanometers there is a fixed coil and a movable magnetic needle, in this galvanometer the coil is movable and the magnet a massive compound horse-shoe of steel is fixed. Fig. 31 represents the instrument itself, as manufactured by P. Jolin & Co., of Bristol. The steel magnet, made of three thin horse-shoes, each magnetised as strongly as possible, is firmly fixed to a metal base, with its poles upwards. Between the poles hangs the coil, which is rectangular in form and weighs only a few grains ; it is held in its place by a thin silver wire above and another thin silver wire below. The coil is made by winding the wire on a continuous rectangular frame made of copper or silver as thin as possible ; this frame, by the reactive effect of the induced currents which the move- ment of the coil sets up, causes the latter when deflected to come rapidly to rest. To reinforce the magnetic field, a strong compound magnet of cylindrical shape is arranged so that the laminations are in a horizontal direction, and so that its north pole comes opposite the south pole of the horse-shoe magnet ; it is placed in the hollow of the suspended rectangular coil without touching it, and is firmly fixed ; the coil is then free to turn in the very narrow space between the compound magnet core and the external magnet-poles ; and it need hardly be added that this contrivance produces a very intense magnetic field. The current is led in by one of the silver suspension-wires, and 62 HANDBOOK OF ELECTRICAL TESTING. leaves the coil by the other. So far the arrangement precisely resembles that adopted in the well-known " siphon- recorder " of Sir W. Thomson, invented several years ago for the purpose of cable-signalling. A small mirror of 1 metre focus is fixed FIG. 31. to the suspended coil ; a brass spring at the bottom keeps the suspending wires adequately stretched ; and a screw-head at the top of the instrument serves both to regulate the tension in the wires and to let the coil down to a position of rest on the central iron cylinder, whenever the galvanometer is to be dismounted for removal to a distant place. The resistance of the coil is made from 150 to 750 ohms in the ordinary pattern of instrument. As there is no suspended needle, no external magnetic forces affect the zero of the instrument ; and, since the position of the coil is determined solely by the elasticity of the suspending wires and the magnetic action of the fixed magnet on the current in the coil, it can be used in any position, and is inde- pendent of the earth's magnetic field. It can even be placed quite near to a dynamo-machine. The intensity of the magnetic GALVANOMETERS. 63 field in which the coil is situated is such that whenever the galvanometer-circuit is closed even through a considerable resistance the motion of the needle is dead-beat. It takes less, than one second to come to rest at its final position of deflection, and when it returns to zero it does so with the most complete absence of oscillations. Altogether, the form of instrument is an extremely satisfactory one. THOMSON'S MARINE GALVANOMETER. 58. This instrument is specially constructed for use on board ship, where the rolling of the vessel and the constant movement of masses of iron about would render an ordinary reflecting galvanometer quite useless. Fig. 32 shows a side view of this instrument, the upper part being drawn in section. FIG. 32. C C C C are the coils, which are similar in form to those employed in the ordinary Thomson's galvanometer; there is r however, but one set, of two coils, instead of two sets as in the latter instrument. The mirror, with the magnetic needle fixed to its back, is strung on a cocoon fibre in a brass frame. The fibre is fixed at one end, and at the other is attached to a spring, which draws 64 HANDBOOK OF ELECTRICAL TESTING. the fibre tight. The frame slides in a groove between the coils, so that it can be drawn out for the purpose of repairing the fibre. A powerful directing horse-shoe magnet (not shown in the figure) embraces the upper parts of the coils, and serves to overpower the directive effect of the FIG. 33. earth's magnetism. This latter effect is still further rendered harmless by en- closing the whole system in a massive soft-iron case A A A A, a little window B being left through which the rays of light reflected by the mirror enter and return. For obtaining exact adjustment of the spot of light to zero, two little magnets, n and s, as broad as the mirror magnet is long, are provided; by turn- ing the pinion p these little magnets can be made to advance or retreat, and so act on the mirror magnet to make it turn in one direction or the other, as it is required. The resistance of this form of galvanometer (which is shown in general view by Fig. 33) is usually as high as 30,000 or 40,000 ohms. 59. The angle of maximum sensitiveness in a Thomson re- flecting galvanometer is, it is perhaps unnecessary to say, the largest deflection we can obtain, as the angle of deflection is but a very few degrees and, consequently, the true maximum angle can never be reached. 60. By turning the controlling magnet of the instrument so that the needle is turned through a large angle, the normal zero becomes at a considerable distance off the scale, and the sensi- tiveness of the galvanometer to changes in the current strength producing a deflection, can be made very great. Thus, supposing the needle to be normally at the ordinary zero, and suppose that a current caused it to deflect to 350 divisions, then an increase in the current of say 1 per cent, would increase the deflection to _ }1 , or, 353 5 ; that is, would increase it 3 5 divisions. If now the working zero had been 350 divisions to, say, the left of the ordinary zero, and if the current had been strong enough to produce a deflection of 350 divisions to the right of the ordinary zero, then the deflection would be equivalent to 350 -f 350, or 700 divisions, and an increase in the current of 1 per cen+ would increase the deflection to GALVANOMETERS. 65 700 X Qr 70^ flyft - g t gaVj an i ncrease O f 7 divisions. If, lastly, the controlling magnet is turned so that the needle has a zero equivalent to, say, 2000 divisions to the left of the ordinary zero, that is an inferred zero, as it is called, of 2000, then, if the needle were deflected to the right by a current sufficiently strong to bring the deflection on the scale, and to give it a value of 350 to the right of the ordinary zero, the deflection representing the current would be 350 + 2000, or 2350 divisions, and an increase in the current of 1 per cent. OOKA V 101 would increase the deflection to - UVJ * , or, 2373-5, that is to say, an increase of 23 5 divisions. In actual practice it is often possible to use an inferred zero considerably greater than 2000, and with corresponding advantage. FIGURE OF MERIT OF GALVANOMETERS. 61. The "figure of merit " of any galvanometer may best be defined as the reciprocal of the strength of current which will produce one division or degree of deflection.* In order to find this current, we have simply to join up the galvanometer in circuit with a battery of a known electromotive force, and a resistance of a known value, and then note the deflection obtained ; from this we can easily calculate the current required to produce 1 degree of deflection ; thus, for example, if we had a tangent galvanometer which gave a deflection of 50 with a 10-cell Daniell battery, that is, with an electromotive force of 10 volts approximately, there being in circuit a total resistance of 1000 ohms, then the current producing this deflection would be 10 1000 = ' 01 amp re ' The current which would be required to produce a deflection of 1 would obviously be 01 x tan 10 = 01 x ' 175 = -OOOU6 ampere; tan 50 1-198 consequently the figure of merit is -, or, 6849. * It is preferable to define the figure of merit as being the reciprocal of the current rather than the current itself, inasmuch as by so doing we avoid the apparently contradictory statement, that a galvanometer with a high figure of merit is one which requires a low current to produce a perceptible deflection. r 66 HANDBOOK OF ELECTRICAL TESTING. In the case of a Thomson galvanometer, we have simply to divide the current by the deflection obtained with the latter, since the deflections are approximately in direct proportion to the currents producing them. If we require to determine the figure of merit of a galvano- meter whose deflections throughout the scale are not pro- portionate to any ordinary function of the degrees of those deflections, then it is best to employ a sufficiently low electro- motive force and high resistance in circuit to obtain a few degrees of deflection only, and then to divide the current by this number of degrees; for on every galvanometer the first few degrees of deflection are almost exactly proportional to the currents producing them, although the higher deflections are not so. The " figure of merit " of a galvanometer has a considerable bearing upon the question of the degree of accuracy with which it is possible to make electrical measurements, as will be seen hereafter. SENSITIVENESS OF GALVANOMETERS. 62. A galvanometer with a high " figure of merit," that is, a galvanometer whose needle will deflect from zero with a very weak current, is not necessarily a highly sensitive instrument ; by a sensitive galvanometer we mean one whose needle when deflected under the influence of a current will change its deflection perceptibly with a very slight change in the current strength. In many tests it is far more important that the galvanometer used be one of great sensitiveness rather than one with a high figure of merit. As a rule it is rarely that an instrument with a compass suspended or a pivoted needle is highly sensitive, unless indeed the pivoting is exceptionally good. Practically, it may be taken that for high sensitiveness the needle must be suspended by a fine fibre so that its movements may be perfectly free. 63. In order to check the oscillations of the galvanometer needle when the latter is either deflected under the influence of a current, or when it recoils after the current is taken off, Mr. J. Gott suggests that a small coil of wire should be placed under the galvanometer in circuit with a small battery and a key, the coil being in such a position that when a current passes through it a deflection of the needle is produced ; by a proper manipula- tion (easily acquired) of the key, it will be found that the oscillations of the needle can, with such a contrivance, be checked in a few seconds, and much time (an important item in some tests) saved. CHAPTER IV, FIG. 34. SHUNTS. 64. IN making certain measurements we sometimes find that owing to the sensitiveness of the galvanometer, we are unable to obtain a readable deflection, from the needle being deflected up to the stops. We may reduce this sensitiveness by the insertion of a Shunt between the terminals of the instrument. This arrangement is shown by Fig. 34. If it is required to reduce the strength of current which ordi- narily passes through the galvano- meter to any proportional part of that current, we must calculate, from the resistance of the galvano- meter, what the resistance of the shunt should be to effect that purpose. Now if we call C the current passing through the gal- vanometer without a shunt, then on introducing the shunt, C will divide between the two resistances, the greater por- tion of the current going through the smaller resistance, and the smaller portion through the greater. Thus if we suppose the total current, which passes from one terminal of the galvanometer to the other, to consist of G + S parts, then ^ . of these parts will go through the shunt, and S G + S p j ~ parts through the galvanometer; that is to say, the current going through the shunt will be G G + S' and the current going through the galvanometer, S C G + S F 2 68 HANDBOOK OF ELECTRICAL TESTING. If, in this last quantity, we put S = G, then current going through galvanometer will be r Q -g. G + G 2 Again, if we make S = , current going through galvanometer will be Once more, if S be made equal to , current going through o galvanometer will be G 3 C : 4* Finally, if S be made equal to , current going through galvanometer will be G 711 From this it is evident, that to reduce the current flowing through the galvanometer to its -th part, we must insert a shunt whose resistance is th part of the resistance of the n I galvanometer. 65. In many galvanometers three shunts are provided,* which enable us to reduce the strength of current flowing through the- same to its T Vth, -yjyth, or y^V^th part. From what has been said, it will be evident that the resistances of the shunts neces- sary to produce these results will have to be respectively the, th, and -g-g-^th part of the resistance of the galvanometer* * Page 59. SHUNTS. 69 We are thus enabled to reduce the sensitiveness of the galvano- meter to any one of these three proportions we wish. 66. Suppose now, in making a measurement, we placed a re- sistance box for a shunt between the terminals of the galvano- meter, and then adjusted it until we obtained a convenient deflection for the purpose we required ; what deflection should we get on removing the shunt ? Let us call C, as before, the current which passes through the galvanometer when no shunt is inserted, and let C 1 be the current which flows through it when the shunt is inserted, then the current which flows through the shunt will be C-G!. Now the two currents will flow through the shunt and galvano- meter in the inverse proportion of their resistances, that is, <&:.0-Q|.::8.:-G, therefore, r-r * G + S c - G! x -- Or expressed in words, we should say that the current which would flow through the galvanometer, when the shunt was removed, would be Galmt*r + Shunt times ^ ^ rf bhunt the current which flows when the shunt is inserted. This pro- portion is called the multiplying power of the shunt. 67. It will be noticed in a circuit like that shown by Fig. 34, that when a shunt having a resistance equal to that of the galvanometer is introduced between the terminals of the latter, it will not exactly halve the current passing through the instrument. If we used a tangent galvanometer, we should find, if the deflection without the shunt was 40 divisions on the tangent scale, the introduction of the shunt would not bring the deflection down to 20, but to some deflection greater than 20. The reason of this is, that the introduction of the shunt reduces the total resistance in the battery circuit, and consequently increases the strength of the current passing out of the battery. It is this increased current, then, which splits between the galvanometer and shunt, and not the original current. If it is required to make up for this decreased resistance caused by the introduction of the shunt, it is neces- sary to add in the battery circuit a compensating resistance equal in value to the amount by which the original resistance has been reduced. In order to obtain this, we must first consider the law of 70 HANDBOOK OF ELECTBICAL TESTING. The Joint Resistance of two or more Parallel Circuits. 68. If we have several wires whose resistances are Kj, E 2 , E 3 . . . . respectively, then conductivity being the in- verse or reciprocal of resistance, their conductivities may be represented by , , .... Now the joint conductivity of any number of wires is simply the sum of their respective conductivities. Thus, two wires of equal conductivities, when joined parallel to one another, will evidently conduct twice as well as one of them; and in like manner, three wires will conduct three times as well as one. Similarly, two wires, one of which has a conductivity of 2, will, when combined with one which has a conductivity of 1, produce a conductivity of 2 + 1 or 3, for this is simply the same as joining -up three wires, each having a conductivity of 1 ; and so with any number of wires. Therefore the joint conductivity of the several resistances, or of the multiple arc as the combination is called, will be 111 and conductivity being, as we have said, the reciprocal of resist- ance, the resistance of the wires will be the reciprocal of this sum, or -. That is to say, the joint resistance of any member of wires joined parallel to one another is equal to the reciprocal of the sum of the reciprocals of their respective resistances. A particular case of these combinations is that of the joint resistance of two resistances, thus 1 B^ _!_ , i "iv+ir; Kj R 2 or, the joint resistance of two resistances joined parallel to one another is equal to their product divided by their sum. 69. Applying the foregoing law, the resistance between the terminals of the galvanometer before the introduction of the shunt being G, that on the introduction of the shunt will be C* Q . Or, as S is usually made some fractional value of G, SHUNTS. 71 say the -th part (which value would be used in reducing the sensitiveness of the galvanometer to th), this combined resistance will be G G n- 1 n - 1 G w- 1 1 + [13 n- I The resistance therefore to be added to the battery circuit will be G _G = G ^. [2] For example. It was required to reduce the sensitiveness of a galvanometer, whose resistance was 100 ohms (G), to ^th. What should be the resistance of the shunt and of the compensating resistance ? Kesistance of shunt equals 100 X -z - 7 = 25 ohms, 5 1 and compensating resistance equals 5-1 100 X = 80 ohms. It would be as well if the shunt boxes provided with galvanometers had compensating resistances connected with them, as calculation would be con- siderably simplified thereby in a large number of measurements. Fig. 35 shows how a set of shunts and compen- sating resistances may be adapted to any galvano- meter ; we will consider how their values may be determined. Let S, S T , S 2 , be the shunts which can be con- nected to the galvanometer by inserting plugs at A, B, or C. , Let r lt r 2 , r lt be the compensating resistances, / and let r i + r 2 + r i R i r 2 + r, = K 2 . FIG. 35. 72 HANDBOOK OF ELECTKICAL TESTING. Now, what we have to do, is to find what values of S, S lf S 2 , and r x , r 2 , r h are necessary, so that when a plug is introduced either at A, B, or C, the resistance between D and E shall always be the same, whilst the necessary portion of the current is shunted off from the galvanometer. Let us first consider the shunt S and the compensating resistance which, in this case, will be "R^ When the shunts and compensating resistances are not in use, the resistance in circuit is of course G, and this value must always be preserved between D and E. Let the value of the shunt S required be -th, then we know (page 68) that the resistance of S necessary to give this, is and from [2] (page 71) that the value of E x must be We next have to consider what value to give to S x and B 2 . Let it be required, by means of these resistances, to reduce the deflection by th, then the value to be given to S x will be n i to solve which we must know the value of r x . Now the combined resistance of the shunt and G -f r, we can Bee from [1] (page 71) is therefore the value required to be given to E 2 , in order to preserve the resistance between D and E, equal to G, when S L is connected, will be or E 2 + r- 1 = G^F- 1 ; W SHUNTS. 73 but from [3], [4], and [5] (pages 71 and 72) E 2 + r 1 = E 1 = G^L 1 ; [7] n therefore, subtracting [7] from [6], we have that is, 1 - n, ni - n T i - = (jr - ! ! or n - " , = G consequently the value of S x will be In like manner it could be shown that the resistance necessary to give to S 2 and r -f r 2 , to reduce the deflection to its - tn P art would be s * = } (., - 1)*' and or Finally we have from [3] and [5] n = RI - (^ + r 2 ) = alZl- ( r , + r2 ). To summarise then, 74 HANDBOOK OF ELECTRICAL TESTING. = eiiX and for any other shunt S p The compensating resistances between the shunts will be and also we have or The Zas< resistance r, beyond the last shunt will be For example. It was required to provide a galvanometer with T Vth, and Tff V?r shunts, and with corresponding compensating re- sistances arranged according to Fig. 35. What should be their values ? We have n = 1000, n = 100, w 2 = 10 ; therefore, n - 1 = 999, Wj - 1 = 99, w 2 - 1 = 9. Then and SHUNTS. 999 X 100 -.f-v.oiom 1000 x 99 X 99 ~ -10(!? 9 X X 9X =QX - iaM88; 75 from which r 2 = G (-11 - -Q090909) = G X -1009091 ; also 999 = G (* 999 - ' n ) = G X <889 ' If the resistance of the galvanometer, for which these shunts and compensating resistances are to be provided, is 5000 ohms, then S = 5000 X -001001 = 5 -005 ohms. 51 = 5000 X -010193 = 50-965 5 2 = 5000 x '123333 = 616-655 r x = 5000 x -0090909 = 45-455 r 2 = 5000 X -1009091 = 504-545 r, = 5000 X -889 = 4445-000 Fig. 36 shows how an ordinary Thomson galvanometer shunt box would be arranged with compensating resistances. The plug hole -, when it has a plug inserted in it, connects the top left-hand brass block to the bottom left-hand block, and so leaves the galvanometer connected to the terminals of the shunt box without any additional resist- ance in its circuit. The connection between FIG. 36. 'breed size these brass blocks is shown by the dotted line in Fig. 35 (page 71). 70. The accurate adjustment of ordinary shunts is often a somewhat troublesome operation, in consequence of the nume- rical values of the resistances of which the shunts are composed not being whole numbers ; thus, supposing the resistance of the 76 HANDBOOK OF ELECTRICAL TESTING. galvanometer to be 5000 ohms, then the resistance of the ^th shunt would have to be 5000-4-9, or 555*56; and, practically, this could not be adjusted to a greater degree of accuracy than one decimal place. Similarly, the ToVkh shunt should have a resistance of 5000 -4- 99, or 50 505, and the 10 1 00 th shunt a resistance of 5000 -4- 999, or 5-005, both of which numbers are somewhat awkward to adjust exactly. Now on page 71 (equation [1]) we saw that the combined resistance of the galvanometer and its shunt was , conse- 7 quently to adjust the T yth shunt we may connect it to its gal- vanometer coil, and adjust it until the joint resistance of the two becomes equal to 5000 -4-10, or 500 ohms. Similarly, the TDir* n shunt would be adjusted by connecting it to the galvanometer coil, and adjusting it until the joint resistance was found to be 5000 -4- 100, or 50 ohms ; lastly, in like manner we should adjust the 10 1 00 th shunt until the combined resist- ance of the two became 5000 -4- 1000, or 5 ohms. 71. We have shown in a previous chapter (page 48) that the deflections on the scale of a Thomson galvanometer, except when they are nearly equal, are not directly proportional to the current strengths which produce them, and that to compare them a formula must be used. If we wish to avoid the use of this formula we must adopt some method of avoiding widely different deflections. This we can do by using a variable shunt for the galvanometer, and with it obtaining either equal, or nearly equal, deflections for all measurements made in one set of tests. The graduated scale of any galvanometer, it should be recol- lected, is not only for the purpose of enabling the strengths of two or more currents to be compared by different deflections, but is also for the purpose of enabling any deflection which may be obtained to be reproduced when required. 72. It is best to obtain as high a deflection as possible, for then not only will a slight variation from the correct resistance of the shunt produce a greater number of degrees of variation from the deflection required, than would be the case if a low deflection was used ( 60, page 64), but also a higher resistance being required for the shunt, a greater range of adjustment is given to it. 73. By the help of the points we have just considered we can graduate or calibrate ( 43, page 46) the scale of a galvanometer. To do this, first calculate from the known resistance of the galvanometer, the resistance of shunts required to reduce the amount of current passing through the galvanometer when no SHUNTS. 77 shunt is inserted, to ^, , J, ^, &c., the amount passing when a shunt is inserted, then the resistance of the shunts necessary to reduce the current to 4, i j, ---- ith will, as we have shown, be of the resistance of the galvanometer. Now, as we have also shown, the insertion of shunts reduces the resistance of the circuit in which the galvanometer is placed ; we must therefore also calculate the resistances necessary to be inserted in the circuit in order to compensate for the reduction of resistance which takes place when a shunt is inserted. These resistances will be respectively of the resistance of the galvanometer. The shunts and their compensating resistances being calcu- lated, to calibrate the galvanometer we proceed as follows : The galvanometer, a resistance box, and a battery are joined up in circuit. The J shunt, that is, the shunt equal in resist- ance to the galvanometer, is then inserted, together with the corresponding compensating resistance in the resistance box. Sufficient resistance is now added in the latter to bring the deflection down to, say, 1 ; the shunt and compensating resist- ance are then removed, and as the resistance in circuit is the same as before, and also the whole of the current passing in the circuit now passes through the galvanometer, the strength of current affecting it is exactly double that which deflected the needle originally; the deflection of the needle, therefore, now represents a strength of current double that of the previous experiment. We next insert the ^ shunt and its compensating resistance, and by again adjusting the resistance coils, obtain a deflection of 1 ; on now removing the shunt and compensating resistance we get three times the strength of current passing through the galvanometer; the deflection obtained therefore will represent that strength, and so by inserting shunts one after another, and repeating the process described, we can get the deflections corresponding to strengths of current equal to 1, 2, 3, 4, . . . . n, and the scale can be marked corre- spondingly ; or these deflections and the corresponding currents producing them can be embodied in a table, so that by referring 78 HANDBOOK OF ELECTKICAL TESTING. to the latter we can at once see the relative powers of various currents giving different deviations of the needle. 74. By the help of this method of calibrating a galvanometer we can determine its angle of maximum sensitiveness ( 28, page 23). All we have to do is to obtain various deflections of the galvano- meter needle with various shunts and their corresponding com- pensating resistances, and in each case to increase the deflection of the needle slightly by reducing the compensating resistances by the same amount ; then the required angle will be the one at which the diminution of resistance produced the greatest increase of deflection. 75. It is evident that if, in making a measurement, we want to reduce the deflection of our galvanometer to a readable value, we can do so, either by placing a large resistance in the circuit of the instrument or, by introducing a shunt between its terminals. It is possible also, in certain cases, to produce the same effect by connecting a shunt between the poles of the battery, but this is not always advisable, as it interferes with the constancy of the latter. If the resistance of the battery and galvanometer in a simple circuit be very high it requires a very considerable increase of resistance in the circuit to produce an alteration in the deflec- tion of the galvanometer needle, whereas just the reverse is the case if a shunt be used to produce that effect. This fact is an important one, as it has a considerable bearing upon the accuracy with which measurements can be made. ( 79,) CHAPTER V. MEASUKEMENT OF GALVANOMETER RESISTANCE. HALF DEFLECTION METHOD. 76. THE simplest method of determining the resistance of a galvanometer is perhaps the one we have already given on page 5 ( 8). In this method it will be seen we joined tip the galvanometer, whose resistance (G) was required, in circuit with a resistance p, and a battery of very low resistance, and having obtained a certain deflection we increased p to E, so that the current passing in the circuit became halved in strength, the resistance (G) of the galvanometer was then given by the formula G = E - 2p. If we were measuring the resistance of a tangent galvanometer, the deflections obtained should be such that the tangent of one deflection is half the tangent of the other, the precaution against having the deflections too high or too low being duly taken ( 28, page 23). For example. With a tangent galvanometer whose resistance (G) was to be determined, and a battery whose resistance was very small, we obtained with a resistance in the resistance box (as the set of resistance coils is sometimes termed) of 10 ohms (p) a deflection of 58, and by increasing the resistance to 120 ohms (E) the deflection was reduced to 38 J (tan 38i = J tan 58) ; what was the resistance of the galvanometer ? G = 120 - 2 x 10 = 100 ohms. 77. In measuring the resistance of an ordinary galvanometer by this method it would be necessary to know what ratio the deflections bore to the current strengths producing them, so that the resistances may be adjusted accordingly. A convenient arrangement is to employ a tangent gal- vanometer of a known low resistance in circuit with the 80 HANDBOOK OF ELECTRICAL TESTING. galvanometer whose resistance is required, and to take the readings from the tangent galvanometer, the resistance then obtained from the formula will evidently be the resistance of the two galvanometers together. If, then, we subtract from this result the known resistance of the tangent galvanometer, we get the resistance we are trying to obtain. If we have not a tangent galvanometer at hand, and if moreover we cannot tell what ratio the deflections bear to the current strengths producing them, we must of course employ a different method of testing. 78. In this, and indeed in all tests, it is important to consider what resistances and battery power should be employed to make the measurements, so that the greatest possible accuracy may be ensured. If we employ very high resistances to measure a low resist- ance, a considerable alteration in the former would produce but little alteration in the current flowing through the gal- vanometer, for the electromotive force being constant, this current, and consequently the galvanometer deflection, is dependent upon the total resistance in the circuit, and an alteration of several units in a large total practically leaves its value the same, but then a few units too much or too little inserted in a formula may make the result appear very much greater or less than its true value. Thus, in the test we have been considering, suppose the battery power had been such that we found it necessary to have the resistance p = 2000 ohms, and that to halve the deflection we found it necessary to increase p to 4100 ohms (R), this would make the resistance of the galvanometer to be, as we saw before, G = 4100 - 2 X 2000 = 100 ohms. Now, practically, if the resistance R had been made 4200 ohms the deflection would have been halved ; whatever differ- ence there was would scarcely be appreciable. If now we work the result out from the formula we get G = 4200 - 2 x 2000 = 200 ohms, or double what it ought to be. It is possible indeed that the error might be greater than this. The test, in fact, would be quite useless. In order to have the best chance of accuracy we should make our resistances as low as possible, for then a small change or error in the latter produces the greatest increase or decrease in the current, and consequently also in the deflection of the galvano- MEASUEEMENT OF GALVANOMETER RESISTANCE. 81 meter needle, and, on the other hand, it produces the smallest error in the value of G, when the latter is worked out from the formula. In order to make R as low as possible it is evident that we must make p as low as possible. 79. What degree of accuracy is attainable in making the test? This is dependent upon the " total possible percentage of error which may exist in the second deflection" ( 41, page 43). We have then to consider what error in the value of G the total error in the second deflection will cause. The error in G must be occasioned by the value of R being obtained incorrectly, this wrong value of R being due to an error made in reading the magnitude of the second deflection. If in the formula G = R-2 P we make a mistake of, say, \\ per cent, in R, then the resulting T> percentage error, Y, in G will be X' = X\ Now the accuracy with which we can adjust R is directly dependent upon the accuracy with which we can adjust (G + R) for the latter is the total resistance in the circuit of the galvanometer, and therefore any change or error made in the value of the galvanometer deflection (the second deflection) must be in direct proportion to the change or error made in (G -f R) ; consequently if we are liable to make an error of y' per cent, in the value of the second deflection, and an error of X\ per cent, in R, then we must have y : \\ : : R : G + R or but or, and G = R - 2p, or, R = G + 2p, therefore hence 82 HANDBOOK OF ELECTRICAL TESTING. For example. In measuring the resistance of the galvanometer in the example given in 76, page 79, it was known that the "total possible percentage of error (/) which could exist in the second deflection " could not exceed 1 7 per cent. (Example 2, page 45). What would be the percentage of accuracy (A') with which the value of G could be determined ? A' = 2 fl + ~\ 1-7 = 3-7 per cent. A single cell of a battery is the lowest electromotive force that can be practically employed in making the test, but we may find that this one cell gives too low a deflection with the lowest value we can give to p, that is 0, and two cells too high a deflection ; we should have, therefore, to employ two cells and then increase p until the proper deflection is obtained. Now on pages 80 and 81 it was pointed out that it is best to make p of a low value, so that the deviation of the needle from its correct position, when E is not correctly adjusted, may be as great as possible ; but equation [A] (page 81), which represents the relative values of the errors A' and y', although it shows that the error A.' is smallest when p is as small as possible, at the same time shows that we gain but little by making p very much smaller than G, for A' is only twice as great when p = 0, as it is when p = G. 80. Practically we may say therefore that the Best Conditions for making the Test are to make p a fractional value of G ; and in the case of a tangent galvanometer the two deflections obtained should be as nearly as possible 55 and 35J ( 32, page 28). Also as regards the Possible Degree of Accuracy attainable. If we can determine the value of the deflection of the galva- nometer needle to an accuracy of y' per cent., then we can determine the value of G to an accuracy of 2 I 1 -f- -^ ) y per C6 If P is very small, then At f) I = 2 7 ; MEASUREMENT OF GALVANOMETER RESISTANCE. 83 so that even under the best conditions for making the test the accuracy with which the value of G could be determined would be only one-half of the accuracy with which the deflections could be observed. 81. It must be understood that the resistance of the testing battery can only be neglected when it forms a small percentage of the total resistance in its circuit. If, then, the galvanometer to be measured has a low resistance, inasmuch as E will have to be proportionately small, the battery resistance can no longer be ignored without introducing an error; moreover, if E is made small, its range of adjustment becomes very limited. The test, therefore, is not suitable for measuring galvanometers whose resistance consists of a few units only. EQUAL DEFLECTION METHOD. 82. The theory of this method is as follows : The galvano- meter whose resistance G is re- quired, a resistance p, and a battery E of very low resistance, are joined up in circuit, as shown by Fig. 37, a shunt S being between the ter- minals of the galvanometer ; a de- flection of the galvanometer needle is produced. Let be the cur- rent flowing out of the battery, then FIG. 37. C - E This current divides into two parts, one part going through S, and the other part through the galvanometer. It does this in the inverse proportion of the resistance of those circuits, the part, GU going through the galvanometer being E S ES GS G G + S The shunt S is now removed ; this causes the deflection of the galvanometer needle to be increased, p is now increased to E, so that the deflection becomes the same as it was previous to the removal of the shunt or, in other words, so that the strength G 2 84 HANDBOOK OF ELECTEICAL TESTING. of the current passing through the galvanometer is C 1? then E r - OI -KTG' therefore ES E By dividing both sides by E and multiplying up, we get SR+GS= GS-f Sp + Gp; therefore Gp = SR-Sp, from which G= S R - p . P For example. A galvanometer whose resistance (G) was required, was joined up in circuit with a resistance of 200 ohms (p), a shunt of 10 ohms (S) being between the terminals of the galvano- meter. On removing the shunt, it was necessary in order to reduce the increased deflection to what it was originally, to increase p to 2200 ohms (K). What was the resistance of the galvano- meter ? 83. In making this test practically, we should proceed thus : Join up the instruments, as shown by Fig. 38, taking care that the two infinity plugs are firmly in their places. Plug up the three holes between B and C, and remove the necessary plugs between D and B. Next remove plugs from between D and E, so as to introduce the resistance p. On the right-hand key being depressed the deflection of the galvanometer needle is obtained. The galvanometer should be gently tapped with the finger in order to see that the needle is properly deflected and is not sticking, as it is very liable to do, especially when a compass suspended needle is used. The oscillations of the needle may be arrested by a skilful manipulation of the key ; slightly raising it when the needle swings under the influence of the current and again depressing it when it recoils. The needle being steadily deflected, and the precise resist- MEASUREMENT OF GALVANOMETER RESISTANCE. 85 ance (p) in the box noted, the left-hand infinity plug must be removed, and the resistance between D and E increased until the deflection becomes the same as it was at first, and the resistance (E) being noted, the formula is worked out. FIG. 38. 84. What are the best values of S and p to employ in making a test like this? Should we make S and p of low, high, or medium values ? The answer to these queries has an important bearing upon the accuracy with which the test can be made; and as we shall more than once have to consider questions of a similar kind, we shall in the present instance enter at some length into the problem. There are two quantities whose values we have to determine, viz. S and p ; let us first consider what S should be, supposing K to be a given quantity and p to vary along with S. If we examine the formula we shall see that if we make S small, then an error ofone or two units in the correct value of R will make a much gnmtor difference in the formula than would be the case when there is the same number of units of difference with S large ; thus to take a numerical example, suppose we had the following values in the formula : and suppose we made K 120 units too large, we should then have n ' 620 - 20 _ G = o - - = loO chms; 86 HANDBOOK OF ELECTRICAL TESTING. or an error of 150 120 = 30 ohms. Next let us suppose we Lad the following values : i 120 ohms, and as before let there be an error of 120 units in E, we then have or an error of 264 120 = 144 ohms, and if S and p had been higher still we should have seen that the error would have been still greater. To put the case in another way ; in the last example let us suppose the error in E had been, not 120 units, but 25 units ; that is, make E = 500 + 25 = 525 ; we then find that G = 480 525 ~ 40 = 150 ohms. 400 The error in G, in fact, in the former case, where E was 120 units too large, was no greater than it was in the latter ease, when the excess in the correct value of E was but 25 units. From this it must be evident that it is highly advantageous to make S as small as possible. Let us, however, put the matter in algebraical form ; thus, let X be the error in G, and let $ be the excess in the value of E which causes this error, then we have p p p and Q E- P SE G = S , or , P =^-^; therefore by subtraction, From this we see that with a constant error $, made in E, the corresponding constant error X, made in G, will be as small as possible when S is very small, as indeed we before proved ; but we also see that we gain but little by making S a very small fiactional value of G, for the error would be only twice as great MEASUREMENT OP GALVANOMETER RESISTANCE. 87 with S = Gr as it would be if S were very nearly = 0. It would not do, however, to make S greater than Gr, for Gr -f- S increases very rapidly by increasing S. Practically, therefore, we may say : make S a fractional value of G. We have next to determine what is the best value to give to p, supposing S to be a fixed quantity. Now if we put the equation in the form we can see that whatever value p has, E will have an exactly proportional corresponding value ; thus to take the example we first had, viz. : 500-20 /500 if in making the test we had made p = 2 x 20 = 40, instead of 20, then the value to which E would have required to have been adjusted would have been 2 x 500 = 1000, instead of 500, Further, if E had had this value, then an error of 20 units in E would have produced the same error in Gr as would the 10 units in the first case, when E was 500. At first sight then it might appear that it would not matter what value we gave to p. Let us, however, consider in what way the adjustment of E is effected. The means by which we adjust E is by observing the deflec- tion of the galvanometer needle, and seeing whether we have brought it to the deflection it had when p and S were the resist- ances in the circuit ; when this deflection is correct we know that E is correct. But the accuracy with which we can adjust E evidently depends upon the divergence of the needle from its correct position being as large as possible when E is not exactly adjusted, and if this divergence is greater when we alter E from 1000 to 1020 ohms than when we alter it from 500 to 510 ohms, then it is better so to arrange the value of p that E shall be 1000 ohms. Or in other words, if the error in E, corresponding to a con- stant error in G, produces a greater divergence of the needle from its correct position when E is large than when it is small, then it is better to have E large than small. 88 HANDBOOK OF ELECTKICAL TESTING. Now the current C producing the deflection of the galvano- meter needle is p E STo' and if we suppose there to be a diminution c, in C, caused by an error $, in E, then we have E . -R + < + G> or E ~B + * + G 5 but we know that therefore E E E

e = wo and then we get _y_ = X'GE 100 - 100 (G - that is to say, For example. In measuring the resistance of the galvanometer in the example given on page 84, it was known that the possible error y in the current, due to the deflection being incorrect, would not exceed 88 per cent. (Example, page 43). What would be the percentage of accuracy (V) with which the value of G oould be determined ? 86. The practical results, then, that we have arrived at from these investigations are, that to obtain the Best Conditions for making the Test : First make a rough test to ascertain approximately the value of G. Having done this, insert a shunt (S) between the termi- nals of G, of a fractional value of the resistance of G. Next join up p in circuit with G and its shunt S, making p as large as possible, but not larger than ~ R ; E being the highest resistance that can be obtained. Insert in the circuit sufficient battery power of low resistance to bring the deflection of the galvanometer needle as nearly MEASUREMENT OF GALVANOMETER RESISTANCE. 91 as possible to the angle of maximum sensitiveness (page 23), ad- justing p, if necessary, so that this angular deflection becomes exact, and note the exact value of p. Now remove the shunt and increase p to E, so that the increased deflection becomes the same as it was at first. Note E, and then calculate G from the formula. Possible Degree of Accuracy attainable. If we can determine the value of the galvanometer deflection to an accuracy of y per cent., then we can determine the value of G to an accuracy (A') of +y per cent If S is very small, and E very large, then Xi r = y, so that under the best conditions for making the test, the accu- racy with which the value of G could be determined would be the same as the accuracy with which the value of the deflection could be observed. 87. In the practical execution of the test, inasmuch as there are only three resistances between D and B (Fig. 38) our choice of a shunt is limited from this source, but these three will usually be sufficient for most purposes. 88. The method we have described of making the test may be modified by making S or p the adjustable resistances instead of E, but in either of these cases it can be shown, by an inves- tigation precisely similar to the one we have made, that the proper values of the resistances should be those we have indicated. The test could also be simplified by making S = p, in this case we get such an arrangement, however, would not give the conditions for obtaining maximum accuracy. FAHIE'S METHOD. 89. If in the last test we make S the adjustable resistance, and make E = 2 p, we get 92 HANDBOOK OF ELECTKICAL TESTING. that is, the resistance of the shunt will be the resistance of the galvanometer. 90. The connections for making the test with the set of resist- ances shown by Fig. 38 would have to be so arranged that the resistances between D and E form the shunt, and those between D and the resistances p and E. This arrangement, however, in consequence of there being so few plugs between D and C, is not a satisfactory one, as some difficulty would probably be found in adjusting the battery power and resistance E so as to obtain the deflection of maximum sensitiveness. With two sets of resistance coils, however, the test can easily be made. As in the previous method, it is best to make the resistance E as high as possible, for then any small change in the value of S produces the greatest movement of the galvanometer needle. The possible degree of accuracy attainable is the same as in the last test. 91. In order that satisfactory results may be obtained in the foregoing tests, it is necessary that the galvanometer be a sensi- tive one (page 66), otherwise even a moderate degree of accuracy cannot be assured. It is also very advantageous to arrange the resistances in connection with a key, as shown by Fig. 39. The key, K, it will be observed, in its normal position short FIG. 39. circuits the right-hand resistance a, so that p is the only re- sistance in circuit; when however the key is depressed the short circuit becomes opened and p consequently becomes in- creased to E, whilst at the same time the shunt, S, becomes connected to the galvanometer ; in practically making the test, therefore, what we have to do is to adjust p (or S, if p is a fixed quantity) until the deflection on the galvanometer remains the same, whether the key is up or down. As a break occurs when the key passes from the top to the bottom contact, during which break a slight movement of the galvanometer needle may take place, a preferable arrangement MEASUREMENT OF GALVANOMETER RESISTANCE. 93 is that suggested by Prof. Moses Farmer.* The key in this arrangement (Fig. 40) consists of two levers, L and I; the latter is normally in contact with a stud joined to the junction of the two resistances p and a. When L is depressed it makes contact with Z, and at the same moment moves away the latter from its contact stud, thus practically no break takes place. THOMSON'S METHOD. 92. Join up the galvanometer g with resistances a, b, and d, and a battery of electromotive force E and resistance r, as shown by Fig. 41, and let a key be inserted between the points E and B, so that by its depression these points can be connected together. First, let us suppose the key to be up and the points con- * ' Electrical Review,' Sept. 24tb, 1886, page 316. 94 HANDBOOK OF ELECTEICAL TESTING. sequently disconnected. The current C l flowing through the galvanometer will then be (g + V) (d + g) ^ a+b+d+g ~T" ,, ' /, ' / i , [1] Next, suppose the key to be depressed and the points E and B thereby to be connected together, then the current (0 2 ) flowing through the galvanometer will be E a = ~ ~ X ~ = E a (6 + d) bd(a + g)' Qt Of 1 Ci = C 2 , Further, let us suppose the adjustment of the resistances to be such that we then get E (a + b) a (6 + d) _ '> by multiplying up and arranging the quantities we get (b ad +[rf(& + <7) + &<;]( + &); therefore that is, , , ad ad = bg, or, g = . A great advantage of this test is the fact of its being entirely independent of the battery resistance. It is also very easily made, as must be evident. MEASUEEMENT OF GALVANOMETEK KESISTANCE. 95 In making the test practically, the connections would be made as shown by Fig. 42. The terminals E and Bj would be joined by a short piece of thick wire. The other connections are obvious. FIG. 42, The left-hand key (which is not shown in the theoretical figure) being first depressed and then kept permanently down, the right-hand key must be alternately depressed and raised, the resistance d, that is the resistance between A and E, being at the same time adjusted until the deflection of the galvano- meter needle remains the same whether the key is up or down. 93. We will now determine the best arrangement of resist- ances for making the test. What we have to do is to suppose that in the equation ad there is a small but constant error in g, caused by a correspond- ing error in one of [the other quantities, let us say d, and then find what values of d and say, a, will cause the alteration of the deflection of the galvanometer needle produced on raising and depressing the key, to be as large as possible. Let \ be the difference between the exact value of g and the value given it by the formula when we have d too large, and let the increased value of d be d. We then have 96 HANDBOOK OF ELECTRICAL TESTING. therefore We next have to determine what the alteration in the strength of the current passing through the galvanometer, produced by raising and depressing the key, is equal to. If in either equation [1] or equation [2] (page 94) we put b g equal to a d, or b equal to , then the resulting equation will give the current, C, which would flow through the galvanometer when the adjustment is exact ; by doing this we get c= When the adjustment is not exact, the currents produced on raising and depressing the key will be obtained by equations [1] and [2] (page 94), and the difference between these two cur- rents relative to the current produced when exact equilibrium is obtained will give the relative current producing the altera- tion in the deflection of the galvanometer needle ; hence we find 0,-Ci + &) (g+a) + d x 6 (g+a) b + a '0*i + g + b + a) + (d l + g) (b + i-lg) {r (d, + b) + d, (6 + a) } (r (a + g) + a(d + g) r(d 1 +g+b+a)+(d 1 +g) (6+ ) but since a d 1 is very nearly equal to b g, we may without sensible error put ad t = ad = b g, or b = , except where dif- ferences are concerned ; in which case we get C 2 - Cj g(fd 1 - Ig) ~ a and since a d l = b g + b X, and - = -, we 9 ~~' [ A ] MEASUREMENT OF GALVANOMETER RESISTANCE. 97 From this it is evident that, in order to make 2 ~ - 1 as large as possible, we must make d as large, and a as small, as possible. o o It is evident also that, as regards increasing ? -, it is useless making d very much larger, or a very much smaller, than g. If we make d about ten times as large, and a ten times as small, as g, we shall have good conditions for ensuring accu- racy, though as regards our power of adjustment, it would be advantageous to make d larger still if possible. From the equation bg = ad we see that g being a fixed quantity, and a as small as possible, we can make d as large as we like by making b as large as possible. 94. It may be pointed out, that when a is small and d and b large, we have the battery connecting the junction of the two greater with the junction of the two lesser resistances. 95. What degree of accuracy is attainable in making the test ? This we can determine from equation [A]. Let us then, in the latter, substitute percentages for absolute values, that is, let i_ '' 100 and let TOO, * then we get y X f g 100 (a that is, For example. In measuring the resistance of a galvanometer by the fore- going method, the values of a, 6, and d were 10, 100, and 300 ohms respectively. What was the resistance of the galvano- meter, and what was the possible degree of accuracy attainable ? The smallest change in the value of the galvanometer deflection which it was possible to observe,* was -88 per cent. ( 40 page 42). * This is synonymous with "the degree of accuracy with which the value of the galvanometer deflection can be read " (page 42). H 98 HANDBOOK OF ELECTRICAL TESTING. 10 X 300 = = 30 ohms. To sum up, we have Best Conditions for making the Test. 96. Make a not greater than T Vth of =-- approximately. Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of ^th of a division, then we can determine the value of G to an accuracy, A.', of ., (D + d)100/ 1 er cent - If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of -Hh of a degree, we can determine the value of G to an accuracy, X f , of (tan D 8 2 + tan d 8,) 100 X = tan f (to D - to>) where Si = tan Df rt - tanD, and, 8 2 = tan^ - tan d. DIMINISHED DEFLECTION SHUNT METHOD. 108. Keferring to Fig. 43, this method is as follows : The galvanometer G, whose resistance is to be determined, is joined up with a resistance E, a battery E, and a shunt Sj ; the deflection obtained is noted ; let this deflection be due to a current Cj, then (page 83) we have MEASUREMENT OF GALVANOMETER RESISTANCE. 109 or The resistance of the shunt is now reduced to S 2 , so that the galvanometer deflection is also reduced ; let this new deflection be due to a current C 2 , then we must have FIG. 43. therefore C 2 G (S 2 + E) + C 2 S 9 B _ G! Q- (Si + E) + d S 1 E S 2 S! that is G [C 2 S x (S 2 + E) - C x S 2 (Si + E)] = S x S 2 E (d - C 2 ), from which we get S.S.EfO.-O,)' or C 2 S 1 (S 2 +E)-C 1 S 2 (S 1 -|-E) J p ft G = [A] In the case of a tangent galvanometer, if the deflections, D and d, are read from the tangent scale, then we should have G = T>-d [B] 110 HANDBOOK OF ELECTRICAL TESTING. (1.) For example. With a tangent galvanometer whose resistance, G, was required, and a battery of very small resistance, we obtained with a shunt of 200 ohms (S^, a deflection of 60 divisions (D) 011 the tangent scale of the instrument; when the shunt was reduced to 25 ohms (S 2 ), the deflection was reduced to 20 divi- sions (d). The resistance, E, was 400 ohms. What was the resistance of the galvanometer ? G = -- ~ ^- -- -- = 100 20 (25 + 40o)- 6 (200 + 400 If the deflections are read in degrees, then in equation [B] we must substitute tan D and tan d for D and d respectively; we then get __ tan D - tan d _ ~' (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the instrument, and deflections of 50 (D) and 2l (d) respectively were obtained. The values of S w S 2 , and E were 200, 25, and 380 ohms respec- tively. What was the resistance, G, of the galvanometer ? tan 50 = 1-1918, tan 21 J = -3990, therefore 1-1918 - -3990 G = - -- -- = 100 ohms. 109. If we make the test by having no shunt inserted when the first deflection is observed, that is to say, if we have S { = oo , or, = 0, then equation [B] becomes - - D - d EDI /I 1\_D \s: + sJ ~ E MEASUREMENT OF GALVANOMETER RESISTANCE. Ill and equation [C] tan D - tan d (jT = : : =r^. B Further still, if we make E a very high resistance, that is, if in equations [D] and [E] we make = 0, then we get the simplifications and , /tan D 110. In order to determine the "Best conditions for making the test," and also the " Possible degree of accuracy attainable," let us write equation [A] (page 109) in the form, G d - C 2 Now this equation is similar in form to equation [B] (page 102) in the last test (Diminished deflection direct method), the only difference being that we have ^ instead of G, and ( ^- -f- ) and (jc \b 2 JK/ ( ^- + p J instead of B and p, respectively ; and inasmuch as an A.' per cent, error in -^ is an A.' per cent, error in G (though of the opposite sign), we can see that the value of X' must be expressed by an equation of the same form as equation [H] (page 107), that is to say, we must have We can see, therefore, from the investigations in the last test that we must have 112 HANDBOOK OF ELECTRICAL TESTING. Best Conditions for making the Test. 111. Make Si and K as large as possible * ( 107, page 108). Make S 2 of such a value that when the deflections, D and d, are in divisions, then approximately; and when the deflections are in degrees on a tangent galvanometer, then approximately. Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of -^th of a division, then we can deter- mine the value of G to an accuracy, A/, of If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of |,th of a degree, we can determine the value of G to an accuracy, A', of . , (tan D S 2 + tan d 6\) 100 X = tan *> (tan D - tan .ff) where Sj = tan D^ - tan D, and, S 2 = tan d - tan d. 112. It may be remarked, that in the foregoing methods unless the galvanometer under measurement has a high degree of "sensitiveness" (page 66), then even a moderate degree of accuracy in making the tests cannot be assured. * The investigations in the case of the last test prove that we should make (5- +^ ) as small as possible ; this, of course, is equivalent to making Vbj K/ B! and R as large as possible. ( 113 ) CHAPTER VL MEASUREMENT OF THE INTEKNAL EESISTANCE OF BATTEKIES. HALF DEFLECTION METHOD. 113. On page 5 a formula is given for determining the re- sistance r of a battery, viz. : where G is the resistance of the galvanometer employed to make the test, p a resistance which gave a certain current through the galvanometer, and R a larger resistance which caused the strength of this current to be halved. As this, though a simple, is a very good test, and is one which is very frequently made use of, a numerical example may prove of value. For exai With a galvanometer whose resistance was 100 ohms (G), and a battery whose resistance (r) was to be determined, we obtained with a resistance in the resistance box of 150 ohms (p), a deflection representing a current of a certain strength, and on increasing p to 600 ohms (R), we obtained a deflection which showed the current strength to be halved. What was the resistance of the battery ? r = 600 - (2 X 150 -f 100) = 200 ohms. To avoid mistakes, it should be carefully observed that in working out the formula we " First double the smaller resistance ; to the result add the resistance of the galvanometer) and deduct this total from the greater resistance." 114. A very common method of making this test is to employ & galvanometer of practically no resistance, and to take the first deflection with no resistance in the circuit except that of the battery itself. In this case (2 p -f G) = 0, so that or the added resistance is the resistance of the battery. i HANDBOOK OF ELECTEICAL TESTING. 115. If we compare the first method ( 113) with the test for determining the resistance of a galvanometer described on page 79 ( 76), we can see that the two are almost identical. In the one case we determine the resistance of the galvanometer, and in the other we determine the resistance of the battery plus the galvanometer, and then from the result deduct the value of the galvanometer. This being so, we can see that the Best Conditions for making the Test are obtained by making p + G a fractional value of r ; to do which we should require a galvanometer of low resistance. As regards the possible degree of accuracy attainable, we can see from the galvanometer test referred to, that that is to say : Possible Degree of Accuracy attainable. If we can be certain of the value of the galvanometer deflec- tion to an accuracy of y' per cent., then we can be certain of / C^ \ the accuracy of the value of r within 2 M -j J y per cent. Or if we employ a galvanometer of low resistance, then we can be certain of the accuracy of the value of r within 2 y per cent. If the galvanometer deflection be too high, i.e., above about 55 (page 28, 32), with the lowest value we can give to p, then the galvanometer must be reduced in sensitiveness by being shunted, and the value of G in the formula will then be the combined resistance of the galvanometer and shunt, that is, the product of the two divided by their sum (page 70). . THOMSON'S METHOD- 116. Fig. 44 shows the theoretical, and Fig. 45 the practical methods of making this test. The theory of the method is as follows : The galvanometer G, a resistance p, and the battery whose resistance r is required, are joined up in simple circuit with a shunt S between the poles of the battery ; a deflection of the galvanometer needle is produced with a resistance pin the resistance box. The shunt is now removed ; this causes the deflection to become larger ; INTERNAL RESISTANCE OF BATTERIES. 115 p is then increased until the deflection becomes the same as it was at first. Let the new resistance be K, and let E be the electromotive force of the battery and C the current passing through the galvanometer. FIG, 44. FIG. 45. In the first case we have E r\ S (P + G) S + p + G ES and in the second case C = E r 4. K therefore E ES 5y multiplying up and cancelling, r = S P + G' i 2 116 HANDBOOK OF ELECTKICAL TESTING. For example. A battery whose resistance (r) was required, was joined up in circuit with a resistance of 200 ohms (p) and a galvanometer of 100 ohms (G), a shunt of 10 ohms (S) being between the poles of the battery. On removing the shunt it was necessary, in order to reduce the increased deflection to what it was originally, to increase p to 3200 ohms (E). What was the resistance of the battery ? 3200 - 200 r = 10 200 + 100 = 10 117. The investigation for determining the best resistances to employ in making this test would be conducted in precisely the same manner as that given on page 85, et seq. For the equation is the same as _ (B + G)-( P + G) P + G which is the same kind of equation as the one in the test we have referred to, viz. : a-s5^ ; P and as in this case we proved that S was to be as small and E as large as possible, so from the preceding equation we should prove that S should be as small, and E + G as large, as possible. In order, therefore, to obtain the Best Conditions for making the Test, 118. First make a rough test to ascertain approximately what is the value of r. Having done this, insert a shunt (S) between the poles of the battery, of less resistance than r. Next join up p in circuit with G, with the battery, and with a its shunt S, making p + G not larger than -^ (G + E) ; E being the highest resistance that can be inserted in the circuit. The galvanometer needle being obtained at the angle of maximum sensitiveness, note the value of p. Now remove the shunt and increase p to E, so that the increased deflection becomes the same as it was at first. Note E and calculate r from the formula. INTERNAL RESISTANCE OF BATTERIES. 117 Possible Degree of Accuracy attainable. From the galvanometer test referred to, we can see that if we can determine the value of the galvanometer deflection to an accuracy of y per cent., then we can determine the accuracy of r to an accuracy of 119. As we cannot in this test vary the resistance of the galvanometer so as to obtain the deflection at the angle of maximum sensitiveness, we must, if the deflection be too high with the highest resistances we can put in the circuit, reduce its sensitiveness by means of a shunt between its terminals ; the value of G in the formula will then be the combined resistance of the galvanometer and its shunt. The constancy of a battery being much impaired by its being on a circuit of low resistance, it is not advisable to reduce the deflection of the galvanometer by making S very small. In fact S, although it should be lower than the resistance of the battery, should not, in this test, be made lower than we can help. Thus, if the resistance of the battery were about 200 ohms, it would be preferable to make S 100 rather than 10 ohms. Should the deflection of the galvanometer needle be too low, the only thing to be done is to use another which has a higher figure of merit. 120. A Thomson galvanometer answers very well for tests like this, as its figure of merit can always be made sufficiently low by placing a shunt made of a short piece of wire between its terminals. 121. If we adjust p in the first place so that together with G- it equals S, we get the simplified formula iat is, the added resistance is the resistance of the battery.* Again, if we commence with no other resistance in the galvanometer circuit beyond that of the galvanometer itself, we get the simplification r-Bg; * Sabine's ' The Electric Telegraph,' p. 314. 118 HANDBOOK OF ELECTRICAL TESTING. Lastly if we make S = Gr, then we get r = E. . If we arrange the tests, however, so as to use these simplified formulae, we are obliged to employ an arrangement of resist- ances which would not be at all advisable if we wish for accuracy, and it is very questionable whether any advantage is gained by adopting a simplification of a formula, in itself simple, at the expense of accurate testing. The arrangement of keys described in 91, page 92, may obviously be applied to the foregoing tests with advantage ; in fact, the key suggested by Professor Moses Farmer * was first applied by that gentleman to the last test mentioned, viz., that in which the resistance of a battery is given by the formula v = B. SIEMENS' METHOD. 122. Fig. 46 shows the arrangement of resistances, &c., for determining the resistance of a battery by Siemens' method. A C is a resistance on the slide principle ( 17, page 15), E a resistance connected to the junction of the galvanometer G- and the battery whose resistance r is required. The other end of E is connected to the slider B. Now it will be found that if B be moved towards A or towards C from a certain point midway between A and C, the current flowing through the galvanometer will be increased. It follows from this that if we put B near A and obtain a certain deflection, we can also obtain this same deflection by sliding B to a point near C. Let B and B x be these points, and let a be the resistance between A and B, I the resistance between B x and C, and p the resistance between B and B r Also let E be the electro- motive force of the battery, and r its resistance, and let C be the current deflecting the galvanometer needle. * 'Electrical Review,' i age 316, Sept. 24th, 188G. INTERNAL KESISTANCE OF BATTEEIES. 119 when the slider is at B E B * ER - (r + a) (B + p + 6 + G) + E ( p + 6 + G) ' and when the slider is at B x P __ EE _ . - (r + a + p) (B + 6 + G) + E (6 + G)' therefore (r + a) (R + p + 6 + G) + E ( P + & + G) = (r + a + p) (R + & + G) + E (6 + G); therefore (r + a) p + E p = p (B + & + G) ; from which r + a = & -f G or r = G 6 - a. In making this test, then, what we have to do is to note what are the values of A B (a) and B x C (6) when the same deflec- tions are obtained on the galvanometer, then from these values and the resistance of the galvanometer we can determine the resistance of the battery. 123. Another way of making the test is to find the point be- tween A and C which gives the least deflection ; then ,a and b will be the resistances on either side of this point. 124. Let us now consider what are the "Best conditions for making the test." The points to be considered are, what are the best resistances to make E and A C, and also, at what point should we place the slider to commence with, that is, should we place it near one of the ends of A C or at some point nearer the middle of the latter ? From the equation r = G+&_ it is clear that any error made in b or a will make an exactly corresponding error in r ; in considering the problem, therefore, we have simply to determine what arrangement of resistances, &c., will cause any slight error in a or Z>, that is any slight movement of the slider, to make the greatest possible alteration in the current, that is in the deflection of the galvanometer needle. 120 HANDBOOK OF ELECTEICAL TESTING. Let us suppose the slider was at B for the first observation, and let us suppose that when the slider was at that point, a current C flowed through the galvanometer, and that when the slider was moved to B! the current was also C. Further, when the slider was moved a distance X beyond B towards, say, A, let us suppose the current was increased to C -f- c. We have then to determine what arrangement of resistances, &c., will make ^ as large as possible. \j Now E E (r + a) (B + P + 6 + G) + R(p + b + G)' and we know that r + a = b + G; consequently c = _ E-B _ (r + a) (B + p + r + a) + E ( P + r + a)' and by putting a X for a, and p + X for p, we get n _L c = _ E R _ _ n (r + _ A) (E + p + r + a) + E (p + r + a) or, c = Ci-0; therefore 1 - 1 - i - C " C therefore c_ = _ X (E + p + r + a) _ C (r + a - X) (E + p + r + a) + E (p + r + a)' or, since X is a very small quantity, we may say _ = _ X(B+P + * + ) _ O (r + a) (R + p + r + a) + B (p + r + a)' or . _ r I I h E + (p + r + a) We will first determine at what point the slider should be placed to commence with. Now if we show at what point it should be placed near A, we determine the point at which it should be placed near C, for INTEKNAL KESISTANCE OF BATTEKIES. 121 r + a must equal G + & What we have to do then is to- determine the best value to give to a. To do this we must suppose the resistance A C to be constant, or since r and Gr are naturally constants, we must have r + a+ p + b + G; that is, r + a + p+r + a, equal to a constant, say, K ; therefore p + r + a = K - (r + a), therefore, by equation [A], we get c C (r + a) (R + K - (r + a)) + K (K - (r + a)) K-(r+o)) From this we see that the smaller we make (r -f- a) the larger will be the numerator of the fraction. Also if r -f- a he less jr than (which it must be in the test), the smaller we make it the smaller will be the denominator of the fraction ; * con- (r + a) (K - (r + a)) = (r + a)K - (r + a) 2 =- - * This may be proved as follows : (r + a) (K - (r + a)) = (r + a If in the latter expression \ve make - then which makes the expression as small as possible. ' But if wo make r + a either larger or smaller than -, then Ur + a) - j does not equal 0, but it has a plus value which increases in proportion as we make either (r + a) larger than , or larger than (r + a) ; for although ( (r+ a) -- j in one case will have a positive, and in the other case a negative value, still ( (/ + a)J is positive in both cases. If, therefore, we make (r + a) smaller than , the value of the expression" iferred to, and consequently the value (r + a) (K - (r + a) ), will increase ' proportion. 122 HANDBOOK OF ELECTRICAL TESTING. sequently the smaller we make (r -f )> and therefore a, the larger will be. c It is best, therefore, to place the slider to commence with as near to one end of A C as possible. Next we have to determine what value we should give to A C. This- we shall do if we determine what value p should iave. If we write equation [B] (page 120) in the form A R p -f r -f a s* we can see that r, a, and E being constant, ~ is made as large as \j possible by making p as small as possible; but we can also see that there is but little use in making p much smaller than r-f-a, or, as a ought to be small, in making it much smaller than r. Lastly we have to find what value it is best to give to E ; this we can also determine from the last equation. We can see from the latter that, r, a, and p, being constant quantities, -^ is L/ made as large as possible by making E as small as possible ; but we can also see that we gain but very little by making E much .smaller than r -f- a, or, as a ought to be small, by making it smaller than r. Actually of course we could not make E extremely small, for the reason that the battery and galvano- meter would then be practically short circuited and a readable deflection could not be obtained. Since r + a = G -f- 6, a can only be made small by having Gr small; it is therefore best to have a galvanometer of as low a resistance as possible, or rather of a resistance not exceeding r. "We proved that the slider should be as near one end of A C as possible. The end we can place it nearest to must evidently be the end to which the greatest resistance is connected ; therefore, whichever value of r or Gr happens to be the greatest, .at the end to which that larger value is connected should the .slider be placed, to commence with. In order to determine the " percentage of accuracy attain- able " we must in equation [B] (page 120) put percentages V and y for the absolute values A. and c, that is to say, we must have \ ' ' X = ofr, and, c INTERNAL EESISTANCE OP BATTERIES. 123 in which case we get per cent. 1 a r To summarise the results, then, we have Best Conditions for making the Test. 125. The slider at commencing should be as near as possible to that end of A C to which is connected the greatest of the values r and G. The value of A should be not less than the value of the greater of the two quantities r and G. R should be lower than the greater of the two quantities r and G. The galvanometer resistance should not exceed r, and the deflection should be obtained at the angle of maximum sensi- tiveness. This can be done by varying R ; but inasmuch as the latter should be lower than r, it is desirable to use a galvano- meter of such sensitiveness that R can be made sufficiently small without reducing the deflection too low. Possible Degree of Accuracy attainable. If we can be certain of the galvanometer deflection to an accuracy of y' per cent., then we can be certain of the value of r to an accuracy, A.', of percent. r If E, a, and p are very small compared with r, then we get \ i r X = y. 126. As in previous tests, we should first determine the value of r roughly and then more exactly with the resistances properly arranged. 127. We have hitherto supposed AC to be & slide resistance, but it is not absolutely necessary that it should be so; the test can very well be made in the following manner : Referring to the figure, and supposing r to be greater than G, let the resistances p and b be ordinary ones and both capable of variation, and let the resistance a be done away with. Having connected R to B, that is, to the pole A of the battery, plug up all the resistance in b and adjust p and R till the deflection of maximum sensitiveness is obtained on the galvano- 124 HANDBOOK OF ELECTEICAL TESTING. meter. Care must be taken that the adjustment of p and E is so made that E is less and p greater than G. If the galvano- meter has a sufficiently high figure of merit, there will be no difficulty in doing this. Next shift the connection of E from B to Bj and proceed to adjust b and p until the original deflection is reproduced, the adjustment being made in such a manner that the same resist- ance is plugged up in p that is unplugged in b ; then r = G + b. It must be noted that of the two quantities G and r the one which has the greatest resistance must be connected to p at B. In the case we have considered we have supposed that r was the larger quantity, but if G had been the larger of the two the position of G and r would have had to have been reversed, and the resistance of r would have been given by the formula r = G - b. The modus operandi of the test would, however, be precisely the same in the two cases. Two sets of resistance coils are evidently necessary to make this test, as it cannot be made with a single set of the ordinary kind (Fig. 6, page 13). MANGE'S METHOD. 128. This test is of a very similar nature to Thomson's method of determining the resistance of a galvanometer given on page 93. Fig. 47 shows the theoretical method of making the test. In the theoretical figure, a, &, and d are resistances, g a galvanometer, and E the battery whose resistance r is required. A key is inserted between the junctions of a with I and d with r. By depressing this key the junctions are connected together. Let us first suppose the key to be up, then the current C^ flowing through the galvanometer will be E a + b r+d+ ( g + 6 )g + & + E (a 4. 6) INTERNAL RESISTANCE OF BATTERIES. 125 Text suppose the key to be pressed down ; then the current C 2 lowing through the galvanometer will be C,= E bd bd bd b + d~*~ 9 ~* g(a + r)(6 + d) + bd(a + r) + ar (6 + d)' Now if the resistances be adjusted so that the deflection of the galvanometer needle remains the same whether the key is depressed or not, then equations [1] and [2] are equal ; that is E (o + 6) E (b + d) a Now if we refer to " Thomson's galvanometer resistance test " on page 94, we can see that this equation is similar to equa- tion [3] on that page, with the exception that r and g are interchanged. It must therefore be obvious, by the same development of the equation as that given on the page referred to, that m ad ^_ b' 126 HANDBOOK OF ELECTEICAL TESTING. 129. The great advantage of this test is that the electro- motive force of the battery need only be constant during the very short interval of time occupied in depressing and raising the key. 130. In making the test practically the connections would be made as shown by Fig. 48. Terminals E and B' would be joined by a short piece of thick wire ; the other connections are obvious. FIG. 48. The left-hand key puts the galvanometer on ; this key must be depressed and held permanently down, and the right-hand key then alternately depressed and raised and the resistance d, that is the resistance between A and E, at the same time ad- justed until the deflection of the galvanometer needle remains the same whether the key is up or down. 131. Again referring to Thomson's galvanometer resistance test; it must be clear, by substituting r for g in the equations, that to obtain the Best Conditions for making the Test, Make a as low as possible and b as high as possible, but not so high that d when exactly adjusted would exceed all the resistance we could insert between D and E (see Fig. 48). Adjust d approximately and then, if necessary, adjust the resistance of the galvanometer shunt (which it will be necessary to employ) so that the final deflection is as nearly as possible that of maximum sensitiveness, and then, having exactly adjusted d, calculate r from the formula. INTERNAL RESISTANCE OF BATTERIES. 127 Possible Degree of Accuracy attainable. If we can determine the value of the galvanometer deflection to an accuracy of y per cent., then we can be certain of the- value of r to an accuracy of(l+ -) (1+77) 7 P er cent. 132. In the practical execution of the test with the set of resistance coils shown by Fig. 48, the lowest value we could give to a would be 10 units, unless we improvised a resistance of less value, which it might be necessary to do. MANGE'S METHOD WITH THE SLIDE WIRE BRIDGE. 133. Mance's test is sometimes made by having a -f- 6 a slide- wire resistance, d being a fixed resistance ; in this case the slider would be moved along between A and until the point is found at which the depression or raising of the key makes no alteration in the deflection of the galvanometer needle. For practically executing the test the apparatus known as the " Slide Wire " or " Metre Bridge " may be used. This apparatus, which is shown by Fig. 49, is described in Chapter FIG. 49. VIII. (The Wheatstone Bridge). The slide wire, a + 6, which is 1 metre long, is stretched upon an oblong board (forming the base of the instrument) parallel to a metre scale divided throughout its whole length into millimetres, and so placed that its two ends are as nearly as possible opposite to divisions and 1000 respectively of the scale. The ends of the wire are soldered to a broad, thick copper band, which passes round each end of the graduated scale, and runs parallel 128 HANDBOOK OF ELECTRICAL TESTING. to it on the side opposite to the wire. This band is interrupted lay four gaps, at m v r, d, and m 2 . On each side of these gaps are terminals. In making the test under consideration, the gaps, m l and m 2 , are closed by thick copper straps. The slider S makes contact with the slide wire by the depression of a knob on S. The battery, r, a resistance, d, and a galvanometer, g, being joined up as shown, the slider S is moved along the scale, the knob being depressed at intervals, until the point is reached at which the depression makes no change in the permanent deflec- tion of the galvanometer needle. When this is the case, then, as in Thomson's galvanometer test (page 93), we have (,1000 - a)' For example. In the foregoing test, equilibrium was produced when d was 1 ohm, and a, 450 divisions ; what was the resistance, r, of the battery ? 450 450 1 1000- 450 550 134. The best conditions for making the test are similar to those required for " Thomson's galvanometer test " (page 93), namely, we should make d larger than r, but not greater than about 10 times r. As a rule the complete slide wire bridge is furnished with but four resistance coils of 1 ohm each, so that the choice of a resistance to insert in d is limited, and it may not be possible to follow out the rule of " making d about 10 times as large as r" In this case the possibility of an accurate measurement becomes proportionately reduced below the highest possible standard, so that on the one hand a cell whose resistance is much less than one-tenth of an ohm, or, on the other hand, a cell whose resist- ance exceeds 4 ohms, cannot be measured with the highest possible accuracy. Strictly speaking (as has been pointed out) in order to ensure accuracy it is necessary that the resistance of the portion of the slide wire, a, be less than the resistance of the battery to be measured ; but as the resistance of the whole length of the wire will not exceed one-tenth of an ohm, the resistance of the length, a, will practically be less than the resistance of the battery, unless, of course, this resistance is extremely low. INTERNAL RESISTANCE OF BATTERIES. 129 The possible degree of accuracy attainable we can see from Thomson's galvanometer test (page 93) must be given by the equation 1000008 X = - /1AAA --- x per cent. a (1000 ay where 8 is the degree of accuracy in divisions to which the slider, S, can be adjusted. If we can adjust to an accuracy of 1 division, then 8=1. For example. In the last example, what would be the degree of accuracy, A', with which the value of r could be obtained, supposing that the position of the slider could be determined to an accuracy of 1 division (S) ? 100000 x 1 , 450 (1000- 450) 135. The facility and accuracy with which all the foregoing tests (except the half-deflection test) can be made may be greatly increased by the following arrangement : Use a galvano- meter with a high " figure of merit " (page 65), and instead of making the test with the needle brought to the "angle of maximum sensitiveness " (page 23), make it with the needle brought approximately to zero by means of a powerful per- manent magnet set near the instrument ; under these conditions the galvanometer needle will be highly sensitive to any small change in the current strength. Another arrangement which may be very conveniently adopted is to employ a galvanometer with a high "figure of merit," and wound with two wires. One of these wires would be joined in circuit with the battery under test, &c., in the usual way; the other would be connected in circuit with a small battery and a set of resistance coils, the connections being so made that the currents through the two coils oppose one another. When the deflection due to the battery under test is obtained, the second battery and resistance coils are connected up, and then this battery is adjusted until the needle is brought to zero as nearly as possible. The test is then made, as in the case where a permanent magnet is used. 136. In the case of Mance's test with the slide-wire bridge, if the test is made either by using a permanent magnet in the way described, or by using a galvanometer wound with a double wire, it is best to make d as nearly equal to the resistance of the battery as possible (it should not be made less), as in this case, since the slider, S, will have to be set near the centre of 130 HANDBOOK OF ELECTKICAL TESTING. the scale, a greater range of adjustment is given to it, for 5 divisions near the centre portion of the scale (500 division mark) are equivalent to only 1 division near the 100 division mark. It is true the arrangement is not quite so sensitive as it would be if the slider were set towards the end of the scale ; but still, if we can employ a galvanometer with a high figure of merit, this small loss of sensitiveness is more than compensated for by the increased range which can be obtained on the scale. 137. In order that satisfactory results may be obtained in any of the foregoing tests, it is very necessary that the galvano- meter used be a " sensitive " one (page 66), otherwise even a moderate degree of accuracy cannot be assured. DIMINISHED DEFLECTION DIRECT METHOD. 138. This method, which has been generally described in Chapter I. ( 6, page 4), is as follows : The battery whose resistance, r, is required, a galvanometer of resistance, G, and a resistance, p, are joined up in simple circuit ; the deflection obtained is noted. Let this deflection be due to a current, 0^ then calling E the electromotive force of the battery, we have C, = . . . p . or, or ' The resistance, p, is now increased to K, so that a new deflection due to a current, C 2 , is produced, then we have E r + G + hence Ci (r + G) + C lP = C 2 (r + G) + 2 K, (r + G)(C 1 -C 2 ) = C a E-C lP ; therefore C 2 B - Cj p r -f U = p p , l/j U 2 that is _ C 2 B-C lP Ci-0 a If a tangent galvanometer is employed for making the test then if the deflections, D and c7, are read from the tangent scale INTEKNAL EESISTANCE OF BATTEEIES. 131 of the instrument, those deflections can be directly substituted for the quantities, Cj, C 2 , for D : d : : d : C 2 ; in this case, then, we have ;; (1.) For example. With a tangent galvanometer whose resistance was 10 ms (G), and a battery whose resistance, r, was required, a deflection of 60 divisions (D) on the tangent scale of the instrument was obtained, when a resistance of 10 ohms (p) was in circuit ; when the latter resistance was increased to 230 ohms (R) the deflection was reduced to 20 divisions (d). What was the resistance of the battery.? 20 X 230 - 60 X 10 r = 60 = 20 10 = 9 0hmS - If the readings are made from the degrees scale, then we must substitute the tangents of the deflections for the deflections themselves ; the formula then becomes _ tan d R - tan D p _ T ~ tan D - tan d L J (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the galvanometer, and deflections of 50 (D) and 21f (d) respectively were obtained with resistances of 10 ohms (p) and 229 ohms (R) in the circuit. The resistance of the galvanometer was 10 ohms (G). What was the resistance, r, of the battery ? tan 50 = 1-1918, tan 21f = -3990, therefore 139. If in equations [B] and [C] we have p = 0, that is to say, if we make the test by having at first no resistance in the K 2 132 HANDBOOK OF ELECTRICAL TESTING. circuit except that of the galvanometer and the battery itself, then we get and r = B, * an< * .-G. [E] tan D t- tan d 140. In order to determine the " Best conditions for making the test," and also the " Possible degree of accuracy attainable,'* let us write equation [A] in the form _ \->l v^2 Now this equation is similar to equation [B] (page 102) in the "Diminished deflection direct method" of determining the resistance of a galvanometer, except that in the latter method we have the quantities R and p in the place of the quantities (R + G) and (p + G); consequently we can at once see from the investigation in the test referred to that we must have Best Conditions for making the Test. 141. Make p as small as possible. Make R of such a value that when the deflections, D, d, are j in divisions, then i.-s 'I approximately; and when the deflections are in degrees on a tangent galvanometer, then , tan D tan d = - o approximately. Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of ^th of a division, then we can determine the value of r to an accuracy, X', of INTERNAL RESISTANCE OF BATTERIES. 133 If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of ^th of a degree, we can determine the value of G to an accuracy, X', of (tan D 8 2 - tan d 8.) 100 / p + G tan d tan D" - tan f ( where . (tan D" - tan f) X = tan D^ - tan D, and, 8 2 = tan d~ - tan d. DIMINISHED DEFLECTION SHUNT METHOD. FIG. 50. 142. This method is shown by Fig. 50. The battery, r, whose resistance is to be determined, is joined up in circuit with a resistance, E, a galvanometer, G, and a shunt, S l ; the deflection obtained is noted; let this deflection be due to a current C 1} then calling E the electro- a |_| JJ [_ I motive force of the battery, we have /^~ [ | ^ (page 115) Tn n C,= or r (Si + K + G) + +B + G)' The resistance of the shunt is now reduced to S 2 , so that the galvanometer deflection is also reduced ; let this new deflection be due to a current C 2 , then we must have (S 2 C 2 S 2 (B S 2 therefore C 2 r(S 2 +R+G)+C 2 S 2 (B+G) _ S 2 that is, _ E; from which we get = - 00 C 2 8^(8, + B + G) - dS^S! + R + G)' 134 HANDBOOK OF ELECTRICAL TESTING. or r _ _ GI "" C 2 G/ In the case of a tangent galvanometer, if the deflections, D and d, are read from the tangent scale, then we should have (1.) For example. With a tangent galvanometer whose resistance was 10 ohms (G), and a battery whose resistance, r, was required, we obtained with a shunt of 200 ohms (S } ), a deflection of 60 divisions (D) on the tangent scale of the instrument ; when the shunt was reduced to 25 ohms (S 2 ) the deflection was reduced to 20 divisions (d). The resistance, E, was 710 ohms. What was the resistance of the battery ? r = _ _ 6 ~ 20 _ _ 90 ohms. l 1 20 (.25 + 710-J-10J "~ 6 \200 710 + If the deflections are read in degrees, then in equation [B] we must substitute tan D and tan d for D and d respectively, we then get tan D - tan d (2.) For example. In a measurement similar to the foregoing the readings were made from the degrees scale of the galvanometer, and deflections of 50 (D) and 21f (d), respectively, were obtained. The values of S w S 2 , R, and G were 200, 25, 655, and 10 ohms, respectively. What was the resistance, r, of the battery ? tan 50 = 1-1918, tan 21 1 = -3990. INTERNAL RESISTANCE OF BATTERIES. 135 therefore 1-1918 - -3990 r = - -- - - -- - - - = 90 ohms. 143. If we make the test by having no shunt inserted when the first deflection is observed, that is to say, if we have Sj = co , or, =0, then equation [B] becomes l E + G/ E + Gl- and equation [C] _ tan D - tan d " 2 K + GJ E + G Further still, if we make E a very high resistance, that is, if in equations [D] and [E] we make ~ = 0, then we get the simplifications and 'tan D c /tan D \ = Hte!^ / 144. If we refer to the " Diminished deflection shunt method " of determining the resistance of a " galvanometer " we can see that equation [A] (page 109) in that test is almost precisely similar to equation [A] (page 134) of the present test, the only difference being that in the latter we have n in the place It -f~ \x of , consequently we must have Best Conditions for making tJie Test. Make S x and E as large as possible. Make S 2 of such a value that when the deflections, D, cZ, are in divisions, then 136 HANDBOOK OF ELECTKICAL TESTING. approximately; and when the deflections are in degrees on a tangent galvanometer, then tanD approximately. Possible Degree of Accuracy attainable. If the deflections are in divisions, and if we can read their value to an accuracy of ^th of a division, then we can deter- mine the value of r to an accuracy, X', of If the deflections are in degrees on a tangent galvanometer, then if we can read their value to an accuracy of ,^th of a degree, we can determine the value of r to an accuracy, X', of (tan D &, + tan d8Q lOOp, /I : "tand(tanD-tanV) [* + r % where Sj = tan Di - tan D, and, 8 2 = tan , and, C 2 =, ELECTKOMOTIVE FORCE OF BATTERIES. 145 therefore E x : E 8 : : C x : C 2 , or since d and d. 2 are directly proportional to C x and C 2 , we must have E! : E 2 : : d r : d 2 . For example. With a tangent galvanometer, whose resistance was 100 ohms, and battery B lf whose resistance was 70 ohms, we obtained, with a resistance of 1830 ohms (total, 100 + 70 + 1830 = 2000), in the resistance box, a deflection of 50 divisions on the tangent scale of the galvanometer ; and with battery E 2 , whose resist- ance was 50 ohms, we obtained, with a resistance of 1850 ohms (total, 100 + 50 + 1850 = 2000, as before), in the resistance box, a deflection of 40 divisions ; then E! : E 2 : : 50 : 40, or as 1*25 to 1, If the deflections are read on the degrees scale of the tangent galvanometer, then d and d 2 must be the tangents of the deflections. In cases where the resistances of the batteries whose electro- motive forces are to be compared are very small, we may, by using a very high resistance, practically regard the total resist- ance in circuit as being the same, whatever battery we use. The deflections then obtained with any number of different batteries will represent their comparative electromotive forces. The galvanometer will, in this case, of course have to be one with a high figure of merit (page 65). 159. The "Best conditions for making the test," and the " Possible degree of accuracy attainable," are almost obvious ; they are Best Conditions for making the Test. Make the resistances in the circuits as high as possible. Possible Degree of Accuracy attainable. If we can be certain of the value of the two deflections to accuracies of 8\ and 8' 2 per cent, respectively, then we can be certain of the relative values of the two electromotive forces to an accuracy of B\ + 8' 2 per cent. L 146 HANDBOOK OF ELECTRICAL TESTING. EQUAL DEFLECTION METHOD. 160. Join np as in last method, and having noted the deflec- tion and total resistance in circuit (Rj) with battery E lt remove it and insert battery E 2 in its place. Now readjust resistance between D and E, until the deflection of the galvanometer needle becomes the same as it was at first. Note the resistance in circuit (R 2 ) ; then calling C the current, = J, and, C = | 2 , K! Itj that is, Ej : E 2 : : R! : R 2 , or the electromotive forces of the batteries are directly as the total resistances that are in circuit with the respective batteries. For example. With a galvanometer whose resistance was 100 ohms, and a battery E x whose resistance was 50 ohms, we obtained, with a resistance of 2350 ohms (total, 100 -f 50 + 2350 = 2500), in the resistance box, a deflection of 40 ; and with a battery E 2 , whose resistance was 70 ohms, it was necessary, in order to bring the galvanometer needle again to 40, to have a resist- ance of 1830 ohms (total, 100 + 70 + 1830 = 2000), in the resistance box ; then E! : E 2 : : 2500 : 2000, or as 5 to 4. An advantage in this test is that it can be made with a gal- vanometer the relative values of whose deflections are unknown. The Best Conditions for making the Test and the Possible Degree of Accuracy attainable are the same as in the last test. WIEDEMANN'S METHOD. 161. In Fig. 45 (page 115) join the zinc pole of battery E x to D, as shown, and the other pole to the zinc pole of battery E 2 , whose other pole in turn is to be joined to C. Adjust the ELECTKOMOTIVE FORCE OF BATTERIES. 147 resistance so as to obtain a high deflection on the tangent scale of the galvanometer. Let the current producing this deflection be C ; then where E is the total resistance in the circuit. Now reverse battery E 2 (the weaker one) so that the two batteries oppose one another, we shall then get a smaller deflection due to a current C, ; then From these two equations we get Ej C - E 2 C = E x C, + E 2 C,, that is, E! : E 2 : : C + C, : C - C,, or, substituting deflections d, d,, for current strengths C, C p E! : E 2 : : d + d, : d - d,. For example. Two batteries E x and E 2 being joined up together in simple circuit, we obtained, by adjusting the resistance in the resistance box, a deflection of 72 divisions (d) on the tangent scale of the galvanometer; and with the same resistance in circuit we obtained, on reversing battery E 2 , a deflection of 8 divisions (><#,) ; then E! : E 2 : : 72 + 8 : 72 - 8, :: 80 : 64, or as 1-25 to 1. If the deflections are read on the degrees scale of a tangent galvanometer, then d and d, must be the tangents of tike deflections. 162. In order to make the test as accurately as possible under ihe last conditions, the resistance in the circuit should be so adjusted that the two deflections make approximately equal angles on opposite sides of 45 ( 32, page 29). The more resistance it is possible to place in the circuit of the batteries the better, since the tendency of the latter to polarise is thereby reduced to a minimum. 163. Wiedemann's method is a very satisfactory one since it is absolutely independent of the resistance of the two batteries, L 2 148 HANDBOOK OF ELECTRICAL TESTING. thus one battery might have a resistance of a fraction of an ohm only and the other a resistance of several thousand ohms, yet this would in no way affect the correctness of the results, but to avoid errors due to polarisation it is necessary with gome batteries to include several thousand ohms in the circuit ; if the galvanometer used be one with a high figure of merit (page 65) this can always be done. 164. The " Possible degree of accuracy attainable " in making the test is greatly dependent upon the relative values of the two electromotive forces. Let us first suppose that the deflections are read in divisions, and let us suppose that there is a possible error 8 in both deflections. Now if we take both errors to be of similar signs, then we should have a total absolute error of 2 8 in the quantity (d + d,), but if one error were plus and the other minus, then we should have a total absolute error of 2 8 in the quantity (d d^). But the latter quantity must be smaller than (d + d,), therefore an absolute error 2 8 in its value must represent a greater percentage error in the relative values of E x and E 2 than would be the case if the same absolute error were in (d -j- d,). As we must assume the resultant error to be the greatest possible, we must therefore take the error 2 8 to be in the quantity (d d,). Let, then, X be the error in the relative values of E l and E 2 , E, that is in =i , caused by, say, an error 8 in d, and an error 8 **> in d lt then we have EX _ A , (d + 8) + (d, - 8) _ d + d, E 2 (d + 8) - (d, - 8) d - d, + 2 8' therefore x = i - E 2 d-d, + 2d d-d, d - d A + 2 8 2 8 (d + d,~) (<* + <*,) (<*-<*, + 28)' or since 2 8 is very small we may say , _ 28 (d + d,-) (d - d,y If we put the percentage for the absolute value of A, that is if we have x - 1L & - *L ~ 100 B, 100 d-d," ELECTROMOTIVE FORCE OF BATTERIES, then we get 149 A' d + d, _ 2 8 (d + d ; ) 100 ^d - d. that is to say 28100 [A] example. In the example given on page 147 the deflections could each read to an accuracy of J of a division ; what was the degree Tjl accuracy with which the value of =i could be determined ? 2 x J X 100 = - 72 - 8 = d, is small compared with d, then ' r e can see from equation [A] that unless d, is small compared tith d, the accuracy with which the test can be made will be but small ; for if d, approaches in value to d, then d d, becomes very small, that is A' becomes large. In order that d, may be as much smaller than d as possible, Ej and E 2 must be as nearly equal as possible; the test therefore will not be a satisfactory one unless such is the case. If d, is small compared with d, then X = 2 8 100 or if we put the percentage instead of the absolute value of 8, that is if we have then we get V = 2 8', so that under the best conditions for making the test the accuracy with which the value of =^ could be determined would be but one-half the accuracy with which the higher deflection could be observed. 150 HANDBOOK OP ELECTRICAL TESTING. 165. To determine the degree of accuracy attainable in the case where the readings are made from the degrees scale of a tangent galvanometer, we must in the preceding investigation substitute tangents for divisions of deflections. Thus we have E I _ > - tan 0* + S ) + tan W + 8 ) E 2 " tan (d + 8) - tan (d, + 8)' or A = tan d + tan d, tan (d + 8) + tan (d, - 8) tan d - tan d " tan (d + 8) - tan ffi~Tjfy If in this equation we put ' and " 1 + tan d tan 8' we get , _ 2 tan 8 [(tan d + tan d,) (1 + tan d tan d t ) + X] (tan d - tan d,) (tan d - tan d, + Y) where X and Y are a number of factors of tan 8. But since tan 8 is very small, we may put X and Y equal to 0, in which case we have x = 2 tan 8 (tan d -f tan d,) 1 + tan d tan d t tan d - tan d, X tan d - tan d, _ 2 tan 8 (tan d + tan d,) I tan d - tan d, ~ X tan (d - d,)' If we put the percentage for the absolute value of A, that is, if we have A = *L fS = -*- x tan d + tan d ' 100 E 2 100 tan d - tan d,' then we get 2 tan 8 100 A ~ + QT , /^o 3-5x- L-DJ example. In comparing the electromotive forces of two batteries by Wiedemann's method, the deflections obtained on the degrees scale of a tangent galvanometer were 71 and 18 respectively ; ELECTROMOTIVE FORCE OF BATTERIES. 151 what were the relative electromotive forces of the batteries, and what would have been the degree of accuracy with which the -pi value of =, could be determined ? The value of the deflections ^2 could be read to an accuracy of . E x : E 2 : : tan 71 + tan 18 : tan 71 - tan 18, or as 2 9042 + 3249 to 2 9042 - 3249, that is, as 1-25 to 1; , = 2xtanyxlOO ; 2 x -4363 = tan (71 -18) 1-3270 Like equation [A] (page 149), equation [B] (page 150) shows that unless d f is small compared with d, the test cannot be made with a high degree of accuracy. To sum up, then, we have Best Conditions for making the Test. 166. To obtain satisfactory results, E L and E 2 should be as nearly as possible equal. As much resistance should be included in the circuit as possible. If the readings are made on the degrees scale of a tangent galvanometer, then the resistance in circuit should be so ad- justed that the deflections, as nearly as possible, make equal angles on opposite sides of 45 ( 32, page 28). Possible Degree of Accuracy attainable. When the readings are in divisions, then Percentage of accuracy = -^ -7- where ^ is the smallest fraction of a division to which the de- flections can be read. When the readings are in degrees on a tangent galvanometer, then 2 tan L 100 Percentage ot accuracy = -- - - tan (d d, ) 152 HANDBOOK OF ELECTRICAL TESTING. where ~ is the smallest fraction of a degree to which the deflec- tions can be read. WHEATSTONE'S METHOD. 167. The most elegant method of comparing the electro- motive forces of batteries is that of the late Sir Charles Wheat- stone. Battery E x is joined up in simple circuit with a galvanometer and a resistance ; a deflection of a is obtained. The resistance is now increased by p lt so that a new deflection, /3, is produced. Battery E 2 is next joined up in the place of E 1} and the re- sistance in circuit is adjusted until the deflection obtained is a, as at first. The resistance is now increased by p 2 , so that the deflection is reduced to /3, as in the first instance. Now from the " Equal resistance method " (page 144), we see that the total resistances, B! and B 2 , in circuit, which were re- quired in the two cases to bring the deflections to a, must be in direct proportion to the electromotive forces, E x , E 2 , of the two batteries. Also the total resistances, B x + p lt and B 2 + p 2 , in circuit which were required in the two cases to bring the deflec- tions to /2, must be in direct proportion to the electromotive forces, E!, E 2 . We therefore have E! : E 2 : : B! : R 2 , or E! B 2 = E 2 B 1? and E! : E 2 : : B! + P! : B 2 + p 2 , or E x B 2 + E x p 2 = E 2 B x + E 2 Pl = ^ B 2 + E 2 Pl ; that is E! p 2 = E 2 p 1? or E! : E 2 : : P! : p 2 . In fact, the electromotive forces of the batteries are directly proportional to the added resistances which, in both cases, were required to bring the deflections of the galvanometer needle from a down to /3. For example. With a galvanometer and battery E x we obtained, with a resistance of 1950 ohms in the resistance box, a deflection of ELECTROMOTIVE FORCE OF BATTERIES. 153 54, and by adding 2000 ohms fo), a deflection of 34. Battery E 2 being inserted in the place of E x , a resistance of 1650 ohms was inserted in the resistance box, which brought the galvano- meter needle to 54 as at first, and by adding 1600 ohms (p 2 ), the deflection was reduced to 34 as in the first instance ; then B! :E 2 :: 2000 : 1600, or as 1-25 to 1. 168. In this and the preceding tests we have supposed that the electromotive forces of any two batteries were being com- pared, but it must be evident that by noting the deflections, resistances added, &c., as the case may be, with any number of batteries, their electromotive forces may all be compared. 169. We will now proceed to determine the " Best conditions- for making the foregoing test." There are two points to be determined : first, what should be the resistances in circuit when observing the first deflections, and second, what proportion should the added resistances bear to the original resistances ? When the test is executed, there are two or more sets of observations made, viz., one for each battery. But it will be found, on examination, that the proportion between the electro- motive forces, the original resistances, and the added resist- ances, is the same for every set; consequently, we have only to determine what relative values these quantities should have in any one set, then those in the others will be in the same proportion. It will be convenient to consider first what proportion the added resistance should bear to the original resistance. For this purpose we will suppose p l to be the former resistance. Now p 1 represents the electromotive force of the battery, and therefore in order that the test may be made as accurately as possible, it is necessary that we should be able to adjust or de- termine the value of p x as accurately as possible. In order to- obtain the required value of p 15 we first adjust R! so as to obtain the deflection a, and then we increase R x by p l so as to obtain the deflection /3 ; consequently, the accuracy with which we can obtain p l must be dependent upon the accuracy with which we can read both the deflections, a and /3. Let, then, the first deflection (a) be due to a current, C x , then we have 0,-, or, C^E^EL. 154 HANDBOOK OF ELECTRICAL TESTING. When the current is reduced to C 2 by the addition of P I} then we get | 2 = BI + Pi' r ' 2Rl + 2 Pl = El J therefore or (C 1 \ On / Now this equation is identical with equation [F] (page 111) in the " Diminished deflection shunt method " of determining the resistance of a galvanometer; consequently, we can see from the investigations there given, that p l would be most accurately obtained if c * = approximately; but when this is the case that is to say, the added resistance should be about double the original resistance. As regards the " Possible degree of accuracy attainable," we can see from equation [H] (page 111) in the test before referred to, that the percentage of accuracy, A', attainable must be v _ (0. e. + 0. e.) 100* C 2 (c 1 -c 2 ) - pei As it is the relative electromotive forces of two batteries ~F which have to be determined, that is to say, the value of =~, the ^2 percentage of accuracy with which the test can be made will be double the above. As regards the value for the original resistance there is little to be said. It does not affect the accuracy of the test, except * The expression |~1 + G { + iYj in the equation referred to [H] (page 111) becomes equal to 1 when S, and K are very high; this must be the case when equation [B] (page 109) becomes simplified into equation [F] (page 111). ELECTROMOTIVE FORCE OF BATTERIES. 155 in so far as the power of adjustment is concerned; this is evidently made as favourable as possible by making the resistance as high as convenient. We must have therefore Best Conditions for making the Test. 170. When making the observations with the first battery, make the original resistance as high as convenient, and make the added resistance as nearly as possible double this. Possible Degree of Accuracy attainable. When the readings are in divisions, then p * 1(D-E 2 r 1 = E 2 R 1 -E 1 p 1 . [2] By subtracting [2] from [1] we get E 2 R - E 2 B! - E! p -f- E! Pl = 0, or that is R - or Ej : E 2 : : R R! : p p x , a proportion in which differences of resistance alone appear. In fact (R R!) and (p pj) are merely the resistances which we subtracted from R and p, in order to get equilibrium a second time. For example. Two batteries whose electromotive forces Ej and E 2 were to be compared, were joined up in circuit with a galvanometer and two resistances as shown by Fig. 56, the resistance p being 500 ohms ; in order to obtain equilibrium R was adjusted to 1050 ohms ; p was then decreased to 300 ohms (p x ), and in order to again obtain equilibrium, R had to be reduced to 630 ohms (R x ). What were the comparative electromotive forces of the batteries ? E! : E 2 : : 1050 - 630 : 500 - 300 420 200 ELECTROMOTIVE FOECE OF BATTERIES. 161 or as 2-1 tol. 177. The question now arises what are the best values to give to R x and p l9 or rather to p lt for the value given to the latter will determine the value given to E x . In order to work out the problem let us suppose, in the equation E E E -pi there is a small error X in =^ caused by a definite error < in JSj E 15 that is, let ?! 4. X = -^ ~ ffii "" *) _ -^ ~ -^i I *ft rjj-i E 2 P Pi P Pi P Pi 5y subtracting [A] (page 160) from [B] we get P- Pi This shows that with the definite error 0, A is as small as possible when p l is as small as possible. A. would be very great if p l approached in value to p, but it would be small when ^ is about equal to , and but little less if p l is made very much smaller still. Although, therefore, we should make p x small, there is but little advantage in making it very much smaller than - ; in fact, there is an actual disadvantage, for when p l is Zi very small, E L is proportionately small and its range of adjust- ment is correspondingly limited. From equation [A] (page 158) we can see that in the present case the currents flowing through the galvanometer when equilibrium is not established, in consequence of R and E x being each 1 unit out of adjustment, are C, = and Co = M 162 HANDBOOK OF ELECTEICAL TESTING. respectively ; and from these equations it is evident that if c l is a perceptible deflection when E is 1 unit out, c 2 will be a still more perceptible deflection when E x is 1 unit out, since E L must be smaller than E ; consequently the value we give to E L will not be limited by any considerations with regard to a perceptible deflection being obtained. As in the first test, c x and c 2 are both greatest when E x is larger than E 2 , the batteries should therefore be so arranged that this is the case. With regard to the Possible degree of accuracy attainable with this test, we can see first of all that E cannot be adjusted quite so accurately as in the case where the resistance of the batteries was negligible; we can, however, ascertain the exact degree attainable by putting p + r 2 instead of p in equation [B] (page 158). Thus to take the example given on page 158, suppose the battery E 2 had a resistance of 5000 ohms (r 2 ) approximately, then we should have (p + 5000) [5000 (1 + i) + p + 5000] = 1,000.000,000 X , or (p + 5000) [12,500 + p] = 500,000,000. If in this equation we make p = 14,000, we get (14,000 + 5000) [12,500 + 14,000] = 503,500,000, which is close to the correct value. In other words, if p does not exceed 14,000 ohms, we can be sure of the value of E within 1 unit. The degree of accuracy with which we can determine the Tjl ralue of ~ from the equation J^ 2 Ji/i xt ~~ -tt depends upon the degree of accuracy with which we can adjust both E and E 15 and as the errors in either of them may be either -|- or , the greatest possible total error is that which will be produced by a + error in E, and a error in E 15 or vice versa. Let these errors be both 1 unit and let the corre- -pi spending error in =i be X, then we have EI E + 1 - (E 1 - 1) E - E! 2 __ -- }_ A -- ----- - T -- 5 E 2 p - p z p-pi p-pi ELECTROMOTIVE FORCE OF BATTERIES. 163 and therefore X= 2 . P ft Since we require to know what percentage (A/) of error this represents, we have or A' = iAi = --.. [C] E 2 - ft To take the example we have just considered, we see that the possible percentage of accuracy attainable, supposing p l to equal |, is 4,000! 7000 178. With a Thomson galvanometer of ordinary sensitiveness it is evident from the foregoing investigation, that if we have two batteries, one E 2 having an electromotive force of 1 volt or more, and Ej_ an electromotive force of twice that value or more, we can without difficulty determine their relative electromotive forces to an accuracy of, at least, -015 per cent.; and if the resistance of the batteries be very low we can be certain of the accuracy within, say, 003 per cent. 179. It is possible to get a still greater accuracy by employ- ing a set of resistance coils adjustable to T Vth or y^th of a unit, for in this case we can make both K and E L low without losing the range of adjustment, whilst by making these quantities low we increase the value of the galvanometer deflection when exact adjustment is not obtained ; this is only the case, however, when the resistances of the batteries and of the galvanometer are low. We can easily determine to what extent the degree of accuracy is increased by using submultiples of the units ; first by ascertaining from equation [B] (page 158) what value p can T^ 2 have, =?- being divided by 10 if E is adjustable to T Vths, and M 2 164 HANDBOOK OF ELECTRICAL TESTING. by 100 if E is adjustable to T ^ths ; and second by working out the value of A' from equation [C] (page 163) which gives the required percentage of accuracy. Of course when great accuracy is required, the test must be made by the method in which the resistances of the batteries are eliminated ; it is no use making the test by the first method, since the accuracy attainable by having E adjustable to tV^h or T ^th of an ohm is more than counterbalanced by the error produced by not taking into account the resistance of the batteries. To summarise the results we have obtained, we have Best Conditions for making the Test. First Method. 180. First make a rough test to ascertain the approximate values of E x and E 2 , then make p of such a value that approximately, c being the reciprocal of the figure of merit of the galvanometer, and E x the stronger of the two batteries, EJ and E 2 being in volts. Second Method. Make p of such a value that approximately. If E is adjustable to T Vtn or UTT^n of an ohm, the right-hand "P 2 "P 2 side of the equation should be ^ 2 or 2 respectively, Jii! 1U_C &! LOO C P! should be about equal to - . In both methods E x should be the larger of the two batteries. Possible Degree of Accuracy attainable. First Method. Where resistance of battery is very small, ELECTROMOTIVE FORCE OF BATTERIES. 165 I f\r\ -ri Percentage of accuracy = - - X ^ 2 . P **i Second Method. Percentage of accuracy = 200 P ~ Pi __ a E i* Or, if p l is nearly equal to -, SB 400 E 2 Percentage of accuracy = X S P -^i 181. A great point in these methods of determining the comparative electromotive forces of batteries, lies in the fact that both batteries are working under exactly the same con- ditions ; moreover, if the resistances R and p are high there is but little tendency for them to polarise. If one of the batteries be a constant one, such as a Daniell, then by varying the values of R and p we can test how the other battery behaves when worked through different resistances. POGGENDORFF'S METHOD. 182. In this method one battery is balanced against the other. The method is shown by Fig. 57. In this figure E L and E 2 are the electromotive forces to be compared. R and p are adjustable resistances, r and r t being the resistances of the batteries, is the resistance of the galvanometer. Before equilibrium is obtained we have G- Oi + - c = + R c 3 - = 0. [1] [2] [3] 166 HANDBOOK OF ELECTKICAL TESTING. By substituting the value of c obtained from equation [1], in equation [2], and then again the value of c 3 obtained from equa- tion [2], in equation [3], we shall find that a - K (r, + G + n + p) + (r x + p) (r, + G)' If we put c 2 = 0, we get or that is E x : E 2 : : E + ^ + p : E, [5] or It will be observed that in order to get the ratio of E x to E 2 from this proportion, we must know the resistance r x of the battery E x . If, however, we decrease p to p x and again get equilibrium by readjusting E to E 15 we get a second proportion, viz., E x : E 2 : : E x + r, + Pl : E x , [7] and by combining the two proportions, r is eliminated in the manner shown in the last test (page 160) and we get, E, (E - EQ + ( P - Pl ) E 2 E - E! or B 1 :E,::Ot-B 1 )-J-0>-p 1 ):08-B,), [A] a proportion in which differences of resistance- alone enter. For example. Two batteries whose electromotive forces E x and E 2 were to be compared, were joined up in circuit with a galvanometer and two resistances as shown in Fig. 57. The resistance p being 200 ohms, it was necessary in order to obtain equilibrium to adjust E to 500 ohms, p was then reduced to 100 ohms (p^, and in order again to get equilibrium E had to be readjusted to 400 ohms E, then E! : E 2 : : (500 - 400) + (200 - 100) : (500-400); or as 2 : 1. ELECTROMOTIVE FORCE OF BATTERIES. 167 183. In making this test practically, the connections with the set of resistance coils shown by Fig. 6, page 13, would be as shown by Fig. 58. Having depressed the left-hand key, then, according to the example, we should take out the two 100 plugs between A and 0, and proceed to adjust between D and E. This being done, we should insert one of the 100 plugs between A and and readjust the resistance between D and E. 184. As only one of the batteries (the smaller) in this test has its electromotive force balanced, the other one should be a constant battery, whose electromotive force does not fall off on being worked continuously, such as a Daniell. 185. It is evident that the test can be made either by making p a fixed resistance and E an adjustable one, or by making E fixed and p adjustable. In order therefore to determine the Best conditions for making the test, one point for consideration will be should E or p be the adjustable quantity ? Now by a similar reasoning to that given in 173, page 157, we can see that in either case the value of the adjustable resist- ance should be the highest one in which a change of 1 unit from its correct resistance produces a perceptible deflection of the galvanometer needle. FIG. 58. A \ rr EE~LJJ A' If we refer to equation [6] (page 166) we can see that if E L = 2 E 2 then r l -j- p must be equal E, and that according as E! is greater or less than 2 E 2 , so will r x + p be greater or less than E. It is evident that the larger we make the adjustable resistance the greater will be the range of adjustment of which it is capable, therefore for this reason it follows that if Ej is greater than 2 E 2 then ^ -f p should be the resistance in which the adjustment is effected, whereas if E x is less than 2 E 2 then E should be the adjustable resistance. 168 HANDBOOK OF ELECTRICAL TESTING. Now if E be the adjustable resistance, then inasmuch as the value which it will have will depend upon the value given to p, therefore we must determine the highest value we can give to p. Equation [4] (page 166) shows the current c 2 obtained through the galvanometer, when equilibrium is not produced. If in this equation we put E 1 in the numerator instead of E, and then put E, E + r, -f p E 2 IT T-' or ' E = ^' + o> E7^E 2 ' we shall get the current, e 2 , corresponding to the change of 1 unit in the correct value of E. Thus = __ (E.-E 2 )' ___ 2 ('i + p)[('- 1 + G)E 1 + (r 1 + p)E 2 ], or (E '~ E2)2 . [A] And if in this equation we make c 2 the reciprocal of the figure of merit (page 65) of the galvanometer, then the value of p which satisfies the equation will be the highest value which it should have ; as explained in the last test, p can be obtained by trial. If p be the adjustable resistance, then what we have to determine is the value which E should have. To do this we must put p + 1 * instead of p in the numerator of equation [4] (page 166) and then put E! E + r x + p E (B L - E 2 ) ir ---> or > r i+' = -kr ' we shall then get the current, c' 2 , corresponding to the change of 1 unit in the current value of p. Thus ___ _ a - E [(r, + G) B l + E (E, - E 2 )]' or E [ Xtj therefore 2 (P - Pi) \ _ P -Pi . P - Pi _ " or, since E Ej is very large, _ 2( P - Pl ) _ 2 - Since we require to know what percentage (A.') of error this- represents, we have or In the case where p and Pl are the adjustable resistances, we should get X = P - Pi + 2 _ p - p l 2 E - E! ]R - Ej E - E^ and calling, as before, X' the percentage of error, we get 200 E 2 p v I' E - E! J^i * This is not the case in Lumsden's test. ELECTROMOTIVE FORCE OF BATTERIES. 171 To sum up, then, we have Best Conditions for making tlie Test. 188. First make a rough test to ascertain the approximate values of E x , E 2 , r 15 and r 2 ; then if E x is less than 2 E 2 , make p a fixed resistance, and of such a value that ('1 + P) [(*, + G) B! + (r x + p) EJ = [A] approximately. If E is adjustable to -Hh of an ohm, then the right-hand side- of the last equation should be c being the reciprocal of the figure of merit of the galvanometer, and Ej and E 2 both being in volts. P! should be about equal to . 2 If E x is greater than 2 E 2 then make E a fixed resistance, and of such a value that E G) E (E x - E,)] = 2 - [B] approximately. If /o is adjustable to -^th of an ohm, then the right-hand side- of the last equation should be c being the figure of merit of the galvanometer, and E x and E 2 being both in volts. T> E! should be about equal to ~. Possible Degree of Accuracy attainable. When E and E! are the adjustable resistances, then 200 (E x - E 2 ) 2 Percentage of accuracy = x or if p 1 nearly equals P -Pi * 177, page 163. 172 HANDBOOK OF ELECTRICAL TESTING. 400 (E x - E 2 ) 2 Percentage of accuracy = x * ; When p and p l are the adjustable resistances, then -D 200 E 2 Percentage of accuracy = x ^r J or if E! nearly equals a 400 E 2 Percentage of accuracy = -=- x ^T. K j^ l 189. If the test is made by obtaining the result from formula [6] (page 166), the resistance r l of the battery being very small, then it is not difficult to see, from the investigation given in "Lumsden's test" (page 155) that when E is the adjustable resistance, -0 f l Percentage of accuracy = - x v P - Also we should make p of such a value that c approximately. When p is the adjustable resistance, then _ 100 E 2 Percentage of accuracy = - x ^-. H &! Also we should make E of such a value that approximately. FAHIE'S METHOD OF MEASUKING BATTERY EESISTANCE. 190. It may be pointed out* that the foregoing test also affords a means of ascertaining the resistance r x , of the battery Ej ; thus from equations [5] and [7] (page 166) we can see that E + r x + p : E : : E! + r x + Pl : E! ; therefore B JL B + B 1 r 1 +BiP = BiB + Br 1 -hBp 1 ; * See Sabine's The Electric Telegraph,' p. 323. ELECTROMOTIVE FORCE OF BATTERIES. 173 therefore or thus if we take the example given on page 166, in which we have Ej = 400 E = 500 p = 200 P! = 100 we get (400 X 200) - (500 X 100) _, , V L 500-100 =75 ohms. 191. A resistance test made in this way, however, would not be an accurate one if the resistance r of the battery were small in comparison with the resistance p l (which is in the same circuit with r L ), for in this case the high value of the latter would swamp, as it were, the low value of r x . If, however, as- suggested by Mr. Fahie,* we commence the test by having no resistance at first in circuit with the battery E 1? that is to say, if we have p l equal to 0, then we can obtain more satisfactory results ; in this case we get i - B - B l ' 192. With regard to the Best conditions for making the test according to formula [A], the resistance E L is the resistance required to produce balance in the first instance and it can have but one value; E, however, is dependent upon p, so that what is required is the value which should be given to the latter quantity. Now from formula [A] we can see that the larger we make p the larger will be the value of E, and the larger we make the latter the greater will be its range of adjustment, consequently, as in the electromotive force test, we should give it the highest value in which a change of I unit from its correct resistance produces a perceptible deflection of the galvano- meter needle ; this resistance we shall obtaiu by giving p such a value that (i + P) [(' + G) E I + fa + P) E J = l c/ * See ' Electrical Review,' vol. xii. t p. 203. 174 HANDBOOK OF ELECTRICAL TESTING. approximately,* c being the reciprocal of the figure of merit of the galvanometer. As regards the Possible degree of accuracy attainable, this we shall obtain, as in previous cases, by supposing that there is an error of + 1 in R and an error of 1 in B x , these errors causing a corresponding total error A. in r 1 : thus 4.x i i (B - 1) - (B x +1) " B - B! - 2' and since we get (B x + 1) p B lP pCB " B - B! - 2 " B - B! " (B - B x - 2) (B - B x )' or since B B x is very large, we may say > P(B + BQ. r B1 (B-BO 2 ' but E ^ r B + + r E 2 " B! B or t + p) = "T^ - T^" ~ E! E 2 therefore and by substituting these values of B + B x and B B x in equation [B] we get Or if we call A/ the percentage of error, then or ^' = = ~ J ~ip 2 ( 1~ T 'l ^2 V P } ] * Equation [A], p. 168. ELECTEOMOTIVE FORCE OF BATTERIES. 175 193. The relative electromotive forces of tlie batteries, it may be pointed out, are given by the proportion E! : E a : : (R - RO + P : (R - EJ, which is the same as proportion [A], page 166, except that p 1 is put equal to 0. To sum up, then, we have Best Conditions for making the Test. 194. Make p of such a value that fri + P) [('a + + (i-! + p) EJ = ' + approximately, c being the reciprocal of the figure of merit of the galvanometer. Possible Degree of Accuracy attainable. "fjl TT1 / O "I \ ^= ( | ) 100. -^2 \P r i / Percentage of accuracy = FAHIE'S COMBINED METHOD OF COMPARING ELECTROMOTIVE FORCES AND MEASURING BATTERY EESISTANCE. 195. This is an extremely ingenious and elegant method, and although its application is rather limited it is well worth being noticed. The arrangement is a combination of PoggendorfFs method of comparing electromotive forces (page 165) and Mance's method of measuring battery resistance (page 124). FIG. 59. Referring to Fig. 59, the following is the mode of making the test : E is the stronger battery whose electromotive force is to be compared with the battery e t and whose internal resistance 176 HANDBOOK OF ELECTRICAL TESTING. is to be measured ; d is a variable and a + b a slide, resistance, B being the slider by the movement of which the ratio of a to & can be varied. The key K being open, the resistance d is adjusted until the needle of the galvanometer shows that no current is passing through the latter; when this is the case, then, as in Poggendorff s method (page 165), we have E: e::r + d + a+b: a + b. [1] Balance being thus obtained, the key K is alternately depressed and raised and the slider B moved until the latter is brought to such a position that the movement of the key K ceases to affect the galvanometer needle, as in Mance's test (page 124). Now, inasmuch as the battery e merely acts as a counteracting force to the current which in Mance's test would cause a permanent deflection of the galvanometer needle, it must be evident that when the movement of the key K ceases to affect g, then we must have or d d * d f _i_ d -J~ d = (o& -4- b i. 6 & ; Substituting this value of r + d in equation [1], we get E : e : : ^ (a + &) -f a + 6 : a + &, or E : e : : d + 6 : b. [3] Equation [2], therefore, gives the resistance of the battery E, and equation [3] gives the relative electromotive forces of the two batteries. For example. The key K being raised, balance was obtained on the galvano- meter g by adjusting d to 200 ohms. When the key K was alternately raised and depressed, the balance on g was disturbed until the slider B was moved to the position at which b was equal to 100 ohms ; the total resistance of the slide resistance a _|_ b was 400 ohms, that is to say, a was equal to 300 ohms ; then r = = 600 ohms, 100 ELECTROMOTIVE FOECE OF BATTERIES. 177 and E : e : : 200 + 100 : 100, or as 3 : 1. 196. The conditions for making this test so as to obtain accu- rate results must evidently be similar to those specified in the cases of Poggendorff's test and Mance's test made with a slide resistance. The nature of the method, however, is such that we cannot obtain the conditions which are best for the Poggen- dorff test without impairing the conditions necessary for making the Mance test accurately, so that practically we must arrange the resistances so as to suit the conditions necessary for making the latter satisfactorily ; at the same time it may be pointed out that these conditions are such as to enable the Poggendorff test to be made with a considerable, though not with a very high, degree of accuracy. As in the case of Mance's test with a slide wire (page 127), the conditions required are that d shall be as large as possible, but not so large that the range of adjustment of the slider becomes excessively reduced. Now, practically, a slide resistance would not consist of more than about 100 coils ; consequently if d were of such a value that the slider had to be set so that b was about 10 times as large as a (as would be the case when a slide wire is used), then the accuracy with which the latter could be adjusted would be extremely small, being only about 1 in 10, or 10 per cent. To make the test satisfactorily, therefore, it would be necessary to arrange so that the slider would have to come near the centre of its traverse, even though the sensitiveness of the whole arrangement became reduced in consequence. As long, however, as sufficient sensitiveness is obtained, that is to say, a sensitiveness such that a movement of the slider from its correct position to either of the contiguous coils produces a perceptible disturbance of the balance, then the nearer we can get the slider to the centre the better. It would not do, however, in any case to pass beyond the centre point ; for in this case, although the error made in a by the slider being one coil out of adjustment is small, yet the error made in b becomes comparatively large. Now, in order that we may be able to get the slider near the centre of its traverse, it would be necessary that d should be approximately equal to r, but since, in order to obtain balance in the first instance, we must liave 178 HANDBOOK OF ELECTKICAL TESTING. or e d could not bfr made equal to r unless E 2r ~ ~ ' - Now if E and e were both fixed quantities and were not of such relative values that the above equation held good, then it would be impossible to obtain the conditions necessary for making the test favourably ; the method of testing we are considering, however, would usually be employed for the purpose of measur- ing the electromotive force of a battery in terms of the electro- motive force of one or more standard cells whose number could be varied to suit any particular requirement ; in such a case it would usually be possible to give to e the value which would enable the above equation to be satisfied. Thus, for example, suppose the resistance, r, of the battery E were estimated to be- about 100 ohms, and suppose the slide resistance a -|- b consisted of 100 coils of 10 ohms each, that is, 1000 ohms in all, then we must have - E 1000 _ E 10 " 1000 + (2 x 100) ~ ~12~ ; that is to say, the electromotive forces of the batteries E and e should be in the proportion of 10 to 12. Now, it is e\ 7 ident that if E were a battery of one or two cells only, then it would prac- tically be impossible to give to e the required value ; but if E consisted of a considerable number of elements, 20 or 30 for example, then there would be no difficulty in adjusting e. From these considerations it must be evident that Fahie's method^ although extremely ingenious and elegant, and in some special cases very convenient, is very limited in its application. 197. With respect to the Possible degree of accuracy attainable, this as regards the resistance test is directly dependent upon the accuracy with which we can adjust the ratio of a to b ; thus if a + 6 consisted of 100 coils, then if the ratio of E to e were such that the slider when adjusted stood near the centre position of its traverse, the error caused by the slider being 1 coil out of position would be 1 in 50 in a, and 1 in 50 in &, consequently the total error would be 1 in 25, or 4 per cent. With n coils, in ELECTKOMOTIVE FOECE OF BATTERIES. 179 fact, the Possible degree of accuracy attainable would be 1 in -, 100 X 4 or, per cent. n To determine the degree of accuracy attainable in the electro- motive force test, we must suppose that d is 1 unit, and b 1 Tjl coil, out of adjustment. If we call X the error caused in then e we must have E d-\-l _ d + l _ E e a + fe' a + b e ' b 21 b ' n n and since we get _ h a + b b If X' be the percentage of error, then we have 100 01 7' or ' therefore _ 100 [5 Q + d) + a d] - ' If the test is made under the best conditions, that is, if we have a b, and d = r, approximately, then we get 100[frQ + r) + 6r] 100( + 2r) " or since is large, we may say _ For example. In determining the relative electromotive forces, E and e, of two batteries by Fahie's method, the resistance, r, of E being N 2 180 HANDBOOK OF ELECTEICAL TESTING. approximately 100 ohms, a slide resistance having 100 coils (w) of 10 ohms each was employed. What was the greatest possible degree of accuracy attainable ? ., = 100 [100 + (2 X 100)] _ j_ 100 (500 -f 100) ~ 2 pei To sum up, then, we have Best conditions for making the Test. 198. Make E (a + E) - a + b + 2r approximately, r being the approximate resistance of the bat- tery E. Possible Degree of Accuracy attainable. Percentage of accuracy = If a = &, and d = r, and w is large, then 100 (TO + 2 r) Percentage of accuracy = + r) % in both cases being the number of coils of which the slide resistance is composed. 199. It may be as well to point out that Fahie's test cannot be made (except under very exceptional circumstances, rarely met with in practice) with a slide wire ; for, as a rule, the latter has such an extremely low resistance that it would be impos- sible to obtain equilibrium in the first instance ; the proportion which is necessary for equilibrium, could not, in fact, be satisfied unless the resistance of the battery E and the resistance d were both extremely small ; in which case, moreover, the latter would have to be adjustable to a very small fraction of an ohm. CLARK'S METHOD. 200. This is a valuable modification of Poggendorff's method, and is shown in theory by Fig. 60. a 6, which takes the place of E in Poggendorff's method (page 165), is a slide resistance ; ELECTKOMOTIVE FORCE OF BATTERIES. 181 E 3 is a third battery which is connected to a slider through a galvanometer Gr 3 . Now if we suppose equilibrium to be obtained in both gal- vanometers, we must have from [5], page 166, ! : E 2 : : r, + P + a + 5 : a + 6, and also E i from which we get E : E : : a + 6 : a. If then we take a + & to represent the electromotive force of the standard battery E 2 , a will represent the electromotive force of the battery E 3 . FIG. 60. In making this test practically, the battery E 3 , which would be the trial battery, being disconnected from the slide resist- ance, balance would be obtained with the standard battery E 2 by adjusting p until no deflection is observed on the galvano- meter GI. E 3 would then be connected up and the slider moved until no deflection is observed on the second galvano- meter G 3 . The great advantage of Clark's method is that both the standard and the trial battery are compared under the same conditions, that is, when no current is flowing in either of them; this is a great point, as errors due to polarisation are avoided. 201. It must be evident that if equilibrium is not produced with the trial cell, then the balance in the standard cell circuit will also be disturbed ; it would therefore seem to be possible to 182 HANDBOOK OF ELECTEICAL TESTING. dispense with the galvanometer G 3 , but inasmuch as the current which would flow through the galvanometer Gj would only be a fraction of that flowing out of the battery E 3 , we should not be able to make a measurement with nearly such a degree of accuracy as we could if we employed the galvanometer G 3 , which would be acted upon by the full force of the current. 202. To determine the best arrangement of resistances, &c., for making the test, let us suppose that there is a small error, A, in E 3 , caused by a corresponding small error in a, and let us find what effect this error has upon the current which would flow through the galvanometer G 3 . Supposing then that a L is the new value of a which causes this error, then, keeping in mind that a -f- & being a slide resistance is not altered by changing a, we have or /I? ! ^ f~ I IA [2] We next have to determine what the current flowing through the galvanometer, when equilibrium is disturbed, is equal to. Eeferring to Fig. 61, in which w, n, a, 6, and g represent the resistances, and c lt c 2 , c 3 , K X , and * 2 the current strengths in the various circuits, we have by Kirchoff's laws (page 156) C l + C 2 + C 3 - K l = KJ c 3 K 2 = K 2 Cj_ c 2 = Cj m + K X ! + KZ ^i "" -^i = ^ c 2 n -f- KI i + K 2 &i E 2 = c 3 G 3 + /q ! - E 3 - 0. We know also that E! : E 2 : : m + i + &i : i + &i, and By finding then the value of c x from the first equation and substituting its value throughout the others, and then again ELECTEOMOTIVE FOECE OF BATTEKIES. 183 the value of c. 2 from any other equation, and again substituting throughout and so on, and also substituting the value of E x obtained from the proportion, and again the value of a x + & x , we shall find that C ( mn \ K a i ( &i H -- ; ) V m + nj mn m 4- n If in this equation we substitute the value of % given by equation [2], we shall get or as flj and 6 X are very nearly equal to a and 6, we may say W < l ^J<> "T" On examining this equation we see that to make c 3 as large / mn \ a(5H V W + / as possible we must make *- as small as possible, mn but we also see that it is no use making it much smaller than G 3 , as c 3 is but very little increased by so doing. a (l) Now the quantity ^ - it L i S the resistance a com- .. mn H -- : m n bined in multiple arc with the resistance & plus m and n combined in multiple arc, consequently this quantity can never be greater 184 HANDBOOK OF ELECTKICAL TESTING. than a. As long therefore as a is smaller than G 3 , the highest values that can be given to the other resistances cannot make c 3 less than -- , whilst, on the other hand, however low we (* 3 + a make these resistances, we can never make c 3 greater than =- . ^"3 The value therefore we give to a practically determines the sensitiveness of the system. But as a is only a portion of the slide resistance a + 6, and as it may include the whole of the latter, as for instance when the slider is moved quite to the end of a + 6, the sensitiveness is practically dependent upon the value given to a + & This must then be made as much lower than G 3 as may be desirable. It would not, however, do to have the resistance excessively low, for the following reason : In order to get equilibrium on the galvanometer G 2 , it is necessary that the relation E x : E 2 : : r x + P + a + I : a + 6, or should hold good. This cannot be the case, however, if = - is greater than a + b ; that is to say, if a + 6 is very small =i - - must be very small also.; but to make the latter small - 1 -! E 2 we must make E x large and r x + p small, but since r 1? the resist- ance of E!, will increase by increasing E x it may be impossible to do this. Practically we may say the resistance of a + & should be a fractional value of G 3 . 203. Let us now determine the Possible degree of accuracy attainable by the method. In equation [1] (page 182) we have supposed that an error A. has been caused in E 3 by a being out of adjustment ; that is to say, from the slider being moved a little too far, so that a becomes a r If we call the distance the slider has been moved beyond its correct position, then we have E 2 (a + 0) E 2 a , E 2 -- ELECTKOMOTIVE FOKCE OF BATTEEIES. 185 but therefore that is to say, the distance the slider is out of position re- presents directly the error X in E 3 . The degree of accuracy therefore with which we adjust the position of the slider will be the degree of accuracy with which we can measure E 3 . We have pointed out that if a + b is small, then / mn \ a [ b ^ ) \ m + n ) will be smaller still ; if therefore G 3 is large compared with a + 6, equation [4] (page 183) becomes X Cs = ff ' If in this equation we put the value of X, given above, we have- or a + b E 2 This equation enables us to determine what movement of the slider produces a perceptible deflection on the galvanometer. With a Thomson galvanometer of 5000 ohms' resistance and figure of merit = 1,000,000 (page 65) we should have, sup- posing E 2 to be 1 volt, <> 5000 1 a + b 1,000,000,000 = 200,000' or a movement of the slider equal to -jo-oVo-^th of the length of a -f b would produce a perceptible deflection ; that is to say, we could determine the accuracy of an electromotive force E 3 of about 1 volt to an accuracy of 186 HANDBOOK OF ELECTEICAL TESTING. To obtain this accuracy, however, it would be necessary to nave the wire a + b graduated into 200,000 parts, each of which would be very small, unless indeed the wire were very long. If a lesser number of graduations were employed, we could practically subdivide each of them by noting what the galvano- meter deflections were when the slider stood, first at one division mark, and then at the contiguous mark. Suppose the slider stood at a distance a from the end of the slide wire, and a deflection due to a current c x was produced to one side of zero ; and suppose that when the slider was moved 1 division forward, that is to a -\- 1, the deflection was on the other side of zero, or was produced by a current c 2 . Then we have from equation [3] (page 183), since a and a -f- 1 are very nearly equal, and <* + & a + b * ~ K K therefore Cl E 3 - Cl E 2 j^- - c, - = _ c 2 E 3 + c 2 E 2 -J or therefore + c 2 ) = E 2 ( Cl + c a ) The subdivision of the division beyond a is therefore given by the fraction . We have seen that we could get a C l ~T~ **2 deflection of 1 division on the galvanometer if the slider were moved a distance of ^nAnnrtb ^ e y n( i tne distance required to give equilibrium. If the wire a -j- b were divided into 20,000 parts, then a movement of the slider through 1 part or division would give 10 divisions of deflection on the galvanometer, each division representing a tenth of one of the wire graduations. If in making a measurement we got a deflection of 7 divisions G to the left when the slider stood at a distance a from the ELECTROMOTIVE FOECE OF BATTERIES. 187 end of the wire, and a deflection of 3 divisions (c 2 ) to the right when the slider was moved 1 wire graduation beyond a, then the position of the slider for exact equilibrium would be The galvanometer can thus be made to act as a vernier ; and the greater the deflection produced by a movement of the slider through one division of the graduated wire, the greater will be the accuracy with which a test can be made. The general results that we arrive at from the foregoing investigations are as follows : Best Conditions for making the Test. 204. Let the slide wire a -f 6 be a fractional value of the resistance of the galvanometer G 2 , but not so low that it is less than U The values given to the other resistances and electromotive forces do not affect the sensitiveness of the arrangement. Possible Degree of Accuracy attainable. _, c 3 G 3 100 Percentage ot accuracy = _: . 205. Mr. Latimer Clark employs a platinum-iridium wire of 40 ohms' resistance, wound spirally on an ebonite cylinder, for the slide resistance. The edge of the cylinder being divided into 1000 equal parts, and there being twenty turns to the cylinder, the whole wire is divided into 20,000 equal parts. By employing with this instrument (which combined with the batteries and resistances is called a " Potentiometer ") a gal- vanometer with a high figure of merit (page 65), and a standard battery E 2 of one Daniell cell, a 1 division movement of the slider, after equilibrium has been produced, will produce a deflection of 50 divisions. It is possible, therefore, with the apparatus to measure an electromotive force of one Daniell cell to an accuracy of 1 = ! th. 20,000 x 50 1,000,000 188 HANDBOOK OF ELECTRICAL TESTING. CHAPTER VIII. THE WHEATSTONE BEIDGE. 206. THE theoretical arrangement of the Wheatstone Bridge, or Balance, is shown by Fig. 62. It consists of four resistances a, 6, d, and a, arranged in the form of a parallelogram, a battery occupying the place of one, and a galvanometer the place of the other, diameter. When the four resistances are so adjusted FIG. 62. that equilibrium is produced, that is to say, when no current passes through the galvanometer, then these resistances bear a certain relation to one" another. This relation may be thus determined : When equilibrium is produced, then since there is no ten- dency for a current to flow between the points A and C, the galvanometer may be removed without altering the strengths of current in the other parts of the bridge ; and, further, we may join the points A and C without affecting the strengths. Let us first suppose the points A and C to be separated ; then the joint resistance given by the four resistances between the points B and E will be d' THE WHEATSTONE BRIDGE. 189 If, now, we join A and C, the resistance may be written db dx a 4. b d + x which must be equal to the former expression, that is, (a + a?) (6 + d) _ db dx a+x+b+d ~a+b + d+x By multiplying up and simplifying we get + & 2 a 2 - 2abdx = ; therefore (ad - fce) 2 = 0, from which a x b = ~d' If, now, three of the quantities in this equation are known, the fourth can be determined ; thus : ad : T- In the most general form of bridge, two of the resistances are fixed, and a third is adjustable, the fourth being the resistance whose value is to be determined. As a rule, we should make a and b the fixed resistances, x the resistance whose value it is required to find, and d the adjust- able resistance. In the simplest method of measuring we should make a and b of equal value, in which case x = d; that is to say, the resistance which is between A and E when equilibrium is produced, gives the value of the unknown resistance. It is absolutely necessary that there be some resistance in a and Z>, for otherwise the galvanometer is short-circuited, and equilibrium, as far as the galvanometer is concerned, will be always produced, no matter what resistances we have in the other two branches. 207. Besides using equal resistances in a and 6, we can make one of the two to be 10 or 100 times as great as the other, or, in 190 HANDBOOK OF ELECTEICAL TESTING. fact, any multiple of it we like, but multiples of 10 are those most commonly used. If, when we are measuring a resistance x, we make 6 10 times as large as a, then every unit of resist- ance in d represents T Vtk of a unit in x, for in this case We can, therefore, by this device determine the value of a resistance to an accuracy of T ^th of a unit, although d is adjustable only to units. In like manner, if we make b 100 times as large as a, then every unit of resistance in d represents of a unit in x; for in this case d and we can thus determine the value of a resistance to an accuracy of y^th of a unit. In the first instance, however, the value of d when adjusted would be 10 times that of x ; we could not, therefore, in that case measure a resistance whose value was greater than T yth of the total resistance we could insert in d ; and in the second instance d would be 100 times- as great as x ; we could not, therefore, in that case measure a resistance greater than T -J-^th of the total resistance in d. In fact, the larger we make d the closer will be the degree of accuracy with which a measurement can be made, but, at the same time, the smaller will be the resistance which can be measured, unless extra resistance coils are added in between A and E. There is, however, a limit to the degree of accuracy with which a resistance can be thus measured, which is dependent upon the figure of merit (page 65) of the galvanometer; of this we shall speak hereafter. If, now, we wish to- measure a resistance which is greater than the total resistance we can insert in d, we must make a larger than &. If a be made 10 times as great as b, we can then measure any resistance which is not greater than 10 times the resistance we can insert in d', but as in this case 1 unit in d represents 10 units in #, we can only be certain of the value of x within 10 units. In like manner, if we make a 100 times as great as 6, we can measure any resistance which is not greater than 100 times d t but we can only determine its value within 100 units. 208. The practical method of joining up one form of the bridge (Figs. 6 and 7, pages 13 and 14) is shown by Fig. 63. THE WHEATSTONE BRIDGE. 191 "When the connections are made, and the proper plugs removed from A B (6) and B C (a), the right-hand, key must be pressed down to put on the battery current. Plugs are now removed from E A (d) until we have inserted a resistance, as near as w& can guess, equal to the resistance we are going to measure. The FIG. 63. left-hand (galvanometer) key is next pressed down, and plugs- removed or shifted from E A (d) until no movement of the galvanometer needle is produced upon raising and depressing the key. The connections in the case of the set of coils shown by Figs. 4 and 5 (page 12) would be similar to the foregoing, but separate keys, in circuit with the battery and galvanometer respectively, would have to be employed. 209. If the galvanometer used is a very sensitive one, with a fine fibre suspension, the key must not, at first, be pressed firmly down, but only snapped down sharply ; for otherwise, if equilibrium is not very nearly produced when it is depressed, there is a danger of breaking the fibre of the galvanometer needle by the violent deflection. When, however, after repeated trials, we have very nearly obtained equilibrium, then the key may be firmly depressed, and the final adjustment of plugs made. 210. Fig. 64 shows a plan of the internal connections of the set of resistance coils which were shown in general view by Fig. 8, page 14. The method of joining up these coils to form a bridge would be as follows : The resistance to be measured is connected between C and E, the " Infinity " plug between the two being removed ; the galvanometer is joined 192 HANDBOOK OF ELECTRICAL TESTING. between A and C ; the battery is connected between B and E. The " Infinity " plug between A and D is inserted firmly in its place. Besides the connections referred to, it is necessary to have a key in circuit with the galvanometer, and another ia circuit with the battery. FIG. 64. ettch. 10 Ohm t? each lOhmetrck/ *-iOhm each D In this form of bridge, when balance is obtained, we have Kesistance to be measured = E T - o We may, if we please, insert the resistance to be measured between terminals A and D instead of between C and E, a plug being inserted between the latter, and the plug between A and 6 being removed; in this case, when balance is obtained, we should have Eesistance to be measured = R -. a An advantage of the foregoing set of coils lies in the fact that there are only five plugs to be shifted, for the insertion of these plugs brings the resistances into circuit, instead of short cir- cuiting them, as in the ordinary coils. The reading, also, of the total value of the resistance in circuit is a very easy matter, as must be obvious. Inasmuch as the withdrawal of a plug causes a disconnection, i.e. makes R = oo, great care must be taken that the galvano- meter key is raised previous to shifting a plug, otherwise a violent deflection of the galvanometer needle will be produced ; this fact renders the use of the " Dial " pattern objectionable in certain tests, especially in " fault " testing. CONDITIONS FOR ACCURATE MEASUREMENTS. 211. Besides the method of joining up, as shown by Fig. 63, we may also join up by placing the battery between A' and C, THE WHEATSTONE BEIDGE. 193 and the galvanometer between B' and E ; this, under certain conditions, renders the action of the galvanometer more sensi- tive than "by the common arrangement. What these conditions are, and what should be the general arrangement of the resist- ances in the bridge in order that a test may be made under the best possible conditions for ensuring accuracy, we will now proceed to consider. To investigate these questions it is first of all necessary to find what relation the current which flows through the gal- vanometer when equilibrium is not produced, bears to the different resistances which make up the bridge. In Fig. 65 let a, 6, d^ x, r, and g be the resistances of the different parts of the bridge, also let c lt c 2 , c 3 , c 4 , c 5 , and c 6 be the current strengths in the same, and let E be the electro- motive force of the battery. FIG. 65. r* ^nHNHh^ Applying Kirchoff's laws (page 156), we get the following six equations as representing the connection between the resistances, current strengths, and the electromotive force : *-<-*- o. [i] c 4 - e 6 - Cl = 0. [2] / -L / / A |"Ql C/o ^ 1 Og ^~" Co ~ v I O j c 5 r + c^ + c 2 6 - E = 0. [4] c x a - c 2 6 - c 6 gr = 0. [5] From these equations we have to find the value of c 6 , the current flowing through the galvanometer. By finding the value of c x from equation [1] and substituting 194 HANDBOOK OF ELECTRICAL TESTING. its value in equations [2] and [5] we get rid of c x ; and in like manner, by finding the value of c 2 from equation [3] and substituting throughout, we get rid of c 2 . By adopting the same process with respect to c 3 and c 4 we shall finally get equations [5] and [6] to become c 5 a - c 6 a - c 6 b - c 6 g - (a + & )E-c*J c r - . [5] o -f- a L / j i >. .EJ CW CfiT f\ r c~\ c 5 x + c 6 g - (^ + ) - r-S-j *- = 0. [6] -r a i From these two equations we get from which E (ad l - bx) A rQ1 = TnT' L 8 J B i This equation gives the strength of the current which would flow through the galvanometer if the resistances were arranged as shown by Fig. 65. 212. Suppose now the battery occupied the place taken by the galvanometer and vice versa, or, in other words, suppose the galvanometer connected the junctions of a with 6, and c? x with a, and the battery connected the junctions of a with a;, and 6 with d lt then the current (c 7 ) flowing through the galvano- meter would be (<*-&*> -n=m -6) B 2 If we subtract equation [9] from equation [8] we get c 6 - c 7 = - = ^-^- (B 2 - BJ, -Pi i> 2 1 ^2 and if in (B 2 B x ) we substitute the values of B x and B 2 given in equations [8] and [9], respectively, and then multiply up, cancel, &o., we finally get In this equation, if g is larger than r, and both a and x are respectively larger or smaller than d x and b ; or if r is greater than g and at the same time both a and 6 are greater than d^ and a;, then c 6 c 7 will be a positive quantity, that is, c 6 will be greater than c 7 . THE WHEATSTONE BK1DGE. 195 But c 6 is the current obtained by the arrangement of the bridge indicated by Fig. 65; and on examination it will be found that when the resistances have the relative magnitudes indicated, the greater of the two resistances g and r connects the junction of the two greater with the junction of the two lesser resistances ; consequently, as this arrangement gives the greatest current through the galvanometer when equilibrium is not produced, it must be the best one to employ. In practice it is almost always the case that the galvanometer has a higher resistance than the testing battery. 213. We have next to consider what should be the relative values of a, 6, d, and &, in order that the bridge test may be made under the best possible conditions. There are several different considerations involved in these questions, but we will investigate the problem from a general point of view first. Equation [8] shows the relation between the current and the resistances. In this equation, as equilibrium is very nearly produced, we may, except where differences are concerned, put -i i ^ , a d a a x = a d = b x t or, b = , d being the adjusted resistance when equilibrium is exactly produced. We then get s = 7 / ; T"; 7i~~; \T^I 71 N ; T~/ ; vf M |# C a ~f* x ) 4- a (d + #)} {*" (d -t- a?) + d (a -f- #)} Let us now suppose that the error in d^ which causes the current c 6 , produces an error X in x, or that #-f-X = -i, or, oa x I J I TT From this equation we can see that if d has the value necessary to make X a minimum, then as long as we do not make a less than - we cannot possibly increase X to more than 2 X. But then g + x * the question arises Suppose we have already increased X by 200 HANDBOOK OF ELECTKICAL TESTING. making d as great as g -f- x, under these conditions what will be the effect of also decreasing a to g+* If we refer to the last equation, we can see from the investi- gation made in the case of equation [4] (page 196), that the value of a which makes X a minimum must be and this value is one which makes X a minimum whatever be the value of d, though to make X an absolute minimum we must also have r x Now, if we increase d, we can see that to make X a minimum we shall have to decrease the value of a, for by increasing d we decrease both [r + x + ) and ( _ ) ; consequently a d / \g + x -f- a/ decrease in a after d has been increased will tend to decrease again the increased value of X. We cannot, however, bring back X to its original absolute minimum, although we may bring it near to it ; for after a certain point the decrease in the value of a causes X to increase again ; as long, however, as we avoid making a less than ~ this increase cannot be great. As the value which d has must depend upon the value given to 6, therefore after we have determined what values to give to a and d, we must ascertain the value of b from the equation i a d For example. It being required to measure exactly a resistance x whose value was found by a rough test to be about 500 ohms, a ten-cell Daniell battery (E = 10-7) whose resistance was 200 ohms (r) was used for the purpose, and also a galvanometer whose resist- ance was 5000 ohms (#) and figure of merit 1,000,00.0,000 (-). \c 6 / What resistances should be given to the arms a and 6 of the bridge in order that the test may be made under the most THE WHEATSTONE BRIDGE. 201 favourable conditions, also what percentage of accuracy would be obtainable under these conditions ? x = 500 g = 5000 r = 200 ; therefore 5000 + 500 d = VOTOOO + 500) goo go = 890 ohms; also we must have 6 = ^890 =1000 ohms. In practice we could make d as high as 5500 ohms (g + x), and a as low as 450 ohms ( ), without seriously increasing X. Supposing, however, we actually gave a and d their best values, then by equation [3] (page 196) we should have 1,000,000,000 X 10-7 =0014 that is to say, we may be '0014 units out when we measure x exactly ; this is equivalent to an error of = 0003 out) per cent, approximately. In order to make the test as accurately as this, it would be necessary that d be adjustable to a small fraction of a unit ; if we call the value of the latter, then we should have X -f- A. = 7 = \- and - ?*. therefore 202 HANDBOOK OF ELECTKICAL TESTING. We therefore have showing that d ought to be adjustable to *003 of an ohm or less. If we make it adjustable to 001 or yoVo th of an ohm therefore, we shall be able to make the test properly. 215. The facts we have arrived at by the foregoing investiga- tion are these, that with a 560 ohms and b 1000 ohms, then when equilibrium is exactly produced, an alteration in the value of d equal to 003 of an ohm (which quantity would mean an error, A, of 0014 units, or 0003 per cent, approximately, in x) would produce a perceptible deflection (1 division) on the gal- vanometer. We have, then, Best Conditions for making the Test. 216. First make a rough test to ascertain approximately the value of x. _ / rUc Make d not greater than g -f- a;, or less than/^/ (#+#) and preferably make it as near to the latter quantity as possible, provided the range of adjustment of d is not reduced to too great an extent by so doing. Make a not less than and not greater than (r -f- x~) , and preferably make it as near to the latter quantity as possible in the case where d is made nearly equal / y oc to \J (g + a?) ; but if d is made more nearly equal to> g + a?> then a should preferably be made more nearly equal to g + x It is clearly advantageous that E should be as large and r as- small as possible. Possible Degree of Accuracy attainable. A 100 Jr ercentage ot accuracy = - , where c 6 being the reciprocal of the figure of merit of the galvanometer. THE WHEATSTONE BRIDGE. Z()& In order to obtain this percentage of accuracy, d must be adjustable to not less than units, or - - th of a unit. 217. In the foregoing investigation we have considered the exact conditions required for a maximum degree of accuracy, and we have seen that in order to attain this it is necessary that d be adjustable to a fraction of a unit. At the commencement of the chapter ( 207, page 189), however, we saw that if dis only adjustable to units, then in order to obtain the greatest possible accuracy we should make d as much larger than x as possible, as by so doing we get a great range of adjustment. But, as we also stated, there is a limit to thus increasing d, for unless we are able to adjust d accurately, we can gain nothing by having the range of adjustment so large. Now to adjust d we note the deflection of the galvanometer needle, and when this becomes we know that d is adjusted exactly right ; but if an alteration of several units produces no perceptible effect on the deflection we may just as well have d of a smaller value. Thus, supposing we have b 10 times as great as a, that is d 10 times x; then if an alteration of 10 units in d only just affects the galvanometer needle, it is evident that we cannot adjust d to a closer accuracy than 10 units, and consequently we cannot obtain the value of x to a closer accuracy than 1 unit. If we have b equal to a, that is, d equal to a?, then if we can adjust d within 1 unit, we shall in this case obtain the value of x to an accuracy of 1 unit, that is, with just as much accuracy as we could in the first case, when d was 10 times x. It is even possible that we could obtain the value of x more accurately in the latter case, for it may be that an alteration of 1 unit in d when b equals a may produce a much greater movement of the galvanometer needle than does the alteration of 10 ohms when b is 10 times a. Whether this is so or not is a point we have to determine. We have also to find what should be the absolute values of a and 6. We have seen that in order to obtain accuracy it is necessary to make d as high as possible, but the highest useful value we could give to d would be that which produces the smallest perceptible deflection when it is 1 unit out of adjustment. Now if A. be the error in x caused by d being 1 unit out of adjustment, we must have 204 HANDBOOK OF ELECTRICAL TESTING and since ad x = therefore x = r-, or, = - , 6 b d ad ad x b o d or We have, then, from equation [2] (page 195) From this equation we have to determine the highest value we can give to d ; this will be limited by the " figure of merit " of the galvanometer, and also by the value of a. Let us write the above equation in the form g+X + d pi Now since is a fixed quantity, therefore in order that d 9 C 6 may have as large a value as possible we must give a such a value that x + T --\ \ + ~^~' \ ^+x^d) is as small as possible. From the investigation given in 213 (page 195) we can see that if we make a as low as possible, but not lower than, say, , then 9+x+d will be very close to its minimum value, no matter how high d may be. THE WHEATSTONE BEIDGE. 205 For the purpose of determining the actual numerical value which d can have, let us write equation [B] in the form c 6 (r + x + a) ' this equation, being an ordinary quadratic,* would enable the value of d to be obtained in terms of the other quantities in the usual manner, but inasmuch as we only require to determine the value of d within, say, 10 per cent., it is a much simpler and shorter operation to adopt the " trial " method ; that is to say, to give d different values until we arrive at one which approximately satisfies the equation. For example. Suppose, as in the last example, E = 10, x = 500 (from a rough test), g = 5000, r = 200, r .. i . 6 1,000,000,000 then make a = 500 ; * The solution of the quadratic equation is as" follows : Let = B, r + x + a v. ~ = K, * + a) then we get therefore d 2 + d (A + B) = K - therefore V4 K + (A - B) 2 A + B _ V4 K + (A - B) 2 - (A + B) 2 2 2 206 HANDBOOK OF ELECTRICAL TESTING. we then get J ,7 r nnn _i_ *nn , 5000 X500 If 200 X 500 10 X 500 X 1,000,000,000 200 + 500 + 500 ~' or, {d + 10,500} {d + 83-3} = 4,170,000,000. If we make d = 60,000 we shall very nearly satisfy the equation, for {60,000 + 10,500} {60,000+83-3} = 4,236,000,000. As the value which d will have will depend upon the value given to &, the latter must be made equal to 6 = 500 X 60.000 = 60)()00 _ oUO As regards the Possible degree of accuracy with which the test can be made, we have seen on page 204 that we therefore have which equals 0083 X 100 600- '0017 per cent.; this compares unfavourably with the result obtained when the test was made with d of a low value and adjustable to y^^th of a unit, the percentage of accuracy in the latter case being 0003 per cent. To summarise the results of the investigation, we have Best Conditions for 'making the Test, i as low as possible, but not lower Make d as high as possible, but not so high that ci or 218. Make a as low as possible, but not lower than ^ THE WHEATSTONE BEIDGE. 207 is greater than 6 (r + SB + a)' c 6 being the reciprocal of the figure of merit of the galvanometer. Possible Degree of Accuracy attainable. P t , l xl 100 .Percentage ot accuracy = - = - - . x d If we make d adjustable to any particular fraction of a unit, we can tell the degree of accuracy with which x could be measured, for if in equation [A] (page 203) we put instead of 1, we get "*'-.(<; ',;,';.:;" and equation [B] (page 204) becomes If in this last equation we give to the fractional value to which d is adjustable, we determine the degree of accuracy with which we can make the test. For example. Suppose d was adjustable to T Vth of a unit t-\ then we have (giving to x, a, g, and r the values used in the previous examples) {d + 10,500} {d + 83-3} = 417,000,000. If we make d = 16,000, we shall very nearly satisfy the equation, and the percentage of accuracy, X', with which x would be measured would be fy% ^d x 10 100 100 X' = -- = - = - ___= -00062 per cent. x n d 10 X 16,000 219. At the commencement of the chapter ( 207, page 189), we saw that by making & 10 or 100 times as great as a, and con- 208 HANDBOOK OF ELECTRICAL TESTING. sequently d 10 or 100 times as great as x, we were enabled to measure x to an accuracy of -^ih or y-J-^th of a unit, although d was adjustable to units only. Every unit in d, in fact, repre- sented y^th or yjj-oth f a un i* ^ n x - But to measure to an accuracy of T i^th of a unit, with the forms of bridge shown in Chapter II., pages 12, 13, and 14, the resistances in a and b have to be 10 and 1000 respectively, we have no other choice. In the investigation we have made, we have seen that a Cl *7* should be not less than , but in the bridge as usually arranged, if we wished to have a and b in the proportion of 1 to 100, so that we could measure to the accuracy of T ^th of a unit, we might find that we should have to very considerably transgress the rule of not making a smaller than - , unless, 9 ~r x indeed, x were a low resistance ; for inasmuch as we could adjust the resistances in the bridge so as to theoretically measure a resistance of 100 ohms to an accuracy of r^th of a unit, if the resistance were as high, or nearly as high, as 100, it might be 10 times, or nearly 10 times, as high as we could make a. Under these conditions, then, the bridge is not in a favourable condition for ensuring an accurate test. We say it is not in a favourable condition for ensuring accuracy, but it does not follow therefore that we cannot measure a resistance of 100 ohms accurately to an accuracy of j-^jth of a unit with such an arrangement. A galvanometer if it has a high figure of merit may, although the conditions are unfavourable, still give a sufficient deflection to enable us to exactly adjust. What, then, it may be asked, is the practical value of the results we have theoretically arrived at ? The value is this : if we find we have not got sufficient sensitiveness to obtain a good test, then we can see what may be the cause of it, and therefore how we can remedy it. The results further show that the values given to a and b in the bridges as ordinarily arranged are such that only certain resistances can be measured under the best conditions for ensuring accuracy. 220. It should not be overlooked that the conditions for obtaining a good test are, to a very great extent, dependent upon the resistance of the galvanometer used, since the value which a must have is dependent upon both g and x. But it must not therefore be imagined that we can make these condi- tions anything we please by employing a galvanometer of a low resistance, for such galvanometers have a low figure of merit, and consequently what is gained in one direction by having g THE WHEATSTONE BRIDGE, 209 low, is more than counterbalanced by having the figure of merit low. It must be evident, then, that the whole question of the accuracy with which a bridge test can be made is dependent, in the first instance, upon both the resistance and figure of merit of the galvanometer, and, as we shall see, in certain cases it is absolutely necessary that the resistance be very low, although the figure of merit has consequently to be low also. MEASUREMENT OF A KESISTANCE WHEN EXACT EQUILIBRIUM CANNOT BE OBTAINED. 221. It very often happens, especially when measuring small resistances, that exact equilibrium cannot be obtained in the bridge ; thus one unit too much in d may give a deflection to one side of zero, and one unit too little, a deflection to the other side of zero, and as no nearer adjustment can be made, the exact value of x is not directly determinate. If, however, the values of the deflections be noted, the true value of x can be obtained very closely. On page 194 we have an equation [8] which gives the value of the current (c 6 ) passing through the galvanometer when equilibrium is not produced. Let, then, c' be the current which produces, say, a left-hand deflection of the galvanometer needle, and let this current be caused by d being too small ; also let c" be the current which produces a right-hand deflection, and let this current be caused by d being too large. Then if d' and d" be the smaller and larger resistances respectively, we have two equations, viz., where B r and B" are quantities corresponding to B x in equa- tion [8]. Now, since d' and d" are very nearly equal, B' and B" may be taken as being equal without sensibly altering the relative values of c' and c" ; therefore we may say c' ad' - bx c" ad" - bx that is, c'ad" - c'bx = c'bx - c"ad', or _ a(c'd" 210 HANDBOOK OF ELECTRICAL TESTING. But as d" would be only 1 unit larger than d', that is, as therefore = a ( C 'd f + c' + c"d') _ a (d' 6(c' + c") ' + c") + being 10 and 1000 ohms respectively, when d' was adjusted to 156 ohms a deflection of 15 divisions (c f ) was obtained to one side of zero, and when d' was increased to 157 ohms, a deflection of 20 divisions (c") to the other side of zero, was observed. What was the exact value of a;? x = 10 1000 SLIDE BESISTANCE COILS BEIDGE. 222. Instead of fixing a and b and varying cZ, we may make a a fixed resistance, and b + d a slide resistance, and vary the ratio of b to d. Either a slide wire or a set of slide resistance coils, such as that indicated by Fig. 9 (page 15), may be used. The former would be employed if b -f- d is required to be a low resistance, the latter if a high resistance is necessary. A set of coils allows of but few different ratios being given to b and d, unless indeed the number of coils is very large, which would be both a cumbersome and an expensive arrangement. The late Mr. Yarley, by means of a movable derived circuit, THE WHEATSTONE BKIDGE. 211 reaching across two of the coils, devised a means of subdividing each of the latter. This arrangement is shown by means of Figs. 66 and 67. Eeferring to Fig. 66, let us suppose that equilibrium is produced so that no current circulates through the galvanometer. This being the case, the points C and A may be joined without altering the current strengths in the various circuits. Let us suppose this junction to be effected ; then, by applying Kirchoff'w laws (page 156), we have the following relations existing between the current strengths and the resistances in the system : c 2 - c 3 - c 4 = 0. C l a 4 C 2 b C 4 Pi = 0- c x c 2 d c 4 p 2 = 0. C ~ C * =0. By substitution we get If we divide one equation by the other, then we have _ ft (Pi + P2 + K ) + * Pi x d( Pl + p 2 + K) + /cp 2 * Now if in this equation we make K = p l + p 2 we S e ^ I . Pi a 26 + + + 2& + ^2 x " 2 d ( Pl -j- p 2 ) + (P! + p 2 ) p 2 2 ^ This equation shows that if the slide resistance Pl + p 2 be made equal to the portion K of the slide resistance b -}- K -{- d which it encloses, then the values of the resistances between the points B A and E A will be to one another, as the resistance & plus half the resistance p lf is to the resistance d plus half the resistance p 2 . If, therefore, we have b + K + d formed of 101 coils of, say, 1000 ohms each, and Pl + p 2 of 100 coils of 20 ohms each, that is, 2000 ohms ( Pl + p 2 ) in all, and further, if the slider -> (Fig. 67) bridges across two of the 1000-ohm coils so as to enclose a resistance of 2000 ohms (K), then a movement of slider p 2 212 HANDBOOK OF ELECTRICAL TESTING. ! from one contact to the next represents an alteration of 10 ohms in the ratio of B A to E A, whilst a similar movement of the slider s 2 represents an alteration of 1000 ohms. We can thus, by means of the 201 coils, 101 of 1000 ohms each and 100 of 20 ohms each, obtain 10,000 ratios of BA and EA, each differing from the next by 10 ohms. FIG. 67. We could, if required, have a second slider like s. 2 to move along p!+p 2 (Fig. 66), and connected to a third set of coils along which the slider s x would move ; by this means the differences of 10 ohms could be subdivided into differences of yV^ n f an nm - In fact, we could have any number of sets of coils with sliders, each carrying out the subdivision to any required degree. When we come to make very small subdivisions, such, for instance, as subdividing T Vth of an ohm into 100 parts of -nnnrth of an ohm each, it would be inconvenient to employ a set of small resistances, as they are difficult to adjust exactly ; slide wires ( 18, page 15) may therefore be employed with advantage for the purpose. 223. Fig. 68 shows a convenient arrangement of the Slide Resistance Coils Bridge ; the coils in this case are arranged in a circle instead of in a straight line as represented by the theoretical diagram Fig. 67. The left-hand dial contains the contacts and double slider for the 1000-ohm coils, and the right- hand dial the contacts and single slider for the 20-ohm coils. Fig. 69 shows a theoretical arrangement of the foregoing Slide Resistance Coils Bridge ; the connections in this diagram differ from those shown in Fig. 67 in so far that the relative THE WHEATSTONE BRIDGE. 213 positions of the battery and galvanometer are reversed, but this reversal is not essential to the principle, as either arrangement can be employed. FIG. 68. FIG. 69. SLIDE WIRE OR METRE BRIDGE. 224. The simple slide wire bridge is a very useful arrange- ment, as a very close adjustment can be made by means of it, and great accuracy of measurement thereby be obtained. It is especially useful for measuring small resistances accurately. 214 HANDBOOK OF ELECTKICAL TESTING. A form in which this description of bridge is very generally constructed is shown by Fig. 70. The slide wire, which is 1 metre long and about 1 5 mm. in sT FIG. 71. ml diameter, is stretched upon an oblong board (forming the base of the instrument) parallel to a metre scale divided throughout its whole length into millimetres, and so placed that its two ends are as nearly as possible opposite to the divisions and 1000 re- spectively of the scale. The ends of the wire are soldered to a broad, thick copper band, which passes round each end of the graduated scale, and runs parallel to it on the side opposite to the wire. This band is interrupted by four gaps, at wij, a, x, and m. 2 . On each side of these gaps, and also at B, C, and E, are terminals. In the ordinary use of the apparatus (Fig. 71), the wires from the battery are attached to the terminals B and E, and the galvanometer is connected between and the slider A ; by pressing down a knob this latter is put in contact with the wire. The conductor whose resistance has to be measured, and a standard resistance, are placed in the gaps at x and a respec- tively. The two gaps at m l and m 2 (Fig. 70) can either be bridged across by thick copper straps, or resistances of known values can be inserted in them ; it is easy to see that these resistances are simply ungraduated prolongations of the slide wire. THE WHEATSTONE BEIDGE. 215 225. If we have no resistance in these gaps, then when we have equilibrium, x d d As- is merely a ratio, we do not require to know the absolute values of d and 6, but only their relative values, that is to say, we only require to know the lengths of the portions on either side of the slider A, and not the resistances of those portions. The length Jc of the slide wire is constant, that is, I + d = &, or, d = k &, therefore x = a but Jc = 1000 millimetres, and b is usually called the scale reading, therefore we have / t 1000 \ x =o( . - T . -- 1 L [A] \scale reading f For example. The standard resistance a being 1 ohm, equilibrium was obtained when the scale reading was 510 ; what was the value of the unknown resistance x ? x = 1 l~7r- 1 ) = 961 ohms. \510 / 226. It has been pointed out by Mr. Martin F. Roberts that equation [A] is the same as x 1000 - i) and that consequently, by the use of a table of reciprocals, cal- culations can be considerably simplified in working out the value of x. 227. Equation [A] is only true if the resistances between the ends of the slide wire and the terminals B and E are zero. But, although it may not appear so, it is by no means easy to make these resistances inappreciable ; even the careful soldering of the ends of the wire to the copper straps introduces a resistance which is sufficient to affect very accurate tests. Referring to 216 HANDBOOK OF ELECTEICAL TESTING. Fig. 70, in which n and w 2 are these resistances, we know that strictly speaking x d -f n 2 ~a = 6 + / or that , / 1000 + rc 1 + K 2 A \scale reading + % / To make a strictly accurate test, then, we must know the values of n^ and n 2 in terms of the equivalent length of the slide wire. These may be obtained in the following manner : Having bridged across the gaps at m 1 and m 2 with thick copper straps, taking care that the surfaces in contact are scraped bright, insert known resistances at a and x, a being rather larger than x ; then having obtained equilibrium, we have a (d + 2 ) = x (b + Wl ) ; now reverse a and x, and again obtain equilibrium. Let the new scale readings be b : and d ; we then have x (dj_ + w 2 ) = a (&i_ + nj. By multiplying up and arranging the quantities, we have an z = xb + i ad and x n 2 = ab : -j- a % a? cZj ; therefore a_ xb -\- xn 1 ad x a~b -\- a n^ x d^ that is a 2 Wj a 2 % = a; 2 & a x d a 2 ~b r + a a eZj , therefore In a similar manner we should find _ ax(b b) - Or since b + d = b 1 + d^ = 1000, that is, d = 1000 - 6, and d l = 1000 - 6 X , THE WHEATSTONE BKIDGE. 217 we have _ a x (b Z^) + # 2 Z> a 2 \ _ b x b l a 0? ic 2 a x and _ (1000 - li)x - (1000 - 6)a a a; ^or example. In order to determine % and w 2 , resistances were inserted at a and cc equal to 3 and 2 ohms respectively. Balance was obtained when the scale reading 6 was 603. On reversing a and #, balance was obtained when the scale reading 6 X was 399. What were the values of % and w 2 ? i 3 2 % = (1000 -399) 2 -(1000 -603) 3 = The value of x, then, would be given by the equation _ / 1000 + 9 + 11 _ A = / 1020 \ scale reading + 9 / \scale reading + 9 228. Although perfectly satisfactory results may be obtained with the metre bridge when the latter is properly made, and when the measurements are carefully carried out, yet consider- able trouble is often occasioned to inexperienced persons by results being obtained which are obviously erroneous. One most frequent cause of error is that occasioned by imperfect con- tacts ; great care should therefore be taken that the important connections, viz. those at the gaps, should be well made ; this should be ensured by having the various surfaces in contact made clean and bright by scraping. Good contacts are best assured by having mercury cups at the gaps instead of screw terminals ; care should be taken that the mercury in these cups is in good metallic contact with them, that is to say, it should wet the metallic surfaces. The mercury should also, of course, be in similar good contact with the ends of the wires or rods (the latter are usually attached to the standard resistances), which may be dipped into the cups. The amalgamation of the metallic surfaces is best effected by scouring the latter with emery paper, and then moistening them with a solution of nitrate of mercury. 218 HANDBOOK OF ELECTEICAL TESTING. = 3) 229. A form of bridge in which mercury cups are used in the place of terminals for the more important connections, is show T n by Fig. 72. This apparatus is also provided with a commutator for reversing the resistances placed at a and x. This commuta- tor is formed of four mercury cups (seen in the centre of the figure) forming the corners of a square. These cups can be connected by means of the connector shown in the upper part of the figure. This con- nector is simply a short bar of ebonite with short copper rods at its extremities and at right angles to the latter ; the ends of these rods are bent down so that they can dip into the cups when the arrangement is placed over the latter. If the connector is placed over the cups so that the ebonite bar is in the position shown by the dotted line, 1-2, then it will be seen that the left- hand cup at a is connected to the right-hand cup at ?%, and the right-hand cup at x to the left-hand cup at m 2 ; if, however, the ebonite bar is in the position shown by the dotted line, 1-2,, then the left-hand cup at a is connected to the left- hand cup at m 2 , and the right-hand cup at x to the right-hand cup at m x . Lj I ' Even if good contacts be assured, correct results cannot be obtained if the standard resistances are in- correct, or if the slide wire is not uniform in its resistance throughout its length. A metre bridge to be really useful, therefore, requires to be very carefully made. THE WHEATSTONE BRIDGE. 219 230. The form of standard resistance generally used with the metre bridge is similar to that shown by Fig. 73. The ends of the brass rods to the left of the figure dip into the mercury cups ; the FIG. 73. resistance itself is enclosed in a brass box and bedded in paraffine wax. The particular pattern shown is an arrangement devised by Professor Chrystal to show whether the temperature of the interior of the brass box is the same as that of the surrounding air. It consists of a thermo-electric couple with one junction outside and one junction inside the box; by connecting this couple (whose terminals are seen on the upper part of the box) to a galvanometer of low resistance, no deflection would be produced if the two junctions, that is, the paraffine inside and the air outside the box, are at the same temperature. 231. The accuracy with which a test can be made, as in the ordinary form of bridge, depends upon the values of the various resistances, and amongst these upon the value given to k. In order to be able to vary the value of this quantity, the gaps at TO! and m 2 are provided. As the resistances placed in these gaps are simply prolonga- tions of the slide wire, it is necessary that their values should be known in terms of equivalent lengths of the slide wire ; that is, we must know how many millimetres of the wire they are equal to. This is best done in the following manner : Close the gaps at m l and m 2 with the thick copper straps, and place resistances of known values at a and x. Adjust the slides so that equilibrium is produced, then 6 + TC! or x (b + O fe a (1000 + n 2 - fc). Now insert one of the resistances, whose equivalent length m : in millimetres is required, at the left-hand gap, and again obtain equilibrium ; calling the new scale reading b l we then have (&! + n l +m l ) = a (1000 + w 2 - & L ). By subtracting the one equation from the other we get x (b 6 X ) xmj^ = a(b l 6), 220 HANDBOOK OF ELECTRICAL TESTING. that is, (6 - & x ) ( + ) = m^, or For example. It being required to know how many millimetres of the slide wire a resistance m l was equal to, the scale reading 6, with the two gaps closed, was 500 mm., and the scale reading b 19 with m l inserted, was 480 mm., the resistances at a and x being 6 and 4 ohms respectively. What was the value of m l ? m l = (500 - 480) -^^ = 50 mm. If we have a and x equal, we get the simplification mi = (& - 6 X ) 2. There are other methods of determining the value of m 19 but the one given, besides being extremely simple, is very accurate, as it is independent of the quantities n l and w 2 . The millimetre values of the resistances to be placed at m z and m 2 being thus determined, the value of x is given by the equation x = a f 1000 + i + "2 + MI + "2 __ \ scale reading -f i 4- ??h 232. Let us now consider the Best arrangement of resistances, &c., for making a test with the metre bridge, under favourable conditions. Now a mistake of a millimetre in the position of the slider will make a much greater error in the result of x worked out from the formula, when the slider is near the ends of the wire than when it is near the middle. Thus, for example, suppose x was 1 ohm and a was also 1 ohm, then we should have the slider standing exactly at 500 if it were properly adjusted. Suppose, however, it was 1 millimetre out, then the apparent value of x would be /innn \ -1)= -996, that is, we make x, I -996, or -004 ohms, too small. Next suppose a = .9 ohms, then for equilibrium the scale read- THE WHEATSTONE BRIDGE. 221 ing would be 900, and if we make a mistake of 1 millimetre we should have that is, we make x, 1 -990, or -010 ohms, too small. Lastly let us suppose a - ^ ohm, then the scale reading for exact equilibrium would be 100, and supposing there to be an error of 1 millimetre, we have 1/1000 x = - - 9V101 that is, we make a?, 1 '989, or 'Oil ohms, too small. To summarise the results, then, we see that with a larger than a, error was -010, or 1 per cent. equal to * 004, or f ,, smaller than,, 'Oil, or 1 The error, in fact, was smallest when the slider was at the middle of the wire. We must, however, determine whether the middle is really the point at which the error is least. Calling Jc' the resistance of the slide wire and its prolonga- tions m lt m 2 , and b f the scale reading plus the prolongation m 1 , let there be an error A in x caused by an error 8 in &', then But or> a = therefore X = x V - 8 - 1 - 1 = x k'S or since S is a very small quantity, we may say, ' X = x [A] V (k - V) Now we have to make X as small as possible ; this we shall do, since x and k' are constant quantities, by making 6' (&' 6') as large as possible. 222 HANDBOOK OF ELECTRICAL TESTING. But V (V - V) = - (~ - and to make this expression as large as possible, we must make 2~ V as small as possible ; that is, since V must be positive, we must make it equal to 0, or |-Z' = 0; that is, b f = |; which proves the truth of the supposition. To obtain the slider as near to the middle of the wire as possible when equilibrium is produced, we must make a as nearly as possible equal to x. If in equation [A] we put \ i -jj \ = r of x, and, V = > we get ., 4003 * = jy~> so that if when the slider is near to the centre of k' we can adjust the slider to an accuracy of 1 division (8), then if k' con- sisted of 1000 parts (as would be the case if there were no prolongations m lt w 2 ), we could measure the value of x to an accuracy of 400 x 1 233. In order to make a measurement in this manner, as we have seen, it is necessary for a to be approximately equal to x. Now in many cases there would be no difficulty in arranging that such should be the case. Thus, for example, suppose it was required to measure the conductivity of a sample of wire, then in this case we should take a sufficient length of the wire to give a resistance approximately equal to a, and then having measured the exact length taken, we should ascertain its exact resistance by adjusting the slider until equilibrium was ob- tained. 234. If we wish the measurement to be made to a higher percentage of accuracy than can be made with the slide wire k alone, then we must add equal resistances, m l and m 2 , at each end of the wire so as to increase the value of k. THE WHEATSTONE BRIDGE. 223 Since %l 400 8 X = -T- therefore so that if we wish to measure x to an accuracy, say, of 1 per cent., then we must make &' equal to that is to say, we must add resistances m 1 and m 2 at each end of &, each equivalent to 1500 millimetres of the wire k. It must "be recollected, however, that there will be no advantage in thus increasing the length of &, unless the figure of merit of the galvanometer employed is sufficiently high to enable a movement of the slider to a distance of 1 division from its correct position, to produce a perceptible movement of the needle. If the resistance to be measured is not one which admits of adjustment, then in order to obtain a satisfactory measurement we must add a resistance on to one or other of the ends of &, according as x is larger or smaller than a ; or wo may add resistances to both ends, their values being unequal. If in equation [A] (page 221) we put %-v> or ' * ' - b '~ m then we get (a + a?) 8 = T. / > b or if we put V we have 100(a + ,)8 ^G+O 8 X' = ^2-, or, 6= x - From this equation we can see that no matter what are the relative values of a and x still 6' can always have a value which will enable x to be obtained to any percentage of accuracy X f ; that is, of course, provided the figure of merit of the galvano- meter be sufficiently high for the purpose. 224 HANDBOOK OF ELECTRICAL TESTING. For example. It is required to measure the exact value of a resistance x, whose approximate value is five times that of the resistance a what must be the value of b' in order that the measurement may be made to an accuracy of 5 per cent. ? The adjustment of the slider can be determined to an accuracy of 1 division. 100 Q + l)l 6' = V5 , ' = 240. * From equation [1] (page 223) we get k' = 240 (l + 5) = 1440, consequently since Jc consists of 1000 divisions we must add a prolongation m 2 equal to not less than 440 divisions, on to k. We may of course make the prolongation larger than 440 ; in fact, in practice we should have to do so unless we had a resistance available of the exact required value ; but it must not be too large, otherwise the position of balance for the slider would be at some point on m 2 instead of on the wire k. / ' r In fact, w 2 must not be greater than If it should happen that in order to obtain a particular per- centage of accuracy it is necessary that b' should exceed &, then in this case it would be necessary to have a prolongation m x in addition to the prolongation m 2 ; the latter quantity in this case must not exceed (Jc -j- m i) In the last example we have supposed x to be less than a. If, however, x is greater than a, then b' will probably have to be greater than k, in which case of course we should have to add the prolongation m-^ in the place of the prolongation m 2 , the value of m-L being such that it does not exceed Jc -, unless we also add a prolongation w 2 in addition to ro lf in which case m l must not exceed (Jc + w 2 ) - We have seen that by means of m l and m 2 the values of which can be determined in the manner shown in 231 (page 219) we can theoretically arrange that the value of x can be assured to any required degree of accuracy, no matter what the relative values of x and a may be. This, however, can only be the case THE WHEATSTONE BRIDGE. 225 provided the figure of merit of the galvanometer is such as to enable the slider to be adjusted to an accuracy of 1 division. The figure of merit of the galvanometer, therefore, as in other tests, is the limit to the " Possible degree of accuracy attain- able." This limit can be determined from equation [2] (page 195) in the following manner : Let A be the error in #, caused by b' being ^th of a unit out of adjustment, then we have and since a d' = V x, and is a very small quantity, we get . ! a+x *-:nr s bx We have then from equation [2] (page 195) by putting d' = , b' c 6 = In order, therefore, that b' may be able to have the value necessary to ensure x being measured to the required degree of accuracy, the value of c 6 must not be less than that given by the above equation. As the values of g, d, x, and r are mostly easily obtained in ohms, the value of b' corresponding to the number of divisions of which it would consist must be in ohms also ; , likewise, will have to be the resistance, in the fraction of an ohm, correspond- ing to 1 division (or fraction of a division, if the slider can be adjusted to a closer accuracy than 1 division) of the wire Tc. For example. In the last example it was required to be known whether a galvanometer whose resistance was 1 ohm (g\ and the reciprocal of whose figure of merit was -0002 (c 6 ) would be suitable for the purpose of making the measurement in question. The resistance of the slide wire, which was divided into 1000 divisions (&), was 5 ohms; the resistance a was 1 ohm, and the resistance x, 5 ohms approximately. The actual value of the prolongation added to k was such as to make V equal to 1560. The resistance Q 226 HANDBOOK OF ELECTKICAL TESTING. of the battery was 5 ohms (r), and its electromotive force 2 volts (E) approximately. Since k = 1000, therefore \ = = -0005. Also (from equation [1], page 223) we have ,, ale' 1 x 1560 . . -5 X 260 = Z+-* = TIT" ~- ' 13 ohms; therefore 2 X '0005 X^ C R = = -0008, (11 -65) (49 -46) which is greater than 0002, the reciprocal of the figure of merit of the galvanometer in question, consequently the latter instru- ment is well suited for the purpose for which it is required. 235. The resistance of the galvanometer employed in making a bridge test is an important point, especially as regards the measurement of small resistances. In the case of the ordinary bridge test, we can adjust within 1 unit, and in the case of the slide wire bridge, we can adjust within 1 millimetre of the wire ; if then the galvanometers em- ployed in these cases are such that when we are 1 unit or 1 millimetre from exact equilibrium we obtain perceptible deflections of the needles, then we have what we require, what- ever the resistances of the galvanometers may be. In the ordinary form of bridge, where the adjustable resist- ances are not capable of being adjusted to a greater accuracy than 1 unit, a Thomson's galvanometer, such as that described in Chapter III. (page 46), and which has a resistance of about 5000 ohms, gives, under all circumstances, a very large deflection when the adjustment is only 1 unit from equilibrium. In the case of the slide wire bridge, however, where to be 1 millimetre from exact equilibrium means to be only ^th of an ohm, or even less, out, a galvanometer of such a high resistance as 5000 ohms would not be found to give a perceptible deflection. The reason of this is, that such a galvanometer is practically short circuited by the very low resistance it has between its terminals. THE WHEATSTONE BRIDGE. 227 The question of galvanometer resistance is considered at length in Chapter XXV., and it is there shown that it is best that the instrument should have a resistance not more than about 10 times, or less than about ^th, ^ . Of course a -}- x in practice we cannot adjust the resistance to meet every par- ticular case, but the limits given are sufficiently wide to enable an instrument to be made which would prove satisfactory for most purposes for which the metre bridge is adapted ; moreover, if a particular galvanometer does not prove to be suitable for a particular purpose, we can ascertain, by the help of the above rule, whether the cause is due to its resistance being too high or too low. It should be clearly understood that when we speak of the resistance of the galvanometer we mean the resistance of the instrument itself, and not the resistance in its circuit ; thus, if according to calculation it were proved that the galvanometer resistance should be 1 ohm, then it would not be carrying out the rule if we took an instrument having a resistance of, say ^ of an ohm, and added a resistance of f of an ohm in its circuit, for this J of an ohm would be an addition to the external circuit, and not an addition to the galvanometer itself. Under no conditions should the battery be joined between A and C, and the galvanometer between B and E, for in such a case the battery current in passing from the slider to the wire would be liable to injure the surface of the latter. To sum up, then, we have Conditions necessary for making the Test to any required Degree of Accuracy. 236. The number of divisions of which b' must consist in order that x may be measured to an accuracy of A' per cent, must be not less than 100 - 8 being the number of divisions, or the fraction of a division, to which it is possible to adjust the slider. If prolongations are necessary, then m 1 must not exceed and m 2 must not exceed ( Jc + m 1 J - Q 2 J-A I./XVXVJJ. ( k+m ^)l< 228 HANDBOOK OF ELECTRICAL TESTING. The reciprocal of the figure of merit of the galvanometer must be not less than E8. a + x where E is in volts and all the other quantities (including V and d'~) are in ohms. Possible Degree of Accuracy attainable. Percentage of accuracy = -=-p ( - + 1 ) 8. o \x / MEASUKEMENTS BY CAREY FOSTER'S METHOD. 237. This method, devised by Prof. Carey Foster,* consists in determining the value of the unknown resistance in terms of an equivalent length of the slide wire; this is effected in the following way : FIG. 74. FIG. 75. The resistance, x, whose value is to be determined, is placed in the left-hand gap (Fig. 74), and resistances r^ r 2 , the ratio of whose values does not diifer from unity more than does that of the resistance to be measured and the resistance of the whole slide wire, are placed in the two centre gaps ; the right-hand gap is closed by a conductor without sensible resistance. The slider is now adjusted until equilibrium is obtained, and the reading Z> is noted, x is then transferred to the right-hand gap, and the left-hand gap is closed by a conductor without sensible resistance (Fig. 75) ; the slider is again adjusted and the reading & L noted. Calling n the resistance of the portion of the copper strap between B and the left-hand end of the slide wire, and n 2 the * Journal of the Society of Telegraph Engineers,' vol. i. page 196. THE WHEATSTONE BKIDGE. 229 resistance of the portion of the strap between E and the right- hand end of the slide wire ; also calling ^ the total resistance between the points B and C, and r 2 the total resistance between the points C and E; finally, calling b and b t the respective resistances of the portions of the slide wire, in the two tests, and calling I the total resistance of the slide wire, we have r 2 I- and also r 2 I - &! + w 2 + ^ ' therefore therefore X l-* 1+ n+ l = therefore Z 6 + w 2 I b^ -{- n 2 -\- x therefore I - 6 + n 2 = I -b l + n 2 + x, or x = b^-b. In order to make this formula useful we must know the resistance per millimetre of the slide wire, since b l and b on the scale represent not resistances but lengths. The simplest method of doing this is to take a test in the foregoing manner, giving the resistance x a known value, 1 ohm for example ; in the latter case, since the difference between the two scale readings multiplied by 10 gives the number, v, of millimetres corresponding to 1 ohm resistance, and therefore when we make a test to determine an unknown resistance, #, we get The accuracy of the test depends upon the conductor with which the unknown resistance, #, is interchanged having prac- 230 HANDBOOK OF ELECTEICAL TESTING. tically no resistance ; it should, therefore, be made of as massive and short a piece of copper as possible, and the connections should be made by means of mercury cups.* The great merit of Professor Foster's method lies in the fact, that the measurements are independent of the resistances of the various parts of the copper band. 238. Professor Foster points out that inasmuch as by his method the value of a resistance, x, can be determined in terms of a certain length of the slide wire, therefore if x be made a known resistance and the slide wire itself be formed of a portion of wire whose resistance per unit length is required, this latter resistance can easily be determined. Such a method would give very accurate results, and is as good as " Thomson's Bridge " method, which was devised by Sir William Thomson for the same purpose, and is as follows : THOMSON'S BRIDGE. 239. The arrangement of this bridge is shown by Fig. 76; its object is the accurate measurement of the resistance of a FIG. 7G. portion of a conductor of low resistance, lying between two points, errors due to imperfect connections being avoided. In the Fig., B F is the conductor, the resistance b of the corre- sponding length, 1-2, of which requires to be determined. F E is a standard slide wire whose resistance per unit length is * As a rule the cups at each side of a gap are too small and are not put close enough together, the consequence being that a conductor used for bridging over a gap is comparatively long and has a sensible resistance. The cups ought to be of terge dimensions and so close together as almost to touch, the bridge piece could then be made so massive and short as to be practically of a negligible resistance. The ends of this piece should be quite flat, so as to lie closely in contact with the bottom of the cups. THE WHEATSTONE BRIDGE. 231 accurately known. Now when we have equilibrium we see from equation [A] (page 211) that we have = & (Pi + p a + K) + K Pl . by multiplying up and arranging we get Pi + p 2 Now if we iiave a a? , . = -, that is, x p l = a p 2 Pi P2 we get a d b x = 0, or from which we see that the value of 6 is independent of the resistance of any of the connections, provided the contacts at the points 1, 2, 3, and 4 are small compared with the resistances a, p 15 p 2 , and a, which, by making these resistances high enough, will practically be the case. The pointa 1, 2, 3, and 4 should be knife edges, so that the exact distance between 1 and 2, and between 3 and 4 can be properly determined. MEASUREMENT OF THE CONDUCTIVITY RESISTANCE OF A TELEGRAPH LINE. Direct Method. 240. When, by means of the bridge, Fig. 63 (p. 191), we are measuring the conductivity resistance of a wire whose further end is not at hand, we should join one end to terminal C, put the further end to earth, put terminal E to earth, and then measure in the usual way. Loop Method. 241. It is always as well, however, when possible, to measure without using an earth, by looping two wires together at their further ends, :he nearer ends being joined to terminals E and C respectively ; this gives the joint conductivity resistance of the two. Errors consequent from earth currents, or a defective earth, &c., are thereby avoided. We cannot, however, by this means, 232 HANDBOOK OF ELECTEICAL TESTING. obtain the conductivity resistance of each wire separately. If, however, we have three wires at hand, we can by three measure- ments obtain the conductivity resistance of each wire, without using an earth. This is effected as follows : Let the three wires be numbered respectively 1, 2, and 3. First loop wires 1 and 2, at their further ends, and let their resistance be Bj. Next loop wires 1 and 3, and let their resistance be E 2 . Lastly, loop 2 and 3, and let their resistance be B 3 . Supposing the respective resistances of 1, 2, and 3 to be r lt r 2 , and r 3 , we get r l + r 2 = E! r, + r 3 = E 2 r 2 + r 3 = E 3 . Now, since each of the wires is looped first with one and then with the other of the other two, it is evident that the sum of the three measurements will be the sum of the individual resistances of the three wires taken twice over, and conse- T? -I- ~R _L- T? quently ^ must be the sum of the resistances of 2t the three wires. If, then, we subtract B T from this result, the remainder must be the resistance of r 3 . Similarly, if we subtract E 2 from the same, the remainder will give us r 2 ; and lastly, by subtracting E 3 , we get the value of r 1 . For example. The conductivity resistance of each of three vires, Nos. 1, 2, and 3 was required. Nos. 1 and 2 being looped, the resistance (E x ) was found to be 300 ohms. Nos. 1 and 3 looped gave a resistance (E 2 ) of 400 ohms. Lastly, Nos. 2 and 3 looped gave a resistance (E 3 ) of 500 ohms. Then : added resistance of the three wires will be 300 -f 400 + 500 . , - ! = 600 ohms ; 2i therefore, Besistance (r^) of No. 1 wire = 600 - 500 = 100 ohms. (r a ) 2 = 600 - 400 = 200 (r 3 ) 3 =600-300-300 By this device, then, we are enabled to eliminate all sources of error without making a greater number of measurements than would be required if we measured each wire separately, by using an earth. THE WHEATSTONE BRIDGE. 233 MEASUREMENT OF THE EESISTANCE OF AN EARTH. 242. By means of a method very similar to the foregoing we can, if we have two wires at our disposal, measure the resist- ance of the earths at the ends of the lines. The following is the way in which this can be done : Let the two wires be numbered respectively 1 and 2. First loop the two wires at their further ends, and let the measured resistance of the loop be E x . Next have No. 1 wire put to the earth at its further end, and measure the resistance, which will be that of the wire and earths combined ; let this total resistance be E 4 . Lastly, have wire No. 2 put to the earth at the distant station, and measure the total resistance, which we will call E 5 ; then by adding E x , E 4 and E 5 together, and dividing the result by 2, we get the sum of the resistances of the two wires and the earth ; by subtracting from this result the resistances of the two looped wires the remainder will be the resistance of the earths. 243. By means of a test made in this manner we can deter- mine not only the resistance of an earth, but also the in- dividual resistance of two wires ; for if we subtract E 4 from T? -4- T? [ , the result will be the resistance of wire No. 2, 2 and if we subtract E 5 instead of E 4 , then the result will be the resistance of wire No. 1. Such a test, -however, although it eliminates errors due to defective earths, does not eliminate errors due to earth currents. But inasmuch as it is a test which is applicable when only two wires can be had, it is useful, since the earth current errors can be eliminated by a method which we shall investigate. MEASUREMENT OF THE INSULATION EESISTANCE OF A TELEGRAPH LINE. 244. In measuring the insulation resistance of a wire, the con- nections would be the same as for conductivity resistance, except that the further end of the wire, instead of being put to earth, would be insulated. 245. It sometimes happens that we require to find the insula- tion resistance of two sections of one wire, but we can only test from one end. Now, if we join several wires together, one in front of the other, it is evident that the total insulation resistance of the combination will diminish according to the number of the wires and according to the insulation resistance of each of them. 234 HANDBOOK OF ELECTRICAL TESTING. The law for the total resistance, in fact, will be the same as that for the joint conductor resistance of a number of wires joined up in multiple arc (page 70). That is to say, the total insulation resistance of any number of wires joined together will be equal to the reciprocal of the sum of the reciprocals of their respective insulation resistances. As a matter of fact, it is immaterial whether the wires be joined together one in front of the other or all be bunched together; the law of the joint insulation resistance is the same in both cases.* ABC Suppose, then, A C to be the wire which is required to be tested for insulation resistance from A in two sections, A B and B C. Let a be the insulation resistance of the section A B, and 6 the insulation resistance of the section B C ; and suppose x to be the insulation resistance of the whole wire from A to C, then we have aft x = -, a-j- b from which 6= a x All we have to do, therefore, supposing we are testing from A, is first to get the end C insulated and to measure the insulation resistance ; this gives us x. Next get the wire separated at B, and the end of the section A B insulated. Again measure the insulation resistance; this gives us a. Then from the two results b can be calculated. For example. The insulation resistance (x) of the whole wire, from A to C, was found to be 6000 ohms, and that from A to B (a), 24,000 ohms. What was the insulation resistance (6) of the section B ? 24,000 X 6000 24,000 - 6000 246. To obtain the conductivity resistance of one section of a wire when the resistance of the other section, and also of the whole wire, is known, we have only to subtract the resistance of the one section from the resistance of the whole section. The truth of this is obvious. * This is not the case if the insulation resistances are very low, as the resistance of the conductor then comes into question and modifies the result. THE WHEATSTONE BKIDGE. 235 MEASUREMENT OF THE CONDUCTIVITY RESISTANCE OF WIRES TRAVERSED BY EARTH CURRENTS. 247. When the conductivity resistance of a line of telegraph is measured by having the further end of the line put to earth, the presence of earth currents, that is to say, the currents set up by electrical disturbances over the surface of the earth, and also currents due to the polarisation of the earth plates, renders the formula x = d r-, when equilibrium is produced, incorrect. To obtain the true value of the resistance of the wire, therefore, a different formula is necessary. Equilibrium Method. 248. In Fig. 77 let E be the electromotive force of the testing battery, Ej the electromotive force of the earth current, whose value will be -f- or according to its direction, and let a, 6, d, x 9 and r be the resistances of the various parts of the bridge ; then FIG. 77. c i C 2 C a C 4) an ^ c 5 being the current strengths in the different branches, we have by Kirchoff's laws (page 156), when equi- librium is produced, the following equations connecting the resistances, current strengths, and electromotive forces : c 2 c 3 = c 5 - c 3 - c 4 = c a c 2 6 = c 4 x c 3 d cr cd = E. 236 HANDBOOK OF ELECTKICAL TESTING. By elimination we obtain two values of c 4 , one in terms of the battery E 15 and the other in terms of E, thus + " . B. and E c. = 249. If we equate the two values of c 4 we can get the relation between the two electromotive forces E and E 1? and thus obtain a method of determining the relative electromotive forces of the batteries, for we have E bx-ad g + r)-| J E 250. From the latter equation we find a d E! -& E To make this equation useful it is necessary that ^ and E be known. If, however, we reverse the testing battery and again obtain equilibrium by readjusting d to d lt we get a second equation, viz., E x b x a d l - E a (di + r) + b (a + r) ' we therefore have bx ad bx ad^ = 0. By multiplying up we get + b x [a (d + r) + b (a + r)] a d^a (d + r) + b (a + r)] = 0; that is _ a d [a (d l + r) + b (a + r)] + J = 690-2 ohm* It may be pointed out that the quantity j^ ^ l r-^ in \Cl + KJ + ( CL-^ + KJ equation [A] is the harmonic mean of the quantities (d + Jc) and (d 1 + k). Various abbreviations of formula [A] have been suggested, but none of them are satisfactory except under certain condi- tions, and inasmuch as the formula is only required occasionally, the advantage of a simplification which at the best is only an approximation is a doubtful one. Mance's Method.* 251. This method, devised by Sir Henry Mance, consists in making the observations as in the last test, but without reversing the current, the first observation being made with resistances a and b in the arms B C and B A of the bridge, and the second with these resistances changed to a^ and b l9 In the first case, then, we have E! ' bx ad ~E~ = a (d + r) + b (a + r)' in the second case E x __ frj x - a 1 d l * Journal of the Society of Telegraph Engineers,' May 8th, 1886. 238 HANDBOOK OF ELECTRICAL TESTING. therefore b x ad b l x a l d r a(d + r) + 6 ( + r) a, (d, + r) + b, (a, + r) By multiplying up, and extracting a;, we get a. = <*[<*i (<*i + r) + b, K + r)] - q^JXd + r) + 6(g + r)] 6 [!(<*! + r) + &!(! + r)] - b, [a(d + r) + 6(0 + r)] _ CT # [(a x + 6Q r + i &i] - ^ <> + & ) + (& + di] = ~ ^6 (& 9 - d i g + b x + b dj + E (d 1 + a?) - E! (b + d,) or say - c 6 *' + E(^ + X ) - E x (6 + d,) THE WHEATSTONE BEIDGE. 239 that is c, = k" K + k' Now, supposing tlie electromotive force E is removed without altering r, and suppose at the same time that c 6 and the other quantities remain unaltered, then we have that is k" K + *' therefore E(^ + a;) - E x (6 + dj - E(a + b)K = - therefore Or giving the value [A] of K, we have r(t.+0 d > 4 } rCa + ^ therefore r (a + 6) (^ + x) + a (b + dO (^ + *) = r(a + 6)(d x + a;) + a? (6 + ^) ( + &); therefore a^ + oj) = a; (0 + 6), or .-*. If therefore we have a key so arranged that on depressing it a resistance equal to that of the battery is inserted in the place of the latter, then on adjusting the resistance d^ until it is found that the deflection of the galvanometer needle is the same whether the key is up or down, we get the value of x at once from the above equation. In the practical execution of the test it would be necessary to short circuit the galvanometer at the moment when the battery key is depressed or raised, otherwise a violent movement of the needle would be produced by the static discharge from the cable. 240 HANDBOOK OF ELECTRICAL TESTING. 253. When the battery connections for measuring conduc- tivity are made, as shown by Fig. 63 (page 191), then in order to put the zinc current to line, we should put the cable or line to C and the earth to E. To put the copper to line we can either reverse the battery or put the cable to E and the earth to C, whichever is most convenient to the experimenter. MEASUREMENT OF THE CONDUCTIVITY EESISTANCE OF A SUBMARINE CABLE. 254. When we are measuring the conductivity of a submarine cable, which requires to be carefully done, the best method to adopt is the following : Put on the battery current for half a minute by pressing down the right-hand key (Fig. 63, page 191) ; at the expiration of that time, proceed to adjust the plugs, pressing down the left-hand key as required until equilibrium is produced; con- tinue to adjust, if the needle does not remain at zero, and at the expiration of half a minute note the resistance. Now reverse the battery connections, put on the current for half a minute ; again measure, again reverse and measure, and so on until about a dozen measurements with either current have been taken. It will usually be found that about half the measurements made with the negative current are the same, and also half the mea- surements made with the positive current ; these results may be taken as the correct measurements for d and d r 255. In order to reverse the current through the cable, we can either reverse the battery, or the line and earth, connec- tions ( 253). There is an advantage in doing the latter, as by this means the galvanometer deflection due to, say, too much resistance being inserted between D and E (Fig. 63, page 191), is always on the same side of zero, although the direction of the current through the cable is reversed. Thus it is easy to see at a glance in every case, and without chance of a mistake, whether balance is out in consequence of too much or too little resistance being inserted. 256. The presence of earth currents can be detected when the line, galvanometer, and earth are joined to the resistance box, by pressing down the left-hand key alone. This will cause the galvanometer needle to be deflected if there are any currents present. A line is seldom, if ever, quite neutral in this respect. 257. It is almost immaterial what battery power is used in measuring conductivity ; sufficient, however, should be used to obtain a good deflection on the galvanometer needle when equi- librium is not exactly produced. About 10 or 20 cells is a THE WHEATSTONE BRIDGE. 241 convenient number to employ. There is no danger of heating the resistance coils with such a power if the battery be a Daniell charged with plain water, or even a Leclanche, as their internal resistances are considerable. It would not be advisable, however, to use a Grove or a Bunsen battery, or a Daniell charged with acidulated water, as their heating power is great in consequence of their small internal resistances, ELIMINATION OF THE EESISTANCE OF LEADING WIRES. 258. In order to determine the exact resistance of the con- ductor of a cable, or coil of cable core, for example, it is of course necessary that the resistance of the wires leading from the testing- room to the tank in which the cable or core is placed, should be deducted from the total measured resistance. This involves a calculation which, although slight, still might be avoided with advantage, especially if a large number of measurements have to be made. At Messrs. Siemens' works, at Charlton, a very simple device is adopted which enables the resistance of the leading wire to be eliminated, thus rendering any deduction unnecessary. For this purpose a small supplementary slide wire resistance ( 18, page 15) is connected in the arm A E of the bridge (Fig. 62, page 188); the leading wires (when con- nected to the bridge) being looped together at their further ends, and all the plugs being inserted in A E, the slide resist- ance is adjusted till balance is obtained on the galvanometer. The leads are now connected to the cable or core to be tested, and then balance is again obtained on the galvanometer by removing plugs from A E in the usual manner. This being done, the resistance unplugged in A E (allowing for the ratio of the arms A B, B C, of the bridge, if the two are unequal) obviously gives the exact value of the resistance required, since the resistance of the leads is balanced by the slide resistance. MEASUREMENT OF BATTERY RESISTANCE. 259. The resistance of a battery which consists of a large number of cells may in many cases be measured with a con- siderable degree of accuracy by means of the Wheatstone bridge, in the following manner : Divide the battery into two equal parts, and connect the two halves together so that their electromotive forces oppose one another ; under these conditions the battery may be treated as an ordinary resistance, and measured as such. 242 HANDBOOK OF ELECTKICAL TESTING. CHAPTEE IX. LOCALISATION OF FAULTS. 260. The theoretical methods of testing for the localities of faults are comparatively simple, but their practical application presents some difficulties. LOCALISATION OF A FULL EARTH FAULT. 261. The simplest kind of fault to localise is a complete fracture where the fault offers no resistance, and the conduc- tivity resistance at once gives its position. Thus, a line which was 100 miles long, and in its complete condition had a resist- ance of 1350 ohms, that is to say, a resistance of WT? = 13*5 ohms per mile, gave a resistance of 270 ohms when broken. Then distance of fault from testing station was 270 = 20 miles. LOCALISATION OF A PARTIAL EARTH FAULT. 262. When the fault has a resistance, the localisation becomes somewhat difficult. The following are the theoretical methods generally adopted (Fig. 78). Earth, BLAVIER'S METHOD. 263. Let A B be the line which has a fault /at C, A being the testing station. A first gets B to insulate his end of the LOCALISATION OF FAULTS. 243 line. He then measures the resistance, which we will call Z, then therefore /=i-. [i] Next, B puts his end to earth, and A again measures. Let the new resistance be Z lf then '.. [2] Calling L the resistance of the line, we have also a + 6 = L; therefore b = L - a. [3] From these three equations we have to determine a. Sub- stituting in [2] the values of /and b obtained from [1] and [3], we get (;- L 4. i _ 2a therefore a 2 - 2aZ x = LZ - L Z x - ZZ X ; from which, since a must be less than l lt and the root conse- quently negative, a = i - For example. A faulty cable, whose total conductivity resistance when perfect was 450 ohms (L), gave a resistance of 350 ohms (Z) when the further end was insulated, and 270 ohms (Z x ) when the end was put to earth. What was the resistance of the conductor up to the fault ? Resistance = 270 - J (350 - 270) (450 - 270) = 150 ohms. If the length of the cable were 50 miles, then conductivity per mile equals M = 9 ohms, and distance of fault from testing station consequently equals 1J-2 - 16|- miles. OVERLAP METHOD. 264. Two measurements are made, one by station A, and the other by station B, A and B, insulating their end in turn. Thus resistance measured from A when B insulates, as before, is a+f=l. [1] E 2 244 HANDBOOK OF ELECTEICAL TESTING. Kesistance measured from B when A insulates *,, [2] also a + 1) = L. [3] Subtracting [2] from [1] a b = I Z 2 , and adding [3] 2 a = L -f 1 - 1 2 ; therefore .For example. A faulty cable, whose total conductivity resistance when perfect was 450 ohms (L), when measured from A with the end at B insulated, gave a resistance of 350 ohms (7) ; and when measured from B with the end A insulated, a resistance of 500 ohms (Z 2 ). What was the resistance of the conductor from A to the fault ? Eesistance = ^50 + 350 - 500 = 15 265. In making the foregoing test it is often found advan- tageous to introduce a set of resistance coils at the end of the cable nearest the fault, and to vary this until it is found that the measurements made at the two ends give the same results. The advantage of this arrangement is that if the same amount of battery power be used at the two stations the test current flowing out at the fault will be the same in both cases, conse- quently the fault is likely to remain constant and more uniform results be obtained. It is obvious that if r be the added resist- ance, then the resistance from either end (the resistance r being taken as forming part of the cable) will be - , L being as in 2i previous cases the total conductor resistance of the perfect cable. PRACTICAL EXECUTION OF TESTS. 266. So far the testing is simple ; the practical application, however, presents some difficulty. This is owing to the varia- tion of the resistance of the fault when the testing current is put to the cable, in consequence of this current acting on the copper conductor, and through the agency of the sea water LOCALISATION OF FAULTS. 245 covering it with a salt, which besides increasing the resistance of the fault, also sets up a current opposing the testing current. To make a proper test, then, it is necessary so to manipulate the testing apparatus and battery as to get rid of the polarisa- tion and resistance set up by the salt formed on the fault, and to measure the resistance at the moment this is done. The following is known as : LUMSDEN'S METHOD. 267. The further end of the cable being insulated, the con- ductor is cleaned at the fault by applying a zinc current from 100 cells for ten or twelve hours, the current being occasionally reversed for a few minutes. A rough resistance test is then made with a copper current. A positive current is now applied to the cable for about one minute, using two or three cells for every 100 units of resist- ance which have to be measured. This coats the conductor with chloride of copper. The cable is now again connected to the resistance coils, and the battery and galvanometer connections made as shown by Fig. 63 (page 191), the zinc pole being to terminal B' and the copper to terminal E. The cable must be joined to C, and earth toE. Both keys being depressed, the galvanometer needle is care- fully watched and plugs inserted and shifted unit by unit, so as to keep the needle at zero; for the action of the negative current is to clean off the chloride of copper, and thereby to reduce the resistance of the fault. At a certain point this decomposition becomes complete, and the needle of the galva- nometer flies over with a jerk, showing that the disengagement of hydrogen has taken place at the fault, which enormously increases its resistance. The resistance in the resistance coils at that moment is the required resistance. The fault being once cleaned by the application of the 100 cells for ten or twelve hours, it is unnecessary on repeating the measurement, which should always be done, to apply the battery for so long a time ; ten or twenty minutes, or even less, will generally suffice. When the measurement is made with the further end of the cable to earth, the same process of preparation can be employed. The rate at which the decomposition of the salts at the fault takes place, depends to a very great extent upon the strength of the current flowing out at the fault ; now, if the latter be very near the end at which the test is being made, the resistance 246 HANDBOOK OF ELECTRICAL TESTING. between the testing battery and the fault will be so small that the changes at the latter will take place with great rapidity, and it would be a matter of great difficulty to adjust the resistance in the bridge quickly enough to follow up the change of resistance at the fault as it takes place. To avoid this difficulty the best plan is to insert a resistance between the bridge and the end of the cable ; this will retard the changes by reducing the strength of the current flowing in the circuit. The value of this resistance will depend entirely upon circum- stances, and will be a matter of judgment with the person making the test, but in any case it should not be out of proportion to the actual conductor resistance of the cable. The amount of battery power used is also a matter dependent upon circumstances, but the higher the power it is found possible to use, the less will the effect of earth currents influence the accuracy of the test. The resistances employed in the arms A B, B C of the bridge (Fig. 63, page 191), will, to some extent, modify the rate at which the changes at the fault take place, and here again discretion must be used, as no definite rule can well be laid down. It might be imagined that a " slide resistance " (page 14) would be very advantageous for making a test of this kind, but practical experience shows that the plug resistances are preferable in many cases. The galvanometer with which this and the following test must be made, must be an ordinary astatic one (page 18) with fibre suspended or pivoted needles. A Thomson's reflecting galvanometer is quite useless for the purpose. Before making the test, A must of course arrange with B, or vice versa, at what time and for how long he is to insulate, put to earth, &c., his end of the cable. FAHIE'S METHOD. 267. Mr. J. J. Fahie, in a paper read before the Society of Telegraph Engineers,* has given the results of some very careful experiments and tests which he has made, bearing upon the subject of testing for faults. His method contains many valuable points, and is, in the author's words as nearly as possible, as follows : The cable-current is eliminated by sending into the line the current of the opposite sign to that coming from it, and * 'Journal of the Proceedings of the Society of Telegraph-Engineers,' Vol. III., page 372. LOCALISATION OF FAULTS. 247 arranging the strength and duration of this current to suit the strength of the one from the cable. Thus, if the latter be strong and negative, put (say) sixty cells positive to line for a couple of minutes, and then note the condition of the cable- current ; if it be still negative, but weaker, put the battery on again for a short time, and continue to do so until the galvano- meter needle indicates a weak positive current from the fault. If the latter be now left to itself and the cable put to earth through a galvanometer the needle will steadily, and as a rule leisurely, fall to zero and pass over to the other side, indicating a negative current again from the fault. While the needle is on zero the line is free and in a fit state for the subsequent test. If the cable-current be positive, put sixty cells negative on until the fault is depolarised ; the effect in this case is more brief than in the other, the needle falling quickly to zero and crossing to its original position. Having once eliminated the current from the fault (and the operation very rarely exceeds ten minutes in the most obstinate cases) the cable can always be kept free by momentary applica- tions of the necessary battery pole. Thus, if the needle begin to move off zero in the direction indicating a negative current from the fault, a positive current applied for a moment will bring it back, and vice versa. In practice it is best to repolarise the fault slightly in the opposite direction, as a little time is thereby gained to arrange the bridge for a test. Having shown how to prepare the cable, the test will now be described. The bridge is arranged as shown by Fig. 79. P is the infinity plug ; when this plug is removed the connection between the branch coils 6 and the resistance d is severed ; K 2 is an ordinary key for putting the line to earth through the galvanometer G 2 or to the bridge as may be required. The rest needs no explanation. First ascertain by an ordinary test the approximate resistance of the faulty cable and leave it unplugged in d. Next allow the line to rest for a few minutes in order that it may recover itself from the effects of the current employed in this pre- liminary test, and then depress K 2 , and observe the cable- current on the galvanometer G 2 ; let it be positive, open the key Kp remove the plug P, and send a negative current from the testing battery of (say) sixty cells into the cable via the branch coils a, which should be plugged-in to avoid heating. When the cable-current has been repolarised a fact which may be ascertained by putting the cable to earth at intervals through G 2 arrange the bridge, close the key K x , and, keeping the cable to G 2 , watch till the needle comes to zero : at that 248 HANDBOOK OF ELECTRICAL TESTING. moment let K 2 fly back, and send a negative current through the bridge system, observing the instantaneous effect on the galvanometer G x . If d be too great the needle will be deflected in a direction (say to the right) indicative of this, but imme- diately after it will rush across zero and up the other side of the galvanometer (to the left), showing that the cable current has again set in. If d be too small the needle will pass to the left, at first slowly, but immediately after with a bound, d is now adjusted, resistance is inserted or removed as required, and the eliminating process begun again. As d more nearly resembles the resistance of the cable, the first and instantaneous deflections after battery-contact become smaller ; and, when d and the cable resistance are equal, the needle trembles over the FIG. 79. Cable zero-point for a moment, and then rushes over to the left under the influence of the cable current. Should the current given off by the fault be negative, having arranged the bridge as before, repolarise the fault with a positive battery current, and, waiting till G 2 shows the cable free, proceed to test as before, but using a positive current instead of a negative. Should d be too great the needle of G], will be deflected in this case, first to the left and then to the right. Should it be too small the needle will move to the right, at first slowly, but immediately after with a rush. The galvanometer Gr l must always be ready, and not short circuited, else the first and instantaneous deflections after battery-contact will not be perceived. In practice it is found that when the cable current is positive it is easily eliminated by a negative current, but that when LOCALISATION OF FAULTS. 249 it is negative the operation with a positive current is more difficult. Indeed, it is better not to employ a positive testing current at all, except for a moment when it is required to eliminate a weak negative cable current. A positive current applied for a few seconds in this manner has only time to depolarise a fault, but when continued longer it seems to actu- ally coat the exposed wire with badly conducting substances, by which the total resistance is increased. It will be noticed that when the fault is depolarised by a positive current of any duration it does not recover itself for a long time. If a galvanometer be joined in circuit, its needle will remain at or near zero for a considerable time, occasionally oscillating feebly. The depolarisation by a negative current, on the other hand, lasts only a few moments. The whole of the foregoing observations do not appear to be applicable to every fault. Thus, when the fault has consider- able resistance in itself, or when more faults than one exist, it is not always possible to eliminate the cable current. Again, when the fault possesses resistance, the direction and strength of the cable current, when the distant end is alternately insulated and put to earth, do not always coincide. For example, a fault occurred on a six-mile piece of shore-end cable, which reduced the insulation resistance to about 2000 units absolute. Now, when the further end of this piece was to earth, a strong nega- tive current was often obtained, but when it was insulated the cable current was slight and positive. Again, when the fault is further off than about 150 miles, and the intervening cable perfect, the charge current interferes with the test. 269. The principal obstacle found in testing for faults is the presence of earth currents. If it were not for these there would really be but comparatively little difficulty in making satis- factory tests. But even earth currents would not create any serious difficulties, .provided they kept constant in strength and direction for any length of time ; this, however, is unfortunately seldom the case, and it is often only by patient watching that a few seconds can be obtained when the cable is in a quiescent condition, and a test of correct value made. The earth current difficulty is especially met with in long cables, and it is not uncommon for days to pass without a satis- factory test being made. MANGE'S METHOD. 270. This method, devised by Sir Henry Mance, has for its object the elimination of the effects of an earth current in a 250 HANDBOOK OF ELECTRICAL TESTING. cable when making a resistance test. The general principle of this method has been described on page 237. As compared with the ordinary " Equilibrium Method " (page 235) it has the advantage that the polarisation current does not become changed, as it is liable to do when reversed currents are sent from the testing batteries; moreover, as the test can be made with a negative current only, the resistance of the fault does not alter materially, as it is liable to do when a positive current is applied. In making the test practically the inventor considers that the simplest plan and the one giving the best results is to have the resistances a and 6 (Fig. 77, page 235) of equal value; the 100 and 1000 pairs of proportion coils in the ordinary bridge would be used generally for the purpose. The test is com- menced by observing the resistance d with the smaller pair of coils, continuing the test until the resistance of the fault appears fairly steady, when, balance being obtained by adjusting d, the galvanometer is short circuited for an instant whilst changing the 100 coils to 1000, and then balance is again obtained by re-adjusting d to d r This operation should be several times repeated, and the pair of readings which seem most likely to be correct are then used for determining x from the formula. In working the method care should be taken that the battery is in good condition and that its resistance is not high. If the con- ductor is not broken and the fault is a small one, sufficient resis- tance should be added at the end nearest the fault to bring the latter near the centre ( 265, page 244). The tests from either side will then compare well with each other. In arranging this, the resistance of the batteries must not be over- looked, and it is therefore desirable that all stations should use similar batteries with approximately the same internal resistance. When testing with the 1000 to 1000 proportion coils, the observations will generally, but not invariably, be higher than when using the 100 to 100 branches. This will depend on the earth currents existing at the time. The corrected result will, however, be approximately the same, although the readings may indicate an alteration of several hundreds of units in the resistance tested. The daily variations in the tests to a fault may of course be due to alterations in the fault itself, especially if it is a small one. The application of the correction will, however, at once show how much is due to the fault, and to what extent the tests are affected by other disturbing influences. Should the alterations be caused by the latter, there will be no material change in the corrected results. LOCALISATION OF FAULTS. 251 271. For the purpose of applying the test with ease and certainty Sir Henry Mance has devised a form of bridge specially adapted for the purpose. Iii this apparatus, which is shown by Fig. 80, a switch is provided for rapidly changing FIG. 80. the proportion coils from 100 to 1000, and vice versa; a set of single ohm slide resistances (page 15) is also added for the purpose of adjusting the main resistance (d and d^) with rapidity. KENELLY'S LAW OF FAULT KESISTANCE. 272. When a cable which has become broken has its resistance measured in order to determine the locality of the break, the value of this resistance represents the resistance up to the fault plus the resistance of the fault itself. Now although by Lums- den's method (page 245) it is often possible to nearly eliminate the resistance of the fault, yet this cannot always be done. In a recent paper read before the Society of Telegraph-Engineers and Electricians,* Mr. A. E. Kenelly has pointed out as the result of numerous experiments, that when the current flowing does not exceed 25 milliamperes ( T ff^- ampere) the resistance of the fault in a broken cable varies inversely as the square root of the current passing, that is to say, for example, if we quad- ruple the current we halve the resistance. As a consequence of this law, it is shown that it is possible to determine what is the resistance of the cable up to the break, independent of the resistance of the break itself. Let r be the resistance of the broken cable up to the fault, and /]_ and / 2 the resistances which the fault has when the * Proceedings of the Society of Telegraph-Engineers and Electricians/ Vol. XVI., page 86. 252 HANDBOOK OF ELECTRICAL TESTING. currents passing are c l and c 2 respectively, then by the law stated we have /i :/ 2 : therefore A Let E! and E 2 be the total measured resistances when the currents c x and c 2 are passing respectively, then we have B, = r+/i E 2 = therefore /. or therefore E 2 Jc 2 r *J c 2 = R! J c r therefore r (v^Cj - Vc 2 ) = R! V c i - ^2 or r = RI V^i - R^V^ For example. The measured resistance of a broken cable when the current passing was 25 milliamperes (c x ) was 435 ohms (EJ, but when the current was reduced to 9 milliamperes (c 2 ) the measured resistance was found to be 445 ohms (E 2 ) ; what was the resistance (r) of the cable up to the break ? V 9 = 3, and, V^ = 5, therefore 435 x 5 -445 X 3 ._. , r = i -- - = 420 ohms. It is obvious that the values of c x and c 2 might be determined by placing a low resistance galvanometer in circuit with the cable whilst the tests are being made, and noting the deflections LOCALISATION OF FAULTS. 253 obtained in the two cases. The strengths of the current could be varied either by changing the battery power or by changing the resistances in the arms of the bridge, as in Mance's test. Mr. Kenelly prefers to adopt the latter method and to calculate the strengths of the current passing, instead of having a gal- vanometer in the cable circuit as suggested. In order to eliminate the effects of earth currents he balances to a false zero ( 252, page 238). 273. Practice is required before any of the foregoing tests can be satisfactorily made. An artificial line, however, can easily be formed with resistance coils to represent the resistance of the line up to the fault, and a short piece of cable core which has been pierced with a needle for the fault itself. This piece of core should be immersed in a vessel of sea-water, using a piece of galvanised iron plate or wire for an earth. By this means a very fair idea of some of the difficulties encountered in testing for faults in cables may be obtained, and good practice made. JACOB'S DEFLECTION METHOD. 274. A disadvantage in using the Wheatstone bridge for measuring the resistances in the foregoing methods is the time it takes to arrive at balance, and the difficulty of seeing what is happening in the way of earth currents, polarisation, &c. ; the determination of the resistance by deflection is, however, as rapid a method as can be desired, and allows of continuous observation of the behaviour of the fault. The only require- ments for the test are, the battery with a reversing switch,* a Thomson mirror galvanometer with a reversing key,* and a set of resistance coils. The battery, galvanometer and cable are first joined up in circuit, one pole of the battery and the further end of the cable being to earth ; and the galvanometer being shunted by a shunt of very low resistance (a short piece of wire answers well for this purpose). The needle of the galvanometer is turned so that it has a large inferred zero ( 60, page 64). The apparatus being thus joined up, the battery is switched on and one of the galvanometer reversing keys depressed so that the needle of the galvanometer turns in the direction necessary to bring the spot of light on the scale ; by adjusting the shunt this deflection is brought to a convenient position. The gal- vanometer reversing key is now released, the battery is reversed by means of its switch, and then the second reversing key of the * Chapter X. 254 HANDBOOK OF ELECTRICAL TESTING. galvanometer is depressed so that the deflection of the galvano- meter needle is in the same direction as it was in the first instance. Since in one case the battery current is in the same direction as the earth current and in the other case it is opposing it, the two deflections will differ, but by a judicious adjustment of the shunt and of the magnitude of the inferred zero it may be arranged that both deflections come well within the range of the scale, the shunt being the same in both cases. These preliminaries being arranged, the shunt and the zero position must not be altered during the series of tests. A number of deflections are now taken with each current, and by a proper manipulation of the short circuiting key,* the oscillations of the needle can be checked so quickly that the value of the deflections can be determined within two or three seconds or less after the battery has been switched on : thus the behaviour of the fault can be carefully observed and the reliability of the readings with either current assured without any great difficulty. After the necessary deflections have been determined, the set of resistances is substituted in the place of the cable, and the deflections obtained are reproduced. Let d l and d 2 be the deflections obtained. Let E and e be the respective electromotive forces of the battery and of the earth or cable current. Let x be the resistance being measured. Let R x and R 2 be the resistances required to reproduce the deflections d l and d 2 . Lastly, let C l and C 2 be the currents producing the deflections d l and d 2 ; and let R be the resistance of the battery and shunted galvanometer. Now when the deflections are taken on the cable we have and When the same deflections are taken with the resistance coils in the place of the cable, then we have Cl = B + R; [3] * Chapter X. LOCALISATION OF FAULTS. 255 and consequently we have E-fe _ E or E-f a; E or 4.1 _^_f_. *" E ~ E + E! ; therefore T? .4. or E K + K L ~ We also have or _ E + a; E ~ E + E 2 '' therefore B + a _ 1=sl _ KH-^ E + E! B + E 2 ' therefore ( K + a; )[raf+TnrR-l = 2 ' |_E + E x E + K 2 J or _ 2(E + E 1 )(R + R 2 ) _ -(B + RJ + CB + B,) that is to say, a equals the harmonic mean of (E + E L ) and (E -f- E 2 ), minus E. In fact we have to add E to both E x and E 2 , take the harmonic mean of the results, and then subtract E from this mean. If E can be made so low as to be negligible, then of course the formula becomes considerably simplified, x being equal to the harmonic mean of E! and E 2 . Although E could be determined by a separate measurement and then inserted in the formula, there is no absolute necessity for doing this, since we have actually all the data requisite to 256 HANDBOOK OF ELECTRICAL TESTING. determine x without knowing the value of R. From equation [3] we have G! R! = E - d R, and from equation [2] C 2 R 2 = E C 2 R ; therefore C x R x + C a R a = 2 E - R (d + C 2 ). Also from equation [1] and [2] we have 2 E therefore C-p I p -p 1 -"I + ^2 K 2 Since the currents d an( i G 2 are represented by the deflections d 1 and d 2 we have an equation which is simpler than equation [A] and which does not require R to be known or to be made negligible, though in order to make the test with the greatest chance of accuracy it is advisable that R should not have a high value, for reasons which have been explained in 78, page 80. If, however, equation [B] is made use of it would be necessary to make the zero of the galvanometer some point on, and not off, the scale, otherwise we should not know what are the true values of the deflections d l and d 2 . By making the zero at the extreme end of the scale the range will be 700 divisions, which will generally enable sufficiently accurate tests to be effected. KEMPE'S Loss OF CURRENT TEST.* 275. In this test, which is shown by Fig. 81, a battery E is permanently connected, through a galvanometer G x , to one end A of the cable, the further end B being connected to earth through a second galvanometer G. * This test was first described by the Author in the second edition of the present work in the year 1881, but it was also independently devised by M. Emile Lacoine and described by him in the ' Bulletin de la Societe Internationale des Electriciens ' for April 1886. LOCALISATION OF FAULTS. 257 Let C s be the current sent through the galvanometer G x , and let C r be the current received on the galvanometer G, then / nr ' / Let the resistance beyond A be Z 3 , then Earth also, as in the previous tests, let a -f- b = L, or, b = L a, then by substitution we get therefore that is or _ L in this equation is known, it being the conductor resistance of the cable when sound. Z 3 is easily determined, when the observations with the cable are completed, by joining up the galvanometer G l and battery E in circuit with a set of resist- ance coils, and then adjusting the latter until the deflection on the galvanometer G x is observed to be the same as it was when the cable was in circuit ; the resistance in the resistance coils then gives the value of l y * * See 3, page 1. 258 HANDBOOK OF ELECTKICAL TESTING. In order to determine C g and C r we must compare the deflec- tions they produce on the respective galvanometers with the deflections obtained on the same galvanometers from a standard current, such, for instance, as that given by a standard Daniell cell (1-079 volts) (page 137), working through 1079 ohms, that is a current of 1 milliampere. Supposing both stations are furnished with standard cells, then each station having noted the deflection obtained when in circuit with the cable, disconnects his galvanometer from the latter, and puts it in circuit with a standard cell and a certain definite resistance, say, 1079 ohms, including the resistance of the galvanometer. The deflection is again noted; then this deflection, divided into the deflection obtained when the cable was in circuit, gives the value of C s or C r , as the case may be. For example. In testing a cable by the foregoing test, the connections being made as in Fig. 81, station A obtained a deflection on his gal- vanometer equivalent to 2800 divisions; station B obtained a deflection equivalent to 1520 divisions. The deflection obtained by A on his galvanometer with a standard cell through 1079 ohms was 100 divisions, and the de- flection obtained by station B with a similar battery working through 1079 ohms was 95 divisions; then "' The value of Z 3 was found to be 280 ohms, and the values of L and G were known to be 345 ohms and 5 ohms respectively. What was the value of a ? = (28 X 280) - [16 x (345 + 5)] = 28 16 If the cable had a conductivity resistance of 10 ohms per mile, then the distance of the fault from A would be ~ - = 18*67 miles. A great advantage which this test possesses lies in the fact, that all the necessary observations with the cable can be made simultaneously, station A arranging with station B that at a definite time the observations are to be made on the galvano- LOCALISATION OF FAULTS. 259 meters ; there is thus no chance of error from the fault changing its resistance between two independent observations, as might occur in the other tests. It has been assumed that this test has been made with Thomson galvanometers, and it is advisable if possible to employ them ; the directing magnets in the instruments would, however, have to be placed very low down and very low shunts employed, otherwise the deflections obtained would be beyond the range of the scale. 276. It will sometimes be found that the cable is traversed by an earth current. The effects of this may be eliminated (as first suggested by Mr. Latimer Clark) by means of a compensating battery of one or two large-sized. Daniell cells, inserted between the end of the cable and the galvanometer. The number of these cells used should be slightly in excess of that required to counteract the earth current, exact balance being obtained by means of a shunt inserted between the ter- minals of the battery. To effect this adjustment, previous to putting on the battery E, we should connect the galvanometer to earth, and then adjust the compensating battery shunt until no deflection is obtained. This being done, the battery E is connected up and the test made as if no earth current existed. It will seldom be found that a larger compensating battery than one or two cells is required to produce a balance, and if these be of a large size their internal resistance may practically be ignored. It is advisable to make the current from the testing battery flow in the same direction as the current which tends to flow from the compensating battery ; thus, if the latter requires to be inserted, so that the zinc pole is connected to one terminal of G! and the copper pole to the end A of the cable, then the copper pole of the testing battery should be connected to the second terminal of G x and the zinc pole to earth. Best Conditions for making the Test. The resistances of the battery E and galvanometers G and G x should be as low as possible. THE LOOP TEST. 277. When a faulty cable is lying in the tanks at a factory so that both ends of it are at hand, or when a submerged cable can be looped at the end farthest from the testing station with either a second wire, if it contains more than one wire, or with a second s 2 260 HANDBOOK OF ELECTKICAL TESTING. cable which may be lying parallel with it, as is often the case, then the simplest and most accurate test for localising the position of the fault is the loop test. This test is independent (within certain limits) of the resist- ance of the fault, thus doing away with the necessity of cleaning and depolarising as would be necessary in the ordinary tests. There are two ways of making this test with the form of apparatus hitherto described. MURRAY'S METHOD. 278. Fig. 82 shows the theoretical and practical arrange- ments, p is the point where the two wires or cables are looped together at the further station, /being the fault. Let x be the resistance from C to the fault, y the resistance from E to the fault. Then B being plugged up and A B (fc) and E A (d) adjusted until equilibrium is produced, b X y = d x x. Let L be the total conductivity resistance of the whole loop, then x + y = L, therefore y = L x. Substituting this value of y in the above equation, we get b (L-a) = d x x, from which "'.' ' ' = L FT-/ ' To obtain L, we should simply join up for the ordinary con- ductivity test, as shown by Fig. 63 (page 191). The fault in this case has no effect upon the test, provided it is not caused by the complete fracture of the cable ; in the latter case the broken ends become covered with salts, which would make the resist- ance appear higher than it really is. When, however, the fault is due to a simple imperfection in the insulating sheathing, the ordinary conductivity test gives the correct result. 279. It is advisable to keep a record of the conductivity resist- ance, so that it can be ascertained without the necessity of making a measurement. 280. In the practical execution of this loop test, the connec- tions being made as shown by the figure, all the plugs between LOCALISATION OF FAULTS. 261 B and C must be inserted; this is necessary, because the galvanometer connection is made on to the terminal B f , which is the same as B, instead of on to C. The test could be made by placing the galvanometer on to C, but in that case we should lose the advantage of the key, which it is always best to use. The plugs being inserted between B and C, and the other plugs being in their places, we should remove, say, the 1000 plug from between A and B, and having pressed down the left-hand FIG. 82. D key, to put the battery current on, which should be a zinc (or negative) one as shown, we should adjust the plugs between D and E, pressing down the right-hand key as required until equi* librium is produced. The different resistances being inserted in the formula, x is found, which being divided by the conductivity resistance per mile of the cable, gives the position of the fault. For example. A cable 50 miles long, whose total conductivity resistance was 450 ohms, that is, 9 ohms per mile, was looped with a second cable, which had the same length and conductivity 262 HANDBOOK OF ELECTRICAL TESTING. resistance as the first cable the resistance of the loop being 450 X 2 = 900 ohms. The adjusted resistance in d to obtain equilibrium was 4000 ohms, b being 1000 ohms, then * = 9 KlOOoT400t)) = 18 hms - Dividing this by the conductivity per mile, which is 9 ohms, we get distance of fault from testing station = if -2- = 20 miles. In making a test of this kind it is advisable to use as high resistances as possible in b and d, because the greater these resistances are the greater will be the range of adjustment. 281. We know that the best galvanometer to employ would be one whose resistance does not exceed 10 times the joint resistance of the resistances on either side of it.* In practice, the resistances b and d would always be greater than the resistance of the looped cables, and the joint resistance of the two resistances would consequently never be more than one-half the resistance of the looped cables ; if, therefore, we do not use a galvanometer with a resistance more than, say, five times the resistance of the looped cables, we may be sure that the con- ditions are very favourable for making an accurate test. The value which d should have depends upon the value given to 6, and since the range of adjustment is large in pro- portion as d is large, therefore for this reason it is advan- tageous to make b as large as possible ; but it is not advisable to make it higher than is requisite to obtain what may be con- sidered to be a sufficient range of adjustment, for by making b and d large the current which passes out of the battery becomes diminished, and consequently the effect on the galvano- meter will also be diminished. This can of course be compen- sated for by adding on extra batteries, but as the number of the latter may have to be inconveniently large, it is as well to avoid doing so, otherwise there is no limit to the values which may be given to b and d. It is possible to avoid making b and d high by making the latter resistance adjustable to a fraction of a unit. If the fault has a very high resistance the employment of high battery power is inevitable, as this high resistance is directly in circuit with the battery. In such a case, however, we may make b and d as high as we like, for, inasmuch as the current flowing out of the battery depends upon the total re- sistance in its circuit, the result of making b and d high is to * Chapter XXV. LOCALISATION OF FAULTS. 263 add but very little to the total resistance, unless indeed they are very excessive, which in practice can hardly be the case. To sum up, then, we have Best Conditions for making Murray's Loop Test. 282. Make b as high as is necessary to obtain the required range of adjustment in d ; if b and d would in this case require to be excessive compared with the resistance of the loop, d must be adjustable to a fraction of a unit. Employ a galvanometer whose resistance is not more than about five times the resistance of the looped cables. Employ sufficient battery power to obtain a perceptible deflection of the galvanometer needle when d is 1 unit, or a fraction of a unit, out of exact adjustment. VARLEY'S METHOD. 283. This is shown theoretically and practically by Fig. 83. In this test B (a) and A B (6) are fixed resistances, and E A (d) is adjusted until equilibrium is produced. Then, x and y being the resistances of the fault from E and respectively, a (d + x) - b y, and y = L - x; therefore a (d + x) = 6 (L - x), from which bL-ad _ Ifb = a, then L - d x = - For example. The two cables being of the same length and conductivity as in the last example, and b being equal to a, equilibrium was obtained by making d = 600 ; then 900 - 600 x = - - - = 150 ohms. 2 264 HANDBOOK OF ELECTKICAL TESTING. 284. It is necessary that the faulty one of the two looped cables be attached to E, or else it would be impossible to obtain equilibrium. If we were testing a looped cable, and after having joined it up we found that we could not obtain equili- FIG. 83. brium, we may be sure that the fault lies between C and p. The cable must then be reversed, and a fresh test made. 285. The conditions for making this test with accuracy are not quite so simple as they were in Murray's test. In this case they are almost precisely similar to what they are in an ordinary bridge test, for the resistance d -j- y takes the place of the resistance d in the latter test, and if we determine the best conditions for finding x we practically determine the best conditions for finding y, as the test is made in the same manner for determining either quantity. It is, however, always best to have the relative positions of the battery and galvanometer as indicated in the figure. For if LOCALISATION OF FAULTS. 265 the galvanometer took the place of the battery, and vice versa, it would be affected by any earth or polarisation currents which might enter at the fault, and this would render adjustment difficult. We have, then, Best Conditions for making Varley's Loop Test. fi or 286. Make a as low as possible, but not lower than - g + x Make b of such a high value that d when 1 unit out from exact adjustment produces a perceptible movement of the galvano- meter needle. A rough test would first have to be made to ascertain approximately the values of x and y, and then if necessary the resistances must be readjusted so that the above conditions are satisfied, and then exact adjustment of E A be made. Best General Conditions for making the Loop Test. 287. Although the loop test avoids errors due to earth currents it does not avoid errors due to cable currents, that is to say, currents set up by chemical action at the fault itself; this action causes a current to flow in opposite directions through the branches of the cable on either side of the fault, in other words, it causes a current to circulate in the loop. This current, although comparatively weak, yet is sufficient to cause errors which it is advisable to avoid if possible. Mr. A. Jamieson states that by balancing to a " false zero " (page 238) the above cause of error may be eliminated and a very considerable increase in the accuracy of the test be obtained. Correction for the Loop Test. 288. It sometimes happens that the resistance of the fault in a cable approaches the normal insulation resistance of the latter ; then the position of the fault indicated by the loop test will not be its true position. The reason of this is, that the current flowing in a faulty cable has two paths open to it : one through the fault and the other through the whole of the insulated sheathing. The cable, in fact, possesses two faults : the actual fault, and the fault due to the conducting power of the insu- lating sheathing. This second or resultant fault, as it is called, in a homogeneous cable is equivalent to a fault in the centre of the cable of a resistance equal to the insulation resistance of the cable itself when in good condition. If the 266 HANDBOOK OF ELECTKICAL TESTING. cable is not homogeneous throughout, this resultant fault will ]ie away from the centre. Its position can be found, however, by the ordinary loop test when the cable is sound. We have then to determine the true position of the fault when the position and resistance of the resultant fault, the insulation resistance of the cable when imperfect, and the position of the fault indicated by the ordinary loop test, are known. The following shows how this may be done approxi- mately : In Fig. 84 let A B be the cable joined up for the loop test, / being the actual fault, i the resultant fault, and /i the apparent position of the fault given by the loop test. Let P equal the resistance of i, that is, the insulation resist- ance of the cable when perfect ; also let I equal the insulation FIG. 81. resistance when the cable has a fault, which resistance is due to the joint resistances of the fault (which we will call c) and the insulation P ; then PC PI 1 = ; whence c = P-I Now it is evident that the position of / x with respect to i and / will depend upon the relative values of P and c : thus if P and c were equal, then / x would lie midway between i and / ; if P were greater than c, then/! would be nearer/; or again, if P were less than c, then / x would be nearer i. This being the case, we have the proportion -p /distance between\ /distance between\ { A and* ) V /i and/ / Let distance A / x = ft and A i = a, therefore distance i fi = ft a; also let distance/!/ = a, then or LOCALISATION OF FAULTS. 267 therefore which gives us the position of the true fault beyond the apparent one. Or the distance of the fault from A will be For example. In a looped cable, whose total length was 100 miles, and total conductivity resistance 900 ohms, the ordinary loop test showed the apparent position of a fault which existed in it to be 700 ohms from A, that is, /3 = 700. The position of the resultant fault given by the loop test when the cable was new was found to be 500 ohms from A, that is, a = 500. The insulation resistance of the cable when new was 3,000,000 ohms, and when faulty 600,000 ohms, that is, P = 3,000,000. I = 600,000. Where was the true position of the fault ? Distance of fault from A (700 x 3,000,000) - (500 x 600,000) 3.000,000 - 600,000 * = 75 hms that is to say, distance of fault beyond distance given by loop test was 750 - ft = 50 ohms. Or, supposing the cable to have a resistance of 9 ohms per mile, the true distance of the fault beyond the apparent distance was ^, or 5 f- miles. If the cable be homogeneous throughout, the resultant fault 268 HANDBOOK OF ELECTRICAL TESTING. will appear in the middle of it. In this case a will equal , 2i where L is the total length of the loop. If we write the equation, Distance of fault from A = -= - , in the form, Distance of fault from A = we can see that if P is very large then = 0, in which case we get Distance of fault from A = /?, as in the ordinary loop test. 289. In order to make this test satisfactorily, it is necessary to know what are the insulation resistances of the cable when good and also when faulty, at the moment when equilibrium is obtained. Now, as will be shown in Chapter XV., the insulation resistance (P) of a sound cable alters in proportion to the time a current is kept on it ; but the rate at which this alteration takes place is definite, and can be obtained by refer- ence to previous tests of the cable made when the latter was sound. The insulation resistance (I) of the cable when faulty cannot, however, be determined by any reference to previous tests ; some plan of enabling it to be measured accurately is therefore necessary. A method suggested by Mr. S. E. Phillips enables this to be done in a very satisfactory manner. The whole of the testing apparatus is carefully insulated by being placed on a sheet of ebonite, or on insulated supports ; the experimenter also stands on an insulated stand or a sheet of ebonite. The battery for making the loop test, instead of being connected directly on to the terminal of the resistance coils, is connected thereto through the medium of a second galvanometer. By noting the deflection on the latter at the moment equilibrium is obtained on the first galvanometer, and comparing it afterwards with the deflection obtained through a known resistance, we obtain the value of I plus the combined resistance of the resistances in the bridge, which quantity will, however, be insignificant compared with I, and need not be taken into account. LOCALISATION OF FAULTS. 269 A note should be made of the time at which the battery is connected to the instruments, and then, when the plugs are adjusted, equilibrium obtained, and the deflection on the second galvanometer observed, the time must again be noted, so that the period during which the battery current has acted may be known and the value of P correctly obtained. The method of determining the value of P will be considered hereafter. INDIVIDUAL RESISTANCE OF Two WIRES BY THE LOOP TEST. 290. Mr. S. E. Phillips has pointed out that the loop test may be made very useful for determining the individual resistance of two wires, the leads in a cable factory, for instance, whose ends cannot be got at to connect to the testing apparatus. To do this, the further ends of the leads would be joined together, and the junction put to earth. It is evident, then, that the loop test applied to the wires would give the resistance of either of them to their junction. 270 HANDBOOK OF ELECTKICAL TESTING. CHAPTER X. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. SHORT-CIRCUIT KEYS. 291. Although the short-circuit plug-hole is convenient to- avoid accidental currents being sent through the galvanometer when the various resistance coils, batteries, &c., are being joined up for making a measurement, yet a key which in its normal condition short circuits the galvanometer, is extremely con- venient and useful. Such a key is represented by Fig. 85. In its normal condi- tion the spring rests against a platinum contact, and, when pressed down, against an ebonite one. FIG. 85. FIG. 86. /4 real size. The two terminals of the shunt are connected to the terminal of the key, which in this and most keys are double, so as to enable the wires leading to the resistance coils, batteries, &c., to be conveniently connected to them. If it is required to keep the key pressed down for a lengthened period, a small piece of sheet ebonite or gutta-percha can be slipped in between the contacts, so as to prevent their making connection when the finger is taken off the key. Some keys of this kind are provided with a catch (Fig. 86), which keeps the spring down when it is depressed. The advantage of the short-circuit key over the short-circuit plug may not seem obvious, but actual practice will soon show its value. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 271 REVERSING KEYS. 292. Besides the short-circuit key, a Reversing Key is usually inserted in the galvanometer circuit, so that the deflections of the needle may always be obtained on the same side of the scale. A form of reversing key very commonly used is shown FIG. 88. FIG. 87. FIG. 89. in elevation and plan by Figs. 87 and 88, and in general view by Fig. 89. The galvanometer terminals would be connected to the two end terminals of the reversing key, or, if the short-circuit key is inserted, to the terminals of the latter. By pressing down one or other of the springs, the current vrill pass through the galvanometer in one direction or the other. The two handles on either side of the two springs are for the purpose of clamping either of the latter down when required. Particular care should be taken, when procuring the key, to see that the terminals, &c., are not fixed on the top of the ebonite pillars by means of bolts running right through them, as in such a case the advantage of the pillars is entirely lost, and the terminals might just as well be screwed direct into the base board. Care should also be taken that the contacts of the keys are clean, as when there are several contacts considerable resist- ance might be introduced into the circuit from their being dirty. 272 HANDBOOK OF ELECTKICAL TESTING. 293. It is sometimes found in this form of reversing key that the springs fail to make the necessary contact when clamped down, owing to the loosening or wear of the cam employed to hold it down. Pell's Patent Self-locking Key, which is shown by Fig. 90, and which was designed by Mr. B. Pell, of the firm of Messrs. Johnson and Phillips, entirely overcomes this difficulty by dispensing with the cam altogether, and introducing a spring latch which, when the key is depressed, automatically catches FIG. 90. and holds it with certainty in position until it is released, the movement, either in depressing the key or in releasing it being effected with one hand. Each latch is released by pressing the corresponding ebonite knob on the insulating pillar, as shown in the figure. The other advantages of this key over the old form, although not of so much importance, will be appreciated by all who take a pride in the appearance of their apparatus. The absence of the cams and their supporting pillars, besides improving the insulation, and allowing of the key being more easily cleaned, makes it look neater, and prevents the lacquered surface of the brass work being disfigured, as is invariably and unavoidably the case when the cam is used. A Short-circuiting Key is also made on the same principle, the spring in this case being somewhat stronger to prevent uninten- tional locking when the key is only gently tapped by the finger. KEVERSING SWITCHES. 294. In addition to the reversing key for the galvanometer, a Reversing Switch for the testing battery is very useful : it need KEYS, SWITCHES, CONDENSEKS, AND BATTEKIES. 273 not, however, be such an elaborate one as that used for the galvanometer. Figs. 91 and 92 represent such a switch. It consists of four brass segments screwed firmly down to an ebonite base. Each segment is provided with a screw, to which to attach the testing wires. FIG. 91. FIG. 92. In some cases each segment is supported on an ebonite pillar, which improves its insulation very much, and, indeed, would be absolutely necessary for some tests we shall describe. The poles of the battery would be attached to two opposite terminal screws, say A and A', and the leading wires to the two other screws, B and B'. To make the current flow in one direction, we should place the plugs between the segments A and B, and A' and B', and to make it flow in the other direction, between the segments A and B', and A' and B. If one or both the plugs are removed the battery current will be cut off altogether. It is always best, in order to do this, to remove both the plugs in preference to one only, for if the battery is not well insulated a portion of the current may still be able to flow out of the battery and disturb the accuracy of a test. Two other pieces of apparatus are necessary to form a very complete set, viz. a "Condenser" and a "Discharge key." CONDENSERS. 295. A Condenser is merely a Leyden jar exposing a large surface within a small space; those constructed for testing purposes are made of sheets of tin-foil placed in layers between thin sheets of mica coated with shellac. The alternate layers of tin-foil are connected together, so that sets are formed corre- sponding to the outside and inside coatings of the Leyden jar. A very convenient form of condenser, manufactured by Messrs. Warden, is shown by Fig. 93 (page 274). The layers of tin-foil and mica are placed in a round brass box with an ebonite top, on which are fixed the connecting T 274 HANDBOOK OF ELECTEICAL TESTING. terminals. These terminals are placed on brass blocks, the ends of which are in close proximity to one another, so that a plug can be inserted between them for the purpose of enabling the FIG. 93. apparatus to be short circuited. This should always be done when the condenser is not in use, so that any residual charge which may remain in it may be entirely dissipated. FIG. 9t. The " electrostatic capacity " of these condensers is usually microfarad, the " farad " being the unit of electrostatic capacity. They are also made, however, so that several capacities can be obtained, by inserting plugs in different holes. Those having five different capacities (Fig. 94), viz. -05, -05, *2, -2, and -5 microfarads, enable any value ftom 05 to 1 to be obtained by inserting one or more plugs. It is very often extremely useful to be able to vary the capacity, so that it is better to have the latter form rather than the former, although it may be a little more expensive. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 275 Fig. 95 shows another form of a divided condenser arranged in a brass box. A good condenser should not lose, through leakage, more than I per cent, of its charge in one minute. FIG. 95. 296. Condensers, like batteries, can be combined for "quan- tity " or in " series," and advantage may often be taken of this power of combination to obtain a large number of capacities from a small number of condensers. When condensers are connected together for " quantity " the capacity of the combination will be equal to the sum of the respective capacities of the several condensers. Thus, if we call Fj, F 2 , F 3 , &c., these capacities, then the capacity of the combination will be F x + F 2 + F 3 + . . . . This may be expressed symbolically thus : "When the combination is made in " series " (corresponding to the "cascade" arrangement of Leyden jars) the joint capacity of the series follows the law of the joint resistance of parallel circuits,* thus : 1 TT\~ T TrT i TfrT T* . See Chapter XXVII. T 2 276 HANDBOOK OF ELECTKICAL TESTING. This may be symbolically expressed thus : F! - F a - F 3 - -....- By following out these laws, if we had two condensers, F L and F 2 , we could obtain four different capacities, viz. F 1? F 2 , I\ + F 2 , and i 2-. With three condensers we could obtain fourteen different capacities, viz. F x , F a , F 3 , F x + F 2 , F x + F 3 , F 2 + F 3 , F x + F 2 + F 3 , FlF2 , FI FS -., F2F J_ ? F! + F2 F y f "p 1 j l 3 "p i . 12 2 I 171 I T71 13 I T71 i TTI Any of these combinations may be expressed symbolically in the manner before shown; thus, for example, to take the ^i ~t~ -F 2 | IT combination, this would be shown thus : ; DISCHARGE KEYS. 297. To enable the discharge from the condenser to be read on a galvanometer a discharge key is necessary. This, like the other pieces of apparatus, is made in a variety of forms. WeWs Discharge Key. 298. Fig. 96 shows a pattern (designed by Mr. F. 0. Webb), which is in very general use. It consists, primarily, of a hinged lever of solid make, pressed upwards by a spring and playing between two contacts. A vertical ebonite lever, hinged at its lower end, is fixed to the base of the instrument in the position shown. This lever has near its upper end a projecting brass tongue, which, when the lever is pressed forward (by means of a spring), hitches over the extremity of the brass lever. The end of the latter is cut away so as to form two steps ; when the brass tongue on the vertical ebonite lever is hitched over the lower step then the KEYS, SWITCHES, CONDENSEKS, AND BATTERIES. 277 brass lever stands intermediate between the top and bottom contacts, and is insulated from both of them, but when the tongue is hitched over the top step then the brass lever is in connection with the lower contact. Again, when the ebonite lever is drawn back the brass lever is freed and springs up FIG. 96. against the top contact step. If we suppose the brass lever to be hitched down on the lower contact step, then by pulling back the ebonite lever a little the brass tongue unhitches from the top step and hitches on the lower one, thus allowing the brass lever to spring up from the bottom contact but not to come in connection with the upper one ; if, however (as before explained), the ebonite lever be pulled completely back then the brass lever rises in connection with the top contact. 299. When using this discharge key for the purpose of measuring the charge in a condenser, the connections to the galvanometer, &c., would be made as shown by Fig. 97 (page 278). On pressing down the key K 2 the two poles of the battery are put in connection with the two terminals A and B of the condenser C, and on releasing the key so that it comes in contact with the top contact, the two terminals of the condenser are put in connection with the two terminals of the galvanometer, which thus receives the discharge current through it. If we so arrange the connections that the top contact of the key, instead of being joined to the condenser through the galvanometer, is connected directly to it, and the galvanometer is placed between the back terminal of the key and the second terminal of the condenser ; then on pressing down the discharge key we get the current charging the condenser through the gal- vanometer, whose needle will be deflected to one side of the zero point ; and then, on releasing the key, we get the discharge deflec- tion, which will be of the same strength as the charge deflection, 278 HANDBOOK OF ELECTRICAL TESTING. but in the opposite direction to it. The first arrangement, given by the figure, is, however, the one generally employed. The discharge deflection on the galvanometer is only momentary, the needle or spot of light immediately returning towards zero. 300. In using the Thomson galvanometer (which is practically the only instrument of any use for the purpose) for measuring the discharge, the adjusting magnet must be put high up, if it is placed with its poles assisting the earth's magnetism, or low down if it opposes it, so as to make the needle swing slowly FIG. 97. enough to enable the deflection to be read on the scale. It is best to avoid making the needle swing very slowly, for then the spot of light will probably not return accurately to zero, but may be three or four divisions out. A little practice will enable a comparatively quick swing to be read to half a division, or even less. Kempe's Discharge Key. 301. A form of discharge key designed by the author is shown in plan and elevation by Figs. 98 and 99, and in general view by Fig. 100. It consists, like Fig. 96, of a solid lever, hinged at one end, and playing between two contacts attached to two terminals. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 279 Two finger triggers, near the other end of the lever, marked " Discharge" and "Insulate," are connected to two ebonite hooks. The height of the hook attached to the finger trigger marked " Discharge " is a little greater than that of the other hook, so that the lever stands intermediate between the two contacts when it is hitched against it. When the lever is pressed down against the bottom contact, the shorter of the hooks hitches it FIG. 98. FIG. 99. down. If in this position wo depress the " Insulate " trigger, the lever is freed from its hook, and springs up against the second hook, thus insulating the lever from either of the contacts. The " Discharge " trigger now being pressed down, the lever springs up against the top contact. To the hook of the " Discharge " trigger there is a small FIG. 100. piece of metal fixed which is broad enough to come in front of the second hook, so that if the " Discharge " trigger is de- pressed first it draws back both the hooks, and thereby, if the lever at starting be hitched to the bottom contact, allows the lever at once to spring up to the top contact. If, however, the " Insulate " trigger be depressed, only the hook attached to 280 HANDBOOK OF ELECTEICAL TESTING. that trigger is drawn back, allowing the lever to spring tip against the second hook and be thereby insulated, as at first explained. Lambert's Discharge Key. 302. The arrangement of discharge key designed by Mr. Lambert and shown by Fig. 101, is a very good one, and possesses the advantage that the principal terminal is highly insulated when the key is in its normal condition, a point of importance in some tests. The two terminals seen at the front part of the key correspond to the top and bottom contacts of the keys previously described. The ends of two spring levers, provided with ebonite finger-knobs, are set over FIG. 101. these contacts; the other ends of the springs are fixed to a brass cross-piece provided with a terminal, the cross-piece being secured to an ebonite bracket fixed at the end of a stout ebonite rod. By this arrangement the terminal connected to the spring levers is insulated by the long ebonite rod as well as by the ebonite bracket by which the rod is supported on the stand. In manipulating the key, the left-hand lever, say, is first depressed, thus putting the back terminal in con- nection with the contact (corresponding to the bottom contact of the other forms of keys) beneath it. This lever is then released, and the right-hand lever depressed, thus putting the back terminal in connection with the contact (corresponding to the top contact of the other keys) beneath it. The only objec- tion to this form of key is the fact that it is possible to press both levers down at once, thus connecting together the back and the two front terminals ; if this is done accidentally, then, as will be seen by reference to Fig. 97 (page 278), a direct circuit is formed by the battery through the galvanometer, which may result in the sensibility of the latter being altered through the violence of the deflection. Such an accident obviously cannot possibly occur in the other forms of keys. KEYS, SWITCHES, CONDENSERS, AND BATTERIES. 281 The Lambert key is often provided with cams similar to those shown in Figs. 87, 88, and 89 (page 271), so that the spring levers can be clamped down if desired. Rymer Jones's Discharge Key. 303. An excellent form of discharge key has been devised by Mr. J. Kymer Jones, and is manufactured by the India Kubber, Gutta Percha, and Telegraph Works Company of Silvertown. The key is so constructed that (like Lambert's key) the principal terminal is left perfectly free during the period of " insulation," as shown in Fig. 102 ; the leakage FIG. 102. Battery. Cable. from this terminal is therefore confined to the ebonite support A B. The form of this support, a vertical section of which is shown by Fig. 103 (page 282), gives a very considerable length of surface over which any leakage must pass, it being in the present case 6J inches in a height of only 2J inches; while, since the portion A screws into the outer cap B, the former may be removed, when important tests are about to be made, 282 HANDBOOK OF ELECTRICAL TESTING. FIG. 103. and scoured with glass-paper, so as to secure the advantage of a fresh surface without disfiguring the outer polished surface. The movements for " Charge," "Insulate," and "Discharge," will be readily understood from Fig. 102. IT are ebonite rods; their brass prolongations c c', which move with them as one piece, have the under surfaces, where they rub against the platinum contacts b and g, tipped with platinum. When I is deflected to the left, the end of the rod r, attached to it, presses against Z' should the latter happen to be turned to that side and carries it over in the same direction, first breaking contact at c' c i E + c i r = E > that is, c { r = E - ^E; but by the law (A) (page 293) we have Va'-V^^r; therefore V/- V^E-^E, or Vi - V; = Cl E - E. [1] Next let us suppose, as in Fig. 110, that we have in E a current, c 2 , caused by an electromotive force, e, in some part of the resistance (or combination of resistances), r, then we have V 2 -V 2 ' = c 2 B. [2] Now if we take the case shown in Fig. 108, where the current C is produced by the two electromotive forces, then the re- spective potentials at the points A and B must be V = V t + V 2 and v = v/ + v a f . MEASUREMENT OP POTENTIALS. 295 Therefore we have V - V = (V, - V/) + (V, - V 2 '), and by substituting the values of Y x V/ and V 2 V 2 given in equations [1] and [2] we get V - V - Cl R - E -f c 2 E = E ( Cl + c 2 ) - E; but we can see that C = c x + c 2 , therefore V - V - C E - E, [3] which is similar to equation [1]. In the case we have taken we have supposed the electromotive forces (and consequently the currents c 1 and c 2 ) to act in the same direction, but we should have obtained an equation precisely similar to [3] had the electro- motive forces opposed one another, provided, however, the current due to the electromotive force e were less than the current due to the electromotive force E ; if, however, the current due to the electromotive force e were greater than the current due to the electromotive force E, that is to say, if the current C acted against E, then we should have V - V = C E + E. [4] 321. The result, then, that we have arrived at by the fore- going investigation is, that (B) The difference of the potentials at two points in a resistance in ichich an electromotive force exists is equal to the product of the current and the resistance between the two points, added to the electro- motive force in the resistance, this electromotive force being negative if it acts with the current, and positive if it opposes it. This law, we have seen, holds good whether the current in question is due only to the electromotive force in the resistance, or to an external electromotive force also. It should be remarked with reference to equation [3], that when e is greater than E then the potential V becomes greater than the potential V, so that in such a case the equation should be written V - Y = C B - E. In the case of equation [4], V is always greater than V. % MEASUREMENT OF BATTERY RESISTANCE BY KEMPE'S METHOD. 322. Besides determining the electromotive force of a battery, we can also determine its internal resistance with great facility by means of a condenser. To do this, first charge the condenser 296 HANDBOOK OP ELECTRICAL TESTING. by means of the battery, and note the discharge deflection which we will call a ; next insert a shunt, S, between the poles of the battery ; again charge and discharge the condenser, and note the new deflection, which we will call j3. Let e be the electromotive force between the poles of the battery when the shunt S is inserted, and let C be the current flowing, then by law (A) (page 293), we have e = G S, or, C = o Also, if E be the electromotive force of the battery, and r its resistance, we have E c = sqb.: therefore e E S " S + r or eS + er = ES; therefore e r = S (E - e), or e but we must also have E : e :: a: /?; therefore r = S--^. [A] To obtain accuracy it is advisable for the value of S to be such that the deflection ft is approximately equal to o For example. A battery whose resistance (r) was required was joined up with a galvanometer, condenser, discharge key, &c., as shown by Fig. 97, page 278. The condenser being charged and then discharged through the galvanometer (by depressing and then releasing the key K 2 ), a deflection of 290 divisions (a) was pro- duced. A resistance of 20 ohms (S) was then joined between the terminals of the battery, and the condenser again charged and discharged through the galvanometer, the value of the * The reason of this will be obvious from a consideration of the investiga- tions given in 105, page 103, and 110, page 111. MEASUKEMENT OF POTENTIALS. 297 deflection obtained being 105 divisions (/3). What was the resistance of the battery ? n 290 - 105 r = 20 - = 35-2 ohms. luo It is evident that if S be adjusted till fi = ^, then r = S. [B] An error in the foregoing kind of test may possibly arise from one measurement being made with the poles of the battery free, when no action goes on in it, and the second being made with it shunted, which may cause a falling off in its electro- motive force, as action would then be taking place ; the accuracy of the test depends upon the electromotive force being constant in both cases. If the shunt S be connected to the battery by means of a key, then the second discharge deflection /5 is best obtained by first pressing down the key K 2 (Fig. 97, page 278), then pressing down the key which connects the shunt to the battery, and then immediately afterwards releasing the key K 2 , and noting the deflection. Thus as little time as possible is allowed for polarisation to take place. MEASUREMENT OF BATTERY EESISTANCE BY MUIRHEAD'S METHOD. 323. A very excellent modification of the foregoing method has been devised by Dr. A. Muirhead ; it possesses the great advantage of being free from the source of error just mentioned. In this test (Fig. Ill) the battery, galvanometer, and condenser are joined up in circuit with a key K 2 . The condenser C being short circuited for a moment, so as to dissipate any charge which may have been accidentally left in it, key K 2 is depressed ; this causes a charge to rush into the condenser through the gal- FlG - vanometer, producing the same deflection as would be produced if the condenser, when charged from the battery direct, were dis- charged through the galvano- meter. The charge deflection (a) being noted, the key K 2 is kept per- manently down, so as to keep the condenser charged. By means of key K x a shunt (S) is now con- nected between the poles of the battery ; at the moment this 298 HANDBOOK OF ELECTKICAL TESTING. takes place the potential at the poles of the latter falls, and a reverse deflection of the needle of the galvanometer is produced. If we suppose this deflection to be due to an alteration of the potential from a to fi (the latter being the same quantity as that given in the previous test), its value, , will be { = a - ft or, ft = a - {. If, then, we substitute this value of ft in equation [A] of the previous test, we shall get a-{ For example. The shunt S having a resistance of 10 ohms, the deflection produced on depressing key K 2 was 310 divisions (a). K 2 being held down, K x was depressed, when a deflection, of 100 divisions () in the reverse direction to a was obtained. What was the resistance of the battery ? As in the previous test, it is advisable to give S such a value that is approximately equal to - . As no polarisation of any extent takes place in the battery till some seconds after the shunt has been connected to the former by the key, and as the deflection takes place immediately the key is depressed, it follows that very accurate results will be obtained by this test. It may be remarked, however, that Pro- fessor Garnett has found that polarisation takes place in a battery in an extremely short space of time in even the T^V^h part of a second; the amount is, however, of course very small. In Muirhead's test the time during which polarisation would tend to affect the accuracy of the test would be that occupied by the galvanometer needle in swinging from zero to the deflection, , consequently the quicker the swing (consistent with accurate reading) the better. MEASUREMENT OF BATTERY RESISTANCE BY MUNRO'S METHOD. 324. A modification of Muirhead's method has been suggested by Mr. J. Munro, which simplifies calculation, inasmuch as it gives the value of a by a single deflection. Key K! is first depressed, and then immediately afterwards MEASUEEMENT OF POTENTIALS. 299 key K 2 is also depressed; this gives a deflection 6, which, is equivalent to the difference between the deflections a and in the last test. Key K x is now raised, leaving key K 2 down ; and as soon as the galvanometer needle becomes steady, K x is depressed again, and the deflection read, then we have As a slight interval of time may elapse between the depres- sion of key Kj and key K 2 , when obtaining the deflection (during which time the battery would be partially short cir- cuited), it would be preferable to make the test in the following manner : Make the connections so that the front contact of key K x is joined on to the lever of key K 2 instead of on to the front contact of the latter, as in Fig. Ill (page 297) ; then in order to obtain 0, depress K x and keep it down, and immediately afterwards depress K 2 ; the deflection observed in this case will be 0. Now raise key K lf keeping key K 2 down, and when the galvanometer needle has become steady, depress K 17 then the deflection obtained will be . Measurement of Polarisation in Batteries. 325. The amount of polarisation which takes place in a battery when the latter is short circuited may, if required, be easily ascertained in the following manner: In Fig. Ill (page 297) let S be a short piece of wire of practically no resistance, then having short circuited the condenser C for a moment, depress key K 2 , and note the deflection d r Keeping K 2 down, depress K 1? and hold it down for a definite time, say one minute ; at the end of the interval, release K x , and note the deflection d 2 ; then the percentage of polarisation in the one-minute interval will obviously be 100 ( E ' = C! i'-i(R therefore c 2 = Now, c' = c x - c 2 , therefore C iri ~ 310 HANDBOOK OF ELECTRICAL TESTING. Or, since S is very small, we may say, From this equation we can see that r should be made small, "but we can also see that there is but little advantage in making it much smaller than G. In fact, there is an actual disadvantage in making r extremely small, for this would necessitate E being made very large, which would be inconvenient. We have next to determine what is the best value to give to ~R. Now, the larger we make the latter, the greater will be its range of adjustment, consequently, as in previous tests, we should give it the highest value such that a change of 1 unit from its correct resistance produces a perceptible deflection of the galvano- meter needle. We have and if in this equation we put 8 = 1 we shall get the current corresponding to a change of 1 unit from the correct value of B, that is or, since r must be small, we may practically say from which ' , K = T- . ' m If then we make c f the reciprocal of the figure of merit (page 65) of the galvanometer, the value of R worked out from the equation will show the highest value that the latter quantity should have. But the value of R depends upon the value given to E ; we must therefore determine what the latter should be. We have Er or E = c i nO MEASUREMENT OF CUKRENT STRENGTH. 311 and substituting the value of E obtained from equation [B], we get E = 5209+3. C v or, as r is small compared with K, that is with ~n=r, we may say c Gcr For example. It was required to measure the exact strength, Cj, of a current whose approximate strength was known to be * 03 amperes. A Thomson galvanometer of 5000 ohms resistance (Gr) was employed for the purpose, its figure of merit being 1,000,000,000 (-\ The resistances of r and r x were 100 ohms and 1 ohm respectively. What should be the value of E in order that E may be as high as possible? E = - C 3 X ^ - = 1-8 volts; 1,000,000,000 X 500 X that is to say, practically, E should consist of 2 Daniell cells. Assuming E to be equal to 2 volts approximately, then (from equation [B]) the value which E would have in order to obtain balance would be OS v 1 E = - . - - - = 6000 ohms X 5000 1,000,000,000 approximately. 345. In order to determine the Possible degree of accuracy attainable, let us suppose E to be 1 unit out of adjustment, and let X be the corresponding error produced in C^, then we have Er Er E ^(K-l+r) -r.CE-l+r) r 1 (E + r) Er 312 HANDBOOK OF ELECTRICAL TESTING. or, since E is large, we may say but therefore E r E r If we call X' the percentage of accuracy, then V = ^of Cl) or, X' If we take the values given in the foregoing example we have approximately w 100 X '03 X 1 A= -2-^Too~ '015 per cent. To sum up, then, we have Best Conditions for making the Test. 346. Make E the nearest possible value above ^ * . where c' c Gr is the reciprocal of the figure of merit of the galvanometer, and Cj is the approximate strength of the current to be measured. The value which R will require to have will be Possible Degree of Accuracy attainable. 100 C, r, Jrercentage of accuracy = DIFFERENCE OF POTENTIAL DEFLECTION METHOD. 347. Fig. 116 shows the general principle of this method. A B is a low resistance through which the current, G lt to be measured passes. A galvanometer, G, in circuit with a high MEASUKEMENT OF CURRENT STKENGTH. 313 resistance, R, is connected between the ends of A B as shown, then, calling V and V x the potentials at A and B respectively, we have by law (A) (page 293) c - v - v '. ^1 " FIG. 116. To determine V V x all we have to do is to note the deflec- tion d on the galvanometer G, and then, having disconnected the latter, together with the resistance R, from A B, to join them in circuit with a standard cell of known electromotive force, E, and to obtain a new deflection d 1 ; we then have or so that EcZ 348. In order that the test may be a satisfactory one the resistance G + R should be very high compared with the resistance r, so that the strength of C x is practically the same whether G + R is connected to A B or not ; also r should be as low as possible, so that it may not appreciably add to the resistance of the circuit in which it is placed. In order, there- fore, that a good deflection may be obtained, the galvanometer G should be one with a high figure of merit (page 65) ; a Thomson galvanometer answers the purpose very satisfactorily. For example. In making a measurement according to the foregoing test the resistance r was T ^th of an ohm, and the deflection obtained on G was 250 divisions (d). When G and R were connected to a standard Daniell cell in the place of being joined to AB, 314 HANDBOOK OF ELECTRICAL TESTING. a deflection of 230 divisions (c^) was obtained ; what was the strength of the current C x ? 1-079 X 250 C >= A X 230 = "'7 *!*"* As it is obviously advisable that the deflections obtained should both be as high as possible, the standard electromotive force E may have to be adjusted for the purpose, that is to say, it may have to consist of several cells. Instead of adjusting E only we may make the latter of any convenient high value, and then adjust R so that the required deflection is obtained; in this case if R x be the resistance when E is in circuit, we must have For example. In making a measurement according to the foregoing test the resistance of r was T Vth of an ohm and the deflection obtained on G was 270 divisions (d) ; the resistances of G and R were 5000 ohms and 1000 ohms respectively. When G and R were connected to a standard Daniell cell, R had to be adjusted to 7000 ohms (R x ) in order to obtain a deflection of 300 divisions (dj) ; what was the strength of the current G l ? n 1-079 X 270 X (1000 + 5000) Cl = A x 800 X (7000 + 5000) ' 6 ^^ Of course if the value of R x is made such that the deflections d and d^ are equal, then E(R + G) 349. From the extreme simplicity of the test it must be obvious that the " Best conditions for making the test " and the " Possible degree of accuracy attainable " must be as follows : Best Conditions for making the Test. Make R and R x of such values that the deflections obtained . are as high as possible. Possible Degree of Accuracy attainable. Percentage of accuracy = 100^ (- + -y J MEASUREMENT OF CURRENT STRENGTH. 315 where -^ is the fraction of a division to which each of the deflections can be read. DIFFERENCE OF POTENTIAL EQUILIBRIUM METHOD. 350. Fig. 117 shows the general principle of this method. A B is a slide wire resistance, s being the slider. A galva- nometer, Gr, and a standard battery, E, are joined up as shown, FIG. 117. so that the latter tends to send a current through r { in a direc- tion opposing the current O r s is then slid along A B until the point is reached at which no deflection of the galvanometer needle is observed; when this is the case, then by law (A) (page 293) we have V-V^C^; and by law (B) (page 295), since no current is flowing through the galvanometer, V-V 1 = E; therefore C^^E, or If the resistance of the whole length of wire A B be r ohms, and if it be divided into n divisions, then if the number of divisions between A and D be w lf the resistance r 1 will be consequently we must have En 1/1 = - rn, For example. The electromotive force E consisted of 1 standard Daniell cell ; the wire A B had a resistance of 1 ohm (r), and was 316 HANDBOOK OF ELECTRICAL TESTING. divided into 1000 parts (w). Equilibrium was obtained when the slider was set at the 750th division (w x ) ; what was the strength of the current C x ? 1-079 X 1000 1= 1X750 = 1 ' 44am P eres - 351. The conditions for making the test in the most satis- factory manner are comparatively simple. The nearer we have the slider to B, that is to say, the larger we make n lt the smaller will be the percentage of error in the latter due to the slider being, say, 1 division out of position. As the position of the slider for equilibrium depends upon the value of E, the latter must be sufficiently great to enable n^ to be as large as possible. The greatest theoretical value which E could have must be that which it would possess when n : = n t in which case we get G! = -, or, E = dr. As it is only possible to adjust E by variations of 1 cell, we must take care that its actual value is less rather than greater than G! r, otherwise it would be impossible to obtain equilibrium. It is also necessary that the figure of merit (page 65) of the galvanometer be sufficiently high to enable a perceptible move- ment of the needle to be obtained when the slider is moved a readable distance, 8, from the position of exact balance. If we suppose the slider to be at D when equilibrium is produced, then the electromotive force which would tend to send a current through the galvanometer, supposing the slider to be displaced a distance 8, would be EX!, consequently the current c r , passing through the galvanometer, will be ,_ E8 _ dr8 ~G! ~ Gn ; if, therefore, we require to adjust the slider to an accuracy of 8, the figure of merit ( -j of the galvanometer must not be less MEASUREMENT OF CURRENT STRENGTH. 317 The percentage of accuracy, A.', with which C l can be obtained must obviously be 1008 or snce Era En therefore _ 1000^8 Era For example. It being required to measure the strength, C 19 of a current whose approximate value is 1*5 amperes, a galvanometer of 500 ohms resistance (G), whose figure of merit is 1,000,000^-, ), is proposed to be employed for the purpose. The resistance of the whole length of the slide wire, which is divided into 1000 divisions (w), is 1 ohm (r) ; the position of the slider can be read to an accuracy of J a division (8). What is the highest value that could be given to E? also to what percentage of accuracy could C L be determined, and what should be the figure of merit of the galvanometer in order that this percentage of accuracy may be attained ? E = 1-5 x 1 = 1-5 volts; therefore we cannot make E greater than, say, 1 Daniell cell (1 volt approximately). 100 X 1'5 X 1 X i .Percentage oi accuracy = - - -- -- = *U7o per cent. To enable this percentage of accuracy to be obtained, the figure of merit ( - J of the galvanometer must not be less than the figure of merit, therefore, of the galvanometer in question is sufficient for the required purpose. To sum up, then, we have Best Conditions for making the Test. 352. Make E the nearest possible value below G l r. The figure of merit of the galvanometer should not be less Gn thau ' 318 HANDBOOK OF ELECTRICAL TESTING. Possible Degree of Accuracy attainable. 100 (V 8 Percentage of accuracy = Era FIG. 118. I B SIEMENS' ELECTRO-DYNAMOMETER. 353. This apparatus, although it can be used for measuring ordinary powerful currents, yet has the special advantage that it enables rapidly alternating currents (such as are employed in the Jablochkoff system of electric lighting, for example) to be measured; such currents would give no indications on an ordinary galvanometer. The principle of the electro-dynamometer is based upon the mutual action of currents upon one another, i.e. upon the fact that currents in the same direction attract, and in opposite direc- tions repel, one another. Fig. 118 shows how the principle is applied. A B C D is a fixed wire rectangle, and abed SL smaller one, suspended by a thread, t, within the larger, so that it can turn freely about its axis ; the planes of the two are at right angles to each other. Now, if the two rectangles be connected together in the way shown, then a current entering at W 1} and passing out at W 2 , will traverse the two, and the current passing from B to C will attract the current passing from a to d, and will repel the current passing from c to b. A similar action takes place with reference to the current passing from D to A, consequently the smaller rectangle, under the influence of the forces, will tend to turn about its axis, in the direction in which the hands of a watch rotate. If the current enters at W 2 , and leaves at W 19 then, inasmuch as the directions of all the currents in the wires are reversed, the small rectangle must still tend to turn in the direction indicated. If one or both of the rectangles consist of several turns of wire, the turning effect for a given current will be proportionally increased. As the turning effect on the coil is produced by the action of the current through the fixed coil acting on the current W) MEASUKEMENT OF CUEEENT STEENGTH. 319 through the movable coil, and as the two coils are in the same circuit, it follows that if the current passing through the fixed coil is doubled, then the current passing through the movable coil is also doubled, consequently we have one doubled current acting upon another doubled current, and therefore we must have a quadruple deflective effect in other words, the deflec- tive force tending to turn the movable coil will vary as the square of the current. The way in which this principle is utilised will be best understood by reference to Fig. 119 (page 320), which shows a general view of the Siemens Dynamometer. The apparatus consists of a rectangle of wire hung from a fibre whose upper end is fixed to a thumb-screw ; the latter is provided with a pointer which can be moved round a graduated dial ; one end of a spiral spring is also attached to the rect- angle, the other end being fixed to the thumb-screw. In this arrangement the number of degrees to which the pointer is directed evidently indicates the amount of torsion given to the spiral spring. To the rectangle also is fixed a pointer, the end of which just laps over the edge of the graduated dial. The rectangle encircles a coil consisting . of several turns of thick, and a larger number of turns of thin, wire ; the two ends of the thick wire are connected to terminals 2 and 3, and the two ends of the thin wire to terminals 1 and 3. Connection is made between the rectangle and the wire coils by mercury cups, into which dip the ends of the wire forming the rectangle. The base-board has three levelling screws ; the level consists simply of a small pointed weight hung at the end of a rod (seen on the right of the figure), the pointed end hangs exactly over a fixed point when the instrument is level. 354. The method of using the instrument is as follows : The wires leading the current whose strength is to be deter- mined are connected to terminals 1 and 3, or 2 and 3, according as a strong or weak current has to be measured. The current deflects the rectangle ; the thumb-screw is now turned in the reverse direction to that in which the rectangle has turned, and torsion being thereby put on the spiral spring the rectangle is forcibly brought back towards its normal position that is, at right angles to the coils, or to the position at which the pointer attached to the rectangle stands at zero on the scale. The number of degrees of torsion given to the spiral spring being then read off, the strength of the current is found by reference to a table supplied with each instrument. To construct this table a current of a known strength is sent through the instru- ment, and then the degree of torsion required to bring the rectangle back to zero is carefully noted. This being done, 320 HANDBOOK OF ELECTEICAL TESTING. FIG. 119. MEASUREMENT OF CURRENT STRENGTH. 321 the currents corresponding to other degrees of torsion are easily calculated. The force of torsion varies directly as the number of degrees through which the spiral spring is twisted, whilst, as has been before explained, the deflective effect of the current varies directly as the square of the latter. In other words, if be the number of degrees of torsion required to bring the rectangle back to zero when it is traversed by a current of C amperes, then if Cj. be the current which will correspond to any other degree of torsion ^j , we have ' or ^A/ For example. If 180 (4>) of torsion were required to bring the rectangle back to zero when it was traversed by 47*57 (C) amperes of current, what current (CJ would be represented by 80 (0!) of torsion ? = A/ )X475X47 ' 5 = 31-7 amperes. 355. Like galvanometers, the Siemens electro-dynamometer is not susceptible of great accuracy when the readings are very low ; in fact the higher the readings are, the more accurate are the results obtainable. Thus, for example, 5 of torsion of the spring represents a current (in the instrument (No. 1009) shown by Fig. 119) of 7* 93 amperes, whilst 5 more, that is 10 in all, represents a current of 11-23 amperes. In other words, a range of 5 of torsion only, represents a difference in the current of (11-23 -7-93) 100 p 7 * i/O If, however, the current had been 66-38 amperes, which corre- sponds to a torsion of 350, then 5 more of torsion, or 355 in all, represents a current of 66*86 amperes, consequently the range of 5 of torsion in this case represents a difference in the current of per cent. ; 66*38 and a greater degree of torsion would have rendered the error still less. Every instrument is supplied with a table which shows the Y 322 HANDBOOK OF ELECTRICAL TESTING. current strengths corresponding to various angles of torsion ; practically this table is different for every instrument, as it is almost impossible (nor is it necessary) to make two dynamo- meters alike. The table supplied with the instrument (shown by Fig. 119 (No. 1009)) is calculated so that the latter can theoretically be used for measuring currents varying from 1 05 to 66-86 amperes in strength. The thin wire coil is to be employed when currents of from 1*05 to 19-87 amperes are to be measured, and the thick wire coil for currents of from 3 54 to 66 -86. The numbers of degrees of torsion representing various currents are all multiples of 5 ; thus the first calculation on the table (thick wire coil) is 1, which represents 3 * 5 amperes of current ; the next is 5, representing 7 93 amperes ; the next, 10, representing 11*28 amperes; and so on. Practically the instrument cannot well be adjusted to a closer degree of accuracy than 5. The thin wire coil, having about three times the magnetic effect of the thick one, requires, for a definite current, that the number of degrees of torsion to bring the needle back to zero be about three times that which is required in the case of the thick coil; in other words, with the thin wire coil we can practically measure currents to about three times the degree of accuracy which is possible with the thick coil ; but, on the other hand, the highest current which we can practically measure with the thin coil is about one-third only of the highest current which can be measured with the thick coil. The lowest current which can be measured consistent with a degree of accuracy equal to 10 per cent, is 5-76, for the next current below this on the table is 5-25, and therefore we have t 5 ' 76 " S' 25 ) 100 per cent. = 10 per cent, nearly. 5 * 2iO If we require to be accurate within 1 per cent., then the lowest current we could measure would be 16-77, as the next current below this on the table is 16-60, and we therefore have Since the percentage of accuracy is equal to ... - (, - .) where C is a particular current, and C x the current next below it on the table, and since C 2 : (V : : < : fa MEASUREMENT OF CURRENT STRENGTH. 323 where < and fa are the degrees of torsion corresponding to the currents C, C 15 therefore Percentage of accuracy = (* /^_ ^ ij 100; and as the smallest difference to which we can practically read is 5, therefore / /JO ~j~ tO v Percentage of accuracy = (*/ y * "^ lj 100 = X f , say. Therefore therefore 5 X' 2 X r ' + ^ 10^00 + l + 50 '* therefore ^ = : 10^00 "" 50 ; or, _ 50,000 = X' 2 + 200 X" which shows us the smallest number of degrees of torsion which must be given to the spiral spring when measuring a current, in order that the latter may be measured to an accuracy of x per cent. For example. It was required to be able to measure currents of 10 amperes and upwards to an accuracy of 1 per cent., by means of an electro-dynamometer ; how many degrees of torsion would the spiral spring be required to make ? 50,000 showing that the electro-dynamometer must be so constructed that when currents of 10 amperes and upwards have to be measured, not less than 248 of torsion have to be given to the spiral spring in order to bring the needle back to zero. 356. From the construction and principle of the electro- dynamometer it must be evident that the accuracy of the Y 2 324 HANDBOOK OF ELECTEICAL TESTING. absolute results obtained by its means must depend entirely upon the torsion of the spiral spring remaining constant. It seems possible that change of temperature and frequent use might alter the value of the torsion, but this point does not appear to have been satisfactorily settled. The instrument might probably be made of more value if its coil were composed of a large number of turns of thin wire, shunted by a thick wire shunt. The latter would be used when measuring the strong currents, whilst the correctness of the instrument could be verified by sending a comparatively weak current through the unshunted coil. It is not often that powerful currents of an accurately known value can be had for the purpose of verifying the correctness of an instrument, though weaker currents are almost always obtainable. ( 325 ) CHAPTER XIII. MEASUREMENT OP ELECTKOSTATIC CAPACITY. DIRECT DEFLECTION METHOD. 357. The simplest way of measuring electrostatic or inductive capacities is, with the same battery power, to compare the dis- charges from the unknown capacities with the discharge from a condenser of a known capacity ; thus we note the discharge deflection a given by the standard condenser F, and then the discharges a l9 2 , &c., given by the cables or condensers whose capacities F 1? F 2 , &c., are required, in which case F : F! : F 2 : : a : ! : 2 . For example. A standard condenser had a capacity of ^ microfarad, and gave a discharge deflection of 300, and two other cables or condensers, F 1? F 2 , gave discharge deflections of 225 and 180 respectively, then J : F! : F 2 : : 300 : 225 : 180 ; that is, 225 FI = * * 300 = * microfarad > 180 = i microfarad. and If we use shunts and obtain the same deflection, then 358. In measuring the electrostatic capacity of a cable by this method, the connections for measuring the discharge from the cable would be made in the manner shown by Fig. 120 (page 326). The arrangements for measuring the discharge from the condenser would be those indicated by Fig. 97 (page 278). jj Then, as before, the capacity of the cable will be to the 326 HANDBOOK OF ELECTRICAL TESTING, capacity of the condenser as the discharge deflection of the one is to the discharge deflection of the other, or obtaining the same deflection by means of shunts, as the multiplying power of the shunts. FIG. 120. 359. The capacity per mile will be the result divided by the mileage of the cable. 360. When a number of capacities of about the same value have to be measured, as, for instance, the capacities of two-knot lengths of cable core, a device may be adopted which consider- ably simplifies the operation. Let F be the capacity of the standard condenser whose discharge is D divisions, and let / be the capacity of one of the lengths of cable, and d the discharge from the same. Then we have or D F Now if we make =r a submultiple of 10, then the value of d read off from the scale will give at once the value of /. Thus MEASUREMENT OF ELECTROSTATIC CAPACITY. 327 if F were a condenser of ^- microfarad capacity, and we so ad- justed the galvanometer that this capacity gave a discharge deflection of a little over 333 divisions, then we should have i j j f = 333^ 1000 so that if the discharge deflection reading from the cable con- sisted of three figures, a decimal point put before the latter would give at once the capacity of the cable ; or if the reading con- sisted of two figures, then we must put a decimal point and a cypher. In the same way, if we had a condenser of 1 micro- farad capacity, we should adjust the galvanometer so as to obtain, a deflection of 100 divisions, for then > ii JL * 100 100* SIEMENS' Loss OF CHARGE DISCHARGE METHOD. 361. The principle of this method of measurement is that of observing the rate at which the charged condenser or cable, whose capacity is required, discharges itself through a known resistance, and calculating the capacity from a formula which we will now consider. The elements with which we have to deal are: capacity (farad), resistance (ohm), quantity (coulomb), time (second), and potential (volt). Let us suppose the cable or condenser has an electrostatic capacity of F farads, and is charged to a potential of V volts, so that it contains Q coulombs (equal to V F) of electricity, and is discharging itself through a resistance of B ohms during one second. The quantity of electricity in the condenser or cable at starting is Q coulombs. If now we take a very short interval of time t, we may con- sider the discharge, which really varies continually, to flow throughout that time t t at the same rate as it had at the com- mencement ; and the smaller t is taken, the more accurate will be the result. Thus, since the quantity escaping is directly proportional to the potential driving it out, and to the time during which the escape occurs, and inversely proportional to the resistance 328 HANDBOOK OF ELECTEICAL TESTING. through, which the escape takes place, the quantity escaping will vary as V t V t -=?- ; that is it equals K, It K where K is a constant to be determined. Now the units are so made that a condenser of 1 farad electrostatic capacity charged to a potential of 1 volt, that is, containing 1 coulomb of electricity, will commence to discharge itself through a resistance of 1 ohm, at the rate of 1 coulomb per second. That is to say, 1 = i~J K, therefore, K = 1. The quantity escaping during the interval of time t in our problem is therefore Vf B* The quantity remaining in the condenser will be Again, since this is the quantity at the commencement of the second interval, that at the end will be and that at the end of the rath interval will be Let these n intervals of t seconds equal T, so that n t = T. Now we have seen that the smaller t is, the more accurate will our results be. Let us therefore make t infinitely small, and n infinitely great, so that n t still = T, we shall then get a perfectly accurate result, and the amount remaining at the end of time T will be where n = oo. MEASUREMENT OF ELECTROSTATIC CAPACITY. 329 To evaluate q put T 1 FE x so that x oo when w = oo ; then ,- T FK when a? = oo ; but when this is the case the expression within the square brackets is known to be equal to e,* thus . therefore T . Q fl" 1 *? therefore T F * " Bfcfcf ' but Q YF Y where v is the value of the potential corresponding to the value q of the quantity, thus F T T V ~ V' Blog e - 2 303 R log, - where, as stated at first, T is measured in seconds, F in farads, and E in ohnis. Since Y and v now appear in the form of a proportion, the unit in which they are measured is immaterial, although they were measured at the outset in volts. In practice E is usually measured in megohms (1,000,000 ohms), and consequently F will, in such a case, be measured in microfarads ( L_ farad). * TocUmnter's Algebra, Fifth Edition, Chapter XXXIX. 330 HANDBOOK OF ELECTRICAL TESTING. For example. ^ A fully charged condenser gave a discharge deflection of 300 divisions (V) ; after being recharged and allowed to discharge itself through a resistance of 500 megohms for 60 seconds (T), the discharge deflection obtained was 200 divisions (v). "What was the capacity of the condenser ? = -295 microfarads. 2 303 x 500 log 362. In executing this test it is advantageous to make V and v bear a certain proportion to one another, for this will cause any small error in reading the value of v to produce as small an error as possible in the value of F when the latter is worked out from the formula. This may be proved thus : Let us assume E to be constant, and let there be an error X in F caused by an error B l in v and an error S 2 in V, the error Sj being plus and S 2 minus, so that the total resulting error is as great as possible ; we then have T T F + A = - - or - A = --- - F J - 4 ' -^fee 1 c\ t/ "T~ Oi *" & ' + 1 but F T T V E ' Rlog.J therefore X - F- li lQ ge F - loge - F V V - 8 2 !, 10g ' V + ! V-S 2 * Inn, V - ** = F V- 8, but if Si and S 2 are very small,* we get 5l X F V " \= * Todhunter's Trigonometry, Third Edition, Chapter XII. 331 If the deflections are taken on a Thomson galvanometer (as would practically be always the case), then Sj^ = 8 2 , so that we get X = F Now the value of v, which makes A. a minimum,* is V v = 3-59' * This may be determined in the following manner : To make \ a, minimum we must make a maximum. Let the above expression equal it, and let V * = > we then get then at a maximum; therefore - n + 1 or 4343. The solution of this equation is best effected by the " trial" method, viz. by giving n various values until one is found which approximately satisfies the equation. If we make n = 3-59, we get 55509 = "I ) '4343 = -55527, \ O'Otf / which is sufficiently close for the purpose. We have therefore V V V = *n. or, * = - = . 332 HANDBOOK OF ELECTEICAL TESTING. y so that practically we may say make v = . 0*0 We need not be particular, however, about making v exactly y equal to -, as we could make it 50 per cent, greater or less 0*0 than this value without materially increasing X. If the rate of fall were comparatively quick, there would be a positive advantage in making v less than , as the greater we make T 3 * 5 the less will any small error in its value affect the correctness of F, as must be self-evident. Now, if E is adjustable, it is clear that by making it large enough, we could make T large without reducing v too much. In the case of a cable, E, being the insulation resistance, is of course a fixed quantity; but when the measurement is being made with a condenser, any value may be given to E that is considered convenient. We therefore have Best Conditions for making the Test. y 363. Make v as nearly as possible equal to -. When it is 0*0 possible to adjust E, make the latter as high as convenient. Possible Degree of Accuracy attainable. 100 (S 1 -f- Percentage of accuracy = F 2-303 log- If the deflections are read on a Thomson galvanometer (as would usually be the case) then 2008 Percentage of accuracy = F - , 2 -303 v log where 8 is the fraction of a division to which each of the deflections Y and v can be read. 364. When it is an ordinary condenser (whose insulation resistance would practically be infinite) that is to be measured, the connections would be the same as those given in Fig. 97, MEASUKEMENT OF ELECTEOSTATIC CAPACITY. 333 page 278, with the addition of the resistance, which would be inserted "between the terminals of the condenser. The instantaneous discharge (V) can be taken without re- moving the resistance ; for, since the latter would be extremely high, there would be no time for any of the charge to have leaked out through it during the small interval occupied by the lever of the key in passing from the bottom to the top contact. To take the discharge after the interval of time, having charged the condenser by pressing down the lever of the discharge key (Fig. 100, page 279), we should depress the " Insulate " trigger, which would take the battery off but not discharge the con- denser; then, after the noted interval of time, we should depress the " Discharge " trigger, which would allow the charge remaining to flow out, the deflection obtained from which gives us v. 365. To measure the capacity of a cable by this method, the connections would have to be those given in Fig. 120, page 326, and the way of making the test would be the same as has just been explained. K in this case would be the insulation resistance of the cable, which in this and the following method would have to be determined beforehand in the manner de- scribed in Chapter XV., page 368. Inasmuch as R in a cable is a variable quantity and is dependent upon the time a charge is kept in the cable, a mean value only can be given to it, and therefore this and the following test can only give the value of F approximately. SIEMENS' Loss OF CHARGE DEFLECTION METHOD. 366. If the two terminals of a condenser are connected by a high resistance in the circuit of which a galvanometer is placed, and if the two terminals be also connected to a battery, then the condenser will become charged up, and the permanent deflection obtained on the galvanometer will represent the potential of the charge. If now the battery be taken off, a current will flow from the condenser through the resistance and the galvanometer, which current will continually decrease in strength as the condenser empties itself. But the current flowing at any particular moment will be represented by the deflection obtained at that moment, and this deflection will be the same as that which would be obtained if the condenser were kept continuously charged to the potential it had at that moment. The deflection obtained therefore on the galvanometer when the battery is connected to the condenser indicates the potential which the latter has when fully charged, and the deflection 334 HANDBOOK OP ELECTRICAL TESTING. after any interval of time after the battery has been taken off indicates the potential of the charge remaining ; the capacity therefore is given by the formula 2-303 E log [A] in which D is the deflection obtained when the battery is on, and d the deflection obtained after T seconds, the battery being off during that time. E is the resistance through which the charge flows. It may be remarked that the deflection obtained when the battery is on is not affected by the presence of the condenser ; it would be the same whether the condenser were connected up or not. 367. The connections for making a test of this kind would be as follows : Eeferring to Fig. 97, page 278, the terminal of K x , which is connected to the top contact of K 2 , should in the present case be connected through the resistance E to terminal A of the condenser ; the other connections remain the same. 368. In the case of a cable where the flowing out of the charge takes place through the insulating sheathing, a galvanometer cannot be put in the circuit of the flow. To enable the fall of charge to be observed, therefore, a high resistance in circuit with the galvanometer is connected to the cable, and through this resistance a part of the charge passes. As it is only the rate at which the fall takes place that is required, it is quite sufficient, in order to observe this fall, that a part only of the charge be allowed to flow through the galvanometer. If we call E L the insulation resistance of the cable, and E 2 the resistance connected to it, then the total resistance through which the charge flows will be E"O 1 -^2 B! + E; This quantity must be substituted in the place of E in- equation [A], so that we have T The resistance E 2 , it may be remarked, includes the resistance of the galvanometer. MEASUREMENT OF ELECTROSTATIC CAPACITY. 335 As in the first test, it is necessary that E 2 , through which the discharge has to pass, be sufficiently great to prevent the flow from being too rapid. For example. A cable 30 knots in length being connected up, for making the test just described, with a galvanometer, and a resistance R 2 , of 4 megohms, the deflection obtained was 300 divisions D). On taking off the battery the deflection after 30 seconds T) fell to 100 divisions (d); the mean insulation resistance of the cable was 10 megohms. What was the electrostatic capacity (F) of the cable? or 9* 55 - = "318 m.f. per knot. 369. The connections for making this test would be as follows: Eeferring to Fig. 120, page 326, the terminal of key KU instead of being connected to the top contact of the discharge key, would in the present case be connected to the cable through the resistance E 2 . 370. A great advantage which this test possesses over the first method (page 326) lies in the fact that it is correct either for long or short cables. Discharge deflections from long cables, or cables coiled in tanks, do not correctly represent their capacity, in consequence of a retardation which takes place in them and which causes the deflection of the galvanometer needle to be less than it would be if this retardation did not exist. By adopting the fall of deflection plan we avoid this cause of error; but, as we pointed out at the conclusion of the last test, since E! can only have a mean value, the value of F obtained from the formula will only be approximate. THOMSON'S METHOD. 371. This is a very good method, and it can be applied to long cables, &c., with very accurate results. The following is its principle : If we have two condensers containing equal charges of opposite potentials, and we connect the two together, the two charges will combine and annul one another, and if we then connect the two condensers, so joined, to a galvanometer, no deflection will 336 HANDBOOK OF ELECTKICAL TESTING. be produced, there being no charge left in either of the two. If, however, the charge in one condenser exceeds that in the other, then the union of the two condensers will not entirely annul their charges, but an amount will remain equal to the difference of the two quantities. This quantity will deflect the needle if the joined condensers be now connected to the galvano- meter, the deflection being to the right or left, according as the charge in the one or other of the condensers had the preponder- ance in the first instance. If then we know the capacity of one condenser, and we so adjust the potentials of the two that no charge remains when they are joined together, we can determine the capacity of the other condenser. Let Q! and Q 2 be the charges in each ; then where Fj and F 2 are the capacities of the two, and V x and V 2 the potentials of their charges. When Q! = Q 2 then or FIG. 121. Earth 372. An important element in this test is the adjustment of the potentials V x and V 2 . Fig. 121 shows a method of making the test when it is a cable whose capacity has to be measured. MEASUREMENT OF ELECTROSTATIC CAPACITY. 337 The poles of the battery are joined together by two resist- ances, E! and E 2 , connected to earth as shown. Then the poten- tials at the points of junction of the battery with the resistances will be in the proportion Vi: Y 2 ::E! : E 2 ;* and since V 2 therefore F 1 = gF 2 . [A] 373. In making the test practically, E x and E 2 are first ad- justed as nearly as can be guessed in the proportion of F x to F 2 , keys & x and & 2 are then depressed by means of the knob K ; this charges the cable and the condenser. K is now released so as to allow & x and & 2 to come in contact with their upper stops; as the two latter are joined together, the cable and condenser become connected to each other. Key Jc is now pressed, which allows any charge which may remain uncancelled to be discharged through the galvanometer G. If no deflection is produced, then E L and E 2 are correctly adjusted, but if not they must be readjusted until no discharge is obtained ; F x is then calculated from the formula. For example. A cable 500 knots long was joined up with a condenser of 20 microfarads capacity, and with resistance coils, according to Thomson's method of measuring electrostatic capacities. When E! and E 2 were adjusted to 500 and 4400 ohms respectively, no charge remained in the cable and condenser when the two were connected together. What was the capacity of the cable? -r-t rETTwU _ or 1 7 A - = 352 m.f. per knot. 374. Fig. 122 shows a very convenient form of key, designed by Mr. Lambert, which enables the test to be made with the * Page 285. 338 HANDBOOK OF ELECTRICAL TESTING. greatest facility. By pushing forward key button K the two keys &p Jc 2 (Fig. 121) are depressed, so that 'F l and F 2 become charged, and upon drawing K back, & x and & 2 are allowed to rise, thus causing the charges to mix; finally, by depressing k the galvanometer is brought into circuit. In the most recent form of this piece of apparatus, on the depression of key Jc the cable F x becomes disconnected, so that only the condenser F 2 becomes connected to the galvanometer. By this arrangement any disturbing force which may cause the charge in the cable to vary slightly, and consequently to affect the galvanometer is prevented from acting. 375. If it were the capacity of a condenser that was to be measured, then the connections would be similar to those in FIG. 122. Fig. 121, with the exception that the points there put to earth would in the present case be connected to the second terminal of the condenser. The resistances E x and E 2 may be formed of a slide resistance, the slider being to earth in the case of a cable test, or connected to the second terminal of the condenser in the case of a condenser test. 376. As in the " Direct deflection method" (page 325), the test F can be considerably simplified if we make ^ (equation [A], page 377) a submultiple of 10, for then the value of E 2 read off from the resistance box will at once give the value of F r Thus if F 2 were a condenser of, say, 5 of a microfarad, and if E x were 5000 ohms, then the capacity of F 2 can be read off directly from B 2 to four places of decimals. 377. When a long cable has to be tested by this method Mr. A. MEASUREMENT OF ELECTROSTATIC CAPACITY. 339 Jamieson recommends that K be depressed for five minutes to charge, and then raised for ten seconds for mixing previous to depressing Jc. It is also advisable to take the mean of several tests made alternately with zinc to line and copper to line. 378. With regard to the " Best conditions for making the test " it is advisable that the capacity of the condenser F 2 be as nearly equal to F x as possible, so that the potentials to which the two have to be charged may not differ to any very great extent. For if a long cable has to be tested, then inasmuch as the latter would have to be charged to a potential of at least 5 Daniells so as to swamp, as it were, any local charge, the potential to which the condenser (if small) would have to be charged would be very great ; this would be liable to cause an error, from the fact that with a very high potential a certain amount of the charge becomes absorbed, and this charge would cause a deflection of the galvanometer needle over and above that due to the simple inequality between the actual free quan- tities in the two capacities. This abnormal deflection might of course be mistaken as being due to an incorrect adjustment of E x and E 2 . If F 2 is about a fifth of F x it will not be too small for the purpose of the test. The values given to E L and E 2 should be as high as possible so that their range of adjustment may be sufficiently wide. The battery power should be sufficiently high to enable a perceptible discharge deflection to be obtained when E 2 (the larger of the two resistances) is 1 unit out of exact adjustment ; this is best determined by experiment. We have therefore Best Conditions for making the Test. 379. Make F 2 as nearly equal to F x as possible. Make E x and E 2 as high as possible. Possible Degree of Accuracy attainable. Percentage of accuracy = -=- . GOTT'S METHOD.* 380. This method, devised by Mr. J. Gott, is shown by Fig. 123 ; it is executed as follows : E! and E 2 are first adjusted as nearly as can be estimated in * ' Journal of the Society of Telegraph Engineers,' Vol. X., p. 278. This method, although independently devised by Mr. Gott, is practically identical with that of Sir William Thomson described in Vol. I., p. 397, of the same Journal. z 2 340 HANDBOOK OF ELECTRICAL TESTING. the proportion of F } to F 2 . The key K is then depressed and clamped down; this causes both the cable and condenser to become charged, since they are connected together in " cascade." After an interval of five seconds key Jc is depressed, and if a deflection is observed on the galvanometer G, this key is raised, key K is undamped so that the latter is put to earth, and the condenser is short circuited by means of its plug for a few seconds. E! or E 2 is now readjusted, and the foregoing operations again gone through. When finally it is found that no deflection on the galvanometer is observed on depressing key &, then ! : F 2 : E or 'S** It is obvious that we must have Best Conditions for making the Test. 381. Make F 2 as nearly equal to F x as possible. Make E x and E 2 as high as possible. Possible Degree of Accuracy attainable. Percentage of accuracy = -=- FIG. 123. Gott's method is a very satisfactory one, and it possesses the advantage over that of Thomson of not requiring a well-insulated MEASUREMENT OF ELECTEOSTATIC CAPACITY. 341 battery. The method is almost exclusively employed in the Cable Department of Messrs. Siemens' Telegraph Works, Charlton, a slide resistance of 10,000 ohms adjustable to 1 ohm p being employed to give the ratio ^ DIVIDED CHARGE METHOD. 382. If a charged condenser has its two terminals connected to the two terminals of a second condenser which contains no charge, then the charge will become distributed over the two ; and if the condensers be then separated, the quantities held by them will be directly proportional to their respective capacities. Thus, if Q 2 be the charge contained in a condenser whose capacity is F 2 , then if it is connected to a condenser or cable whose capacity is F I} the quantity Q which will remain in F 2 will be F 2 From this we get Tjl Tjl ^2 ^_ fAl If therefore Q 2 be the discharge obtained from a condenser F 2 when full, and Q the discharge obtained from it when, after being charged from the same battery, it is connected for a few seconds to F 15 then the capacity of F x is given by the above formula. For example. A condenser of ^ microfarad capacity (F 2 ), when fully charged, gave a discharge of 300 (Q 2 ). After being recharged and con- nected a few seconds to a piece of cable whose capacity Fj was required, the quantity of charge remaining gave a discharge of 140 (Q). What was the capacity of the piece of cable ? 300 - 140 *\ = i-X-^ =-381m.f. 383. The capacity which the condenser F 2 should have in order that the test may be made as accurately as possible, may be thus arrived at : Let there be an error X in F^ caused by an error S in Q and an error + S in Q 2 , so that X is as great as possible ; we then 342 HANDBOOK OF ELECTRICAL TESTING. Lave F 4- X F Q 2 + S - (Q - S) F Q 2 - Q + 2 8 Fl + X = F2 -- Q=J- - = F * Q - 8 ; "but we know that -Ci -HI Q2 "" Q TTI T71 Q . F I- P -Q-' or ' F2 = Fl ^TQ' therefore T?J_X ^ Q v .Q 2 -Q + 28. FI + X==FI QT^Q X Q^-S- that is, x - -F I Q Y Q 2 - Q + 2 8 1 _ (Q 2 + Q)8 MQ 2 -Q Q-S 1 J" - 1 (Q 2 -Q)(Q-"8J > or, since S is a very small quantity, we may say x _ (Q. + Q)8. -(Q,-Q)0 We have then to find the value of Q which makes X as small as possible. Now (Qs + Q)8 8 J Q 2 - Q 2 Q 1 (Q 2 -Q)Q CM Q_ Q 2 -Q 8 Qs - Q f, QV2- 2 and to make the latter expression as small as possible we must make as small as possible ; that is to say, we must make or therefore or MEASUREMENT OF ELECTROSTATIC CAPACITY. 343 that is j 2 + 1 - It was pointed out, however, in a similar investigation which we made in ( 105, page 103), that practically we may say, make Q = where 8 is the fraction of a division to which each of the deflec- tions Q and Q 2 can be read. 344 HANDBOOK OF ELECTRICAL TESTING. 386. By a modification of the foregoing method, due to Dr. Siemens, a comparatively small condenser may be used for measuring the capacity of long cables, or of condensers of high capacity. It may be called SIEMENS' DIMINISHED CHARGE METHOD. If we connect a condenser to a charged cable, the latter loses the amount which the condenser takes up, and if the condenser be discharged and then again connected to the cable, and again discharged, and this process be repeated several times, the quantity in the cable can be definitely diminished as much as we like. The quantity removed each time, however, is not the same, but becomes less and less after each discharge. Let Q 2 be the quantity contained in the condenser, and Q 1 the quantity contained in the cable, when the two are charged full from the same battery. Then or Supposing now the cable to be completely charged, and the battery taken off, and the condenser to be empty, then, on con- necting the condenser to the cable, the charge the former will take will be whilst the quantity remaining in the cable will be Qi FI On discharging the condenser and connecting it a second time to the cable, the charge it will take will be F x F 2 Fj Fj F 2 1 1 ~\~ -^2 2 1 i 2 consequently, after the wth application, the charge Q it will take will be MEASUEEMENT OF ELECTKOSTATIC CAPACITY. 345 therefore from which Y Q s V "- .For example. A condenser of 1 microfarad capacity (F 2 ), when full, gave a discharge equal to 300 (Q 2 ). A cable whose capacity was required was charged from the same battery which was employed to charge the condenser. The latter was then alternately connected to the cable, removed and discharged 16 times (n) ; on the sixteenth occasion the discharge was noted, and it was found equal to 83 (Q). What was the capacity of the cable? = 11-97 m.f. 16/300 V 83 " 387. In order to make this test as accurately as possible when it is applied to a cable, the repeated charges and discharges must be made with as little loss of time as possible, as during that time a leakage of the charge will be going on through the insulating sheathing of the cable; the accuracy of the test depends upon this leakage being nothing, or at least very small. 388. The connections for making the test would be similar to those employed in the foregoing one, and the practical execution would be the same with the exception that the trial condenser or cable, and not the standard condenser, would be charged from the battery, and in taking the repeated discharges the galvanometer would have to be short circuited. Best Conditions for making the test. ' 5552372 389. Make n equal to : -= - -^rr approximately.* * This may be proved as follows : In order to determine F l as accurately as possible from the equation F 1 = - - - , that is. F 1 = -- ?? - , 346 HANDBOOK OF ELECTKICAL TESTING. Possible Degree of Accuracy attainable, Percentage of accuracy = - . ( Q 2 + Q)Q 2 * 100 f Q(Q 2 -Q) where 8 is the fraction of a division to which each of the deflec- tions Q and Q 2 can be read. we must determine ( -^j" as accurately as possible. Let (7^)" equal-, and let there be a small plus error 5 in Q 2 , and a small minus error 5 in Q, and let there be a corresponding error ^ in It. that is, let therefore Now " = ' or ' = **' or> Q> = Q* Jl "-- therefore *,=(!^!r'-*=*ih*- but since 5 is very small, wo get Q 2 n ake A, a minimum w Let " A; ~ * -4- 1 " To make A, a minimum we must make - a minimum. n -*-+!. n then at a minimum ; therefore fc' l = 0, or log, Jt n + 1 + &* = MEASUREMENT OF ELECTROSTATIC CAPACITY. 347 390. It may be remarked that when a cable is tested for electrostatic capacity at the factory, it is immaterial whether the test be made by charging the cable positively or negatively ; but in the case where the cable is laid, it is advisable to make two tests (or sets of tests), one with a positive and the other with a negative charge, and to take the arithmetic mean of the two results. It is rarely, however, that the two latter differ to any material extent. or log & + (! + &") -4343 = 0. The solution of this equation may be obtained by the " trial " method, i.e. giving fe n various values until one is found which approximately satisfies the equation. If we make k n equal to 27846 the equation will be very nearly satisfied, for log -27846 = 1-4447628 = - -5552372 and (1 + -27846) -4343 = 5552352. Now therefore (F + V )* = u " = ' 27846; hence _ log -27846 _ - -5552372 _ -5552372 For example. It being required to measure the exact electrostatic capacity of a cable whose capacity was 12 microfarads (F,) approximately, a condenser of 1 micro- farad (F 2 ) was used for the purpose. How many times should the con- denser be applied to the cable in order that the test may be made with the greatest chance of obtaining an accurate result ? 5552372 -5552372 _ .. 0347622 348 HANDBOOK OF ELECTEICAL TESTING. CHAPTEK XIV. THE THOMSON QUADKANT ELECTROMETER. 391. This is a most valuable and useful instrument for accurately measuring potentials. DESCRIPTION. Fig. 125 (page 349) gives a general view of the instrument. In the small figure to the right, n n is a thin needle of sheet aluminium, shaped like a double canoe-paddle. It is rigidly fixed at its centre to an axis of stiff platinum wire Jc (Fig. 124), in a plane perpendicular to it. At the top end of the wire a small cross-piece i is fixed, to the ex- tremities of which single cocoon fibres are attached. These fibres are fixed to small screws c and d, by the turning of which the length of the former can be altered. The small screws a and 6 enable the screws c and d to be shifted either to the right or left. Finally, by turning e, the screws a and b can be parted more or less, thereby separating the threads of suspension, and rendering the tendency of the needle to lie in its normal position more or less powerful. A little below the cross-piece t is fixed the mirror m, whose movements are reflected on a scale, as in a Thomson galvanometer (page 46). The platinum wire below the mirror passes through a guard tube t (Fig. 125), to prevent any great lateral deviation of the needle and its appendages, which might cause damage should the instrument receive any rough usage. The guard tube itself is fixed to the framework from which the needle is suspended. It will be seen in the figure that the needle is suspended, apparently, beneath four quadrants (#), A, B, C, and D. There are, however, four quadrants also below the needle, united to the top ones at their circumferences. The arrangement is in fact a round, flat, shallow box, cut into four segments. The alternate segments are connected together by wires as shown in the figure. THE THOMSON QUADRANT ELECTROMETER. 349 Now, if the needle is electrified and the quadrants are in their normal unelectrified condition, and are placed sym- metrically with reference to it, no effect will be produced on the needle. That is to say, the spot of light on the scale will be stationary exactly at the centre line. FIG. 125. : reu.L SLT.C, But if the quadrant D, and consequently A, be electrified, then an attraction or repulsion will be exerted on the needle, causing it to turn through an angle proportional to the potential of the electricity. As the angular movements are very small, the number of divisions of deflection on the scale will directly represent the degree of potential which the quadrants possess. 350 HANDBOOK OF ELECTRICAL TESTING. We can also connect another electrified body to C and B; the needle will then move under the influence of both forces. To render the instrument of practical value, several conditions must be assured. Let us suppose the needle to be electrified. We stated that, at starting, the ray of light should point to the centre line on the scale. To ensure this, the quadrants must be symmetrically placed. This can be roughly done by hand, as means are provided for enabling the quadrants to slide backwards or forwards, and to be fixed by means of small screws, shown in the large figure. For obtaining the final position, one of the quadrants (B) is provided with a micro- meter screw (^), which enables a fine adjustment to be given to it. We must also have means of keeping the needle at one uniform potential for a considerable time. The needle itself could only contain a very small amount of electricity, and a slight escape of this would seriously lower the potential, and make comparative measurements useless ; for it is evident that the whole principle of the instrument depends upon the potential of the needle remaining constant during the time a set of experiments are being made. To get over this difficulty a large glass jar, like an inverted shade, is provided, partially coated with strips of tin-foil (/) outside. Inside the jar, to about a third of its height, strong sulphuric acid is placed. This answers a threefold purpose. It enables the air inside it to be kept quite dry, thereby very perfectly keeping those parts insulated which require to be so ; secondly, it holds a charge of electricity (acting as the inner coating of the jar) ; and thirdly, it allows the charge to be communicated to the needle without impeding its movements. This latter is effected by means of a fine platinum wire, which is attached to the lower end of the thick wire which supports the needle and mirror. The fine wire dips into the acid, whose charge is thereby communicated to the needle. To keep this wire from curling up out of the acid, and also to steady the movements of the needle, a small plummet of plati- num is attached to the end of the wire, as will be seen in the figure. A thick platinum wire, fixed to the lower extremity of the guard tube t, and reaching nearly to the bottom of the jar, is for the purpose of enabling the latter to be charged, in a manner to be explained. So far, the jar answers the purpose of keeping the needle THE THOMSON QUADRANT ELECTROMETER. 351 supplied with electricity; but although this may prevent the potential from falling very rapidly, it will not prevent its doing so entirely. The HeplenisJier. 392. As the instrument is extremely sensitive to very slight changes of potential, some means are requisite by which any small loss can be easily supplied without there being any fear of putting in too much. This is effected by means of the " replenisJier" whose principle we can explain by the help of the small cut to the left, in Fig. 104. A and B are two curved metal shields, one of which (say A) is connected to the acid in the jar and the other, B, to the framework of the instrument, and through it to the foil outside the jar. b and b are two metal wings insulated from one another by a small bar of ebonite, which is centred at s, so that it turns in a plane represented by the paper. The spindle is represented in the large figure by s, other parts being omitted for simplicity. It will be observed that the wings curve outwards. This is done in order that they may make a short contact in their revolution with springs c c and e e. c and c are connected together permanently, but are insulated from the rest of the apparatus, e and e are connected to the. shields A and B respectively. Now let us suppose the wings to be rotated in the reverse direction to that in which the hands of a watch turn. As soon as the left-hand wing comes in contact with the spring c, at the lower part of the figure, the right-hand wing comes in contact with the other spring. The two wings being thus connected together, and under the influence of the shields, the electricity in A, which we will call positive, draws negative electricity to the wing close to it, and drives the positive to the other wing. On being rotated a little farther the wings clear the springs, and being thus disconnected, each retains its charge. Continuing the rotation, the right-hand wing, which had the positive charge communicated to it, comes in contact with the spring e of shield A, and the charge is communicated to the jar, the negative electricity in like manner on the other wing running to the outer coating of the jar. The shields are now in a neutral condition, as at first, and on continuing the rotation the process is repeated. Thus every turn increases the potential of the charge in the 352 HANDBOOK OF ELECTRICAL TESTING. jar, and by continuing the rotation we can augment this as much as we please. By reversing the motion we can diminish the charge, if we require to do so. The axis of the replenisher projects above the main cover, and is easily turned by the finger. The Gauge. 393. But we still require some arrangement by which we can see whether we have kept the potential constant. This is done by means of a small " gauge." The gauge consists of two metallic discs having their planes parallel and close to each other. The lower of these planes, which will be seen dotted at the upper part of the figure, is in electrical connection with the acid of the jar from which it takes its potential. The upper disc is perforated with a square hole immediately over the centre of the lower disc. A light piece of aluminium, shaped like a spade, has the part corresponding to the blade fitting in this square hole. At the point where the handle would be joined to the blade this spade is hinged, by having a tense platinum wire fixed to it, which runs at right angles on each side of the handle and blade, and lies in the same plane as the latter. When the lower plate is electrified, it would attract the blade, thereby raising the end of the handle. So that if we notice the position of the end of the handle with respect to a mark, and see that it moves above or below it, we know that the elec- tricity of the lower plate is either overcoming the tendency of the light platinum wire to keep it up, or is unable to do so. If then we charge our jar to such a potential that the handle is situated close to the mark, and we keep it so, we know that the potential of the jar is constant. When we notice the handle sinking below the mark, we know that the potential of the electricity in the jar is falling ; but a few turns of the replenisher will bring it up again. In the actual arrangement, the rung of the handle is formed of a fine black hair. Inside the handle there rises a small pillar, with two black dots on it. The sign of division -f- represents this, the line being the hair which, by the movement of the spade blade, rises above or below the two dots, which of course would be almost quite close together. To enable the hair and spots to be seen distinctly, a plano- convex lens is placed a little distance off. Care must be taken, THE THOMSON QUADKANT ELECTKOMETER. 353 in order to avoid parallax error, to keep the line of sight a normal to the centre of the lens. We spoke of the lower disc, which becomes electrified by the jar, and which acts on the spade blade. Now it is evident that if the distance between the plates be always the same, and the elasticity of the platinum axial wire be also the same, to get the hair between the two spots is to obtain the jar at a particular fixed potential. But we may require to get this potential, although the same whilst a certain set of experiments are being made, yet different for different series of experiments. This is provided for by enabling the lower disc to be lowered by screwing it round. The Induction Plate. 394. To enable high potentials to be measured, an " induction plate " is added. It consists of a thin brass plate, smaller in area than the top of the quadrant beneath it, and supported from the main cover by a glass stem. It is provided with an insulated terminal I. The use of the plate will be explained later on. 395. A flat brass plate covers the mouth of the jar, and is secured to it so as to be air-tight and prevent the entrance of moisture. A kind, of lantern rises from the middle, which covers the mirror and its suspending arrangements, and above this a box with a glass lid protects the gauge. The front of the lantern is of glass, which allows the ray of light to fall on the mirror and be reflected back on the scale. Terminal rods or electrodes, in connection with each set of quadrants, pass through ebonite columns to the outside of the case, and have terminals attached to them. These electrodes can be pulled up and disconnected from the quadrants if necessary. A charging rod (seen in Fig. 125, page 349, to the left of the left-hand quadrant terminal) also is provided, which can be turned round on its axis. It has at its lower end a small spring, fixed at right angles to it. By turning this terminal rod round, the spring can be brought in contact with the framework from which the needle is suspended, and thereby, through the medium of the guard tube and the platinum wire attached to it, the acid in the jar can be charged. When this is done, the spring is moved away, so that no accidental leakage can take place through it. Various insulating supports are provided inside the jar and 2 A 354 HANDBOOK OF ELECTEICAL TESTING. lantern. One supports the guard tubes and the adjusting screws of the needle ; others support the quadrants, The whole arrangement is supported by a kind of tripod on a metal base, to keep it steady. There are also levelling screws, p 126 and a level on the brass cover, to enable the instrument to be properly levelled, so that the axis of the needle may swing clear of the guard tube. H is a screw-capped opening through which acid can be introduced into the glass jar. Reversing Key. 396. Fig. 126 represents a reversing key which is specially adapted for use with the instrument. To SET UP THE ELECTROMETER. 397. In setting up the instrument for use the following instructions * should be followed : The cover being unscrewed and lifted off and supported about 18 inches above the table, it will be observed that the stiff platinum wire to which the needle is attached just appears below the narrow guard tube enclosing it in the centre of the quadrants, and terminates in a small hook. The loop at the end of the fine platinum wire is to be slipped over this hook, so that the fine wire and plummet may hang from it. The wide guard tube, when in its proper position, forms a continuation of the upper guard tube, so as to enclose the fine platinum wire just suspended. It must therefore be passed upwards over the suspended wire, and neck foremost, until the neck embraces the lower part of the upper guard tube, where it must be fixed by the screw pin provided for the purpose ; this pin is screwed in by means of one of the square-pointed keys, supplied with the instrument, fitting the square hole in its head. This being done, replace and fasten the cover, place the instrument on a sheet of ebonite or block of paraffin wax so as to insulate it, and level up by means of the circular spirit level on the cover. Next unscrew and lift off the lantern and, if necessary, adjust the four quadrants so that they hang properly in their places, with their upper surfaces in one horizontal plane. The needle and mirror which have been secured during transit by a pin passing through the ring in the platinum wire just above the * From instructions drawn up by the late Mr. "W. Leitcli. THE THOMSON QUADKANT ELECTKOMETEE. 355 guard tube, and screwed into the brass plate behind, must now be released by unscrewing this pin with the long steel square- pointed key, and placing it in the hole made for it in the cover just behind the main glass stem to prevent its being lost. The needle will now hang by the fibres. The two quadrants in front of the mirror should now be drawn outwards from the centre as far as the slots allow, by sliding outwards the screws from which they hang, and which project above the cover of the jar with their nuts resting upon flat oblong washers ; a better view will thus be obtained of the needle. The surfaces of the latter ought to be parallel to the upper and under surfaces of the quadrants, and midway between them. This will be best observed by looking through the glass of the jar just below the rim. If the needle requires to be raised or lowered, it is done by winding up or letting down the suspending fibres, that is, by turning the proper way the small pins c, d (Fig. 124, page 348). The suspending wire which passes through the centre of the needle should also be in the centre of the quadrants. This is best observed when the quad- rants have been moved to their closest position. The fourth quadrant is moved out or in by the micrometer screw g (Fig. 125, page 349), with the graduated disc overhanging the edge of the cover. A deviation of the suspending wire from its proper central position, as was explained at the beginning of the chapter, may be corrected by means of the small screws a, 6, c, and d (Fig. 124, page 348). When proper adjustment is attained the black line on the top of the needle should be parallel to the transverse slit made by the edges of the quadrants when these are symmetrically arranged. The sulphuric acid may now be put into the jar. For this purpose, the strongest sulphuric acid of commerce is to be boiled with some crystals of sulphate of ammonia, in a florence-flask supported on a retort-stand over a jet of gas or other convenient source of heat. It is recommended to boil under a chimney, so that the noxious fumes rising from the acid may escape. To guard against the destructive effects of the acid in the event of the flask breaking by the heat, there should be placed beneath it a broad pan filled with ashes, or it should stand above a fire- place containing a sufficient quantity of cold ashes. A little sand put into the flask will lessen the risk of breaking, The object of boiling the acid is to expel the volatile acid impurities which will otherwise impregnate the air inside of the jar and tarnish the works. When cool, the acid may be best poured into the jar through a glass filler with a long stem inserted through the screw opening H (Fig. 125, page 349) provided for the purpose. 2 A 2 356 HANDBOOK OF ELECTKICAL TESTING. The stem of the filler should reach the bottom of the jar, to avoid splashing upon its sides or upon the works, and in removing it care should be taken that it is drawn out without its end touch- ing any of the brasswork. The acid may be poured in till the surface is about an inch below the lower end of the wide brass tube which hangs down the middle of the jar. It must at least reach the three platinum wires hanging from the works. 398. The instrument thus adjusted and charged with acid should be allowed to rest for some little time so that any films of moisture on the insulating portions of the apparatus may become absorbed. The scale should now be placed at the proper distance so that the reflected image is sharply defined and stands at the middle of the scale, that is, at 360 ; for the electrometer scale (unlike that of a galvanometer) is graduated from to 720, 360 being the middle point. Care must be taken that the two ends of the scale are equidistant from the centre of the mirror. Next connect together the two electrodes of the quadrants and the induction plate electrode, by means of a piece of thin wire joined to the cover of the jar ; also turn the charging rod so that it touches the framework of the platinum wire of the needle. Now charge the jar positively by means of a few sparks from a small electrophorus, the frame of the instrument being put to earth for the purpose, and afterwards disconnected. When the proper potential is reached, it is indicated by the lever of the aluminium balance rising; the charging rod should then be turned so as to disconnect the latter from the needle. The replenisher must now be used to adjust the charge exactly, so that the hair may stand between the black spots when observed through the lens. When the lever carrying the hair is at either extremity of its range, it is apt to adhere to the stop ; in using the replenisher to bring it from either limit, therefore, it is necessary to free it from the stop by tapping the cover of the jar with the fingers. If the charge has caused the reflected image to be deflected from the middle of the scale, it may be brought back to that position by turning the micrometer screw which moves the fourth quadrant, and, if necessary, sliding out or in one or more of the other quadrants. The small percentage of the charge lost from day to day may be recovered by using the replenisher. Under ordinary conditions this loss will not amount to more than J per cent, per day. The charge may suffer loss from several causes, the most prevalent being the presence of dust on portions of the appa- ratus inside the jar. Every portion should be carefully dusted THE THOMSON QUADEANT ELECTROMETER. 357 with a camel-hair brush, and especially the round induction plate beneath the aluminium balance. Loss may occur by shreds inside of the quadrants drawing the charge from the needle. It should be ascertained whether this takes place. Insulate alternately each pair of quadrants by raising the corresponding electrode, while the other pair are connected through their electrode with the cover. If the re- flected image in either case keeps moving slowly along the scale, for instance over 20 scale divisions in half an hour, the charge in the jar being at the same time kept constant by the use of the replenisher if necessary, the insulated pair of quad- rants is receiving a charge from the needle. In that case the inside of the quadrants may be brushed with a light feather, or camel-hair brush, after sliding them outwards as far as the slots allow, and securing the needle in the position in which it was fixed during transit ; care being taken not to press upon the needle so as to bend it or the suspending wire. Without secur- ing the needle, each quadrant may be drawn outwards and brushed, while the needle is deflected away from it by the screws a. b (Fig. 124, page 348), or by any obvious means of keeping the needle deflected, care being taken not to strain the fibres. Another possible source of loss of charge is want of insulation over the portion of the glass jar above the acid. If the per- centage of the charge lost from day to day be so considerable as to require much use of the replenisher to recover it, the glass should be cleaned with a wet sponge, rubbed with soap at first, or with a piece of hard silk ribbon, wet and soaped at first, then simply wet with clean water, which may be drawn round the glass to clean every part of it. The ribbon being dried before a fire, may be used in the same manner to dry the glass. If everything fails to make the apparatus keep its charge, the cause is probably due to a defective glass jar, -and this can only be remedied by the manufacturers. 399. The good insulation of the instrument being satisfac- torily accomplished, the symmetrical suspension of the needle by the fibres should be tested. The conditions sought to be realised are, that in the level position of the instrument the needle may hang with equal strain on the two fibres, and in a symmetrical position with regard to the four quadrants. It is plain that if these conditions be fulfilled the deflection produced by the same electric force in the level position of the instrument, will be less than it will be in any position of the instrument which throws the greater part of the weight on one fibre, or brings the needle nearer to any part of the inner surface of the quadrants than it is in its symmetrical position, which is its position of greatest 358 HANDBOOK OF ELECTRICAL TESTING. distance from all the quadrants. To make the test, the two quadrant terminals should be connected to the two poles of a single-cell battery, and the deflections produced upon the scale compared, while the instrument is set at different levels, by screwing one or more of the three feet on which it is supported. At each observation the extreme range, or difference of readings got by reversing the battery, should be noted. If the range diminishes as one side of the instrument is raised, the sus- pending fibre on that side must be drawn up, by turning very slightly the small pin c or d (Fig. 124, page 348), round which it is wound, and another series of observations taken in the same manner, beginning with the instrument levelled. Instead of drawing up one fibre, the other may be let down, to keep the needle midway between the upper and under surfaces of the quadrants, and after each alteration of the suspension it will be necessary to readjust the screws a, b (Fig. 124, page 348), to make the black line on the needle hang exactly midway between the quadrants when the needle is undisturbed by electricity. It will be observed also that the charge of the jar is lost by touch- ing these screws, unless the insulated key is used. They are reached without taking off the lantern by screwing out a vul- canite plug in the glass window in front of them. In deflecting the instrument much from its level position, the guard tube may be brought into contact with the wire hanging from the needle, and the movements of the latter be thus inter- fered with by friction. When the needle vibrates freely, it will be observed that the image comes to rest in any position to which it may be deflected, after vibrating with constant period and gradually diminishing range on each side of this position of rest. The occurrence of friction is shown by the needle coming to rest abruptly, or vibrating more quickly than proper. The reading obtained under these circumstances is, of course, of no value. The quicker vibrations obtained in using the induction plate must not be mistaken for vibrations indicating friction, from which they may be easily distinguished by their regularity. If, as may possibly happen, the process of observing the deflections at different levels, and drawing up the fibre 011 that side which is being raised while getting less sensibility, should only lead the operator to draw up one fibre till it bears the whole weight, while the other is seen to hang loosely, he should adjust them as nearly as he can by the eye to bear an equal share of the weight, and examine the position of the needle by looking through the glass of the jar just below the rim, the two quadrants in front of the mirror being drawn out, and the lantern taken off to let in plenty of light. He will probably THE THOMSON QUADKANT ELECTROMETEK. 359 find that the needle leans slightly downwards relatively to the quadrants on that side which he was drawing up while getting smaller deflections. To correct this is a delicate operation, which should only be attempted by a very careful operator. Though perfect symmetry of suspension is aimed at, it is not essential to the utility of the instrument. If it be desired to make the correction, first secure the needle as during transit ; take off the cover, and while it is held by a careful assistant, or properly supported in a position in which it may be levelled, remove the lower guard tube (the wide brass tube hanging down the centre) after screwing out the small pin in its neck. It will be observed that the upper and narrower guard tube consists of two semi-cylindrical parts united. The part in front may now be removed by taking out the two screws which fasten it at the top, and the platinum wire which carries the needle may be examined. If it has got bent it must be straightened ; if not, it may be bent carefully just above the needle, so as to raise that end of the needle which was observed TO hang lowest. If the cover be supported so that it may be levelled, the needle may be set free, and the operator may observe whether he has suc- ceeded in making it hang parallel to the surfaces above and below it. The needle must not, however, be allowed to hang by the fibres, while bending the platinum wire, or while re- moving or replacing the guard tubes. The works being replaced, the process of observing the deflections at different levels and adjusting the tension of the fibres should be repeated, with the view of getting minimum sensibility in the level position. The two unoccupied holes bored through the cover and flange of the jar are intended to receive the square-pointed keys, when not in use. GRADES OF SENSITIVENESS. 400. There are several ways of making the connections to the terminals of the quadrants, frame, and induction plate, so as to get various degrees of sensitiveness for measuring potentials of various strengths. \st Grade. The following is the most sensitive arrangement, such as would be used for measuring the potential of a Daniell cell : One pole of the battery would be connected, through the medium of a reversing key (Fig. 126, page 354), to one quadrant terminal, and the other to the frame of the instrument and to the second quadrant terminal. This, by reversing the key, 360 HANDBOOK OF ELECTKICAL TESTING. would give about 50 divisions on either side of the 360, equal to 100 in all. 2nd Grade. Leaving one pole of the battery to the frame, the next degree of sensitiveness is obtained by disconnecting the pair of quad- rants that are connected to the frame, the electrode being raised for the purpose ; the other connections must be the same as in the last case. By this arrangement the needle is acted upon by one pair of quadrants only. 401 By using the induction plate we may still further diminish the sensitiveness of the instrument. For instance, when we connect the pole of the battery to a pair of quadrants, those quadrants take the potential that it has ; but if we connect it to the induction plate, then the charge in the quadrant below is only an induced one, and, since there is an interval between the plate and the quadrant, this induced charge will be small, and the effect on the needle proportionally small. Again, if we disconnect one pair of quadrants, and connect the wire from the battery to the induction plate and to the corresponding quadrants, then the charge will be partially bound. The effect on the needle will therefore be less still. The actual number of grades of sensitiveness with the induction plate are as follows : 3rd Grade. One pair of quadrants connected to one pole of battery. Induction plate and second pole of battery connected to frame. Second pair of quadrants disconnected by raising electrode. 4th Grade. One pair of quadrants connected to one pole of battery, and also to induction plate. Second pole of battery connected to frame. Second pair of quadrants disconnected by raising elec- trode. 5th Grade. Induction plate connected to pole of battery. One pair of quadrants and second pole of battery connected to frame. Second pair of quadrants disconnected by raising electrode. 6$ Grade. Induction plate connected to pole of battery. Second pole of battery connected to frame. Both pairs of quadrants discon- nected by raising the electrodes. THE THOMSON QUADRANT ELECTROMETER. 06 1 402. We can in each of these cases interchange the terminals of the quadrants, that is to say, we can use the left terminal where we used the right, and vice versa. 403. There is one more point to mention in connection with the instrument, and that is, that it may be found, on raising- one of the electrodes to disconnect it from the quadrants, that the act of doing so causes the image on the scale to deviate a few degrees from zero in consequence of a charge being induced thereby. In the most recent form of instrument there is a small milled vulcanite head provided, by turning which the quadrants are connected to the frame, and the charge being thereby dissipated, the image returns to zero. When this is done the milled head must be turned back before commencing to test again. THE USE OF THE ELECTROMETER. 404. The electrometer can be used in every test where a con- denser is usually employed. In using the condenser we have to charge it, and then note its discharge on the galvanometer, which gives the potential. With the electrometer we have simply to connect to its terminals the wires which would be connected to the condenser, and the permanent deflection on the scale gives us the potential, which can be observed at leisure. Thus in measuring the resistance of a battery by the method given on page 295 ( 322), we should first connect the battery wires to the electrometer (through the medium of the reversing key is best), note the deflection, then insert the shunt, again note the deflection, and calculate from the formula. The great value of the electrometer, however, lies in the fact of its enabling us to notice the continuous fall of charge in a cable, and not, like the condenser method, merely to determine what the potential has fallen to after a certain time. We can see with unfailing accuracy when the charge has fallen to one- half, or any other proportion we please. We see, in fact, exactly what is going on in the cable at any moment. The connections for such a test could not well be simpler. We charge the cable, connect it to the electrometer, the frame being to earth, and then notice the deflection as it gradually falls down the scale. We do not even require a battery, as we can charge the cable with a few sparks from an electrophorus. The degree of sensitiveness necessary for any particular cable we can, of course, only tell by experience. 362 HANDBOOK OF ELECTRICAL TESTING. Measurements from an Inferred Zero. 405. When very high, resistances, such, for instance, as short lengths of highly insulated cable, are measured by the ordinary fall of charge method, the fall, even in a considerable time, would be so small that the test would be an unsatisfactory one, for the difference between the deflection at the beginning of the test, and that after the interval of time, could only be a small fraction of the whole length of the scale ; and if the deflections are not accurately noted, still less can we be satisfied of the correctness of our result when worked out from a formula. By means of a plan suggested by Professor Fleeming Jenkin, however, such high resistances can be measured by the fall of charge method with considerable precision. Professor Jenkin's improvement consists in virtually prolong- ing the scale and counting the divisions from an inferred zero. An explanation of the method of making the test will best show what an inferred zero is. One pole of the battery being to earth, the other pole is connected to one pair of quadrants and to the framework of the instrument. The second pair of quadrants is connected to the cable. By joining for an instant the two pairs of quadrants together, the cable and quadrants take the same potential ; therefore, at the moment of disconnecting them, the needle will be at zero. The potential, however, of the cable, and the quadrants connected to it, will fall, and the needle be deflected. Suppose, now, one cell connected to the electrometer gave 100 divisions deflection, and suppose the battery which charged the cable was 100 cells, then if the cable lost 1 per cent, of its charge, the charge remaining would be 99, and as the other .quadrant, being permanently connected to the 100 cells, has the potential of 100, the difference between the two is 100 99 = 1 cell, which, as we have said, gives 100 divisions. The 2 per cent, loss would give 200 divisions, and so on, whereas by the method mentioned on the last page, if we get 300 say, at first, then I per cent, loss would only move the image down to 297, and 2 per cent, would move it down to 294. When all the charge is lost, the deflection would evidently be 100 x 100 = 10,000, which is the inferred zero. To obtain this zero for any particular battery, we should have to get the deflection from 1 cell and then determine, by the method given on pages 287 ( 316) and 300 ( 327), what the electromotive force of the testing battery is in terms of the 1 cell. Then by multiply- ing the 1 cell deflection by this value we get what we require. THE THOMSON QUADKANT ELECTKOMETEK. 363 The numbers representing the potentials we must evidently get by subtracting the deflections on the scale from the inferred zero. To obtain the full range of the scale we should, at starting, get the image on the actual marked zero, which is, as we have before said, at the end, and not at the middle of the scale. 406. It is possible to use the electrometer without having the acid of the jar charged. For this purpose one pair of quadrants should be connected to the needle ; by this arrangement the needle becomes charged by the same electricity that charges the quadrants to which the needle is connected. It will be seen, however, that with this arrangement the deflections will not be directly proportional to the potentials producing them, as the action is similar to that which takes place in the case of an electro-dynamometer (page 318); the deflections, in fact, will be proportional to the squares of the potentials. The special advantage of the foregoing method of using the instrument is that it enables rapidly alternating potentials to be measured, as in the case with rapidly alternating currents through the electro-dynamometer. 364 HANDBOOK OF ELECTRICAL TESTING. CHAPTER XV. MEASUBEMENT OF HIGH KESISTANCES. 407. The highest resistance which it is possible to measure by means of the Wheatstone bridge described at the commence- ment of Chapter VIII. (page 188), is 1,000,000 ohms. It is true that some bridges have another set of resistances in the top row, which will enable the ratio 10 to 10,000 to be used, and consequently a resistance of 10.000x10.000 m 1 to be measured ; but this is not often the case, and the values of resistances much greater than this frequently require to be determined. For this purpose a modification of the deflection method given in Chapter I., page 5 ( 9), must be adopted. 408. Provide a single, and also about 100 constant cells. Find their respective electromotive forces by the discharge method given on pages 287 ( 316) and 300 ( 327). Thus, suppose the discharge taken from the 1 cell, which, as we have explained, should be taken first, gave a deflection of 300, the galvanometer shunt (S 2 ) being adjusted for this purpose to 560 ohms. Sup- pose also that the discharge from the 100 cells in the place of the 1 cell, gave a deflection of 302, with a shunt (S x ) of 6 ohms ; then by multiplying the 302 by we get the deflection we should have had if no shunt had been used ; this will represent the electromotive force of the 100 cells. In like manner, by multiplying the 300 by we get a number representing the electromotive force of the 1 cell. Taking the resistance 8 ( + ') 100 Percentage of accuracy = - p -- ; [C] where 8 is the fraction of a division to which each of the deflec- tions Y, v, and v' can be read. Loss OF POTENTIAL METHOD. 435. In Chapter XIII., page 329, an equation 2-303 E log - was obtained, where F was the electrostatic capacity, in micro- farads, of a condenser, or cable, the potential of whose charge MEASUREMENT OF RESISTANCES BY POTENTIALS. 381 fell from Y to v when it was discharged during T seconds through a resistance of E megohms. Now if F is the known and E the unknown quantity, then 2- 303 F log - so that we can determine the value of a resistance by a capacity and loss of charge measurement. 436. The connections for making such a test would be pre- cisely similar to those given for determining electrostatic capacities by loss of charge ( 364, page 332). If we were measuring the resistance of a short cable by this method, the discharge deflection V, compared with the discharge deflection obtained with the same battery from a standard condenser, would give us the value of F. For long- cables, however, as we have before explained, this does not give correct results, so the capacity must be determined by other methods, Thomson's for example (page 335). 437. From ( 362, page 330) it is obvious that we must have Best Conditions for making the Test. y Make v as nearly as possible equal to O " > Possible Degree of Accuracy attainable. Percentage of accuracy = E - 2-303 v log - where 8 is the fraction of a division to which each of the deflec- tions Y and v can be read. GOTT'S PROOF CONDENSER METHOD. 438. An excellent method of determining the relative values of Y and v in the foregoing test has been suggested by Mr. J. Gott. This method avoids the necessity of discharging the cable, and consists in applying what may be termed a "proof" condenser to the latter, and then measuring the discharge from the same. This condenser should be of small capacity, so as not to remove an appreciable portion of the charge from the cable ; if this is the case, it is obvious that the discharge obtained from the condenser, after it has been connected for a few seconds to the cable at any particular time, will represent the potential which the cable has at that time. 382 HANDBOOK OF ELECTRICAL TESTING. 439. When the insulation resistance of a cable is measured by the foregoing methods, the result obtained is a mean of the resistances which the cable has at the commencement and at the end of the test, as electrification ( 414, page 369) goes on the whole time the charge is falling. 440. Experimental results show that in the case of a cable whose core is insulated with gutta-percha, if the cable be charged 10 seconds before taking the discharge V, and again 10 seconds before insulating it preparatory to observing the discharge v, then the value of R after 1 minute, obtained from the formula, agrees with that obtained by the constant deflection method given in the last chapter ( 414, page 369). 441. If we know the potential which the cable has when fully charged, and also its potential after a certain time, we can determine the potential it will have after any other time, in the following manner : A charged cable loses equal percentages of its charge in equal times, that is to say if, for example, 5 per cent, of its charge were lost during the first second, then five per cent. of what remained would be lost in the second second. Let V be the potential at first ; v after 1 sec. ; i ., *i sees. ; V 1 * 5 ? t-2 J and let us suppose the charge loses -th of its potential during the first second ; then the potential at the end of first second will be and the potential at end of second second will be but from equation [1] we get V therefore, substituting this value in [2] the latter becomes -^, which equals V ( j MEASUREMENT OF RESISTANCES BY POTENTIALS. 383 and consequently the potential at the end of t l seconds will be Also we must have therefore and t. = o" y that is, , 3 , V log- log- log^ log- .For example. The potential at first was 300 (Y), and after 20 seconds (^) it fell to 200 (^). After what time (t. 2 ) would it fall to 100 (v. 2 ) '? 300 2-4771213 ^ g oo 2-0000000 300 X 2 = 2-4771213 X 2 = 200 2-3010300 442. It being usually required to know the time the charge in a cable will take to fall to half charge, the formula becomes 30103 443. The formulae we have given are capable of various modifications, which, however, are more of a fanciful than of an actual and practical value. 384 HANDBOOK OF ELECTRICAL TESTING. Thus the formula} E = may be simplified if we make v = , for in this case log. ^ = log. 2 = -693; therefore To obtain experimentally the time occupied in falling to half charge, repeated trials would be necessary, and the time taken in doing this would hardly compensate for the advantage of using a simpler formula. The object of obtaining the time of fall to half charge is to get a convenient unit for comparison with other cables, and this lime of fall is easily calculated from the formula before given, in which the potential after any time may be used, this being obtained by one observation only. 444. A useful formula is that suggested by Mr. W. H. Preece, which is obtained in the following manner : In the equation _ -30103 2 ~ ~ ' l let n = percentage of loss in time / lt then (Y - PI ) 100 . therefore _ 100 - n l ~ ~Too~ Substituting this value of v 1 in the above equation, we get 30103 -30103 1 ~ ~l 100 ' l ~ 2 -000 -log (100 -) ' 1 ' g 100 - For example. If a cable lost 20 per cent, of its charge in 5 minutes ; in how many minutes would it fall to half charge ? MEASUREMENT OF RESISTANCES BY POTENTIALS. 385 '30103 _ *- 2 -000 -log (100 -20) 445. From tlie equations _ , * we can find what would be the potential, 2 , after a certain interval of time, 2 2 , the potential at first, and the potential v l9 after a time, t v being given. Thus we have from the above equations /"iV* /*Vi. W W ' therefore i 1 9 = This formula we should have to work out by the aid of logarithmic tables. For example. The potential of the charge in a cable when full was 300 (V). After 20 minutes (^) the potential fell to 200 (v^. What would be the potential v 2 at the end of 30 minutes ( 2 ) ? log 2= -3010300 log 3 = -4771213 1-8239087 3 4717261 1-7358631 log 300 = 2-4771213 2 -2129844 = log of 163-3. 446. In connection with the foregoing tests it may be men- tioned that, in testing cables, it is very usual to make fall of charge measurements, but not to work out the results by any of the foregoing formulae. The general practice is to simply calculate and record the percentage of fall. 2 c 386 HANDBOOK OF ELECTEICAL TESTING. CHAPTER XVII, LOCALISATION OF FAULTS BY FALL OF FOTEOTIALS. CLAKK'S METHOD. 447. In Fig. 129 (page 377) in the last chapter, if & c were a portion of a cable making full earth at c, then by the method described for determining b c we should find the position of the break. Supposing, however, a cable had a fault which did not make full earth, then the potential would not fall to zero at that point, but would have a value depending upon the resistance of the fault. The potential, however, would be the same as the potential at the further end of the cable, provided that end were insulated. If we can determine the value of this potential, we can readily localise the position of the fault. FIG. 130. In Fig. 130 let be be the cable which has a fault at c, th end of the cable at e being insulated ; and let Rbe a resistance between the battery and the end of the cable 6, then V vi : v ! : : R + a? : a; ; therefore or therefore LOCALISATION OF FAULTS BY FALL OF POTENTIALS. 387 that is, , = E : . [A] 448. If, as explained in the last test ( 430, page 378), we at once determine the value of Y v, by connecting the wires from the condenser, &c. (or from the galvanometer and high resistance), to the points a and 6, then if we call v' this difference of potential, we get * = E. [B] 449. In order to determine the relative values of the poten- tials at the two ends of the cable, their values with reference to some standard of potential or electromotive force must be obtained. For this purpose any of the standard cells mentioned in Chapter VII. (page 137) may be used. The way in which such standard cells would be employed for making the test we have been considering, would be as follows : The electrician at a charges a condenser from one of the standard cells, and notes the discharge deflection on his galva- nometer. This deflection, then, represents the potential of the cell. The wires from the standard cell are now disconnected, one wire is connected to earth, and the other to a, and again a discharge reading is taken ; then this reading, divided by the reading obtained with the standard cell, gives the value of V in terms of the standard cell. The wire at a is then disconnected and joined to 6, and another discharge measured, which result divided by the standard discharge gives the value of v in terms of the standard cell. The electrician at the other end, e, of the cable makes a similar test, and thus determines the value off?!. Since the standard cells at the two stations are exactly equal in electromotive force, the relative values of Y, ?, and v l will be obtained exactly. The capacities of the condensers at the two stations, it may be observed, need not be alike. For example. The discharge deflection obtained from a condenser at station e with a standard cell, was 180 divisions ; and the potential ;,, measured from the same condenser, gave a discharge deflec- tion of 360 divisions ; therefore 2 c 2 HANDBOOK OP ELECTEICAL TESTING. At the other end of the cable, from a standard cell of the same electromotive force as the one employed at station e, a discharge deflection of 150 divisions was obtained from a con- denser. The potentials V and ?, measured with the same con- denser, gave deflections equivalent to 2550 and 1050 divisions respectively ; therefore v = = 17-0. loO 1050 B was equal to 1000 ohms. What was the value of a? 7 2 x = 1000 = 500 ohms, showing that the fault was 500 ohms distant from the end b of the cable. If the length of the cable were, say, 80 knots, and its total conductivity resistance 800 ohms, or 10 ohms per knot, then the distance of the fault from b would be , or 50, knots. The value of v t when obtained at e would be telegraphed to fc ; this could be done, since the cable would not be entirely broken down. If the potentials are measured by observing the permanent deflections obtained through a high resistance (314, page 286), the observations with the standard cells must be made in the same manner. 450. In making the test we are liable to make errors in V, u, and 0J, and these errors will produce the greatest total error in x when the errors in V and v l are minus, and the error in v is plus ; let each of the errors be 8, and let A. be the total error produced in a, we then have ( + 8) - ( Pl - 8) p -. Pl + 2 8 or ^ v Irat * = E T^' or> E = x t -^> LOCALISATION OP FAULTS BY FALL OF POTENTIALS. 389 therefore but, since 8 is small, we may say \ = x I or Now if we regard (V f^) as a constant quantity, then in order to make A as small as possible we must make the deno- minator of the fraction as small as possible ; from ( 432, page 378) we can see that in order that this may be the case we must make that is to say, we must make R approximately equal to x. In the case of formula [B] (page 387) the conditions for making the test in the most satisfactory manner, are slightly different from the foregoing ; for since (V v) in this case is obtained by a single measurement, v' y there can be but one error, 8, in it. We have, in fact, 8(2 Y --..) __ Vl - V_,-8 ^J ~ ~ (v - O (V - v - 8) * but, since 8 is very small, we may say 8(2 Y -.-.) ( v _ - ^ /: ^ ^^ or = Now this equation is of the same form as equation [F] (page 104), consequently the investigation there given may be applied to the present case. In the latter, the coefficients of ( V i) and (w v x ) are 2 and 1 respectively ; if therefore, in equation [G] (page 105) we substitute J for &, and also if we 890 HANDBOOK OF ELECTRICAL TESTING. substitute a; and E, for C 2 and C 15 respect the conditions we require ; we have then substitute a; and E, for C 2 and C 15 respectively, we shall obtain th = (0 -00 1-7071; that is to say, we must have K = 1-7071 x. Practically we may say, make E - 2x approximately. We have therefore Best Conditions for making the Test. 451. In the case of formula make R approximately equal to x. In the case of formula make E approximately equal to 2 x. Possible Degree of Accuracy attainable. In the case of formula [A] _ S (V - O 200 Jrercentage 01 accuracy = ^ 5 - > il - In the case of formula [B] 8 (2 i/ + t> - f7j) 100 r ercentasre of accuracy = - ; - ^-^- - ( - 00 0' where 8 is the fraction of a division to which each of the deflec- tions can be read. SIEMENS' EQUAL POTENTIAL METHOD. 452. In Fig. 131 let B E be the cable which has a fault at c, x and y being the distances on either side of the fault, and z the LOCALISATION OF FAULTS BY FALL OF POTENTIALS. 891 equivalent length of the latter. Suppose that one pole of a battery is connected at B, the other pole being to earth, then if the end of the cable at E is insulated we shall have, as in the last test, the potential at E to be the same as the potential at FIG. 131. the fault. Next suppose that the battery at B is removed, and that that end of the cable is insulated ; then, if a battery is connected to E, of such a strength that the potential at the fault, and therefore at B, is the same as was the potential at E in the first case, then V 2 will be the new potential at E. Now, Vi - i : V 2 - ! : : ^ : y, therefore 9 V 2 - If I be the length of the cable, then l = x + y, or, y therefore l-x that is, or For example. In a faulty cable 500 knots (Z) long, after adjusting the potentials according to the foregoing method, the values of the same were found to be 392 HANDBOOK OF ELECTRICAL TESTING. Y! = 200, Y 2 = 300, v l = 40. What was the distance (a?) of the fault from B ? 900 4-0 = 5 (300 - 40) + (200 - 40) = 19 ' 5 453. In making the test practically, the following course would be pursued : Station B first connects one pole of a battery direct on to the cable, the other pole being to earth, whilst E insulates his end of the cable. This being done, B notes the potential V lt and E the potential v r When B thinks that sufficient time has elapsed for E to have taken his observation, he removes the battery and insulates his end of the cable. E noting that his potential has fallen to zero, connects up his speaking apparatus, and B having done the same, E communicates to B the result he has obtained* Station E now connects up his battery to the cable, taking care that the pole connected to the latter is similar to that employed by B in the first instance. The latter observes the potential at his end of the cable, and if it is not the same as that previously obtained at E, he informs the latter, by means of signals agreed upon, that such is the case, whereupon E increases or decreases his battery power, and regulates it by varying a resistance in its circuit until the potential at B is made the same as it was at E on the first occasion. The potential Y 2 is then noted by E, and the result being; reduced to terms of a standard cell,* is communicated to B. The latter station, having also reduced his results to terms of a standard cell, then works out the formula, and thus determines the position of the fault. 454. For localising faults in long cables this method is more accurate than the previous one, as it is not so much influenced by the resultant fault f produced by the conductive power of the insulating sheathing, more especially if the fault is near the middle of the cable. It must be understood that both tests are only accurate in cases where the total insulation resistance of the cable is very high compared with the resistance of the fault, for in such cases the fall of potential is practically represented by a straight line, and the formulae are constructed on this assumption. * See page 387. t See page 265, 288. LOCALISATION OP FAULTS BY FALL OF POTENTIALS. 393 When, however, the cable is very long and the total insula- tion resistance consequently comparatively low, then the potential cannot be regarded as falling regularly from end to end, but must be graphically represented by a curve, and the potential at the fault is less than that indicated in the straight line diagram, and the potential at the extreme end is lower than this still. The exact formulas for these tests are considered in Chapter XXII. 455. From the nature of the test it must be evident that there are no particular conditions which enable a maximum degree of accuracy to be obtained, except in so far that the battery power employed should be sufficient to enable high deflections to be produced. Possible Degree of Accuracy attainable. , 200 S Percentage 01 accuracy = == , V v l where 8 is the fraction of a division to which each of the deflections can be read. SIEMENS' EQUILIBRIUM METHOD.* 456. If two batteries have their opposite poles connected to the ends of a perfect cable, their other poles being to earth, then the fall of potential along the cable is continuous and cuts the latter at a certain point. The position of this point can be varied by altering the relative electromotive forces of the batteries, or by adding in resistances between the batteries and the ends of the cable. In the case of a faulty cable, if the fault is at this point, then no current passes from the batteries to earth, consequently any alteration in the resistance of the fault does not affect the values of the potentials at the different points along the line of fall. By observing what arrangement of resistances and electro- motive forces is necessary to bring the zero point to the fault, the position of the latter can be accurately determined. In Fig. 132 (page 394) let x and y be the portions of the cable on either side of the fault, and let r, r be equal resist- ances connected to either end of the cable, also let E x and B 2 be resistances whose values can be varied at pleasure. Now, in making the test we have to adjust R x and E 2 so that the potential at the fault shall be zero, and consequently that * Journal of the Society of Telegraph Engineers,' Vol. V., page 61. This method, it should be remarked, is substantially the same as that devised by M. Emile Lacoine, and described in Vol. IV., page 97, of the same journal. 394 HANDBOOK OF ELECTKICAL TESTING. A B shall be a straight line. To obtain this result we must have or : v. 2 : : x : y, and also or : v l : : r 4- # : FIG. 132. v. 2 r v r r . 2 " " 2 = T~ ~*~ ; from which we get that is to say in order that AB may be a straight line the differences of the potentials on either side of r, at both ends of the cable, must be the same. 457. To obtain this result in practice only one of the resist- ances Bj and E 2 need be adjusted. The best way of making the test would then be as follows : The two stations should first adjust their galvanometers by means of the movable magnets so that they both give precisely the same deflections when a current from a standard cell through a standard resistance is sent through them. This being done, batteries Ej and E 2 are connected by the two stations on to the ends of the cable, and then the adjusted galvanometers are severally connected on each side of the LOCALISATION OF FAULTS BY FALL OF POTENTIALS. 395 respective resistances r and r at the two stations, there being in the circuit of each galvanometer very high, but equal, resistances. Station A, say, now adjusts B x and watches the effect on his galvanometer; B also watches the effect on his own galvanometer, and from time to time signals to A the deflection he obtains ; this signalling is easily done by having the front contact of a well-insulated key connected to the end of the cable, and the back contact connected to earth, whilst the lever of the key is connected to one terminal of a small condenser whose second terminal is to earth. By pressing down this key a small quantity of the charge in the cable will rush into the condenser, and a momentary movement of the galvanometer needle at station A will be produced; by arranging then that so many movements shall represent a particular deflection, B can easily communicate his results to A. When exact adjustment is obtained, that is to say, when (YJ Vj.) and (V 2 tf 2 ) are equal, the galvanometers are dis- connected from either side of r and r, and the potential v 1 is measured ; x is then obtained from the formula where v equals (V T t^), as in the " Fall of Potential Method" of measuring a resistance, page 377. 458. To make the foregoing test as accurately as possible it is advisable, for the reason explained in 432, page 378 (after the value of x has been obtained by a rough test), to adjust r and r so that they shall each be approximately equal to x. With regard to the " Possible degree of accuracy attainable," we are liable to make an error in obtaining the value of Vj v 19 but inasmuch as V x v l may itself contain an error due to V 2 v 2 being incorrectly measured, the actual total error which may exist in x must be twice that given by formula [C] (page 380) ; consequently we have Best Conditions for making the Test. Make r, r, each approximately equal to x. Possible Degree of Accuracy attainable. _ 8 fo + ') 200 Percentage ot accuracy = - f- - , v^ v where 8 is the fraction of a division to which each of the deflections can be read. 396 HANDBOOK OP ELECTRICAL TESTING. CHAPTER XVIII. TESTS DURING THE LAYING OF A CABLE. 459. The immediate detection of a fault which may occur in a cable during its submersion is a point of great importance. To enable this to be done, a good system of testing is requisite. Whatever the system be, it should be a continuous one, that is to say, the cable should be continuously and visibly under test, so that the moment a fault occurs it may be detected by the ship and traced. SYSTEM FOR COMPOUND CABLES. 460. For laying cables which are not more than 200 miles or so in length, and which have several wires, the method shown by Fig. 133 may be employed. In this system the wires are all connected up in one con- tinuous length as shown. Should there be an odd number of wires, the odd one would have to be coupled on to one of the others in " multiple arc." , FIG. 133. SHIP SHORE In Fig. 133, g l and g 2 are two ordinary " detector" galvano- meters well insulated. The battery e, of one or two cells (also well insulated), keeps a continuous current circulating through these galvanometers and the conducting wires of the cable ; this serves as a " continuity " test, for if any of the wires should break within their insulating sheathing, the circuit becomes TESTS DURING THE LAYING OF A CABLE. 397 interrupted, and consequently the needles of both galvano- meters will fall back to zero. In the case of a cable with an odd number of wires, should the conductor of either of the two which are coupled together become broken, then the needles will only fall back a little way and not back to zero; this, however, will be quite sufficient to indicate that the conductor is fractured. The galvanometer G is of the marine description, shown on page 63, and is connected to one of the wires. The battery E, of about 200 cells, keeps a continuous current flowing through the galvanometer and through the insulating covering of the wires. If a fault occurs in the insulation, the current by escaping direct to earth causes an immediate and very large increase in the deflection of the needle of G. In order to keep up communication with the shore, the current from battery e is reversed after certain equal intervals of time. If the shore perceives that the reversal has not taken place, or that the needle of g 2 is not steadily deflected, he knows that something has gone wrong, or that the ship wishes to communicate with him, and he joins up his speaking apparatus and tries to communicate with the ship. The galvanometers g l and g 2 could be used for this purpose by having treZZ-insulated keys inserted in their circuit at the ship and shore, these keys being so arranged that their depression breaks the circuit ; the movements of the needles could then be worked according to the ordinary Morse code, and communication be kept up without interrupting the insulation test. SYSTEM FOR SINGLE WIRE CABLES. 461. The method just described is only applicable to a cable which has more than one wire, for although with the latter the insulation test would be kept up, there would be no means of communicating with the shore. In such cases the following plan may be adopted: The end of the cable on board the ship is well insulated, and connected to a battery and Thomson galvanometer as in the previous test and as shown by Fig. 134 (page 398). On shore (Fig. 135) a condenser is provided, one terminal of which is connected to a brass lever which plays between two insulated contacts ; one of these contacts is connected to the second ter- minal of the condenser, which latter terminal is also connected, through a Thomson galvanometer, to earth ; the other contact is connected to the conductor of the cable. The battery connected to the cable on board the ship charges the former to a certain 398 HANDBOOK OF ELECTRICAL TESTING. potential, and the value of this potential will "be the same throughout the whole length, provided no fault exists. If the lever on shore be moved against the contact connected to the cable, a portion of the charge in the latter will rush into the con- denser and will charge up the set of plates, to which it is connected, FIG. 134. SHIP. Co-Vlc H'l'l'l Earth to the same potential as the cable ; the second set of plates will become charged to the opposite potential by a charge rushing in from earth through the galvanometer ; this in-rush will produce a throw, or momentary deflection of the needle, the amount of which will represent the potential of the charge in the con- denser, that is, the potential at the end of the cable. If now the lever be moved from the cable contact to the contact con- nected to the condenser, the latter will be short circuited and discharged. The rush of the charge into the condenser when the latter is connected to the cable contact, produces a simul- taneous rush into the cable from the battery on the ship, and as this takes place through the galvanometer on board the ship a sudden throw is produced on the needle. Now if a fault occurs during the laying, the steady deflection on the ship's galvanometer, which is due to the flow of current through the dielectric of the cable, and which is distinct from the throw which takes place when the condenser becomes connected to the cable at the shore end, becomes greatly increased and renders the presence of the fault evident immediately. On the shore the effect of the fault is to reduce the potential at that end of the cable, and consequently the charge which the condenser takes becomes correspondingly reduced; when then the con- denser becomes charged through the galvanometer, a reduced throw is produced, which thus shows the shore the existence of the fault. The lever on shore which charges and discharges the con- denser is moved by clockwork which causes it to act every five minutes, so that every hour twelve throws are observed on each TESTS DUKING THE LAYING OF A CABLE. 399 galvanometer. At the end of every hour the ship reverses the "battery so that the direction of the throws is changed. In order to enable the ship to communicate with the shore, instructions are given that if at the end of the hour the throws do not become reversed, or if they become reversed before the expiration of the hour, it is a sign that the ship wishes to com- municate with the shore; in this case, then, the shore disconnects the cable from the clock lever and connects it with the speaking apparatus, and as the ship does the same, the necessary com- munications can be carried on. If, on the other hand, the shore wishes to call the attention of the ship, he can do so by moving a lever, corresponding to the clock lever, two or three times quickly by hand ; the ship then observing that the throws on her galvanometer take place quickly, instead of at intervals of five minutes, immediately joins up her speaking apparatus, and thus communicates with the shore. The movement of the lever L in the foregoing system of testing is effected, as has been pointed out, by means of a clock, but L may be a hand-worked key, and this is sometimes pre- ferred, as although a clock ensures the discharges being obtained after regular intervals of time, yet the hand method ensures the necessary watchfulness of the electrician on shore, which is a point of importance. WlLLOUGHBY SMITH'S SYSTEM. 462. For long single-wire cables a refinement of the foregoing- method, devised by Mr. Willoughby Smith, has been adopted. This system is shown by Figs. 136 and 137 (page 400). On shore, the cable is connected to a key K, galvanometer G 2 , and condenser C x as in the last method of testing. To the cable there is also connected a resistance in circuit with a gal- vanometer Gr. This resistance is very much greater than the total insulation resistance of the cable, and consequently it does not appreciably affect the potential measured by the key K, whilst it allows sufficient current to pass through the galvano- meter G to produce a sensible deflection of its needle. The high resistance is made of selenium, and it must be care- fully excluded from light, and kept at as uniform a temperature as possible, otherwise it will vary considerably. On the ship the cable is connected to a slide resistance Wheatstone bridge similar to that described in Chapter VIII., page 210. The working of the apparatus is then as follows : On the ship, plugs are inserted at p and p 2 and balance is- 400 HANDBOOK OF ELECTRICAL TESTING. kept on the galvanometer G 4 by adjusting the slides of the slide resistances, the resistance K being preserved constant. This gives the insulation resistance of the cable. I Galvanometer G 5 is kept short circuited under ordinary con- ditions, it being only used occasionally for the purpose of ascer- taining whether the batteries are in good condition. FIG. 136. SHORE. CaJ>U FIG. 137. SHIP. Calk Should it be thought advisable, as a check, to take an ordinary deflection insulation test,* this can be done by removing the plugs p : and p 2 ; the current then passes direct from the battery through the galvanometer G 4 into the cable. Page 368. TESTS DUKING THE LAYING OF A CABLE. 401 On shore the potential at the end of the cable is observed on G 2 by depressing the key K every five minutes. The deflections obtained are carefully noted and recorded. The battery E is reversed every fifteen minutes by the ship, and this is observed on the galvanometer G and shows that the conductor of the cable is entire. If the ship requires to com- municate with the shore, it reverses the battery several times after short intervals; this is acknowledged by the shore by means of the key K ; when this is done, the shore moves over the switch S x and receives signals from the ship on galvanometer G 3 through the medium of the condenser C 2 . The insulation test is not interrupted by this signalling, as the cable remains insulated the whole time. The effect of working the signalling key K 2 is only to add or subtract a little from the charge in the cable through the medium of the condenser, and thereby to produce momentary deflections on the galvanometer G 3 . The same in the case when the shore signals to the ship, the switch S 2 being moved over to key K L for that purpose. Various slight modifications have been, and are, employed in practically using this method, but the general arrangement is that which has been indicated. 2 D 402 HANDBOOK OF ELECTRICAL TESTING. CHAPTEE XIX. JOINT-TESTING. 463. Joints are the weak points in a cable, and it is there- fore essential that they should be not only carefully made but carefully tested. A joint, being a very short length of the core, offers, or should offer, a very high resistance ; it would consequently be impos- sible to test it by a direct deflection method, that is, a method similar to that by which the insulation resistance of a cable is taken (page 368). Even with a very powerful battery, the galvanometer deflection, provided the joint were good, would be quite inappreciable. One or other of the following methods must therefore be adopted. A condenser can be charged through the medium of the joint, and after a noted time the discharge taken, which gives the amount which has leaked through the joint. This is known as Clark's accumulation method. Or a charged condenser may be allowed to discharge itself through the joint, and the amount lost after a certain time noted. In both these methods the discharge deflections are compared with the results obtained with a few feet of perfect core. CLARK'S ACCUMULATION METHOD. 464. A gutta-percha or ebonite trough is provided, which is suspended by long ebonite rods from any convenient hook. The good insulation of the trough is a point of great im- portance, and consequently the suspending rods should be quite dry and clean. The most effectual way of obtaining this result is to well scour the surface of the ebonite with a glass or emery paper ; this is a much better method than covering the surface with hot paraffin wax as is sometimes done. 465. We may here remark that surface leakage is almost the only medium of loss to be feared in electrical apparatus, and this should always be seen to by keeping all surfaces over which leakage is likely to occur, in proper condition. The peculiar formation of ebonite causes minute quantities of sulphuric acid JOINT-TESTING. 403 to form on its surface, so that the latter should be often rubbed over with a dry cloth. Hot paraffin wax painted over the dry surfaces is very advantageous, but, where appearance is im- material, nothing is so effectual as a surface well scoured with glass or emery paper. 466. The trough is rilled with water, and the joint to be tested is immersed and held down in it by two hooks placed at the bottom. The portion of the core on either side of the joint should be carefully dried (not paraffined), for the same reason that the suspending rods were so treated. The connections for the test, shown by Fig. 138, are very similar to those shown by Fig. 97, page 278 ; the only difference FIG. 138. Cable being that the pole of the battery, which in that figure was connected directly to the condenser, is, in the joint test, con- nected to it through the medium of the joint. The battery used should be as large as possible ; 200 Daniell cells is the power very commonly employed. 467. After the joint is placed in the trough for testing, it is necessary to see that the latter is sufficiently well insulated. 2 D 2 404 HANDBOOK OF ELECTRICAL TESTING. To do this the pole of the battery, which for the regular test would be connected to the core, must be connected to the wire attached to the plate in the trough, and the discharge key pressed down ; this charges the condenser ; the battery being then disconnected from the plate, an interval of time (usually one minute) equal to that which would be occupied by the test of the joint, is allowed to elapse, and then the "Discharge" trigger is pressed and the discharge noted ; this should be equal, or very nearly so, to the instantaneous discharge. 468. The good insulation of the trough being satisfactorily obtained, and the connections being made as shown by Fig. 138 (page 403), the short-circuit plug of the condenser must be inserted in its place, the discharge key pressed down, and then the short-circuit plug removed; the battery then charges the condenser through the joint. After a certain time, usually one minute, the discharge deflec- tion must be noted. A similar measurement must also be made, using a length of perfect core in the place of the joint. If, in the latter case, the discharge deflection after the same interval of time is much less than that obtained from the joint, the latter is defective and must be remade. 469. It is a very important point in making the test to insert the short-circuit plug in the condenser previous to depressing the discharge key ; if this is not done, an induced charge is thrown into the condenser by the sudden rush of the battery current into the core when the discharge key is depressed. This induced charge will give a considerable deflection when the condenser is discharged, which deflection is in no way due to leakage through the joint, though it might be mistaken for such. By keeping the condenser short circuited this induced charge is dissipated. 470. If the joint is good, the discharge deflection seldom exceeds two or three divisions. Indeed, the fact that it does not do so is usually a quite sufficient proof of the soundness of the joint, and it is not often the case that a comparison with a piece of perfect core is necessary. DISCHARGE METHOD. 471. This is a reversal of the foregoing, and consists in charging the condenser full and letting it discharge itself through the joint. The connections for making this test would be similar to those employed in measuring high resistances by the loss of potential method given in Chapter XVI., page 380 ( 435), the JOINT-TESTING, 405 one end of the core taking the place of one end of the resistance, and the plate in the trough the place of the other end. The system of charging the condenser through the joint cannot of course be carried out unless one end of the core is at hand to which to attach one pole of the battery. When a joint is made in a cable core at sea, neither end can be got at. The joint, however, could be tested by making the connections as for the last method of testing, only instead of joining the core to the condenser terminal, the latter, and also the cable end, would be put to earth. To carry out the test in this manner, arrangements would have to be made with the shore, previous to the manufacture of the joint, that at a certain time the end of the cable shall be put to earth. The first method could also be adopted for testing at sea, by using an earth in the foregoing manner. As a matter of fact, joints made at sea are never tested, though there seems no reason why they should not be so. 472. We may if we please, in both the foregoing tests, place the galvanometer between the back terminal of the key and the condenser, and join the two terminals from which it was re- moved, by a piece of wire. We should then get a charge as well as a discharge deflection, and there is this advantage, that if the joint is very bad or the trough not well insulated, we should get a permanent deflection after the charge deflection has taken place. 473. The connections should always be so made that the zinc pole of the battery is connected to the core and the copper pole to the plate. 474. It is very advisable to employ a special condenser for making these tests, for if one is used which has been charged at any time with a high battery power, it will often be found that a portion of this charge will have become absorbed, and when the condenser is left to itself, this portion will become free and give a discharge which may be mistaken for an accumulation through the joint. ELECTROMETER METHOD. 475. Although the preceding methods of testing are often the only ones which can be adopted, yet when possible it is best to make joint tests by means of an electrometer, as the results are always more trustworthy than those obtained by the con- denser method, since they are free from the source of error mentioned at the end of the last paragraph. Fig. 139 (page 406) shows the connections for making this test, which is executed in the following manner : 406 HANDBOOK OF ELECTKICAL TESTING. After the insertion of the joint in the trough, the insulation of the latter must be tested ; this is done by pressing down key K! and moving the switch S over to its well insulated contact stop s; this puts the ten-cell battery E : in connection with the quadrants of the electrometer, and thereby charges them and causes a steady deflection of the needle. Key K x being kept FIG. 139. InsuLcutecL Eart switch S is now opened and the deflection watched for two minutes to see whether there is any sensible fall due to the charge on the quadrants leaking to earth through the medium of the trough ; if this loss is only equal to two or three divisions, the insulation of the trough may be considered to be good. Key K! is now released and switch S closed so as to discharge the electrometer. Switch S is now again opened and key K 2 depressed ; this puts the 200-cell battery E 2 in connection with the core of the cable, and the momentary rush of current into the latter causes an induced charge to rush out of the trough and produce a sudden deflection of the electrometer needle ; it is usual to record this deflection, although it is of no value, except to show that the various connections have been properly made, and that the joint has been placed in the trough. JOINT-TESTING. 407 Key K 2 being kept depressed, switch S is now moved over to s (so as to discharge the electrometer), and then again opened. The scale of the electrometer is then watched, as the current leaking through the joint into the trough accumulates and causes a gradually increasing deflection of the needle; the amount of this deflection should be noted at the end of one and two minutes after the opening of the switch. After the observations with the joint have been made, a piece of perfect core must be inserted in the trough and a similar test executed, the results of which should not differ much from those obtained with the joint. It always happens that a joint gives a greater accumulation than an equal length of perfect core, unless indeed the joint has been made several days before being tested, which is seldom, if ever, the case. 408 HANDBOOK OF ELECTEICAL TESTING. CHAPTEE XX. SPECIFIC MEASUREMENTS. 476. In order to compare the relative or specific "Conduc- tivity," " Insulation," and " Inductive capacity " of the materials used in the construction of the core of submarine cables, it is necessary that they should each of them be referred to some standard unit with which the comparison can be made. SPECIFIC CONDUCTIVITY. 477. For the specific conductivity of a wire, the conductivity of the pure metal is taken as the standard. Experiments by Dr. Matthiessen have proved that 1 foot of pure copper wire weighing 1 grain has a resistance of -2262 ohm at a temperature of 24 Cent., which is equivalent to a resistance of -2261 ohm at 75 Fahr. ; consequently, I feet of such wire at the latter temperature will have a resistance of I X *2261 ohms. But I feet of the wire will weigh, not 1 grain, but I grains, and therefore the resistance of Z feet weighing 1 grain must be Z X '2261 X Z, or, Z 2 X *2261 ohms; and, further, if the Z feet weighed w grains then the resistance would be Z 2 X '2261 ohms. w But, again, the resistance of the wire will vary with the temperature, consequently to obtain the resistance at any parti- cular temperature we must correct the same by means of a coefficient k ; we have then Resistance, R, of Z feet of pure copper) _ Z 2 X '2261 wire weighing w grains J w k The numerical value of k for various temperatures is given in Table IV.* Having thus obtained a simple formula which expresses the * The general question of corrections for temperature is considered in Chapter XXI. SPECIFIC MEASUREMENTS. 409 relation between the length, &c., and the resistance, of a pure copper wire, we are in a position to determine the specific con- ductivity of any other wire ; for having measured off a definite length of the latter and ascertained its weight, temperature, and resistance, then the latter compared with the resistance of a pure copper wire of the same length, temperature and weight, gives us, by direct proportion, what we require. For example. Suppose the length of our sample of wire were 20 feet, its weight 500 grains, its resistance *200 ohm, and its temperature 60 Fahr. From Table IY. we get, for 60, ~k = 1-0323, con- sequently the resistance, E, at 60, of a pure copper wire whose length and weight are similar to the sample, will be 20 x 20 X -2261 5QOX1-0323 'W2ohnL Then to get the specific conductivity (#) of the wire sample, we have the inverse proportion 200 : -1752 :: 100 : x, or that is to say, the conductivity of the wire sample is 87 6 per cent, of that of pure copper. 478. In the case of a cable where the weight per knot of the conductor is always known, the calculations are much simpler, as they can be made by reference to Table II., which gives the resistances corresponding to various percentages of conductivity of a conductor 1 knot long weighing 1 lb., and at a temperature of 75 Fahr. The way in which this table would be used is as follows : Supposing we had a cable whose conductor weighed 107 Ibs. per knot (this is a very usual weight for the conductor of a cable), and whose resistance per knot at 75 Fahr. was found by experiment to be 11 '56 ohms, then by multiplying 11-56 by 107 we get the resistance of a knot-pound of copper of a corre- sponding conductivity. 11-56 x 107 = 1236-92 and this resist- ance in the table corresponds to a conductivity of 96-8, which is therefore the percentage of conductivity of the conductor of the cable. 479. In calculating out Table II., the determination of Dr. Matthiessen before referred to, given in the British Association 410 HANDBOOK OF ELECTEICAL TESTING. report on electrical standards, has been taken as the basis ; this determination makes the resistance of a foot-grain of pure copper at 24 C. (75-2 F.) to be -2262 ohm; the latter value appears to be the most trustworthy one yet obtained. 480. It is sometimes required to determine the specific con- ductivity of a wire whose length and diameter (d) are known ; in this case, the determination of Dr. Matthie^sen viz., the resistance of 1 foot of pure copper wire whose diameter is 1 mil (yJ^th of an inch) is 10-323 ohms at 60 F. (or 10-656 ohms at 75 F.) may be taken as the standard. Since the resistance of a wire varies inversely as its sectional area, that is, inversely as the square of its diameter (d), we must have : Eesistance of I feet of pure copper 1 _ I X 10*656 , wire d mils in diameter J " d 2 k For example. The resistance of 50 feet of copper wire, 14 mils in diameter, is found to be 2 746 ohms, at a temperature of 65 F. ; what is the specific conductivity of the wire? For 65, Jc = 1-0214 (Table IV.), therefore Resistance of 50 feet of pure \ 50 x 10 656 copper wire 14 mils inl = = 2-661 ohms; diameter, at 65 P. j 14x14x1-0214 then by inverse proportion we have 2-746 : 2-661 : : 100 : x or 2-661 X 100 x = = 96- 9: 2-746 that is to say, the conductivity of the wire sample is 96-9 per cent, of that of pure copper. 481. In the case of small wires where it is difficult to measure the diameter with great accuracy, it is preferable to test for specific conductivity by weight rather than by gauge, for by taking a sufficient length of wire, we can determine the value of the weight as accurately as we please. 482. Table III.* shows the resistances, &c., of various gauges of pure copper wire at 60 F. * This Table was compiled by Messrs. W. T. Glover and Co., electrical wire makers, of Manchester, and is inserted by permission. SPECIFIC MEASUKEMENTS. 411 SPECIFIC INSULATION. 483. To obtain the specific insulation resistance of any material is not an easy matter, for we have no pure standard material with which to compare it, and even if we had, the resistance would be so enormously high that we could not, as in the case of the wire, get a piece ot a certain length and compare it by measurement with another. We must therefore look for some other method. Now, the form in which gutta-percha is used for submarine cables is that of a cylinder, in which the conducting wire is con- centrically placed ; and to compare the relative resistances of different cores we must first ascertain the law of the insulation resistance of cores whose sheathings have various thicknesses. As this is an interesting problem, we will give it at length. Looking at a transverse section, let us suppose the sheathing; to be divided into a number of concentric circles, such that the resistance of the piece between any two circles equals p. For this to be the case, it is evident that the circles nearer the cir- cumference must be of a greater thickness than those near the- centre, since their circumferences are greater. Let there be n of these circles, so that n p = W (p here cor- responds to the little interval of time t in the loss of charge problem, page 327, 361). Now, if be the internal and external radii or diameters- of any one cylinder, and if the difference r d + l r d is very small, the resistance of the cylinder will be 2<7rlr d when Z is the length of the cable, and s the specific resistance of the insulating material. Now, the smaller we make r d + 1 r d , the nearer will this be true. But in order to do this, we must make p small and n large. Now _ since p equals the resistance of each cylinder ; therefore 2 TT I P \ + p > 412 HANDBOOK OF ELECTRICAL TESTING. Then, as in the problem we have referred to, ,. = r ,( 1 + ^) n where r n , or R, is the external, and r d , or r, the internal radius of the sheathing ; that is, and the larger n is the nearer is this true ; therefore make p = and n = oo so that n p still equals W ; we then get a perfectly accurate result. Let sn x so that x = oo when n = oo . Then 2irlW when x = oo , but when this is the case the expression within the square brackets is known to be equal to e* thus E = e r therefore . R . R slog.- slog- w _ _ i _ _ !_. 2 __ -A Now as -~f - j is a small quantity, its squares and higher powers may be neglected, in which case we get Coefficient = 1 + CORRECTIONS FOR TEMPERATURE. 417 or for the quality of wire in question Coefficient = 1 + = 1 + V X -0022206. A still closer approximation may be obtained by including the 3rd term of the series, viz., Now this expression equals we have therefore or giving the numerical values to the constants we get Coefficient = 1 + <> X -0021141 + n^ X -0000024655. Influence of Conducting Power upon Variation of Resistance by Change of Temperature. 489. The influence of temperature upon the resistance of metals varies according to the conducting power of the metal. According to Matthiessen,* " the percentage of decrement in the conducting power of an impure metal, between 0. and 100 C., is to that of the pure one, between C. and 100 C., as the conducting power of the impure metal at 100 C. is to that of the pure one at 100 C." A numerical example will best explain this law : Supposing we have two wires of the same metal, one of which is pure and the other impure, and we take such a length of each that they both have a resistance of 300 ohms at C. ; and suppose that the relative specific conductivities of the two kinds of metal are as 100 to 90. Now if we found that the pure sample increased its resistance from 300 ohms to 420 ohms, or * Phil. Trans., 1864, p. 167. 2 E 418 HANDBOOK OF ELECTRICAL TESTING. 40 per cent., when the temperature was increased to 100 0. ; then we should find that the impure sample when raised to 100 C. would have increased its resistance to 408 ohms, or 36 per cent., for 100 : 90 ::40:36. From this it is evident that the correction coefficients require to be Taried according to the purity of the metal, but if we know what the coefficients are for the pure metal, and also the specific conductivity of the metal, we can correct the coefficients accordingly. Let E be the resistance of both the pure and impure metals at a temperature , and E L the resistance of the pure metal at a temperature t and let K be the coefficient required to correct E to the latter temperature, that is, let E I= =EK. [1] Let E 2 be the resistance of the impure metal at the tempera- ture t lt and let /q be the coefficient required to correct E to this temperature, that is, let E 2 = -R Kl . [2] Also let C and C l be the specific conductivities of the pure and impure metals. Lastly, let p and p r be the percentages of increase in resist- ance of the two samples respectively, between the temperatures t and t r We then have the following equations : . ;. ^ = ^1100 " ' _.; [a] Pl = ^5?100. [4] and the proportion p-.^-.-.G-.C, or but from equations [3] and [4] we get CORRECTIONS FOB TEMPERATURE. 419 therefore , C R - R t G! ~ R - K 2 ' or, substituting the values of R x and R 2 , obtained from the equa- tions [1] and [2], we get G__ R - RK _ 1 - K 07 ~ R - R *i " r^ 5 therefore l-i = (!-'), that is As the specific conductivity of the pure metal is always taken as 100, the formula becomes For example. From Table IV., the correction coefficient for correcting from 45 to 75 (equal to 30 of difference of temperature) is 1-0657, for pure copper. What is the coefficient for copper whose conductivity is 96 per cent, of that of the pure metal ? 96 '1 = 1 + 100 (i-657-i) = i-063i. 490. Although strictly speaking the foregoing method of obtaining the proper coefficient for correcting for conductor resistance ought to be followed, it is but rarely that it is so. It is usually the case that a table is employed in which the co- efficients have values corresponding to a mean purity of copper, Table V. for example. Inasmuch as the variation from this particular mean value is never very great in practice, no error of practical importance is caused by the use in all cases of such a table. CORRECTIONS FOR INSULATION RESISTANCE. 491. The law of change of resistance by change of tempera- ture for Insulators is the reverse of that for Conductors, that is to say, increase of temperature diminishes their resistance, and vice versa. If, therefore, we obtain our coefficients from the formula 2 E 2 420 HANDBOOK OF ELECTEICAL TESTING. /E Coefficient == where E is the observed higher resistance at the lower tempera- ture, v the observed lower resistance at the higher temperature, n the number of degrees of difference between the two tem- peratures, and n^ the number of degrees of difference for which the coefficient is required, then we must divide by the coefficient when we require to find the resistance at the higher temperature, that at the lower being given, and vice versa. 492. The influence of temperature is very much greater on Insulators than on Conductors ; thus, whereas a wire containing 96 per cent, of pure copper only increased its resistance from 1000 ohms to 1030 ohms by an increase of 15 of temperature, a particular gutta-percha core increased its resistance from 1000 ohms to 9080 ohms by the same amount of decrease of temperature. The amount of the change of resistance by change of tem- perature which takes place in the case of insulating materials is dependent upon the quality of the latter, and, therefore, the correction coefficients- for the same can only be regarded as approximately correct. 493. The range of temperatures required in practice is not large. If we calculate coefficients for a difference of from to 45 it will usually be sufficient. 494. Tables of coefficients for copper and two kinds of gutta- percha, calculated on the foregoing principles, will be found at the end of the book. As it is usual in practice to correct all measurements to 75 Fahr., the coefficient corresponding to the number of degrees of difference between any particular tempe- rature and 75, is placed opposite that particular temperature. 495. It was pointed out on page 416 that if all the coefficients are worked out by the formula " log coefficient = ( 0009209) n" then in order to correct from a lower to a higher temperature it is necessary to multiply by the coefficient corresponding to the number of degrees of difference between the two temperatures, but to correct from a higher to a lower temperature we must divide; now, if in the latter case we employ the reciprocal of the coefficient, then we must multiply as in the first case. By taking advantage of this fact, in Tables IV. and V. the coefficients are so calculated that whether we have to correct from 100 down to 75, or from 32 5 up to 75, in all cases we have to multiply ; in Tables VI. and VII., in all cases we have to divide. If it should be required to correct up to any temperature CORRECTIONS FOR TEMPERATURE. 421 other than 75, then we must first ascertain the number of degrees of difference between the two temperatures, and then the coefficient opposite to the temperature corresponding to 75 minus the number of degrees of difference, will be the coefficient required. Thus, if we want the coefficient for correcting the resistance of a pure copper wire from 45 up to 60, then 60 - 40 = 20, and 75 - 20 = 55, and the coefficient corre- sponding to this temperature in Table IV. is 1 0434, which is the required coefficient. Should it be necessary to correct from 60 down to 45, then in this case the coefficient will be that corresponding to 75 -J- 20, or 95, the value of which is *9586. 496. The exact effect of temperature on Electrostatic Capacity has not, it is believed, been yet determined or published ; it is, however, very slight. DETERMINATION OF THE TEMPERATURE OP A WIRE BY CHANGE OF KESISTANCE. 497. By a reverse process to the foregoing we can tell what the temperature of a wire is, if we know what is its resistance at one temperature, and also its resistance at the unknown temperature. For all we have to do is to divide one resistance by the other, and note with what number of degrees of tem- perature the coefficient so obtained corresponds ; then this result shows the number of degrees the wire has above or below the temperature at which the wire was measured. For example. To take the case on page 415, we found that the wire had a resistance of 2064 ohms at 32, and at the temperature which we will suppose to be unknown, a resistance of 2262, then the coeffi- cient is -7 = 1 096, which corresponds to an increase of 43 2 the temperature of the wire is therefore 32 + 43-2 = 75-2. In this way, if we ascertained the resistance of a cable at a noted temperature before it was laid, and then measured its resistance after it was laid, we could tell the mean temperature of the sea by referring to the Tables. Or if we ascertained the resistance of the cable at two different temperatures before it was laid, then we could deter- mine its temperature after it was laid without the use of Tables ; thus from the formula m ?)-(!)*' 422 HANDBOOK OF ELECTRICAL TESTING, we get therefore therefore For example. The conductor resistance (R x ) of a cable at 60 Fahr. when lying in the tanks at the factory, was 2000 ohms, and at a temperature of 45 Fahr. the resistance (r) was 1941 6 ohms. When the cable was laid the resistance (R) was found to be 1961 ohms. What was the temperature of the sea ? BI = 60 - 45 = 15, 1961 . 2000 ' log !94F6 The temperature of the sea will therefore be 45 + 5 = 50 F. 498. It is very often the case in cable factories that two sections of the cable are in different tanks at different tempera- tures, as, for instance, when several miles of core are added on to the made-up cable in a colder tank. As the whole length must be tested in one section, it may be necessary to know what correction must be applied to the measured resistance of the whole length of cable to correct it to the value it would have at one uniform temperature. CORRECTIONS FOR CONDUCTOR RESISTANCE WHEN Two SECTIONS OF A CABLE ARE AT DIFFERENT TEMPERATURES. 499. Let Z L and Z 2 be the two lengths of the cable in the different tanks, also let r and r 2 be the respective conductor resistances of the two sections at the temperatures of the tanks, and let P c be the resistance of the two together. Also let CORRECTIONS FOR TEMPERATURE. 423 Tc l and 7c 2 be the coefficients by which r x and r 2 must be multiplied respectively in order to reduce them to the values they would have at one uniform temperature, and let E c be the total resistance of the cable at this uniform temperature ; we then have the following equations : PC = r, + r a , K c = r^ -f r 2 & 2 , [1] ?i = r i *a Z 2 r 2 & 2 therefore ^-f r a fe a _ p r i *i + r 2&2 . ' ~ IV r 1 and also we may say 77 <^i = ri*i> or, ty = 7 7 o-/2 = ^2^ 2 or, r 2 = where cr is a constant ; therefore If Z x and Z 2 are the lengths of the portions of the cable in knots, then the corrected resistance per knot (r,.) will be _1 -L _? -1 _L -1 7. ' 7. 7, ~f" 7. 71 1 ^2 K l K 2 For example. At a cable factory there were 15 knots (Z x ) of manufactured cable lying in a tank whose temperature was 50 Fahr. Con- nected to this cable were 5 knots (Z 2 ) of core in a tank whose temperature was 55 Fahr. The total observed conductor resistance of the 20 knots was 215 ohms (P f ). What would be the conductor resistance per knot (r e ) of the cable and core at 75 Fahr. ? From Table V. we have &! = 1-054, & 2 = 1-043; 424 HANDBOOK OF ELECTEICAL TESTING. therefore 15 r c = TT = = 11-30 ohms. lo o 1-054 ' 1-043 CORRECTIONS FOR INSULATION RESISTANCE WHEN Two SECTIONS OF A CABLE ARE AT DIFFERENT TEMPERATURES. 500. Let Z x and 1 2 be the lengths of the two sections, r and r 2 their respective insulation resistances at the temperature of the tanks, P f the combined resistance of the two sections, A^ and Jc 2 the coefficients by which r 1 and r 2 must be divided in order to reduce them to the values they would have at one uniform temperature, also let R f be the combined resistance of the two sections at this uniform temperature ; then we have the follow- ing equations : P - r i r 2 A ~ > rq -i K- r k 2 i\ 1 2 ^*2 1 *l therefore TT = I!' . !% Or > E < = P and also we may say - 7 i If c\\* i* r " 1 ' 2 "'I* U1 '2 ~ 7. or Z 9 o-Z 2 = j-^2, or, r-L - -~ where o- is a constant ; therefore K. - p. T 2 = p. CORRECTIONS FOE TEMPERATURE. 425 If ^ and 1 2 are the lengths of the sections in knots, then the corrected resistance per knot (r<) will be r, = For example. Taking the same lengths and temperatures as in the previous example, let us suppose the total observed insulation resistance of the 20 knots was 160 megohms (P ), what would be the insu- lation resistance per knot (r t .) at 75 Fahr., the insulator " Willoughby Smith's gutta-percha ? From Table VII. we find *! = 6-928, Jc 2 = 4-704, therefore / * f ' tr \ = 516-5 megohms. APPROXIMATE CORRECTIONS FOR COPPER AND INSULATION KESISTANCE WHEN TWO SECTIONS OF A CABLE ARE AT DIFFERENT TEMPER- ATURES. 501. Instead of correcting each section from its actual to the required temperature, in the way shown, we could assume that the whole length had a temperature which was a mean between the two actual temperatures, and correct the resistances by the coefficients (both for conductor and insulation resistance) corre- sponding to that mean temperature. This mean temperature may be calculated approximately on the evident assumption, that its value is closer to the temperature of the longer length in proportion to the proportion which the longer length bears to the shorter length. Let us therefore have tj = temperature of length Z L ^2 = J> > ^2 t m = mean temperature ; then, assuming t to be the higher temperature, we have JO jO.jO / O . . 7 .7. *1 *m f m ~ h ' ' 1 2 l l > therefore /07 f O 7 _/07 __/07 "1 "1 .*ta *1 ^m "2 2 2 or 426 HANDBOOK OF ELECTRICAL TESTING, that is o _ For example. At a cable factory there were 1 5 knots (Z x ) of manufactured cable lying in a tank whose temperature was 50 (^) Fahr. Connected to this cable were 5 knots (Z 2 ) of core in a tank, whose temperature was 55 (^) Fahr. What was the mean tempera- ture (C) of the whole length? 50 x 15 + 55 X 5 1025 L ' 15 + 5 "2F As a rule the temperature coefficients are only given for degrees and half degrees, as in Tables IV., V., YI. and VII. ; conse- quently, in cases where the mean temperature works out to a fraction of a degree other than 5, we should take the coefficient of the degrees which are nearest in value to the calculated temperature ; this is usually sufficient for all practical purposes* In the above example it is obvions that the coefficient is a mean value between the coefficients for 51 and 51 5, consequently in such a case we can by a very simple calculation see what the exact (practically) coefficient would be. Thus, for example, the coefficients in Table V. for 51 and 51-5 are 1-051 and 1-050 respectively, consequently the coefficient for 51 25 is obviously 1-051 + 1-050 , or 1-0505. PRACTICAL APPLICATION OF CORRECTIONS FOR TEMPERATURE. 502. When a cable is in course of manufacture, the insulated conductor (or " core " as it is called), before being covered with the protecting sheathing, is placed in water heated to a tempera- ture of 75 F., and is kept in the same for a period of not less than 24 hours. By this lengthened immersion the core acquires the temperature of the water throughout its mass. Careful tests are then made. After the core has been sheathed it is coiled into a tank and kept covered with water at a normal temperature, and tests are made regularly every day to ascertain its condition. As regards the testing of the conductor, it was usually the practice to take a resistance measurement and then to correct the same to 75 by means of the coefficient corre- sponding to the temperature of the water in the tank in which the cable is coiled ; this corrected result, if the conductor were CORRECTIONS FOR TEMPERATURE. 427 in proper condition, would obviously correspond with the results obtained from the core at the 75 temperature. Now, owing to the slowness with which the gutta-percha covering conducts heat, unless a considerable time has elapsed after the immersion of the sheathed cable in the tank, the temperature of the water in the latter would not necessarily indicate the precise temperature of the conductor, consequently the actual tests at 75 and the corrected observed results might not correspond. The object of a conductor test, made in the foregoing manner, would of course be to ascertain whether the conductor had deteriorated in any way during the course of manufacture; experience has, however, shown that no such deterioration does take place, and that consequently a corrected conductor test, even if it were accurate, is practically of little value. At the present time, therefore, it is very usual to make use of the conductor test for the purpose of ascertaining the internal temperature of the core, so that a correct reduction coefficient may be applied to the insulation test of the insulating covering. The method of obtaining the temperature has been indicated in 497, page 421. In cases where the calculated internal, and the observed external, temperatures do not corre- spond, the reduction coefficient corresponding to the mean of the two should be taken for correcting the insulation resistance. When the manufacture of a cable is quite completed, and the latter has been allowed to remain in the tanks for 24 hours or more, the external observed, and the internal calculated, temperatures will generally correspond very closely. 503. In certain cases it happens that a length of cable may be coiled in two different tanks which are at different tempera- tures ; in such cases the temperature calculated from the con- ductor resistance is, of course, a mean of the two. 504. It should be remarked that the corrections indicated in paragraphs 499, 500 and 501, although not generally used, are still employed in some cases. ( 428 ) CHAPTEE XXII. LOCALISATION OF FAULTS OF HIGH RESISTANCE. FAULTS IN CABLES. 505. In all the tests for localising faults hitherto described, with the exception of the loop test ( 288, page 265), it has been assumed that the insulation resistances of the portions of cable on either side of the fault are infinitely great compared with the resistances of the conductor. Such an assumption practi- cally holds good in cases where the cable under test is short, and also if the resistance of the fault is small, but when we come to deal with long cables having faults of high resistance, the formulae we have obtained are no longer correct. The fol- lowing investigation * is made for the purpose of obtaining a test which shall be correct for cables of any length and having faults of any resistance : Let A B (Fig. 140) be a cable of any length, connected to a battery as shown, and having its further end to earth through FIG. 140. a resistance 5 Q = C + d Let the resistance of X A P = E XAQ = B-ME XAB=E n X A = E = o-. ,, ,, Also let resistance of unit length of conductor = r And sheathing = i. Calling E the electromotive force of the battery, then since the flow of electricity from any point to any other point close to it is from the point of higher to that of lower potential, and is equal to the difference of potential divided by the resistance separating the two points, therefore the current along A B at P is = C. rdx ~ r dx The resistance of the wire P Q is evidently r dx, because it varies directly as the length of the wire, but the resistance of the in- sulating sheath P Q is -= , because it varies inversely as the length. Hence the " leakage " or the current from the surface of the conductor between the points P and Q to the earth where the potential is zero, is i i dx Hence dO _ V dx ~ i 430 HANDBOOK OF ELECTKICAL TESTING, but c = r die' and, therefore, do; r die 2 therefore d 2 Y r V where m 2 - - i e w * The solution of this differential equation, obtained by the well- known method,* is V = A e* + B 6- BW , [1] and Now when a; = w V = V. = E, and = 0,, potential ,, . . , therefore, since resistance = current strength ' R V - E E " = c. = o. ; and similarly when x = Y = Yo, and C = C , and E = o- = -' Taking, therefore, x = n, we get E = Y n = A e nm -f B e~ m \ by [1] -Be ],by[2], * See Boole's ' Differential Equations,' Second Edition, Chapter IX., p. 194. LOCALISATION OF FAULTS OF HIGH RESISTANCE. 431 therefore K = 7T = ~ "'"4- B e -"'""[ '" - B e- J [3] Again, taking x = 0, we have A + B therefore and ^(A-B) ( TO \ T- + I TO 1 \ H. = 2 m -4 g- [4] [5] 506. Let us now see how we can apply this investigation so as to obtain a test which shall be strictly accurate for a cable of any length and resistance. The following, which accomplishes this, is in reality the fall of potential test given on page 386, with the formula corrected. FIG. 141. Earth Let b c (Fig. 141) be the cable having a fault z at c, x and y being the lengths on either side of the fault, and let R be a resistance. Now from equation [1] we have v = A e'" r + B e~ m * [6] 432 HANDBOOK OF ELECTRICAL TESTING. and at c, where x = 0, v x = A + B ; therefore, A = v x - B ; therefore v = or Now from [6] v - 2 B e' w = A e 1 "* - B e~ mx . therefore . [7] Now, calling K rt the resistance beyond 6, we have Y R + Rn v = E n therefore but from [3] r [-Ae- + Be "I K " ~ m LAe--Be J therefore y - v m |_ A e - therefore from [7] y _ v m V + e-" 1 *) - 2 Again, considering the portion of the cablefty, we have ji _ A! e my + B! e" OTy ^- A^B, ' and from [4] LOCALISATION OF FAULTS OF HIGH EESISTANCE. 433 In which, since the end of the cable is insulated, + e^^^). Multiplying the top and bottom of the equation by e m we get V-v _ _ ^ (v - v 2 e-"") + v - v 2 6"* ' therefore (V-y)-(v-v 2 e-"OE- from which * = In this form the formula cannot be practically used, as we require to know r and w, that is . /?, r being the resistance 2 P 434 HANDBOOK OF ELECTEICAL TESTING. per nnit length of the conductor, and i the resistance per unit length of the insulating sheathing. These we cannot deter- mine individually, for the measurement made when the end of the cable is to earth is not that of the conductor alone but that of the conductor diminished by the insulation resistance ; and similarly, when the end is insulated the measurement made is not that of the insulating sheathing alone but of the latter combined with the resistance of the conductor. If, however, we know what these measured values are we can substitute them in the formula in the place of m and r. Let the measured resistance of the cable when to earth at the further end be R e , and when insulated E,.; then This value of B we obtain from equation [5] (page 431)jby putting or = 0, and of E f by putting cr - oo . By multiplying one equation by the other, we get therefore g- 1 . no] r ~ VE7R, Also, we have therefore / Ti / "p / T> ^ ^ C e V JK.,- _ _ m j V ^e therefore e 2ml = ^~=^ ^- = ?. [11] that is 1 2lw i " LOCALISATION OF FAULTS OF HIGH KESISTANCE. 435 and also from [11] we have mi = A/E, V VB< E t . j VE< - v^ ' We have thus determined - , , e*" 1 , and e" mZ , and can consequently insert their values in any equations we may require. 507. Instead of employing the resistance E (Fig. 141, page 431), we may make the test by connecting the battery direct on to b through a galvanometer, so that the resistance E M of the cable can be measured by the ordinary deflection method ( 9, page 5). Then, since V : v : : E + E n : E n , therefore If we substitute this value of V v in equation [9] (page 433) we get For example. A cable 1000 knots (Z) long had a very small fault in it which was required to be localised. When the cable was good its resistance with the further end insulated, after five minutes' electrification, was 700,000 ohms (E t ), and its resistance with the further end put to earth, 5000 ohms (E e ). When the cable was faulty its resistance with the end insulated, after five minutes' electrification was 100,000 ohms (E n ). The potentials at the ends of the cable, after five minutes' electrification, were 300 (v) and 292 (v 2 ) respectively. What was the distance (x) of the fault from the nearer end of the cable ? 2 F 2 436 HANDBOOK OF ELECTEICAL TESTING. m 1 1_ r ~ V5000 x 700,000 ~ 59,161' 2m 1nfr y-7-00,600 + V5QOO Qn V 700,000 - V5000 TV? V 700,000 -j- V5000 = V 700,000- i _ /V 700,000 - V5000 _ e A V 700,000 - Inserting these values in the equation, we get 300+[300 - (292 x 1-0884)] 100,000 x x = 5902-1 x log 300 - [300 - (292 x -9188)] 100,000 X 5Q 161 X 2-3026 = 538 knots. 508. Since in the case of a small fault the difference between the potentials at the two ends is comparatively small it is essential that they should be measured with great accuracy, otherwise a small error made in determining their value will make a considerable error in the value of x. The readings on the scale of the galvanometer or electrometer must therefore be made as high as possible; it is even advisable to extend the length of the scale so that this may be done more effectually. 509. The relative values of the potentials at the two ends of the cable must be determined in the manner described in Chapter XVII. ( 449, page 387). LOCALISATION OF FAULTS IN INSULATED WIRES. WARREN'S METHOD.* 510. This method is adapted for localising faults of high resistance in lengths of cable core which have not been covered with the iron sheathing which forms the complete cable. The length of wire to be operated on is immaterial, provided that the whole or a portion of it can be coiled on an insulated drum, and that between the parts coiled the surface of the core * 'Philosophical Magazine,' No. 314, June 1879. LOCALISATION OF FAULTS OF HIGH EESISTANCE. 437 for a length of 6 or 8 inches can be cleaned and dried so as to prevent conduction. In the first case (when the whole can be coiled on a drum), one-half is coiled off on a second drum, the two drums being carefully insulated. The surface of the core between the drums is well cleaned and dried. The conductor is attached to one set of quadrants of an electrometer, the other set being to earth, and the two drums are connected to earth by an attendant at each drum ; one pole of the battery (whose second pole is to earth) is then connected to the conductor, so that the whole becomes charged; the battery is then disconnected from the electro- meter, and the earth-wires simultaneously taken off the drums. It is best to leave the battery on until the earth-wires are removed from the drums. The insulation of the drums and electrometer should be such that no loss can be perceived after a few minutes, when, if the earth-wire be applied first to one drum and then to the other, the fault will be found on that drum which causes the greatest fall in the electrometer. The wire is coiled from the faulty side to the other, and the test repeated as often as is required. A mile of core with a small fault in it can by a little practice be put right in an hour or two, involving no more waste than a portion of the insulator which can be held between the fingers, and without even cutting the conductor. The position of the fault, when it is obtained between the two drums, can be found more closely by cleaning and drying the surfaces on each side of it, and then touching the place where the fault appears to be, with the earth-wire, and seeing whether there is a fall in the electrometer. In the second case, where the bulk would prevent the whole from being insulated, we should continue to coil the core upon an insulated drum until the fault disappeared that is, until it was coiled on the drum. This is a useful method when dealing with " served core," that is core with its hemp covering only, at a cable factory. 511. By the foregoing method a joint may be tested with great ease by immersing it in an insulated trough of water, and putting the latter to earth, or even by simply touching the moist joint with the earth- wire. 512. The tests can be made with a galvanometer instead of an electrometer, although it is not such a sensitive arrange- ment. In this case the battery would be connected through the galvanometer to the conductor, as in an ordinary insulation test, and then the drums be connected to earth alternately, when the deflection of the needle shows on which drum the fault 438 HANDBOOK OF ELECTEICAL TESTING. exists ; but as the lengths on each drum may be very unequal, and consequently one drum may show a greater deflection simply in virtue of its having a greater length of core on it, the rush of current alone is not sufficient to enable the drum on which the fault is, to be found ; but by carefully watching the electrification, and seeing whether the fall is regular or not, no difficulty will be found in fixing upon the drum containing the fault. The battery-power required will vary with the magni- tude of the fault and the sensitiveness of the instrument, and can only be determined by experience and experiment. JACOB'S METHOD. 513. This method, which is a very satisfactory one, consists in winding the faulty core, no matter what the resistance of the fault happens to be, on to a drum or platform which requires to be only partially insulated, so that a wooden stand or even the floor is often quite sufficient ; the battery with one pole to earth is then applied direct to the conductor of the core the other end of which is insulated, the galvanometer is inserted between the drum and earth ; thus so long as the resistance between the drum and earth along the surface of the core or otherwise is not too small, so as to shunt the galvanometer too much, the current through the fault, if it be on the drum, will flow through the galvanometer to earth, but if the fault is not on the drum the current will pass direct to earth since the tank in which the rest of the core is coiled will be to earth ; the core will be then wound off or on accordingly until the fault is found. Often this fault will not be visible to the naked eye and the exact place can be electrically determined by passing the end of the wire leading from the galvanometer, over the surface of the core, while the battery is connected as above described. If two drums are used, the core on one will have its surface connected to earth, and that on the other connected to the galvanometer, or vice versa, so that it can be seen that the fault has not disappeared in coiling over. It should be added that in this test there is no necessity for cleaning the surface of the portion of core between the two drums, and that any description of core except that covered with a metallic sheathing,* can be so tested. * When a cable is to he laid in seas where the teredo worm abounds, it is now usual to cover the insulated core with a close-fitting lapping of thin brass tape. ( 439 ) CHAPTER XXIII. LOCALISATION OF A DISCONNECTION FAULT IN A CABLE. LOCALISATION OF A TOTAL DISCONNECTION. 514. The localisation of a total disconnection in a cable is a very easy matter. The conductor being broken inside the insulating sheathing, a battery connected to the end of the cable will charge the latter up as far as the fault only, conse- quently if we measure the discharge and compare it with the discharge from a condenser of a known capacity charged from the same battery, we shall obtain the capacity of the portion up to the fault. Also since the capacity per knot of the cable is always known, the observed capacity of the length in question, divided by the capacity per knot, will give at once the distance of the fault. LOCALISATION OF A PARTIAL DISCONNECTION. 515. Partial disconnection faults, although they are seldom met with in cables with gutta-percha cores, frequently occur in those whose insulating material is indiarubber. This arises from the elasticity of the substance ; for when any undue strain is put on the core the conductor breaks, but the indiarubber only stretches, and an earth fault is not made. When the strain is taken off, the two ends of the conductor come together and make contact more or less perfectly. If the break is noticed at the moment the cable is being laid from the ship, its position is of course known. But in some cases a fault of this nature does not develop itself until some time after the submersion; its locality can then only be found by testing. Such faults are difficult to localise, as none of the ordinary tests are applicable to them. The following method, however, devised by the author, is susceptible of considerable accuracy if carefully made. In Fig. 142 (page 440), B K represents the cable with its further end to earth ; R and r are the resistances of the portions of the cable on either side of the disconnection, and y is the resistance of the latter; a, b, and c are the three sides of a 440 HANDBOOK OP ELECTRICAL TESTING. Wheatstone bridge, of which the cable forms the fourth side ; g and g l are two galvanometers, the former being of the ordinary Thomson form, and the latter also a Thomson, but provided with heavy needles, so that its movements are very sluggish. FIG. 142. Connected to the battery, and also to the galvanometers, is a key of a peculiar description ; it is formed in two parts. The ordinary lever Jc of the key has its back-stop connected through the galvanometer g t to the junction of the resistances a, b ; thus when the key is in its normal position the galvanometer g l is connected to earth. The second portion of the key consists of a lever Z, to the underneath part of which is fixed the metal piece Zj, which is insulated from Z. Normally, as shown in the figure, Z x rests on a stop connected to one pole of the battery, the other pole of the latter being connected to B. The point P is connected permanently with l lt whilst the lever I is itself permanently connected to the galvanometer g. Now, the result of this arrangement is, that normally the battery is connected between the points B and P, and the gal- vanometer g^ is connected between the junctions of a and 6 and with earth, that is with the end B of the cable; the whole arrangement, in fact, forms an ordinary Wheatstone bridge. Now, if a, 6, and c are adjusted so that balance is produced, then the needle of the galvanometer g l will stand at zero ; if, when this is the case, the key k be depressed, g l will be discon- nected, and when the lever of k touches the end of Z, g will be put in circuit in the place of g l ; but immediately this takes place Zj will be lifted off its contact and the battery will be cut off; exactly at this moment then the charges in the cable will discharge and divide themselves, portions flowing out at the further end and the other portions flowing out through g, a, and LOCALISATION OF A DISCONNECTION FAULT IN A CABLE. 441 fc, and thence through c to earth. A throw of the needle of the galvanometer will thus be produced. Supposing the key Jc to be in its normal position, so that the battery causes a current to flow through the cable, whilst the resistances a, 6, and c are so adjusted that the galvanometer g l is unaffected, then let V be the potential at the beginning, and v be the potential at any other point of the portion B D. If now the key be depressed, the charges in the cable repre- sented by the areas A B D C and E H K, will flow out at the two ends of the cable in proportions dependent upon the values of the resistances K, y, and r, and the combined resistances of a, b, g, and c. Let v dx be a differential part of the charge AB D C, then this portion will split, and the portions flowing out at the two ends of the cable will be inversely proportional to the resistances on either side of v dx ; thus the portion flowing out at B will be ' where K x is the combined resistance of a, 6, g, and c. Now . V:f>::B + y + r:B + y + r-aj, therefore K+y+r-x " V K + 2 , + r ' that is * - ( R + y + r - *) 2 v (B! + E + y + r) (K + y + r)' and the integral of this between the limits x = K and a; = will give the quantity Q' flowing through the galvanometer, that is Q ' = J V (B 1 + E + y + r) (B + y + r) ^ + B + 2f + r) (R + 2/ + r V (R + y + r) 3 - (y +T) " 3 * (B x + E + y + r) (R + y + r)' -l J 442 HANDBOOK OF ELECTRICAL TESTING. Similarly we should find that the quantity Q" flowing out from the portion r of the cable would be ,, = V _ r* _ ' 3 (B I + B + r + r)(B + y + r) 1 and therefore the total quantity Q flowing through the galvano- meter will be 0' i o" v (B + y + r)3 - (y + r)3 + "* Q rn 8 (B 1 + K + 2, + r)(R + 2 , + ,-)- Q - Now the total quantity Q x which the cable would take if its further end were insulated and the end B maintained at the potential V, would be Q L = V (R + r). Again, if/ be the capacity in microfarads of such a length of the cable as would have a conductor resistance of 1 ohm, then R -f- r) / will be the actual total capacity of the cable, and if be the charge held by a condenser of F microfarads capacity, also charged to the potential V, then Qi:Q a :: (B + r)/:F; therefore or v- F ' Substituting then this value of V in equation [1] we get O = Q sA (B + y + r)*-(y + r)3 + f3 * 3F B B Let R 4- y 4- f - L, therefore, y + r = L B ; B 4. r = L x , therefore, r = TJ I R. Substituting these values in the last equation we get 3F Q 2 / I^-LS- 3F " Q 2 / L, 3 + 3 R (L 2 - V) - 3 R 2 (L - LQ SF ' B LL LOCALISATION OF A DISCONNECTION FAULT IN A CABLE. 443 therefore therefore Tm-j-T^ 8QF(B 1 + L)L-Q./L 1 E^-E(L + L 1 )= - 3 therefore that is L + L! /(L + L L ) 2 SQ^YEi + L)L - Qa/I^ 3 ro1 -V - -SQ.AL-LO Now the quantity Q discharged at B will split between the resistances g, and a -J- 6, the quantity Q 3 passing through the galvanometer being from which -h Q = Qs--; The value of E x , the combined resistance of a, 6, therefore C& + c) a . a c 444 HANDBOOK OF ELECTEICAL TESTING, and therefore Q (B! + L) = QL [g (a + 6) + a (6 + c)] Substituting this value in equation [2] we get T. J_ T. B = T" 3Q 2 /(L-LO In which, as we have before stated, ' Should it be necessary to employ a shunt for the galvano- meter g, of the Jth value say, then the observed deflection will have to be multiplied by n in order to give the value of Q 3 , and also the value of g in the formula will be ;th of the actual resistance of the galvanometer. For example. In localising a partial disconnection in a cable by the fore- going test, the branches a and 6 of the bridge were made 100 ohms each, and balance was obtained on g when c was adjusted to 5000 ohms ; consequently since a and 6 are equal L = C = 5000 . The conductor resistance L x when the cable was perfect was 2000 ohms. The resistance of the galvanometer was 5000 ohms, but when the discharge was noted it was necessary to employ the T ^th shunt, so that in the formula we must put 5000 g = -jo- = soo. The discharge deflection observed on depressing the key was 248 divisions, therefore Q 3 = 248 x 10 = 2480. The discharge deflection Q 2 observed from a condenser of 1 microfarad capacity (F) when charged to the potential V was 202 divisions with no shunt, therefore Q 2 = 202. LOCALISATION OP A DISCONNECTION FAULT IN A CABLE. 445 The cable, having a conductor resistance of 10 ohms per mile, and an inductive capacity of 3 microfarad per knot, the capa- city in microfarads of such a length of the cable as would have . CJ a conductor resistance of 1 ohm, would be = '03 microfarad, that is /= '03; then \ /( "V" 3 X 2480 x p _ 5000 + 2000 (5000 + 2000) 2 100 2 " [500 (100 + 100) + 100 (100 + 5000)] 5000 - 202 x '03 X 2000 3 *3 x 202 X '03 (5000 - 2000) = 3500 - 2996 = 504 ohms. 516. In making this test practically, after c and Q 3 have been obtained the cable must be disconnected from the bridge, and a resistance equal to L be connected between B and F, the potential at the point B will then still be V, and further the galvanometer can be removed without altering this potential ; the condenser and galvanometer must then be joined up in the manner shown by Fig. 97, page 278, the wires, however, which in that figure are shown as connected to the battery, being connected in the present case to the points B and F, Fig. 142 (page 440) ; then the discharge obtained, multiplied by the shunt (if one is employed), gives Q 2 . 517. It will sometimes be found that the cable is traversed by an earth current. The effects of this may best be neutralised in the manner indicated on page 259, 276, Chapter IX., the compensating battery being connected between the cable and the point B, and adjustment effected with the lever 1 1 raised so as to cut the testing battery off; when the galvanometer g is unaffected the adjustment is correct, the lever 1 1 is then let down, and the test made as if no earth current existed. 518. As it would be a matter of considerable difficulty, if not of impossibility, to adjust the bridge balance with an ordinary Thomson galvanometer (g^ in consequence of the latter being greatly affected by slight changes in the earth current, a gal- vanometer with a heavy needle whose movements are very sluggish, and which is consequently unaffected by slight and 446 HANDBOOK OF ELECTEICAL TESTING. sudden changes of current, is necessary for the purpose. For measuring the discharge, however, a highly sensitive instru- ment (#) is necessary, which must be brought into use only at the exact moment required, since it is necessary that its needle be steady at zero at that time. By the arrangement of key shown in Fig. 142 (page 440), this object is completely effected, as the galvanometer g is only brought into use at the moment when the battery is cut off, and the cable discharged. ( 447 ) CHAPTEE XXIV. A METHOD OF LOCALISING EARTH FAULTS IN CABLES. LOCALISATION OF FAULT WHEN CABLE is NOT BROKEN. 519. This test is of the same nature as the foregoing, and possesses the advantage of having all the necessary observations taken simultaneously, and from one end of the cable only. In Fig. 143, E and p represent the resistances of the portions of the conductor of the cable on either side of the fault, and r represents the resistance of the fault itself. As in the previous FIG. 143. Eartfi test, a, &, and c are the three sides of a Wheatstone bridge, of which the cable forms the fourth side, and g and g l are two gal- vanometers, k I Z x is a key, the construction and working of which were fully described in the previous test (page 440), and which it is unnecessary to consider again here. Supposing the key to be in its normal position, then let V represent the potential at the beginning of the cable, v the potential at the fault and also at the further end H, and v the potential at any point between B and D. If now the key Jc is depressed, the charge in the cable, which is represented by the area A B C D, will flow out at B and at D in proportions dependent upon the values of the resistances r, E and the combined resistances a, b, g, and c. 448 HANDBOOK OF ELECTRICAL TESTING. Let v dx be a differential part of the charge. Then the portion of this which will flow out at B will be **- where E x is the combined resistance of , 6, g, and c. Now therefore therefore and the integral of this between the limits x - E and x - will give the total quantity q^ due to the charge A B D C, flowing out at B, that is : .jFT A o.+:-.?...* V &! + B + r) (B + r) V (E x + E + r) (E + V (E + r (B + r - ] Now besides the quantity q^ there will be a quantity q 2 flowing out at B, due to the charge represented by the area C D H I. Let this charge be q', then but V : v :: therefore therefore A METHOD OF LOCALISING EARTH FAULTS IN CABLES. 449 This quantity q' in flowing out at D will divide, the portion q 2 flowing along E and out at B, being 1 B! + E + r (B! + E + r) (E + r) Consequently the total quantity flowing out at B will be V (E + *Q 3 - r 3 r^ 3 '(B! + E + r) (E + r) + V (E x + E + r)(B + r) V CE + r) 3 - r 3 4- 3r = Q j- 11 Now if the cable had no fault in it, and its further end were insulated, and if it had been charged to the potential V, then the quantity Q w which the length B D would contain, would be represented by the equation Qi = YE, Again, if/ be the capacity in microfarads of such a length of the cable as would have a conductor resistance of 1 unit, then E/ will be the actual total capacity of the length B D ; and if Q 2 be the charge held by a condenser of F microfarads capacity also charged to the potential V, then therefore or Substituting the value of V in equation [1] we get _Q2/ (E + r)3-r3 + 3r*p w " 3F "(Ei + E-f r)(E+r)' or Let E - + r = L 2 a 450 HANDBOOK OF ELECTEICAL TESTING. and E+J> = L I; therefore r . = L - K and p - r = L! - L, or, p = L x - L -f r ; therefore (E + r) 3 - r 3 + Zr^p = L 3 - r 3 + 3 r 2 ^ - L) + 3r 3 = LS + Sr^Li -L) + 2r 3 = L 3 + 3(L - R) 2 ^ - L) + 2(L - E) 3 ; therefore (L - E)3 + Jk:il) (L - E). = B Also if Q 3 equals the quairtity discharged through the gal- vanometer, then by substituting this quantity and the combined values of a, 6, c, and #, to which 'R l is equal, in the manner shown on page 443, in the last chapter, we shall have (fr + c)]L JU. If in making the test it is found necessary to employ a shunt with the galvanometer when taking the discharge, then if the value of this shunt be -ith, we must multiply the observed deflection by n in order to obtain Q 3 , and also the value of g in the above equation will be -th of the actual resistance of the galvanometer. From the cubic equation [1] E has now to be determined; this can be done in the following manner : Dividing each side by (L x L) 3 , we get h ?(.LzY _ -o. h 2VL 1 -L/ 2^-L) 3 therefore /L-E , l'\ 3 - 3/L-E VT,. _L + 2/ " 4 1 L, -L 8 ^CL.-L)^ A METHOD OF LOCALISING EARTH FAULTS IN CABLES. 451 therefore /L-E _ A 3 _ 3 /L-E A 1 G VLj-L 2/ 4VL 1 -L + 2/ + 4 2(L 1 -L) 3 ~ that is, Now this equation is of the same form as the identity 4 COS 3 a 3 cos a cos 3 a = . If then we put f) f^\ (L 1 -Ly" 1 = CMl8g ' M we shall have L-E 1 _ or T *D /'T T \ f 1 *\ that is, E = L - (L x - L) (cos a - J). [4] 2 n So that, having worked out the numerical value of -= - 1, and ascertained in a table of cosines to what angle this corre- sponds, then the cosine of Jrd of this angle gives cos a, which value inserted in equation [4] enables the value of E to be obtained. For example. In localising a fault by the foregoing test, the two arms a and b of the bridge were made 100 ohms each, and balance was obtained on g when c was adjusted to 700 ohms; therefore L = 700 ohms. The resistance of the galvanometer was 5000 ohms, but when, the discharge was noted on it the T ^th shunt was inserted, so that g = - = 500 ohms. The discharge deflection observed on depressing the key was 350 divisions; therefore Q 3 = 338 X 10 = 3380. The discharge 2 G 2 452 HANDBOOK OF ELECTKICAL TESTING. deflection Q 2 obtained from a condenser of 1 microfarad capa- city (F) charged to the potential V was 106 divisions with the -jLth shunt ; therefore Q 2 = 106 X 10 = 1060. The capacity / of such a length of the cable as would have a conductor resistance of 1 ohm was 03 microfarad ; and lastly, the total conductor resistance L x of the cable when sound was 1100 ohms. Thus we have a = 100 I = 100 g = 500 c = 700 L = 700 L! = 1100 Q 2 = 1060 Q 3 = 3380 F = 1 / = -03 we then get _ 3 X 3380 X 1 [500 (100 + 100) + 100 (100 + 700)] 700 _ 100 X 1060 x '03 700 3 = 401,770,000 - 343,000,000 = 58,770,000; therefore 20 2 x 58,770,000 >ftq , ft o 1 = ,^ nn -^NO 1 = "8366 = cos 3 a (L x - L) 3 ~ (1100 - 700) 3 = cos of 33 13'; therefore the cosine of which is '9814; therefore E = 700 - (1100 - 700) (-9814 - J) = 507 ohms, which gives the distance of the fault. 520. It may be remarked that the foregoing test is an excellent example of one of those rare cases in which the solution of an equation of the third degree is practically required, and in A METHOD OF LOCALISING EAETH FAULTS IN CABLES. 453 which the application of trigonometrical formulae for the pur- pose is useful.* 521. Now the cosine of an angle can never exceed 1, and it will 2 sometimes be found, on working out the value of -rr - ^ 1, \L*i Li) that its value will exceed unity ; consequently in such a case R cannot be determined by the help of a cosine table, but some other method must be adopted. Let us determine this method. In equation [2] (page 451) let L-B 1 l_ L l -L + 2- y + 4y'> we then have or Let 20 . K . (LpnO- therefore a quadratic equation, from which y 3 can be determined in the ordinary manner. Thus 6 _K 3 . /K\ 2 K 2 1 therefore or and = J { [K + V^ - 1] + K - * See Todhunter's Trigonometry, Third Edition, Chapter XVII., page 202, = * { [K 454 HANDBOOK OF ELECTRICAL TESTING. so that we get K-L-CL-i-LHICK in which K= 20 i and For example. In making the test, suppose the following to have been the numerical values of the different quantities : a = 100 I = 100 g = 500 c = 900 L = 900 L! = 1100 Q 2 = 300 Q 3 = 1230 F = 1 / = '03 therefore 3 X 1230 X 1 [500 (100 + 100) + 100 (100 + 900)] 900 _ 100 X 300 x '03 900 3 = 538,000,000 - 729,000,000 = 9,000,000; therefore therefore 2 x 9,000,000 (UOO - 900)3 1-1.25- - 25* - 1 = '75; from this we get R = 900 - (1100 - 900) J (2* + -6* - 1} = 900 - ?5? {1-2599 + -7937 - 1 } = 795 ohms. A METHOD OF LOCALISING EAKTH FAULTS IN CABLES. 455 LOCALISATION OF FAULT WHEN CABLE is BROKEN. 522. In this case, referring to page 449, the quantity dis- charged at B when the key is depressed will be only ^i instead of q l + q 2 ; consequently equation [A], on the same page, will become 3F or (E , r y r3 _ 0,2 f and putting R + r = L, and r = L R , we get therefore therefore or and by substituting a, 6, c, ^7, and Q 3 , in the manner shown on page 443, we get 3 3Q 3 F[gr(q + 6) + q(6 + c)]L example. In localising a fracture in a submarine cable by the fore- going test, a and 6 were made 100 ohms each, and balance was obtained on g when c was adjusted to 700 ohms. The resistance of the galvanometer was 5000 ohms, but when the discharge was noted, the T Vth shunt was inserted, therefore 5000 g = - = 500 ohms. The discharge deflection observed on depressing the key was 456 HANDBOOK OF ELECTEICAL TESTING. 186 divisions, therefore Q 3 = 186 x 10 = 1860. The discharge deflection Q 2 obtained from a condenser of 1 microfarad capacity (F) charged to the potential V was 120 divisions with the T Vth shunt, therefore Q 2 = 120 x 10 = 1200. The capacity /of such a length of the cable as would have a conductor resistance of 1 ohm was 03 microfarad, then = 700 - 529 = 171 ohms. 523. A great merit in the foregoing methods of testing for faults lies in the fact that the two cable measurements can be made almost simultaneously; thus the moment balance is obtained on g l by adjusting c, at that moment the key is depressed, and the discharge deflection Q 3 noted on the galva- nometer g. The other measurement, viz. that from the con- denser, can be made at leisure. Thus after c and Q 3 are obtained, the cable must be disconnected from the bridge, and a resistance equal to c be connected between B and F, the potential at the point B will then still be Y, and further, the galvano- meter g can be removed without altering this potential; the condenser and galvanometer must then be joined up in the manner shown by Fig. 97, page 278. The wires, however, which in the latter figure are shown as connected to the battery, must in the present case be connected to the points B and F, Fig. 143 (page 447) ; then the discharge obtained, multiplied by the shunt (if one is employed), gives Q 2 . 524. Should earth currents be present when the test is about to be made, they may be neutralised in the manner explained on page 259, 276, in Chapter IX., and also at the end of the last chapter ( 517, page 445). 525. With reference to the foregoing test it should be men- tioned that Mr. J. Gott states that it is often possible to increase the resistance of the fault at the end of a broken cable to such an extent that practically the whole of the discharge may be obtained at the nearer end. For this purpose the charging battery should be of from 7 to 10 volts electromotive force, the zinc pole being connected to earth; the battery should be applied to the cable for some time before taking the discharge. The lower the resistance of the galvanometer consistent with a sufficiently high figure of merit (page 65), the better, as must be obvious. ( 457 CHAPTER XXV. GALVANOMETER RESISTANCE. 526. The question of what resistance a galvanometer should have in order that its figure of merit (page 65) may be high, involves several points, such as " the shape of the coil," " the diameter of the wire," &c. The determination of all these points, however, would be more useful for the purpose of finding what are the most economical conditions under which a gal- vanometer can be made, than (what is more to the purpose of the practical electrician) for showing how any particular gal- vanometer can be arranged so as to enable any particular test to be made with accuracy. The problem we have to solve in the latter case is as follows : Having given a galvanometer with a coil of a certain size, should thin or thick wire be on it in order that any parti- cular test may be made under the most favourable conditions ? Or supposing the coil to be divided into several sections, how should the latter be coupled up ? Eeferring to Fig. 144, which repre- sents a section of a galvanometer coil, let us direct our attention to the 4 turns of wire at A. If these 4 turns be joined up in one continuous length, then calling the resistance of each turn 4, their total resistance will be 4x4, or 4 2 . If, however, the 4 turns be coupled up for "quantity," then their joint resistance will be 1. If we suppose the total current flow- ing to be constant, then in the case where the 4 wires are joined up in one continuous length, the current makes 4 turns round the needle of the galva- nometer, its effect will therefore be equal to 4; but in the second case, v AA where the turns of wire are coupled up for " quantity," the same current only makes 1 turn round the needle, hence its effect can only be equal to 1. FIG. 144. 458 HANDBOOK OF ELECTRICAL TESTING. If instead of 4 turns we have 9 turns, then the relative values of the resistances when joined up in one continuous length, and when joined up for " quantity," will be as 1 to 9 X 9, or 9 2 , whilst the relative effect of the current on the galvanometer needle will be as 1 to 9. In the first case then, where the resistance was reduced 4 2 t imes, the effect on the needle was only reduced 4 times ; and in the second case, where the resistance was reduced 9 2 times, the effect was only reduced 9 times ; or, in other words, the effect varied directly as the square root of the resistance ; consequently for the whole of the galvanometer the effect varies directly as the square root of its resistance. If we replace the 4 wires at A by a solid wire of twice their diameter, then this wire, shown by the dotted lines, will have the same resistance as these 4 wires coupled up for " quantity," and its influence on the magnetic needle will be very nearly the same. As a matter of fact, the effect will be rather less, in consequence of the metal being differently distributed over the area which the 4 wires occupy. But inasmuch as the silk covering with which the wires are insulated is practically of the same thickness for large as for small wires, if the thick wire were wound on the coil the sectional area of that wire would actually be rather larger than the area of the small wires which it takes the place of, consequently we may withoutjany con- siderable error say that the effect varies directly as tj g. 527. This fact enables us to determine what should be the resistance of the galvanometer in order that any particular test may be made under the best possible conditions. Let us take the case of the Wheatstone bridge. On page 195 we obtained an equation which gave the strength of the current flowing through the galvanometer when equili- brium was very nearly produced, viz. : E x (a ^ b x) CQ ~ {9 ( + ) + ( d + *)) (r ( d + *) + d ( a + *))* This equation may be written (a + x){r(d + x) + d(a + x)} = ( _L_ )X K. We have shown that the effect of the galvanometer coil on the needle varies directly as the square root of the resistance of GALVANOMETER RESISTANCE. 459 the former. Its effect must also vary directly as tlie current passing through the coils, consequently the total effect M will where K is a constant dependent upon the shape of the coil, the magnetic strength of the needle, &c. We have to find what value of g will make M as large as possible, and this we shall do, since K K is constant, by finding 7* what value of g will make Jg + = as small as possible. Vg Now and this will be made a minimum by making J g %9 minimum, that is, by making therefore \fg = V& or > g = k; but , a(d + a) . (a + V) (d + x) and ^ - is the same as ^ - rrri - ^> when 6 a; = a a, and this expression is the joint resistance of the resistances on either side of the galvanometer; theoretically therefore we should make g equal to this quantity if we wish M to be as large as possible. This rule, however, although it shows what value g should have in order to make M an absolute maximum, is one which cannot well be strictly followed out. We should rather seek to determine to what extent the exact rule may be violated without seriously diminishing M. 460 HANDBOOK OP ELECTKICAL TESTING. Let us suppose g to be n times &, then we have for an absolute maximum n = 1, that is 1 KK M = 2 X ~7t' * V& Suppose, now, we make g nine times as large as &, that is, make n = 9, then we have KK 1 KK 9T In other words, although g is nine times as great as it should be for making M a maximum, yet M has only been reduced from - down to . Or, to put it in another way : supposing we -i 0*0 were making a bridge test, employing a galvanometer of the exact theoretical value for obtaining a maximum deflection, and supposing that having nearly obtained equilibrium, the deflec- tion of the galvanometer needle was 3 3 divisions, then, if the resistance of the galvanometer had been 9 times the theoretical value, the deflection would only have been reduced down to 2 divisions. It must therefore be evident that, unless we employ a gal- vanometer whose resistance very much exceeds the theoretical value, this resistance will practically be the one required. If it is necessary to draw a limit, we may say avoid making the resistance more than 10 times as great (or as small, as can also be shown) as the theoretical value. 528. It will be found that in all tests in which g has a parti- cular best value, an equation of the form can be obtained. 7c in fact is in reality the resistance external to the galvanometer, so that we have simply to find what this resistance is, and then make g as nearly as possible equal to it. ( 461 ) CHAPTER XXVI. SPECIFICATION FOR MANUFACTURE OF CABLE. SYSTEM OF TESTING CABLE DURING MANUFACTURE. 529. As soon as the laying of a new cable has been decided upon, and the route which it is to take has been selected, &c., the manufacture has to be commenced. The choice of the types of cable to be adopted, the lengths of the " shore ends," " inter- mediate," and " deep-sea " sections are purely matters of expe- rience and discretion with the engineers in charge of the work, and no satisfactory rules for general guidance can be laid down. When the description of cable has been settled upon, a speci- fication has to be drawn up, of which the following is a general specimen. 530. THE TELEGRAPH COMPANY AND TELEGRAPH WORKS. CONTRACT SPECIFICATION for the manufacture of the Submarine Telegraph Cable of the Telegraph Company, to be laid between the coast of _, near , and the Island of The following lengths of cable will be required : Actual distance, 480 knots (each being 2029 yards), or, in- cluding 10 per cent, slack,* 528 knots. A. Main cable 500 knots. B. Intermediate cable 11 C. Shore-end cable 17 CORE. The core of the entire length of cable to be as follows : Conductor. To be formed of a strand of seven copper wires of a conductivity of not less than 96 per cent, of pure copper * The amount of slack required will vary with the length of the cable and with the depth of water it is laid in. 462 HANDBOOK OF ELECTKICAL TESTING. according to Matthiessen's standard,* and weighing one hundred and seven (107) pounds per nautical mile (2029 yards). Insulator. The copper conductor to be covered with three coatings of the purest gutta-percha, a coating of Chatterton's compound being placed next the conductor and between each layer of percha. The insulator to weigh one hundred and fifty (150) pounds per nautical mile, making the weight of the con- ductor, when covered with the insulator, two hundred and fifty-seven (257) pounds per nautical mile. The insulation resistance of each coil to be not less than 250 megohms per nautical mile after having been kept in water, maintained at a temperature of 75 Fahrenheit, for not less than twenty-four consecutive hours, and after one minute's electrifi- cation. Each coil of insulated wire, before being placed in the tem- perature tank for testing, to be carefully labelled with the exact length of wire, the exact weight of copper, and the exact weight of insulator it contains. A margin of 4 pounds over or under the specified total weight (257 Ibs.) will be allowed, but the mean weight of the core for the whole cable must not be under the specified weight. The core during manufacture to be carefully protected from sun and heat, and kept under water. Joints. Every joint to be tested by accumulation, and the leakage from any joint during one minute not to be more than double that from an equal length of the perfect core. Notice to be given to the inspecting officer of the company when a joint is about to be made, so that he may test it. SERVING AND SHEATHING. Main Cable A. Serving. The insulated conductor to be served with the best wet-tanned Russian hemp to receive the sheathing as specified, and to be then kept in tanned water and not allowed to be out of water more than is necessary to feed the closing machine. Sheathing. The served core to be sheathed with fifteen gal- vanised iron wires, each 120 of an inch in diameter. The lay to be 10 inches, no loose threads of hemp to be run through the closing machine, and no weld in any one iron wire to be within six feet of a weld in any other wire. The sheathed core to be finally covered with three coatings of Bright and * See page 400, 479. SPECIFICATION FOK MANUFACTURE OF CABLE. 463 Clark's compound, a serving of tarred yarn made from the best Eussian hemp being placed between each layer of compound, each serving of yarn being laid on in contrary directions.* Intermediate Cable B. Serving to be similar in every respect to that on the Main Cable A. Sheathing to be generally similar to that specified for the Main Cabe A, but the iron covering to consist of ten galvanised iron wires, each '180 of an inch in diameter. The lay to be 10 inches. Shore-End Cable C. The shore-end cable to consist of Cable A complete, and further well served with the best wet-tanned Eussian hemp, and then sheathed with twelve galvanised iron wires, 300 of an inch in diameter. The lay to be 17 inches, no loose threads of hemp to be run through the closing machine, and no weld in any one iron wire to be within six feet of a weld in any other wire. The sheathed core to be finally covered with three coatings of Bright and Clark's compound, a serving of tarred yarn made from the best Eussian hemp being placed between each layer of compound, each serving of yarn being laid on in contrary directions. The completed cable as fast as it is made, to be passed into a tank of water and kept covered with water until shipped. A correct indicator to be attached to the closing machine, and the length of cable to be marked as agreed. QUALITY OF MATERIALS. The wire used in the Main Cable A to be of the best quality of homogeneous wire, galvanised, and having a tensile strength of 50 tons per square inch area, and 850 Ibs. as a minimum breaking strain on a length of 12 inches between the clamps. The wire must elongate not less than % per cent, before break- ing. It shall bend round itself and unbend without breaking. The joints in the homogeneous wires to be of the form decided upon by the company's and contractor's engineers, and, as far as practicable, no one joint to be within six feet of any other joint. * In the place of the tarred yarn and Bright and Clark's compound, two layers of tape, saturated with a mixture of rosin oil and pitch, each layer beirg wound on in contrary directions, are now frequently employed ; this gives ao excellent finish to the cable. 464 HANDBOOK OF ELECTKICAL TESTING. The iron wire to be used in Cables B and C is to be of the quality known as Best Best, free from inequalities, galvanised and annealed, and having a tensile strength of 25 tons per square inch of area. A margin of 5 per cent, will be allowed in weight, provided the average weight is as specified above. The wire for Cables B and C to be capable of being bent round a cj'linder four times its own diameter and unbent without breaking. No wire of brittle quality shall be put into the cables, and the engineers or their assistants shall have power to reject any hanks which break frequently in the closing machine, or are of unsatisfactory quality. No weld shall be made in the B and C cables within six feet of any other weld. The galvanising of the iron to bear four dips of one minute each in a solution of one part by weight of sulphate of copper and five parts of water. Each intermediate cable to be finished off with suitable tapers to be arranged to the satisfaction of the engineer of the company. TESTING ACCOMMODATION. A proper room and all necessary batteries and leading wires to be provided for testing the cable during the whole manu- facture. INSPECTION. The engineer of the company or his agents to have access to the works for inspecting and testing cable and all materials employed, and may reject all materials which are unsatis- factory. PENALTY. The whole of the cable to be completed on or before the time stated in the tender under a penalty of per cent. on the price for each day, or fraction of a day, after the said time, until the day the cable may be actually completed and ready for shipment. The manufacture may not be carried on at night without the written consent of the engineer of the company or his agent. The cable ship or ships are not to leave the wharf with cable on board until the cable has been thoroughly tested in all respects by the engineers or their assistants from the shore, and ample time after the shipment of the last mile to be allowed for this purpose. SYSTEM OF TESTING CABLE DURING MANUFACTURE. 465 SYSTEM OF TESTING CABLE DURING MANUFACTURE. 531. The tests made by the cable manufacturers, although systematic, are not as a rule quite so exact or lengthy as those made by the electrician representing the company for whom the cable is being made. The cable, once manufactured, passes out of the hands of the manufacturer, and the latter has no further interest in the matter ; whereas the company may require at any time to localise a fault, and the more precise the data they possess the more closely will they be able to determine the position of the defect. Besides, when a large number of cables are being made at once at the factory it would be impossible, without a very large staff, to make an elaborate series of tests for each cable ; whereas these can easily be made by the electrician and his assistants when there is only one cable to look after. The methods of working out the tests, and the forms employed for entering down the same, depend upon the individual opinion of the electrician in charge of the work, but the following will give a general idea of the course to be pursued : TESTS OF THE COILS. 532. The core of the cable is usually made in 2-knot lengths approximately, which are coiled upon wooden drums as manu- factured, and then placed in tanks of water heated to a tem- perature of 75 F. to be tested.* After being placed in the tank, the coils should remain there for at least twenty-four hours, so that they may acquire through- out their mass the necessary uniform temperature. At the end of this time the tests may be taken. Sheets A, B, C, and D are employed for entering all the details of the tests as they are made ; the more important of these details are then copied on to Sheet E. The working out of the tests of the coils and cable is shown on corresponding pages. The figures given are such as are often obtained in actual practice. The insulation resistances of the coils are very often considerably higher than those shown, but this entirely depends upon the time which elapses after the manufacture. * At the works of Messrs, Siemens & Co., Charlton, the coils are tested at two different temperatures, viz., at 75 and 50 F. 2 H 466 HANDBOOK OF ELECTEICAL TESTING. I 6 E 1 ,j| 1 I 3 1? : : ? ? rH g S. 3 + 2 r- 1 d >-> >-" e-a s. 1 ^ 1? ? : : S + | : : S g : j. 1 s 0? * rH (M Gq 1 1 II. O p. .TH rH rH to 5^ g rH GO i-H rH rH rH rH S O5 O CO IO IO .!> rH in . . . 2 co co t>- t^ co o o o o o rH i 1 rH rH rH I- ^i-t CO C5 CO CO -HO to to o to 2 i 1 ir ^ CO C^ *^ C^ CO EH & m -CO rH O * (M * ' 1 'S In Knots. 4 rH CN C- IO O O O O O O tl" 1 N rH (M CO rH 10 i - .CO fa *'**'* SYSTEM OF TESTING CABLE DUKING MANTJFACTUKE. 467 CALCULATIONS FOR SHEET (A). Copper. April 3rd. No. 1 Coil. log 213 = 2-3283796 log 1-9946 = -2998558 2-0285238 log of 106-79 No. 2 Coil. log 214 = 2-3304138 lo-2-0074 = -3026339 2-0277799 = log of 106 -60 No. 3 Coil log 215 = 2-3324385 log 2 -0069 = -3025257 2-0299128 = log of 107-13 No. 4 Coil log 214 = 2-3304138 log 1-9990 = -3008128 2-0296010 = log of 107 '05 No. 5 Coil. log 212 = 2-3263359 1-9990 = 3008128 2-0255231 = log of 106 -05 Gutta-percha. No. 1 Coil No. 2 Coil. log 298 = 2-4742163 log 302 = 2-4800069 log 1-9946 = -2998558 log 2 "0074 = -3026339 2-1743605 log of 149-40 2-1773730 = log of 150 -44 No. 3 Coil. log 304 = 2-4828736 log 2 -0069 = -3025257 2-1803479 = log of 151-48 No. 4 Coil. log 299 = 2-4756712 log 1-9990 = 3008128 2-1748584 = log of 149 -57 No. 5 Coil. log 296 = 2-4712917 log 1-9990 = -3008128 2-0174789 = log of 148-07 2 H 2 HANDBOOK OF ELECTKICAL TESTING. 1 atfg, fills CO l> *" CO -l ^i" i-H CM CO -* 4 CO 1 00 p^ SYSTEM OF TESTING CABLE DURING MANUFACTURE. 469 CALCULATIONS FOR SHEET (B). April 3rd. Conductor Resistance. No. 1 Coil. log 22-98 = 1-3613500 log 1-9946 = -2998558 1-0614942 = log of 11-52 log 106 -79 =2-0285238 3-0900180 = log of 1230-3 = 97'3 per cent, pure copper * ' No. 2 Coil log 23-05'P = 1-3626709 log 2-0074 = -3026339 1-0600370 = log of 11-48 log 106 -60 =2-0277799 3-0878169 = log of 1224-1 = 97-7 per cent, pure copper * No. 3 Coil log 23-01 = 1-3619166 log 2-0069 = -3025257 1-0593909 =3 log of 11-47 log 107-13 = 2-0299128 3-0893037 = log of 1228 '3 = 97 - 4 per cent, pure copper * No. 4 Coil. log 22-96 = 1-3609719 log 1-9990 = -3008128 1-0601591 = log of 11 -49 log 107 -05 =2-0296010 3-0897601 = log of 1229-6 = 97*3 per cent, pure copper * No. 5 Coil log 23-22 = 1-3658622 log 1-9990 = -3008128 1-0650494 _ log of 11-62 log 106 -05 =2-0255231 3-0905725 = log of 1231-9 = 97-2 per cent, pure copper * * Table II. See also page 409, 478. 470 HANDBOOK OF ELECTEICAL TESTING. Q fa H B 1 8 gj 1 I | ^iO I-H t- t^ CO t- O CO 1-- .gt^ t> l> t> t> <0 p ~ Ct .2 3~3 111 all- 2 go o "ScO C7i CO i-H O .> t> t> t- CO CO "O 1 3 Dischar Electrifia 1 s Is 3 S R S R > (M "0 11 i 5 10 o o 10 10 ^ CO CO CO t- CO >! g IS 111- | O 1 . CO * 05 on -ti t^- CO Oi O5 c05 Oi 05 C5 O O O) O) NJ ^TH > > 00 &( " 1-1 <* SYSTEM OF TESTING CABLE DURING MANUFACTURE. 471 CALCULATIONS FOR SHEET (C). April 3rd. Inductive Capacity. log 3= -4771213 log 1720 = 3-2355284 log 330 = 2-5185139 - G + S_ 5460 + 330,5790 6-2dll6db Q = ^f: = -QQA log 5790 = 3-7626786 2-4684850 No. 1 Coil log 167 *5 =2-2240148 2-4684850 1-7555298 log 1-9946= -2998558 I- 4556740 = log of -2855 No. 2 Coil log 169 -5 =2-2291697 2-4684850 T- 7606847 log 2-0074= -3026339 1-4580508 = log of -2871 No. 3 Coil log 169 -5 =2-2291697 2-4684850 1-7606847 log 2-0069 = -3025257 4581590 = log of -2872 No. 4 Coil. log 171 -5 =2-2342641 2-4684850 1-7657791 log 1-9990= -3008128 1-4649663 = log of -2917 472 HANDBOOK OF ELECTRICAL TESTING. CALCULATIONS FOR SHEET (C) continued. No. 5 Coil. log 168 -5 = 2-2265999 2-4684850 1-7581149 log 1-9990 = -3008128 1-4573021 = log of -2866 Percentage of Loss. No. I Coil No. 2 Coil 167-5 169-5 76-25 77-25 log 91-25 = 1-9602329 log 92-25 = 1-9649664 log 100 =2- log 100 = 2- 5-9602329 3-9649664 log 167 5 = 2 2240148 log 169 5 = 2 2291697 1-7362181 = log of 54 -48 1-7357967 = log of 54 -42 #0. 3 Coil. No. 4 Coil 169-5 75-75 171-5 78-75 log 93-75 = 1-9719713 log 92-75 = 1-9673139 log 100 =2* log 100 =2- 3-9719713 3-9673139 log 169 -5 =2-2291697 log 171 '5 =2-2342641 1-7428016 1-7330498 = log of 55-31 = log of 54-08 No. 5 Coil. 168-5 77-75 log 90-75 = 1-9578466 log 100 =2- 3-9578466 log 168 -5 =2-2265999 1-7312467 = log of 53- 86 SYSTEM OF TESTING CABLE DURING MANUFACTURE. 473 s 4 I I S e i EH INSULATION TESTS OP COILS AT 75 FAHR. s^ma* S a W daou TO 5 00 i^ IO 00 C1 O ' l SS?SSw?SS I " a? *r ? r ? S2 S S 2 2 | ajnuini puooaa Suunp ,, 10 *- la 1-1 * * o o * oo t- O Is 1 S 2 3 I " |S * v J |l > f . .-,i" . junqg o& - + -jsax o> aiojaq BIJOO raoaj ^naxinQ M j CONSTANT. a S^l 11 s ^ : s : s junqg ss r- t qil-w 'eraqo 000*01 t- |3 ' s s ' i Nt 00 S e5 O & 1 S v* a Ifaj s ^J2 .... 3 ^ ^-~>*.- s 5 s s = s ga 81190 Jo aaqamtf i| s s r s snoojoq^aT |i I I I i *^ ^* W W ^H i-^ SIPOJO-OK -I 01 eo -T m 2 . w l| ^ S = = 474 HANDBOOK OF ELECTRICAL TESTING. CALCULATIONS FOR SHEET (D). April Mi. Insulation Resistance. log 17,000= 4-2304489 log 173= 2-2380461 1 9924028 = log of 98 27 = value of battery log 10,020 X 1000 = 7-0008677 log 152= 2-1818436 11-1751141 = log constant log 780 = 2-8920946 14-0672087 log 6240= 3-7951846 " _ 22 5460 + 780 _ 6240 780 780 No. 1 Coil 10-2720241 log 148= 2-1702617 8-1017624 = log of 126-41 megs. log 1 9946 = 2998558 8 4016186 = log of 252 1 megs. No. 2 Coil. 10-2720241 log 142= 2-1522883 8-1197358 = log of 131-75 megs, log 2 -0074= -3026339 8 4223697 = log of 264 5 megs. #0. 3 Coil 10-2720241 log 144 -5 = 2-1598678 8-1121563 = log of 129-47 megs, log 2 -0069= -3025257 8-4146820 = log of 259-8 megs. SYSTEM OF TESTING CABLE DURING MANUFACTURE. 475 CALCULATIONS FOR SHEET (D) continued. No. 4: Coil. 10-2720241 log 140 -5 = 2-1476763 8-1243478 = log of 133-15 megs, log 1-9990= -3008128 8-4251596 = log of 266-2 megs. No. 5 Coil. 10-2720241 log 138= 2-1398791 8-1321450 = log of 135-56 megs, log 1-9990= -3008128 8-4329578 = log of 271-0 megs. Percentage of Electrification. No. 1 Coil No. 2 Coil 148 142 137 130 log 11 = 1-0413927 log 12 = 1-0791812 log 100 = 2- log 100 = 2- 3-0413927 3-0791812 log 148 = 2 1702617 log 142 = 2 1522883 8711310 -9268929 = log of 7-43 = log of 8-45 No. 3 Coil No. 4 Coil 144-5 140-5 134 132 log 10-5 = 1-0211893 log 8'5 = '9294189 log 100 =2- log 100 =2- 3-0211893 2-9294189 log 144-5 = 2-1598678 log 140'5 = 2-1476763 8613215 -7817426 = log of 7-27 = log of 6-05 476 HANDBOOK OF ELECTEICAL TESTING. .CALCULATIONS FOR SHEET (D) continued. No. 5 Coil. 138 127-5 log 10-5 = 1-0211893 log 100 = 2- 3-0211893 log 138 =2-1398791 8813102 = log of 7- 61 SYSTEM OF TESTING CABLE DUKING MANUFACTURE. 477 I o a z? w 3 H . 02 I II I 5 g S, IsU S^^ P.H S.S II 1 NCE OF CTKIC. INDUCTIV CAPACITY. ajnuira paooas 2aunp uoijeogui $ uaddoo amj qjiM. iAiipnp 2 aj -i s ae 2i 10 C d co -3JS.2 ats SJ ait> t> CO rH IS S S ^ ^ ^ ^ il - rH rH C* t> i ^ ' ' i - CO t*- QO S'C It' t ^ *"( L *^ SYSTEM OP TESTING CABLE DUKING MANUFACTURE. 485 I jjj C3 I |s ll |3 So w -g la i| s *-> s lull ialss-i I2-S233 a' a.g HH ga H e] 1 i9 9 09 csi 13 s < 486 HANDBOOK OF ELECTRICAL TESTING. SECTION A. MAIN TABLE. INDUCTIVE CAPACITY TESTS OF CABLE. 1 "e3 a bp 02 /A CD 5J Tt< 7 V^ si & I o jil 05 * CO go co t- gOO O5 O5 vS |S ' Immediate Discharge after 10 sees. Electrification. 1 s 03 * is s s ^rH i-. li! s xf 5 ? CO SO 1 s s r C co 3 g^ S 5o o o Jw co o ri H Capacity ^ m.f. at" D *? * i 1 i ' r I ^ ^ ^ " J8!J3UIOaBArC) M O O JO s CD O O go! g |s IH - .co o co "o O5 O5 O5 O O5 O5 O5 _i-J CO O JfH ^ - - 1 - .CO t> 00 SYSTEM OF TESTING CABLE DURING MANUFACTURE. 487 I s H ^ 1 I 02 55 HP* |2 P (paqijTOqaon) eaoo (paqjTOqs) ^ 8IPOJO 11=1? - Signatiifft _ n 1!i"S . Slllflg o i!!Pi s ' 00 CO 00 Estimated Resistance of Cable from Tests of Coils at 75 F. IlPi? s OJE: " Q co 10 38*.* OQ P< CO 10 ,S . |! -a X | 85* I *"""* lifeS^ili- s S,^ ^3 Q 1-1 to n "o N I * o lift s 1? .?? Ijll s -g 00 i-H 10 Si 'coA. S rt *" CABLE. Deflection after L m d 2 mins. Electri- -J ' fication. g g 73 j; Suunp jijpaia 5 [uaojaa: Li Oi . V) <# PH iij- is : si , 1 s C^l ' C1 CO 73 Deflection after 1 min. Electrifica- tion. 1 3 ""* rH |H US' IS : il i s p o . o 10 |co 'coco I - O 488 HANDBOOK OF ELECTRICAL TESTING. o O g M : IT . 8 " 2 CQ g n io png Snunp Ci O T^ CO l> CO upnp 80UBJ8193H ut es^aao jo oSZ, o? paonpaa Qi o^ peonpaa ^ 0M3 10U3 aad Xip^d^o ^ 'goes 09 pu 8I3 -8398 01 agqjB esoq; jo SHOO jo aapjQ 9Anno98uoo eo CO 1 SYSTEM OF TESTING CABLE DURING MANUFACTURE. 489 THE_ MANUFACTURE .TELEGRAPH COMPANY. _SUBMARINE CABLE AT. Length -40 '32 knots. CABLE WOBKS. SECTION A. MAIN CABLE. FINAL TEST. June 3rd, 1884. Total observed. Total of Coils at 75. 450 17 ohms 463 68 ohms Conductor Resistance. Temperature. Observed. Calculated. 61i Fahr. 61 Fahr. Per Mile from Coils at 75. 11-50 Inductive Capacity. % m.f. Con- denser Dis. Immediate. Cable. Percentage of Loss. Total. Per Knot. After 1 min. 172 x 10 162 X 5520 + 15 144 X 552 + 15 11-1 ll'585m.f. '287 m.f. 15 lo Insulation Resistance. Constant. Battery 300 volts. G = 5520. 1 Cell thro. 10,000 + 20, S = ^ m 153 def. 1 Cell Dis. 172. Battery Dis. 182 x 552 + 20 S. on Cable, 560 ohms. 20 Time. Zinc to Line. Earth Reading. Copper to Line. Earth Reading. After 1 min. 267 82 300 66 2 233 53 264 40 3 219 42 250 28 4 213 35 241 22 5 207 30 234 18 6 204 229 7 201 225 8 199 222 9 197 218 10 195 215 11 193 212 12 191 210 13 189 208 14 188 206 15 187 205 All readings steady. Zinc to Copper Resistance per knot at nor-) Line, to Line.. mal temp, at min. Zinc to I Do. reduced to 75. not at nor-) end of IbU jine. . .) 1075 mege. Percentage of Electrification) between 1st and 2nd miu. / 370-8 ., Do. 1st and 15th min. . . 43-5 13-1 12-5 48'5 Signature 490 - HANDBOOK OF ELECTRICAL TESTING. CHAPTER XXVII. MISCELLANEOUS. TO DETERMINE THE TRUE INSULATION AND CONDUCTOR RESISTANCES OF A UNIFORMLY INSULATED TELEGRAPH LINE. 536. On page 6 it was pointed out that the rule of multiply- ing the total insulation by the mileage of the wire to get the insulation per mile was not strictly correct. Now, although the leakage on a telegraph line insulated on poles is really a leakage at a series of detached points, and not a uniform leak- age, as in a cable, yet practically, and especially in the case of long lines, it may be considered as taking place uniformly, and consequently the solutions of problems dealing with cables also apply with considerable accuracy to land lines. Wo may there- fore consider the case in question by the help of the equations we have obtained in the investigations made in Chapter XXII. On page 434 we have an equation [12] ! I and on the same page an equation [10] m 1 therefore 2m 2r by substitution and transposition we get lr = Since I is the length of the line, and r is the Conductor resistance per unit length, I r is the Total Conductor Resistance of the line, R e and R t being the respective total resistances of the line when the further end is to earth and when it is insulated. MISCELLANEOUS. 491 Again we have (page 430) therefore therefore by substitution and transposition we get Since Z is the length of the line, and i is the Insulation resistance per unit length, y is the Total Insulation Resistance of the line. / To get the per mile results, we must, of course, in the first case divide the total by the mileage, and in the second multiply it by the mileage. By expanding the logarithm we may obtain approximate simplifications of the foregoing formulae. We have But therefore therefore Todhunter's Algebra, 5th Edition, page 337. 492 HANDBOOK OF ELECTRICAL TESTING. and B, 5 3 E,. 45 VEV 945 \E If R t is not less than 5 times R e , then the formulae are correct within 1 per cent. If, however, E< is not more than 2 J times R e , then it would be necessary to take three of the terms given above in order to be correct within 1 per cent. In such cases the logarithmic formulas would probably be but little more laborious to work out, and would, of course, give exact results. 537. A direct means of ascertaining the Insulation Resistance per mile of an insulated wire is the following : As has been pointed out, on page 430, we have an equation where r is the conductivity resistance per unit length, and t the insulation resistance per unit length, of the line. Also on page 434 we have an equation r 2 where, as before, R e is the total resistance of the line when the further end is to earth, and E.- the total resistance when the end is insulated. By combining these two equations we have or T? e r \~\ = B, T . If we take the unit length to be a mile, then r being the true conductor resistance per mile, i will be the Insulation Resistance per mile. MISCELLANEOUS. 493 It will be seen that the mileage of the line does not come into the equation, this quantity being represented by What we do, in fact, in order to obtain the true Insulation per mile of a line, is to multiply the total resistance of the line when its end is insulated, not by the absolute total conductivity divided by the absolute conductivity per mile, which is the same thing as the mileage, but by the observed total conductivity (i.e. the total resistance of the line when its end is to earth) divided by the true conductivity per mile. For example. The resistance of a line, 200 miles long, when the further end was insulated was 4000 ohms. When the end was to earth the resistance was 2400 ohms. The absolute conductor resistance of the wire, at the time the measurements were being made, was known to be 16 ohms per mile. What was the true insulation per mile of the line ? i = 4000 X = 600,000 ohms, lo The value of i given by the ordinary rule would be i = 4000 x 200 = 800,000 ohms, a result 200,000 ohms, or 33 per cent., too high. 538. It must be evident that what is ordinarily called the conductor resistance of a line is really the true conductivity resistance diminished by the conducting power of the insulators. Conductivity resistance, therefore, in the case of a land line can only be measured accurately in fine weather, when the insula- tion is very high. To obtain, then, the value of r from equation [A] it would be necessary to take a conductivity test in fine weather, and to note the temperature at that time ; and then when an insulation test is made in wet weather, to observe the temperature, and from this correct the value of r previously obtained in the fine weather. 539. In the case of a submarine cable, the insulation resistance (when the cable is in good condition) is always so greatly in excess of the conductivity resistance that the true value of the latter is obtained at once by measuring the resistance of the cable when its end is to earth. Also the insulation per mile is practically equal to the total resistance when the end is insulated, multiplied by the mileage. 491 HANDBOOK OF ELECTRICAL TESTING. TESTING TELEGRAPH LINES BY EECEIVED CURRENTS. 540. The system of daily testing for insulation, described in Chapter I., page 6, and which was in general use on the lines- of the Postal Telegraph Department, has been superseded by a system of testing by received currents, which possesses many advantages over the old method of testing. Every day at a definite time, currents from batteries, each of an approximately definite electromotive force, are transmitted over the different lines, or sections of lines, and the strengths of the currents received at the further ends are measured. It is evident that the strengths of these currents will vary with ih& amounts of leakage on the lines, that is with the state of their insulation ; if then the battery power employed for transmitting the currents be constant, the strengths of the received currents observed from day to day will give an accurate knowledge of the condition of the lines. The way in which this general principle is practically carried out is as follows : Let A B (Fig. 145) represent the section of line to be tested,, then to each end of the latter, resistances, E, E, of 10,000 ohms- FIG. 145. T Ri _ 0000000' "T '(TCWOl? N 10,000 600 | 600 W.COO 320 each are connected, together with a galvanometer Gr (whose resistance is 320 ohms) and a battery E (whose resistance is also approximately 320 ohms), as shown. Although the sections- tested are not all of equal lengths or resistances, yet practically they are such that they may all be assumed to have a mean conductor resistance of 1000 ohms. Now it can be demonstrated mathematically that if the resistances E, E, are very great, then a " resultant " fault * / * See page 265, .288. MISCELLANEOUS. 495 (that is, the total insulation resistance of the line) will produce very nearly the same effect on the current received on the gal- vanometer G, whether this fault is at the middle, at the end, or at any intermediate point on the line. As a matter of fact the fault has the greatest influence when it is at the middle of the line, and the least influence when it is at either of the ends, but when the resistances E, E, are each about 10 times the con- ductor resistance of the line, then the difference in the two cases is practically very small. If then we assume, for convenience of calculation, that the resultant fault is at the middle of the line, we have f, = E /_ E/ H . B,/ *h + x where C r is the current received on the galvanometer G, E the electromotive force of the battery, and E x the total resistance on either side of the fault /. From this equation we get _ _ _ . E ' E _2' 0,- Kl STOTB, Now the battery from which the current is sent consists of 50 Daniell cells, and if we take the electromotive force of a Daniell cell to be 1 07 volts approximately, we have E = 50 x 1'07 = 53-5 volts. We also have E! = 320 + 10,000 + 500 = 10,820 ohms; therefore 53-5 2 10,820 x 10,820 X C r 10,820 . ohms, 00000045698 -.00018484 a where C f is measured in amperes. If now we so adjust the galvanometer G by means of the directing magnet, that one milliampere (nnnrth ampere) of 496 HANDBOOK OF ELECTKICAL TESTING. current gives a deflection of 80 divisions, then, if d be the de- flection given by any other current, we must have 80 x 1000 = d X ' 000125 amperes. From this last equation, then, we can obtain the strength (O f ) of the received current, in amperes, corresponding to any particular deflection; whilst from the previous equation, by inserting this value of C r , we can obtain the corresponding value of/, that is the total insulation resistance of the line. For example. Suppose d = 136 ; then C r = 136 X '0000125 = -0017 amperes; therefore / = -00000045698 ' = U ' 909 hm8 ' "00018484 or 11,900 ohms, 'approximately. 541. In order to save calculation, a table showing the values of C r and / corresponding to the various deflections (d), is pro- vided at each of the different test offices ; this table is arranged as on p. 497. 542. In order that the station transmitting the currents may be able to ascertain whether his 50-cell battery is in proper condition, he can test its electromotive force in the following way: The battery being joined up in circuit with the galvanometer and two of the 10,000 ohms resistances, the deflection is noted. Now if the 50 cells are in proper condition, their total electro- motive force would be 50 x 1-07 = 53-5 volts. Taking then the resistance of the battery to be 320 ohms approximately, and the resistance of the galvanometer being 1070 ohms,* the current deflecting the needle will be 53-5 x 1000 = 2*5012 milliamperes. 320 + 10,000 + 10,000 + 1070 * When this test is being made, the galvanometer resistance is 320 + 750 = 1070 ohms; the 750 ohms is a resistance, connected to the instrument, whose use will be explained in describing the latter. MISCELLANEOUS. 497 CD O II ooooooooo COlOt^-OCMlOGOC^-^H 00^ I- CO C0_ 0_ T* CO CO 04 co co co co co~ co~ co~ co* co~ no n 1U9JJT10 P9A1909H COlQCOCi IGOIOCO flCOOCOtNC< Therefore since the currents both affect the needle in the same direction, the joint effect will be 12 12 24 c + c = c = c 4-8. 5oo "We can therefore obtain degrees of sensitiveness in the proportions 3:4-8:9: 12 500 HANDBOOK OF ELECTRICAL TESTING. or 1:1-6:3:4. These relative values are, however, only approximate. The resistances of the wires are practically nil. An adjusting magnet (as shown in Fig. 13, page 22) is set on the upper part of the instrument. 544. In testing the strength of a current in milliamperes, the standard cell is connected to A and B, and both plugs are removed from the plug-holes ; there is then in circuit a total resistance of 1070 ohms, viz. 750 -f- 320. As the electromotive force of the standard cell is 1 07 volts, the resulting deflection of the galvanometer-needle (which is adjusted by means of the adjusting magnet to 80 divisions on the outer scale) will be due to a current of 1-07 = '001 ampere, or, 1 milliampere, and any other deflection obtained with any particular current, compared by direct proportion with the standard deflection, will give the strength of that current in milliamperes. When the standard deflection is obtained, the standard cell is removed and the circuit from which the received current is to be measured is connected to terminal A, terminal B being put to earth. In order to enable the oscillations of the needle to be checked as quickly as possible, a key is provided, which short circuits the instrument on being depressed. TO DETERMINE THE INSULATION EESISTANCE OF A LlNE WHEN THE STRENGTHS OF THE SENT AND KECEIVED CURRENTS ARE KNOWN. 545. The further end of the line being to earth, and I being the length of the line, we have from equation [2], page 430, by putting x = l t Current sent = C. = j A ^ - B e ' """I ; and from the same equation by putting x = 0, Current received = C, = - A B ; MISCELLANEOUS. 501 therefore C r A-B but from equation [4], page 387, we have m A " 7 r therefore C, O r by inserting the values of e m \ e ~ m \ and , given by equations [10] and [13], pages 434 and 435, we get r \/~~ == ^t ==("==+ i) /~/W = ( = + ^) 2VE.--B. cr VT> J^t The value of B,-, although it could be determined from this equation, would be represented by a somewhat complex fraction ; if, however, we have 0111 Ohms - We are not, of course, necessarily bound to use the -nny^th shunt, but in practice it would almost always have to be employed. 553. The degree of accuracy with which the test could be made would depend entirely upon the values of the deflections d and d 2 ; and as we should endeavour to make them both as high as possible, that is to say, both as nearly equal as possible, the S 900 "Percentage of accuracy" would practically be , where 8 *i is the fraction of a division to which each of the deflections could be read. THE SILVERTOWN COMPOUND KEY FOR CABLE TESTING. 554. This key, designed by Mr. J. Eymer Jones, and which is in general use in the testing rooms of the India Bubber, Gutta Percha, and Telegraph Works Company, Silvertown, is an excellent arrangement, and greatly facilitates the execution of the " Inductive capacity " and " Insulation " tests of insulated wires or of cables ; it is particularly useful when a large number of wires have to be tested. The apparatus (Fig. 152) consists of two keys, of the form shown by Figs. 102 and 103, pages 281 and 282, mounted on one base. Supposing the connections to be made as shown by the figure, then in order to measure the " discharge " from the cable, levers C and D are set in the positions shown. Lever B is now pressed to the left so that its projecting piece n comes in contact with lever A; the brass tongue of lever B is then in contact with b, so that the battery, whose zinc pole is joined to lever B, is connected to the cable. If now lever A is pressed over to the 510 HANDBOOK OF ELECTRICAL TESTING. right, then lever B is also moved and the tongue of the latter consequently leaves b whilst the tongue of A comes in contact with a, and thus puts the cable in connection with the galvano- meter. As the second terminal of the galvanometer is connected to the piece c d, the circuit is completed to earth through d and the tongue of lever D. To measure the discharge from a condenser, one terminal of the former would be connected to the piece a b and the other terminal to earth; the manipulation of the levers would of course be the same as in the case of the cable. FIG. 152. CMo To take the " Insulation" test (p. 368) of the cable, levers A and B would be set over to the right so that the tongue of lever A is in contact with a whilst the tongue of B is disconnected from b. The short-circuit key of the galvanometer being closed, lever C is now pressed over to the right, so that the tongue of lever C comes in contact with c, whilst the tongue of lever D becomes disconnected from d ; the zinc pole of the battery thus becomes connected through c with one terminal of the galvano- meter, and as the other terminal is connected (through lever A and a) with the cable, the circuit is complete. The short-circuit key of the galvanometer is now depressed, and the deflection MISCELLANEOUS. 511 noted in the usual manner (p. 369). As soon as the observa- tions are completed the short-circuit key of the galvanometer is raised, and lever D being pressed over to the left the batteiy becomes disconnected from the galvanometer terminal and the latter is connected to earth, so that the cable discharges itself. Particular care must be taken that the short-circuit key of the galvanometer is raised before lever D is pressed over to the left, otherwise the whole discharge from the cable will pass through the galvanometer coils, and the needles may either be demagnetised, or at least the " constant " of the instrument be altered. 555. The battery power with which the " Insulation " test is taken is much greater than that required for the " Inductive Capacity " test ; consequently after the latter test has been made (with about 10 Daniell cells usually), the battery power has to tie changed to the required larger amount. METHOD OF TESTING BATTERIES IN THE POSTAL TELEGRAPH DEPARTMENT. 556. One form of apparatus employed in the Postal Telegraph Department for battery testing is shown by Figs. 153 and 154. FIG. 153. FIG. 154. It consists of two sets of resistance coils R lf B 2 , the former being in the direct circuit of a tangent galvanometer * G-, and the latter being a shunt between the terminals of the battery x when the shunt plug S is inserted. The values of the resistance coils A, B, C, D, E and F, in B D are 1070, 3210, 4280, 8560, 17,120, and 34,240 ohms, respectively ; that is, A, B, C, D, E, and F are in the proportion of 1 : 3 : 4 : 8 : 16 : 32. * This galvanometer is the same as that employed for making the daily morning tests by received currents (Fig. 13, page 22, and Fig. 146, page 498). 512 HANDBOOK OF ELECTRICAL TESTING. Electromotive Force Test. 557. The principle of the method of testing for electromotive force is as follows : If the standard cell (page 137, 149) were joined tip in circuit with the tangent galvanometer, both plugs being out, then the deflection obtained would be that due to an electromotive force of 1 *070 volts (the approximate electromotive force of the standard cell) acting through a resistance of 1070 ohms. If, say, five Daniell cells were in circuit, and also a total resistance of 5 X 1070 ohms, then the deflection obtained should be the same as that given by the standard cell, provided the total electromotive force of the five cells were five times that of the standard cell, or, in other words, if the average electromotive force per cell were 1 070 volts ; and it is evident that if with a still larger number of cells there were placed in circuit a total resistance as many times greater than 1070 ohms as there are cells to be tested, then if the average electromotive force per cell of the battery were equal to the electromotive force of the standard cell, the deflec- tion obtained would be the same as that given by the latter. If the deflection were less, it would show that the average electro- motive force per cell of the battery must be proportionately less. For example. Suppose the standard cell gave a deflection of 25, then if, say, a 30-cell battery with a total resistance in circuit of 30 x 1070, or 32,100 ohms, gave a deflection of 22, the average electro- motive force per cell of the battery would be 928 volts, thus, = -928,01* Now, if instead of the resistance in circuit being increased in exact proportion to the number of cells tested, it had been in- creased in a less proportion, then the deflection representing an electromotive force of 1*070 volts would be correspondingly higher. For example. If, when the 30 cells were tested, there were in the circuit, not 30 X 1070 ohms, but 12 x 1070 ohms, then the deflection which would indicate that the average electromotive force per cell of the battery is 1-070 volts would be 49^, thus, 30 5 tan 25 X ^ = '466 X = 1-165 = tan 49J. 12 2 MISCELLANEOUS. 513 If, therefore, the total resistance in the circuit of the battery tested is made equal to 1070 x number of cells tested X f , [A] and if 25 is the deflection given by the standard cell through a total resistance of 1070 ohms, then 49 J will be the deflection given by a battery whose average electromotive force per cell is 1-070 volts, and any deflection other than 49 J will (by pro- portion of the tangents of the deflections) represent the actual electromotive force per cell of the battery. For example. If the deflection obtained were 40, then the electromotive force per cell of the battery would be -767 volts, thus, tan 40 -839 1-070 x tan = 1-070 x 1-171 - -767. If the total resistance in the circuit of the battery tested is made equal to 1070 X number of cells tested x f , [B] then the deflections obtained will represent average electro- motive forces per cell which are double those which they represent when the resistance in circuit is that indicated by formula [A]. So that if formula [A] is applied when Daniell cells are tested, and formula [B] when Bichromates are tested, the range of deflections required in the two cases will be the same, since the electromotive force of a Bichromate battery is double that of a Daniell. 558. In order to facilitate calculation, tables constructed on the foregoing principles are employed ; portions of these tables are as shown : TABLE I. Number of Cells to be tested. Coils to be placed in Circuit in Rj. Daniells. Bichromates. Leclanches. 5 3 A 10 *5 6 B _^ 8 C ii> tt 10 A + C 20 10 12 B + C 25 .. 16 A + D 80 15 18 and 20 B + D 35 ... A + C + D 514 HANDBOOK OF ELECTRICAL TESTING. n tJ ^ 000 ^,0 H t" CO CO tO to 1 1 ; J2 "cS '! 1 g "8 1 A Number and Description of Cells tested. Leclanches. fe s" 10 " O rH CM * rtf CO t> CS " O 00 of co o o CS QO CO CO O rH 0 CO CO * I I rH CO 4l rH rH rH Daniells. 5 to 160. O CO -H tO co co Jto to r to to TH T^ ; CO rH t> co -cotoo MISCELLANEOUS. 515 The way in which these tables would be used would be as follows : The 25 constant deflection having been obtained correctly, the standard cell is removed from terminals B, and the battery to be tested joined in its place, resistances having been pre- viously inserted in resistance coils K L , according to Table I. For example, if 35 Daniells are to be tested, the resistances to be inserted would be A, C, and D. The two plugs in the galvano- meter must still remain out so that the resistance of the latter (1070 ohms) is included in the circuit. The deflection obtained being now noted, the electromotive force per cell of the battery is given by Table II. ; thus if the deflection is 45 J, the electromotive force per cell is 930, and the percentage of fall from the normal electromotive force is 13-09. 559. It will be observed that in the case of Leclanche batteries, the resistances to be placed in circuit and the deflections corre- sponding to the various electromotive forces, have to be taken in a somewhat different proportion from that adopted in the case of Daniell or Bichromate batteries, as the cells are made up in sets of 6, 8, and 10, and not in sets of 5, and moreover the normal electromotive force of a Leclanche is intermediate in value between a Daniell and a Bichromate battery ; the general principle, however, upon which the resistances and deflections are arranged is similar to that adopted in the case of the latter batteries. 560. The accuracy of the method of testing electromotive force depends upon the resistance of the batteries being small in pro- portion to the external resistance, and this is attained by making the latter very large, so as to reduce the error beyond sensible limits. Resistance Test. 561. This test is made by the "Diminished deflection shunt method" described in Chapter YL, page 133. The resistance Ej being very high, the resistance of the battery is given by formula [G], page 135, in the test referred to, that is to say we have /tanD \ x = E 2 ( - -; - 1 ) . 1 V tan d ) For example. If by the insertion of a shunt R 2 of 25 ohms, the deflection D of 45i were reduced to 23 (d), then resistance () of battery 2 L 2 516 HANDBOOK OF ELECTRICAL TESTING. would be 35-0 ohms, thus, 562. To facilitate calculation, a table giving values of ( j- 1 ) for various values of D and d, is employed ; hence tan d / it is only necessary to multiply the corresponding quantity by E 2 , and the result is the total resistance of the battery. 563. In exceptional cases where an odd number of cells have to be tested for electromotive force, i.e. a number which is not included in Table I., the resistances inserted in 'R 1 are those cor- responding to the number in the table next above the odd number; thus if 13 Bichromates are to be tested, the resistances corresponding to 15 cells, viz. B and D, are inserted in B 1 . The deflection obtained having been noted, the result corresponding to that deflection in Table II. is multiplied by the even number of cells and divided by the odd number, the result being the electromotive force per cell of the battery. 564. It may be remarked that the range of the apparatus is considerable, it being possible to test from 5 to 160 Daniell cells, or 5 to 80 Bichromate cells, with an equal degree of accuracy, and with equal facility. DIRECT BEADING BATTERY TESTING INSTRUMENT. 565. In order to simplify the method of estimating the electro- motive force and resistance of batteries, and lessen the time necessary for the tests, a new instrument has recently been devised by Mr. A. Eden, which has partially superseded the foregoing apparatus, and which obviates the necessity for any calculation, or any reference to Tables. The theory of the instrument is as follows : Electromotive Force Test. 566. The constant of the galvanometer is so adjusted by means of the controlling magnet that a deflection of 80 divisions * on the skew tangent scale (page 30) is obtained with the standard cell (page 137, 149) connected to the instrument; this deflection then represents the electromotive force of the standard cell. If now we place in circuit n cells, each having an electromotive force * This deflection is taken because it is found to be the highest that can be obtained with certainty on the instrument. MISCELLANEOUS, 517 equal to that of the standard cell, and at the same time we add sufficient resistance in the circuit to make the total resistance n times as great as it was originally, then the deflection will still be 80 divisions, provided the cells are each equal in force to the standard cell. If, instead of increasing the total resistance in circuit to n times its original value, we make it as great, then it is obvious that the deflection given by the n cells will be 100 divisions provided all the cells are in good condition. If the force of the cells is less than their normal value, then the deflection observed will be lower than 100, and the value of this deflection will obviously directly represent the percentage value of the force ; thus, if the deflection were 93 divisions, then this would mean that the cells have but 93 per cent, of their normal power. Since the total resistance in circuit with n cells is times 100 the resistance in circuit with the standard cell, and as this latter resistance is 1070 ohms (page 500, 543), therefore this total resistance is X 1070 = n X 876 ohms. In order that a similar standard deflection (100 divisions) may be obtained with Bichromate and Leclanche batteries, shunts and compensating resistances ( 568, page 519) are connected to the galvanometer when those batteries are being tested, so that the same resistance per cell in the case of the three kinds of batteries can be inserted in circuit. Resistance Test. 567. The general theory of this test is as follows : Let r (Fig. 1 55) be the battery which can be shunted by a shunt, s, and E a high resistance which can be shunted by a shunt, S, and let G be a galvano- meter of negligeable resistance. First suppose both the shunts to be discon- nected, then, since R is very large, the current, C, through the galvanometer will be E FIG. 155. E being the electromotive force of the battery. Next suppose the shunts to be both connected up, then from 518 HANDBOOK OF ELECTRICAL TESTING. equation [A], page 505, we can see that if S and E are very great compared with 8 and r, the current, C', through the galvanometer will be E _ __+_**. SB S + B If the currents in the two cases are the same, then we get E-JL E _ _J_r, B ~ SB S + B or or that is, B S or _ : Now r (the total resistance of the battery) is equal to the resistance per cell, r 1? multiplied by the number of cells of which the battery is composed; if, therefore, we make B directly pro- portional to the number of cells, as we do in the case of the electromotive force test, that is, if we make B = nE lf then we get _ S^B! or '.-I- From this it is obvious that if B x and S are constant quantities, MISCELLANEOUS. 519 then s multiplied by a constant will directly give the value of r L (that is, the resistance per cell of the battery), no matter what number of cells are being tested. In making the electromotive force test we insert in circuit a resistance n E x , E x being of such a value that (as we have seen) the galvanometer deflection is 100 divisions if the battery is in good order. The value which it is preferable to give to S is that equal to the highest value which n E x will have ; this in practice is the case when n is 60. As we have seen, the value of n E x is n x 856 ohms, that is, E d is 856 ohms, therefore the value of S should be 60 x 856, or 51,360 ohms. Since S is 60 times E 15 we get Ej 8 TI * S 60~E; = 60 ; that is to say, the resistance per cell of the battery being tested is -gVtJbL the resistance of the shunt. If therefore the resistances of which the shunt is composed are marked with values which are ^ih of their actual values, then these marked values will give at once the resistance per cell of the battery under rtest. The theoretical values for S and E are only applicable when the battery resistance is inappreciable in comparison with tho external resistance, and when the galvanometer is either of very low resistance, or double wound, so as to admit of one half of the coils being placed in circuit with E, and the other half in the shunt, S. 568. The use of a galvanometer which has a resistance of 320 ohms, and which is outside of the shunt, S, makes it necessary to compensate for the inaccuracies so introduced ; this is done by making 'R 1 equal to 428 instead of 856 ohms, also by making S equal to 25,200 instead of 51,360 ohms, and by shunting the galvanometer by a permanent shunt of 320 ohms, thus reducing the resistance of the instrument to 160 ohms. As a further compensation, when the two shunts are connected up a resistance of 28 ohms is cut out of circuit from the n E x coils. These compensations are not based on any strictly theoretical basis, but are a compromise which, it is found, reduces the general error to practical limits. 569. The joining up of the shunts in the latest form of the apparatus is effected by means of a plunger key, so that the actual manipulation for the resistance measurement consists ia adjusting the shunt, s, until it is found that the galvanometer de- flection remains unaltered when the key is depressed or raised. 570. It must be obvious that as the value of the shunt, s, is 520 HANDBOOK OP ELECTRICAL TESTING. practically the same for every size of "battery, the accuracy with which a test can be made varies according to the number of cells of which the battery is composed, but practicably sufficient accuracy is obtainable in all cases. 571. The actual form of the battery testing instrument embodying the foregoing principles, as most recently arranged, is shown by Fig. 156. In this fig., R are the resistances which FIG. 156. are inserted in circuit in proportion to the number of cells to be tested. B are the resistances for shunting the battery, h is a switch which can be turned to three different positions accord- ing as Bichromate (B), Daniell (D), or Leclanche (L) have to be tested, a is a plunger key which, on being depressed, con- nects up the shunts s and S (Fig. 155). c is a switch which, on being turned to the right, alters the connections in such a way that a half-deflection test * for resistance can be made as a check, if desired. COMBINED RESISTANCES. 572. PROBLEM Required the joint resistance of the resistances , 6, c, d, and g, between the points A and B (Fig. 157). If we call R the resistance of the combined resistances between the points A and B, then what we have to do is to obtain an equation of the form E RB = * This test follows from formula [G], page 135, if we put tan d = tan D for then r = S 2 , the S 2 in this case corresponding to the s in Fig. 155, page 517. MISCELLANEOUS. 521 Now it is obvious that the value of R can be in no way depen- dent upon the value of r, hence in order to simplify the problem we may assume r to be equal to 0. By Kirchoff's laws (page 503) we have the following six equations, showing the connection between the resistances , 6, c, rf, and #, the current strengths c it c 2 , c 3 , c 4 , c 5 , and c 6 , and the electromotive force E : [i] [2] [3] w [5] [6] C 4 - 6 - f 1 = C 3 + c 6 - c 2 = c, o Co 6 c (7 = c 3 d c t x c 6 g = FIG. 157. In order to determine the value of c 5 from these six equations we must first find the value of c x from, say, equation [1], and substitute this value in the other equations, thereby getting rid of c x ; again in like manner, if we find the value of c 2 from, say, equation [3], and substitute throughout, we get rid of c 2 , and so on. As it will be unnecessaiy to show all these substitutions, we shall confine ourselves to one or two only; thus from equation 1 [1] we have C ~ C - C = = C - C therefore we get c 6 - c 5 + C 6 C 2 = a c 2 a c 2 b c 6 g = c d - x - = . [2] [3] [4] [5] [6] 522 HANDBOOK OF ELECTRICAL TESTING. By continuing this process, we at length get c 5 a.- c 6 a - c 6 b - c 6 g - (a + &) -J^f- and therefore and 6 (& + ^ 3 + & a + & d ) = c 5 (6 a? + d a?) + E (d + a;) .. By dividing one equation by the other, c 6 is eliminated, that, is, we get ad + bd-\-bg + dg _ c 5 (a b + a d) - E (a + 6) fc^ + d.^ + fcaj + fedi ~ - c 5 (6 a? + d x) + E (d + x) 9 or E (d+x) (ad + bd+bg+dg) + (a + b) (bg+dg+bx+bd) By dividing the numerator and denominator of the fraction below the thick line by a + x, we finally get E 5 - g [(a + X )(b + d)] + g[(a + x) + (6 + d)] + (a + 6) (d + a?) that is to say, The combined resistance of the resistances, a, b, c, d, x,} and #, between A and B } g [(a + a?) (b + *)] + a b (d + a?) + d x (a + 6) ^ [(a + a?) + (b + d)] + (a + 6) (d + a?) It will be observed that if g = oo , that is to say, if we removes t^" t^* !> !> IT 1 * t*" t^ t^* !> t^ !> IT** t>* C^* t^* 00 00 GO GO GO GO GO GO GO GO OO OCOCOrHl>l>OCOCDOr-COCO(N>OCOOCt>t>COCOCOC0050505050000rHrHrHrH(?qoqcq(:v| Q r-< rH rH r-t rH rH i-H rH rH rH rH rH rH rH rH rH O-*COCrHlCC5 t-rHCOOHHOiCOGOOqCD^HlOOlTHCOCOt-rH rHCN(MCOCOCOT^HHlOOCOCDCOt>t^GOOOO^'_. ^ . ooooooooooooooooooo IOOOOIOOIOOOO1O OO(MiO: ft "* TABLES. 527 OOCOOOQOCOCOGOCC(X)GOCOCOCOOOCX)OOOOGOOOOOGOCOOOOOOOCOOO !OCDOOO(MlOl>OCOCDOCOl>r-(CDOOOCDrHl>COOCDCO ^C^^OOOOT^OrH^^^COCOTHOOCOT-tCOOOH 528 HANDBOOK OF ELECTRICAL TESTING. TABLE II.* KESISTANCE OF A KNOT-POUND of COPPER WIRE of various CONDUCTIVITIES, at 75 FAHB. Percentage of Con- ductivity. Resistance. Percentage of Con- ductivity. Resistance. Percentage of Con- ductivity. Resistance. Percentage of Con- ductivity. Resistance. 100-0 1196-7 97-5 1227-4 95-0 1259-7 92-5 1293-4 99-9 1197-9 97-4 1228-6 94-9 1261-0 92-4 1294-8 99-8 1199-1 97-3 1229-9 94-8 1262-4 92-3 1296-1 99-7 1200-3 97-2 1231-2 94-7 1263-7 92-2 1297-4 99-6 1201-5 97-1 1232-5 94-6 1264-0 92-1 1298-8 99-5 1202-7 97-0 1233-7 94-5 1266-4 92-0 1300-1 99-4 1203-9 96-9 1235-0 94-4 1267-7 91-9 1301-6 99-3 1205-1 96-8 1236-2 94-3 1269-1 91-8 1303-1 99-2 1206-4 96-7 1237-5 94-2 1270-4 91-7 1304-6 99-1 1207-6 96-6 1238-8 94-1 1271-8 91-6 1306-1 99-0 1208-8 96-5 1240-1 94-0 1273-1 91-5 1307-6 98-9 1210-0 96-4 1241-4 93-9 1274-5 91-4 1309-1 98-8 1211-2 96-3 1242-7 93-8 1275-8 91-3 1310-6 98-7 1212-5 96-2 1244-0 93-7 1277-2 91-2 1312-1 98-6 1213-7 96-1 1245-3 93-6 1278-6 91-1 1314-6 98-5 1214-9 96-0 1246-6 93-5 1280-0 91-0 1315-1 98-4 1216-2 95-9 1247-9 93-4 1281-3 90-9 1316-5 98-3 1217-3 95-8 1249-2 93-3 1282-7 90-8 1318-0 98-2 1218-6 95-7 1250-5 93-2 1284-0 90-7 1319-4 98-1 1219-9 95-6 1251-8 93-1 1285-4 90-6 1320-9 98-0 1221-1 95-5 1253-1 93-0 1286-8 90-5 1322-4 97-9 1222-4 95-4 1254-4 92-9 1288-1 90-4 1323-8 97-8 1223-6 95-3 1255-7 92-8 1289-4 90-3 1325-3 97-7 1224-9 95-2 1257-0 92-7 1290-8 90-2 1326-8 97-6 1226-1 95-1 1258-4 92-6 1292-1 90-1 1328-2 Kesistance of " statute-inile-pound " equals j-esistance of "knot-pound" multiplied by -752422. log -752422 = 1-8764614. * See page 409, 478. TABLES. 529 TABLE IV.* COEFFICIENTS for correcting the OBSERVED RESISTANCE of PURE at any TEMPERATURE to 75 FAHR., or at 75 to any COPPER WIRE TEMPERATURE. Tempera- ture in Degrees Fahr. Coefficient. Tempera- ture in Degrees Fahr. Coefficient. Tempera- ture in Degrees Fahr. Coefficient. Tempera- ture in Degrees Fahr. Coefficient. 100 9484 82-5 9842 65 1-0214 47-5 1-0601 99-5 9494 82 9853 64-5 1-0225 47 1-0612 99 9504 81-5 9863 64 1-0236 46-5 1-0623 98-5 9514 81 9874 63-5 1-0247 46 1-0634 98 9524 80-5 9884 63 1-0258 45-5 1-0646 97-5 9534 80 9895 62-5 1-0269 45 1-0657 97 9544 79-5 9905 62 1-0280 44-5 0668 96-5 9554 79 9916 61-5 1-0290 44 0679 96 9564 78-5 9926 61 1-0301 43-5 0690 95-5 9575 78 9937 60-5 1-0312 43 0702 95 9585 77-5 9947 60 1-0323 42-5 0714 94-5 9595 77 9958 59-5 1-0334 42 0725 94 9605 76-5 9968 59 1-0345 41-5 0736 93-5 9615 76 9979 58-5 1-0356 41 0748 93 9626 75-5 9990 58 1-0367 40-5 0759 92-5 9636 75 1-0000 57-5 1-0378 40 0771 92 9646 74-5 0011 57 1-0389 39-5 0782 91-5 9656 74 0021 56-5 1-0400 39 0793 91 9666 73-5 0032 56 1-0411 38-5 0804 90-5 9677 73 0042 55-5 1-0422 38 0816 90 9687 72-5 0053 55 1-0433 37-5 0828 89-5 9697 72 0064 54-5 1-0444 37 0839 89 9708 71-5 0074 54 1-0455 36-5 0851 88-5 9718 71 0085 53-5 1-0466 36 0862 88 9728 70-5 1-0096 53 1-0478 35-5 1-0873 87-5 9738 70 1-0106 52-5 1-0489 35 1-0885 87 9749 69-5 1-0117 52 1-0500 34-5 1-0896 86-5 9759 69 1-0128 51-5 1-0511 34 1-0908 86 9769 68-5 1-0139 51 1-0522 33-5 1-0920 85-5 9780 68 1-0149 50-5 0533 33 1-0932 85 9790 67-5 1-0160 50 0544 32-5 1-0943 84-5 9801 67 1-0171 49-5 0556 32 1-0955 84 9811 66'5 1-0182 49 0567 31-5 1-0966 83-5 9821 66 1-0193 48-5 0578 31 1-0978 83 9832 65-5 1-0204 48 0589 30-5 1-0990 See page 416. 530 HANDBOOK OF ELECTKICAL TESTING. TABLE V.* COEFFICIENTS for correcting the OBSERVED KESISTANCE of ORDINARY COPPER WIRE at any TEMPERATURE to 75, or at 75 to any TEMPERATURE. Temp. Fahr. Co- efficient. Logarithm. Temp. Fahr. Co- efficient. Logarithm. Temp. Fahr. Co- efficient. Logarithm. 100 9491 1-9772950 77 9958 1-9981836 54 1-045 0-0190722 99-5 9501 9777491 76-5 9969 9986377 53-5 1-046 0195263 99 9510 9782032 76 9979 9990918 53 1-047 0199804 98-5 9520 9786573 75-5 9990 9995459 52-5 1-048 0204345 98 9530 9791114 75 1-000 o-ooooooo 52 1-049 0208886 97-5 9540 9795655 74-5 1-001 0004541 51-5 1-050 0213427 97 9550 9800196 74 1-002 0009082 51 1-051 0217968 96-5 9560 9804737 73-5 1-003 0013623 50-5 1-053 0222509 96 9570 9809278 73 1-004 0018164 50 1-054 0227050 95-5 9580 9813819 72-5 1-005 0022705 49-5 1-055 0231591 95 9590 9818360 72 006 0027246 49 1-056 0236132 94-5 9600 9822901 71-5 007 0031787 48-5 1-057 0240673 94 9610 9827442 71 008 0036328 48 1-058 0245214 93-5 9621 9831983 70-5 009 0040869 47-5 059 0249755 93 9631 9836524 70 010 0045410 47 060 0254296 92-5 9641 9841065 69-5 012 0049951 46-5 061 0258837 92 9651 9845606 69 013 0054492 46 062 0263378 91-5 9661 9850147 68-5 014 0059033 45-5 064 0267919 91 9671 9854688 68 015 0063574 45 065 0227460 90-5 9681 9859229 67-5 016 0068115 44-5 066 0277001 90 9691 9863770 67 017 0072656 44 067 0281542 89-5 9701 9868311 66-5 018 0077197 43-5 068 0286083 89 9711 9872852 66 019 0081738 43 069 0290624 88-5 9722 9877393 65-5 020 0086279 42-5 070 0295165 88 9732 9881934 65 021 0090820 42 071 0299706 87-5 9742 9886475 64-5 022 0095361 41-5 072 0304247 87 9752 9891016 64 023 0099902 41 074 0308788 86-5 9762 9895557 63-5 024 0104443 40-5 075 0313329 86 9772 9900098 63 025 0108984 40 076 0317870 85-5 9783 9904639 62-5 1-026 0113525 39-5 077 0322411 85 9793 9909180 62 1-027 0118066 39 078 0326952 84-5 9803 9913721 61-5 1-029 0122607 38-5 079 0331493 84 9814 9918262 61 1-030 0127148 38 080 0336034 83-5 9824 9922803 60-5 1-031 0131689 37-5 082 0340575 83 9834 9927344 60 1-032 0136230 37 083 0345116 82-5 9844 9931885 59-5 1-033 0140771 36-5 084 0349657 82 9855 9936426 59 1-034 0145312 36 085 0354198 81-5 9865 9940967 58-5 1-035 0149853 35-5 086 0358739 81 9875 9945508 58 1-036 0154394 35 087 0363280 80-5 9886 9950049 57-5 1-037 0158935 34-5 088 0367821 80 9896 9954590 57 1-038 0163476 34 089 0372362 79-5 9906 9959131 56-5 1-039 0168017 33-5 091 0376903 79 9917 9963672 56 1-041 0172558 33 092 0381444 78-5 9927 9968213 55-5 1-042 0177099 32-5 093 0385985 78 9937 9972754 55 1-043 0181640 32 094 0390526 77-5 9948 9977295 54-5 1-044 0186181 31-5 095 0395067 * See page 419. TABLES. 531 TABLE VI.* COEFFICIENTS for correcting the OBSERVED EESISTANCE of " SILVER- TOWX" GUTTA-PERCHA at any TEMPERATURE to 75 FAHR. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. 100 1494 1 1744650 77 8589 1-9339572 54 4-937 0-6934494 99-5 1552 1909757 76-5 8922 9504679 53-5 5-128 7099601 99 1612 2074864 76 9267 9669786 53 5-327 7264708 98-5 1675 2239971 ! 75-5 9627 9834893 52-5 5-533 7429815 98 1740 2405078 75 1-000 o-ooooooo 52 5-748 7594922 97-5 1807 2570185 i 74-5 1-039 0165107 51-5 5-970 7760021* 97 1877 2735292 74 1-079 0330214 51 6-202 7925136 96-5 1950 2900399 73-5 1-121 0495321 50-5 6-442 8090243 96 2026 3065506 1 73 1-164 0660428 50 6-692 8255350 95-5 2104 3230613 i 72-5 1-209 0825535 49-5 6-951 8420457 95 2186 3395720 72 1-256 0990642 49 7-220 8585564 94-5 2270 3560827 71-5 1-305 1155749 48-5 7-500 8750671 94 2358 3725934 71 1-355 1320856 48 7-791 8915778 93-5 2450 3891041 70-5 1-408 1485963 47-5 8-093 9080885 93 2545 4056148 70 1-463 1651070 47 8-406 9245992 92-5 2643 4221255 69-5 1-519 1816177 46-5 8-732 9411099 92 2746 4386362 69 1-578 1981284 46 9-070 9576206 91-5 2852 4551469 68-5 1-639 2146391 45-5 9-422 9741313 91 2962 4716576 68 1-703 2311498 45 9-787 9906420 90-5 3077 4881683 67-5 1-769 2476605 44-5 10-17 1-0071527 90 3197 5046790 67 1-837 2641712 44 10-56 0236634 89-5 3320 5211897 66'5 1-908 2806819 43-5 10-97 0401741 89 3449 5277004 66 1-982 2971926 43 11-39 0566848 88-5 3583 5542111 65-5 2-059 3137033 42-5 11-84 0731955 88 3722 5707218 65 2-139 3302140 42 12-29 0897062 87-5 3866 5872325 64-5 2-222 3467247 41-5 12-77 1062169 87 4016 6037432 64 2-308 3632354 41 13-27 1227276 86-5 4171 6202539 63-5 2-397 3797461 40-5 13-78 1392383 86 4343 6367646 63 2-490 3962568 40 14-31 1557490 85-5 4501 6532753 62-5 2-587 4127675 39-5 14-87 1722597 85 4675 6697860 62 2-687 4292782 39 15-44 1887704 84-5 4856 6862967 61-5 2-792 4457889 38-5 16-04 2052811 84 5044 7028074 61 2-899 4622996 38 16-66 2217918 83-5 5240 7193181 60-5 3-012 4788103 37-5 17-31 2383025 83 5443 7358288 60 3-128 4953210 37 17-98 2548132 82-5 5654 7523395 59-5 3-250 5118317 36-5 18-68 2713239 82 5873 7688502 59 3-376 5283424 36 19-40 2878346 81-5 6100 7853609 58-5 3-506 5448531 35-5 20-15 3043453 81 6337 8018716 58 3-642 5613638 35 20-93 3208569 80-5 6582 8183823 57-5 3-783 5778745 34-5 21-74 3373667 80 6837 8348930 57 3-930 5943852 34 22-59 3538774 79-5 7102 8514037 56-5 4-082 6108959 33-5 23-46 3703881 79 7378 8679144 56 4-240 6274066 33 24-37 3868938 78-5 7663 8844251 55-5 4-405 6439173 32-5 25-32 4034095 78 7960 9009358 55 4-575 6604280 32 26-30 4199202 77-5 8296 9174465 54-5 4-753 6769387 31-5 27-32 4364309 See page 419. 532 HANDBOOK OF ELECTRICAL TESTING. TABLE VII.* COEFFICIENTS for correcting the OBSERVED KESISTANCE OP "WlLLOUGHBY SMITH'S " GUTTA-PEBCHA at any TEMPERATURE to 75 FAHR. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. Temp. Co- efficient. Logarithm. 100 1992 1- 2992893 77 8789 1-9439395 54 5-083 0-7061201 99-5 2057 3132343 76-5 9077 9579423 53-5 5-284 7229628 99 2125 3273589 76 9375 9719713 53 5-492 7397305 98-5 2194 3412366 75-5 9682 9859651 52-5 5-709 7565600 98 2266 3552599 IS 1-000 o-ooooooo 52 5-934 7733475 97-5 2340 3692159 74-5 1-039 - 0166155 51-5 6-168 7901444 97 2417 3832767 74 1-080 0334238 51 6-412 8069935 96-5 2497 3974185 73-5 1-123 0503798 50-5 6-665 8238002 96 2579 4114513 73 1-167 0670709 50 6-928 8406079 95-5 2667 4260230 72-5 1-213 0838608 49-5 7-201 8573928 95 2751 4394906 72 1-261 1007151 49 7-485 8741918 94-5 2841 4534712 71-5 1-296 1126050 48-5 7-781 8910354 94 2934 4674601 71 1-363 1344959 48 8-088 9078411 93-5 3030 4814426 70-5 1-417 1513699 47-5 8-407 9246410 93 3130 4955443 70 1-473 1682027 47 8-739 9414617 92-2 3232 5094714 69-5 1-531 1849752 46-5 9-084 9582771 92 3338 5234863 69 1-591 2016702 46 9-442 9750640 91-5 3448 5375673 68-5 1-654 2185355 45-5 9-815 9918903 91 3561 5515720 68 1-719 2352759 45 10-203 1-0087279 90-5 3678 5656117 67-5 1-787 2521246 44-5 10-606 0255516 90 3798 5795550 67 1-858 2690457 44 11-024 0423392 89-5 3923 5936183 66-5 1-931 2857823 43-5 11-460 0591846 89 4051 6075622 66 2-007 3025474 43 11-911 0759842 88-5 4184 6215917 65-5 2-086 3193143 42-5 12-382 0927908 88 4321 6355843 65 2-169 3362596 42 12-870 1095785 87-5 4463 6496269 64-5 2-254 3529539 41-5 13-378 1263912 87 4609 6636067 64 2-343 3697723 41 13-906 1432022 86-5 4761 6776982 63-5 2-436 3866773 40-5 14-455 1600181 86 4917 6917002 63 2-532 4034637 40 15-025 1768145 85-5 5078 7056927 62-5 2-632 4202859 39-5 15-618 1936254 85 5245 7197455 62 2-736 4371161 39 16-235 2104523 84-5 5417 7337588 61-5 2-844 4539296 38-5 16-876 2272695 84 5594 7477225 61 2-956 4707044 38 17-542 2440791 83-5 5778 7617775 60-5 3-073 4875626 37-5 18-235 2609058 83 5967 7757560 60 3-194 5043349 37 18-954 2777009 82-5 6163 7897922 59-5 3-320 5211381 36-5 19-702 2945103 82 6365 8037984 59 3-451 5379450 36 20-480 3113300 81-5 6574 8178297 58-5 3-587 5547314 35-5 21-288 3271349 81 6789 8318058 58 3-729 5715924 35 22-128 3449422 80-5 7012 8458419 57-5 3-876 5883838 34-5 23-002 3617656 80 7227 8589585 57 4-029 6051973 34 23-910 3785796 79-5 7480 8739016 56-5 4-188 6220067 33-5 24-853 3953788 79 7725 8878985 56 4-354 6388884 33 25-834 4121917 78-5 7978 9018940 55-5 4-526 6557145 32-5 26-854 4290090 78 8240 9159272 55 4-704 6724673 32 27-913 4458065 77-5 8510 9299296 54-5 4-890 6893089 31-5 29-014 4626076 * See page 419. TABLES. 533 TABLE VIII.* Of the MULTIPLYING POWER of SHUNTS EMPLOYED with a GALVANOMETER of 6000 OHMS RESISTANCE. Resist- ance of Shunt. Logarithm of Multiplying Power. Com- bined Resist- ance of Galva- nometer and Shunt. Resist- ance of Shunt. Logarithm of Multiplying Power. Combined Resist- ance of Galva- nometer and Shunt. Resist- ance of Shunt. Logarithm of Multiply- ing Power. Combined Resistance of Galva- nometer and Shunt. ohms. ohms. ohms. ohms. ohms. ohms. 1 3-7782236 1-0 75 1-9084850 74-1 950 8642618 818-3 2 3-4772660 2-0 80 1-8808136 79-0 1000 8450980 857-2 3 3-3012471 3-0 85 1-8548402 83-8 1100 8098626 929-6 4 3-1763807 4-0 90 1-8303769 88-7 1200 7781513 1000-0 5 3-0795430 5-0 95 1-8072508 93-5 1300 7493807 1068-5 6 3-0004341 6-0 100 1-7853298 98-4 1400 7231107 1135-7 7 2-9335581 7-0 110 1-7446450 108-0 1500 6989700 1200-0 8 2-8756399 8-0 120 1-7075702 117-7 1600 6766936 1263-2 9 2-8245619 9-0 130 1-6735185 127-2 1700 6560407 1324-7 10 2-7788745 10-0 140 1-6420488 136 1800 6368188 1384-6 11 2-7375504 11-0 150 1-6127839 146-3 1900 6188636 1443-1 12 2-6998377 12-0 160 1-5854607 155-9 2000 6020600 1500-0 13 2-6651493 13-0 170 1-5598348 165-3 2200 5713943 1609-7 14 2-6320441 14-0 180 1-5357118 174-8 2400 5440680 1714-3 15 2-6031444 15-0 190 1-5129244 184-2 2600 5195201 1814-0 16 2-5751878 16-0 200 1-4913617 193-6 2800 4973306 1909-1 17 2-5489296 17-0 220 4513719 212-2 3000 4771213 2000-0 18 2-5241753 17-9 240 4149733 230-8 3300 4499718 2129-0 19 2-5007578 18-9 260 3816024 249-2 3600 4259742 2250-0 20 2-4785665 19-9 280 3508099 267-5 4000 3979400 2400-0 22 2-4373224 21-9 300 3222193 285-7 4300 3793780 2504-8 24 2-3996737 23-9 330 2828939 312-8 4600 3625579 2603-7 26 2-3650572 25-9 360 2461628 340-4 5000 3424227 2727-3 28 2-3330239 27-9 400 2041200 375-0 5500 3182929 2883-1 30 2-3031961 29-9 430 1747574 401-2 6000 3010300 3000-0 33 2-2620237 32-8 460 1461280 428-6 6500 2840019 3120-0 36 2-2244554 35-8 500 1139434 4.61-5 7000 2688353 3230-8 40 2-1789769 39-7 550 0722867 508-0 7500 2552725 3333-3 43 2-1477999 42-7 600 0413927 545-5 8000 2430380 3428-6 46 2-1187276 45-6 650 0131744 582-1 8500 2319536 3517-2 50 2-0827854 49-8 700 9809755 617-6 9000 2218574 3600-0 55 2-0378646 54-5 750 9542425 666-7 9500 2126137 3677-4 60 2-0043214 59-5 800 9294189 705-9 10000 2041200 3750-0 65 1-9699189 64-3 850 9062704 744-5 10500 1962946 3818-2 70 1-9380892 69-2 900 8846085 782-6 11000 1890562 3882-3 * See page 375 ( 424). 534 HANDBOOK OF ELECTEICAL TESTING. TABLE IX.* Of the MULTIPLYING POWER of SHUNTS EMPLOYED with a GALVANOMETER of 10,000 OHMS RESISTANCE. Resist- ance of Shunt. Logarithm of Multiplying Power. Com- bined Resist- ance of Galva- nometer and Shunt. Resist- ance of Shunt. Logarithm of Multiplying Power. Combined Resist- ance of Galva- nometer and Shunt. Resist- ance of Shunt. Logarithm of Multiplying Power. Combined Resistance of Galva- nometer and Shunt. ohms. ohms. ohms. ohms. ohms. ohms. 1 3-0000434 1-0 75 2-1281838 74-4 950 1-0616905 863-6 2 3-6990569 2-0 80 2-1003705 79-4 1000 1-0413927 900-9 3 3-5230090 3-0 85 2-0742570 84-3 1100 1-0039303 982-1 4 3-3981137 4-0 90 2-0496487 89-2 1200 9700368 1061-7 5 3-3012471 5-0 95 2-0264827 94-1 1300 9391350 1140-4 6 3-2221092 6-0 100 2-0043214 99-0 1400 9107769 1228-1 7 3-1552059 7-0 110 1-9633585 108-8 1500 8846065 1304-4 8 3 0972573 8-0 120 1-9259993 118-6 1600 8603380 1379-3 9 3-0461482 9-0 130 1-8916660 128-3 1700 8377370 1453-0 10 3-0004341 10-0 140 1-8599100 138-1 1800 8166095 1525-4 11 2-9590848 11-0 150 1-8303747 147-8 1900 7967934 1596-8 12 2-9213396 12-0 160 1-8027737 157-5 2000 7781512 1666-7 13 2-8866208 13-0 170 1-7768721 167-2 2200 7439371 1803-3 14 2-8544796 14-0 180 1-7524753 176-8 2400 7132105 1935-5 15 2-8245597 15-0 190 1-7294206 186-5 2600 6853972 2063-5 16 2-7965743 16-0 200 1-7075702 196-1 2800 6600520 2187-5 17 2-7702888 17-0 220 1-6670282 215-3 3000 6368221 2307-7 18 2-7455085 18-0 240 1-6300888 234-4 3300 6053377 2481-2 19 2-7220708 19-0 260 1-5961741 253-4 3600 5772364 2647-1 20 2-6998377 20-0 280 1-5648351 272-4 4000 5440680 2857-1 22 2-6585137 22-0 300 1-5357159 291-3 4300 5218675 3007-0 24 2-6208299 23-9 330 1-4955864 319-5 4600 5015951 3150-7 26 2-5861544 25-9 360 1-4590573 347-5 5000 4771213 3333-3 28 2-5540563 27-9 400 1-4149733 384-6 5500 4499690 3548-4 30 2-5241796 29-9 430 1-3848158 412-2 6000 4259687 3750-0 33 2-4829169 32-9 460 1-3567739 439-8 6500 4045705 3939-4 36 2-4452582 35-9 500 1-3222193 476-2 7000 3853509 4117-6 40 2-3996737 39-8 550 1-2828898 521-3 7500 3679767 4285-7 43 2-3683950 42-8 600 1-2471546 556-0 8000 3521825 4444-4 46 2-3392354 45-8 650 1-2144362 610-3 8500 3377528 4594-6 50 2-3031961 49-8 700 1-1842858 654-2 9000 3245111 4736-8 55 2-2620194 54-7 750 1-1563472 697-7 9500 3123110 4871-8 60 2-2244467 59-6 800 1-1303338 740-7 10000 3010300 5000-0 65 2-1899004 64-6 850 1-1060108 784-1 10500 2905G46 5122-0 70 2-1579315 69-5 900 1-0831840 825-7 11000 28082G6 5238-1 See page 375 ( 424). TABLES. 535 TABLE X. STANDARD WIRE No. Diameters. .No. Diameters. Mils.* Differences. Millimetres. ! Mils* Differences. Millimetres. i 0,000,000 500 12-70 23 24 4 610 000,000 464 36 11-78 j 24 22 2 -559 00,000 432 32 10-97 I 25 20 2 -.508 0,000 400 32 10-16 j| 26 18 2 -457 000 372 28 9-45 27 16-4 1-6 417 00 348 24 8-84 28 14-8 1-6 -376 324 24 8-23 >! 29 13-6 1-2 345 1 300 24 7-62 \ 30 12-4 1-2 315 2 276 24 7-01 31 11-6 8 295 3 252 24 6-40 32 10-8 8 274 4 232 20 5-89 33 10-0 8 254 5 212 20 5^38 34 9-2 8 234 6 192 20 4-88 35 8-4 8 213 7 176 16 4-47 36 7-6 8 193 8 160 16 4-06 37 6-8 8 173 9 144 16 3-66 38 6-0 8 152 10 128 16 3-25 39 5-2 8 132 11 116 12 2-95 40 4-8 4 122 12 104 12 2-64 41 4-4 4 112 13 92 12 2-34 42 4-0 4 102 14 80 12 2-03 43 3-6 4 0914 15 72 8 1-83 44 3-2 4 0813 16 64 8 1-63 45 2-8 4 0711 17 56 8 1-42 46 2-4 4 -0610 18 48 8 1-22 47 2-0 4 0508 19 40 8 1-016 48 1-6 4 0406 20 36 4 914 49 1-2 4 0305 21 32 4 813 50 1-0 2 0254 22 28 4 711 * 1 Mil. = ^th of an inch. t This gauge is the only legal standard wire gauge for tlie United Kingdom. 533 HANDBOOK OF ELECTRICAL TESTING. S INDEX. ACCUMULATION joint test, Clark's, 402 Ampere, definition of, 1 Angle of maximum sensitiveness in galvanometers, 23, 78 Arc, multiple, 70 Astatic galvanometer, 18 B. BALANCE, "Wheatstone's (see Wheatstone bridge) Batteries, 283 , Clark's standard, 140 , De la Hue's 143 , Fleming's 139 , Muirhead's 141 , Post Office 137 , Wheatstone's 137 , Leclanche, 284 , Minotto, 283 , Electromotive force of, comparison of (see Electromotive force) , 1 and 100 cells, 300 - of low resistance, measurement of resistance of, 299 -, polarisation in, measurement of, 299 -, Kesistance of, measurement of, 113 , by condenser method, 295, 297, 298, 299 deflection method, 4 diminished deflection direct method, 130 shunt method, 133 electrometer method, 361 Fahie's method, 172, 175 half deflection method, 5, 113 ___ Kempe's method, 295, 299 Mance's method, 124, 127 538 HANDBOOK OF ELECTRICAL TESTING. Batteries, Resistance of, measurement of, by Muirhead's method, 297 Munro's method, 298 Postal Telegraph method, 511, 516 Siemens' method, 118 Thomson's method, 114 Wheatstone bridge method, 211 -, shunted, Pollard's theorem in, 505 Battery resistance, use of table for calculating, 516 testing apparatus, Eden's, 516 Bridge, Wheatstone's (see Wheatstone bridge) CABLES, completed, tests of, 370, 374 , compound, tests during laying of, 396 , conductor resistance of, method of measuring, 240 , corrections for effects of temperature on conductor and insulation resistance of, 414 , earth readings on, 370 , electrostatic capacity of, measurement of, 325 , faults in, localisation of (see Faults) -, final tests of, 479 , insulation of, measurement of, 368 , laying of, tests during, 396, 397, 399 , manufacture of, specification for, 461 , tests during, 465, 478 , single wire, tests during laying, 397, 399 Calibration or graduation of galvanometer scales, 46, 76 Capacity, electrostatic (see Electrostatic capacity) Cardew's method of measuring current strength, 305 Carey Foster's method of measuring low resistances, 228 Cells, standard, Clark's, 140 , De la Rue's, 143 , Fleming's, 139 , Muirhead's, 141 , Post Office, 137 , Wheatstone's, 137 Charge, loss of (see Potential, loss of) Chloride of silver battery, 143 Chrystal's standard ohm, 219 * Clark's accumulation joint test, 402 correction for condenser discharge, 289 electromotive force test, 181 fall of potential fault test, 377 method of eliminating earth currents, 259 INDEX. 539 Clark's standard cell, 140 Coefficient for effect of temperature on conductor resistance, 414 insulation resistance, 419 Coils, for core of cable, tests of, 465 , resistance, 10 , Dial pattern, 14, 192 , for cable testing, 12, 14, 192 , Post Office pattern, 13 , slide, 15 . , Varley's, 210 Combined capacity of condensers, 275, 523 conductivity resistance of parallel wires, 70 insulation resistance of parallel wires, 233 resistances, 520 Compensating resistances for galvanometer shunts, 71 Compound cables, tests during laying of, 396 key for cable testing, 509 Condensers, 273, 523 , battery resistance measured by means of, 295, 297, 298, 299 , connections for discharge from, 278 , corrections for discharge from, 289 , electromotive force measured by means of, 287, 300 , joint capacities of, 275, 523 Conducting power of copper, effect of temperature on, corrections for, 414 Conductivity resistance, by Wheatstone bridge, 231 , correction for effect of temperature on, 414, 422, 425 , elimination of effects of earth currents in measuring, 235, 237, 238, 259 , joint, of several wires, 70 of cables, method of measuring, 240 of three wires individually, 231 of two wires individually, by loop test, 269 per mile of telegraph lines, 490 -, specific, 408 Constant for measuring high resistances, 366 insulation resistances, 366 morning tests, 5 Copper resistance, Mathiessen's standard of, 409 wire, effect of temperature on resistance of, 414 , specific conductivity of, 408 , W. T. Glover's table of, 410 Correction for condenser discharge deflections, 289 loop test, 265 tangent galvanometer, 21, 34 Corrections for temperature, 414 , practical applications of, 426 540 HANDBOOK OF ELECTKICAL TESTING. Coulomb, definition of, 328 Cubic equation, example of practical use of, 452 Current, Resistance, and Electromotive force, between two points in a circuit, relation between, 292 Current strength, measurement of, 301 , by Cardew's differential method, 305 Kempe's bridge 308 difference of potential deflection method, 312 equilibrium 315 direct deflection method, 302 Siemens' dynamometer, 318 -, unit of. 1 Currents, earth, elimination of, effects of, in testing by "Wheatstone bridge, 235, 237 , in testing, 259 , received, table for calculating, 497 , testing telegraph lines by, 494, 500 D. DAILY or morning table for calculating, 8, 497 tests of land lines, 8, 494 Dead-beat galvanometer, D'Arsonval-Deprez's, 61 , Thomson's, 59 Deflections, galvanometer, degree of accuracy attainable in reading, 42 , method of reading, 41 De la Rue's standard battery, 143 Deprez-D'Arsonval dead-beat reflecting galvanometer, 61 Dial pattern of resistance coils, 14, 192 Discharge deflections, connections for measuring, 278 , correction for, 289 key, Kempe's, 278 , Lambert's, 280 Rymer Jones's, 281 F. C. Webb's, 278 test of joint by, 404 Disconnection, partial localisation of, in cables, 439 , total 439 Dynamometer, Electro, Siemens', 318 E. EARTH current, to eliminate, in testing, 259 , by Wheatstone bridge, 235, 237 INDEX. 541 Earth faults, a method of localising, 447 readings, on cable, 370 , table of, 8 , resistance of an, to measure, 233 Eden's battery-testing apparatus, 516 Electric lamps, method of measuring the resistance of and currents flowing through, 506 Electrification, 369 , influence of temperature on, 369 Electrodynamometer, Siemens', 318 Electrometer, Thomson's quadrant, 348 , fall of charge in cable by, 361 , gauge of, 352 , grades of sensitiveness of, 359 , induction plate of, 353 , measurements from an inferred zero, by, 362 , replenisher of, 351 , reversing key for, 354 , tests of joints by, 405 -, use of, 361 Electromotive force, Current, and Resistance, between two points in a circuit, relation between, 292 , measurement of, 137, 144 , by Clark's method, 180 equal deflection method, 146 resistance method, 144 Fahie's method, 175 Law's method, 287 _ Lumsden's or Lacoine's method, 155, 159 Poggendorff's method, 165 Postal Telegraph 512, 516 Wheatstone's 152 Wiedemann's 146 -, table for calculating, 514 -, unit of, 1 Electrostatic capacity, measurement of, 325 . , by direct deflection method, 325 divided charge method, 341 Gott's method, 339 Siemens' diminished charge method, 344 Siemens' loss of charge deflection method, 333 . Siemens' loss of charge discharge method, 327 542 HANDBOOK OF ELECTRICAL TESTING. Electrostatic capacity, measurement of, by Thomson's method, 335 , specific, 413 F. FAHIE'S method of measuring battery resistance, 172, 175 testing for faults in cables, 246 False zero, 238, 265 Farmer's key for galvanometer and battery resistance tests, 93, 118 Fault resistance, Kenelly's law of, 251 Faults caused by disconnection, localisation of, 439 , localisation of, 242 , by Clark's fall of potential method, 386 combined resistance and discharge test, 403 Fahie's method, 246 Jacob's deflection method, 253 Kempe's loss of current method, 256 Loop test, 259 Lumsden's method, 245 Mance's 249 Siemens' equilibrium method, 393 or Lacoine's equal potential method, 390 , in coils of insulated wire, Jacob's method, 438 -, Warren's 436 -, of high resistance, 428 Figure of merit of galvanometers, 65 Final tests of cables, 479 Fleming's standard cell, 139 Foster's, Carey, method of measuring low resistances, 228 G. GALVANOMETER deflections, degree of accuracy attainable in reading, 42 , method of reading, 41 astatic, 18 D'Arsonval-Deprez's dead-beat reflecting, 61 Gaugain's, 36 Helmholtz's, 36 Obach's, 37 sine, 19 tangent, 7, 19, 498 , best conditions for using, 28 , correction for, 21, 34 -, principle of, 20 Thomson's reflecting, 46 , dead-beat form of, 59" INDEX. 543 Galvanometer, Thomson's reflecting, Gray and March Webb's arrangement of, 52 , Jacob's transparent scale for, 55 -, lamp and scale for, 54 -, marine, 63 , portable form of, 54 , resistance of, 54 , scale for, 56 , Silvertown form of, 52 Galvanometers, angle of maximum sensitiveness in, 23, 78 , calibration or graduation of scale of, 46, 76 , figure of merit of, 65 for measuring currents, Post Office form, 49S , method of adjusting, 75 , resistance for best effect from, 457 , Eesistance of, measurement of, 79 , by deflection method, 3 diminished deflection direct method, 82 shunt method, 89 equal deflection method, 83 half 5,79 Phillips' method, 282 Thomson's 93, 98 , sensitiveness of, 66 -, shunts for, 59 Gaugain's galvanometer, 36 Gauge for electrometer, 352 Glover, W. T., table of resistances, etc., of copper wire, 410 Gott's electrostatic capacity test, 339 method of sealing up faults for testing, 456 proof condenser method of measuring resistances, 381 Gray, E. K., arrangement of reflecting galvanometer, 52 Gutta-percha, effect of temperature on resistance of, 369 , electrification of, 369 , specific inductive capacity of, 413 insulation of, 411 H. HALF-CHARGE, fall to, 383 Halving deflection, resistance of battery by, 5, 113 galvanometer, by, 5, 79 Helmholtz's galvanometer, 36 Hi-h resistances, localisation of faults of, 428 , by Jacob's method, 43S 544 HANDBOOK OP ELECTRICAL TESTING. High resistances, localisation of faults of, by Warren's method, 436 , measurement of, 5, 364 , by loss of potential, 380 , Gott's proof condenser method, 381 I. INDIARUBBEE, electrification of, 369 Individual resistance of three wires, 231 two 269 Induction plate of electrometer, 353 Inductive capacity (see Electrostatic capacity) , specific, 413 Inferred zero, 65, 362 Insulated wires, detection of faults in, by Jacob's method, 438 Warren's method, 436 Insulation, correction for effect of temperature on, 419, 424, 425 , joint, of several wires, 233 , measurement of, 5, 7 , by received currents, 494 tangent galvanometer, 8 transmitted and received currents, 500 - Wheatstone bridge, 233 -, of cables, 368 -, by Jacob's method, 375 , of two sections of wire, 234 , per mile of telegraph lines, 490 , specific, 411 , standard of, for land lines, 6 , table for calculating, 8, 497 J. JACOB'S fault test, 253 method of measuring insulation of cables, 375 transparent scale for reflecting galvanometers, 55 Jenkin's method of measuring high resistances, 362 Joint capacities of condensers, 275, 523 conductivity resistance of parallel wires, 70 insulation 233 Joints, testing of, at sea, 405 , by Clark's accumulation method, 402 discharge method, 404 electrometer 405 Warren's 436 Jolin's D'Arsonval-Deprez dead-beat reflecting galvanometer, 61 INDEX. 545 Jolin-Thomson rheostat, 16 Jones, Rymer, discharge key, 281 K. KEMPE, A. B., on the leakage of submarine cables, 428 Kempe's battery resistance test, 295, 299 current strength test, 295 discharge key, 278 loss of current fault test, 256 Kenelly's law of fault resistance, 251 Key, compound, for cable testing, 509 , discharge, Farmer's, for galvanometer and battery resistance tests, 53, 118 , Kempe's, 278 , Lambert's, 280 , for Thomson's capacity test, 338 , Rymer Jones', 281 -, F. C. Webb's, 276 -, reversing, 271 , for electrometer, 354 -, Pell's, 272 , short-circuit, 270 Kirchoff's laws, 156 , proofs of, 503 L. LACOINE'S or Lumsden's electromotive force test, 155, 159 Siemens' fault test, 393 Lambert's discharge key, 280 key for Thomson's capacity test, 338 Lamps, electric, method of measuring the resistance of and current flowing through, 506 Land lines, measurement of insulation of, 6 , standard of insulation for, 6 Laws' test for electromotive force, 287 Laying of cables, tests during, 396, 397, 399 Leading wires, elimination of resistance of, 241 Leclanche battery, 284 Loop method of measuring conductivity resistance, 231 test, 259 , correction for, 288 , individual resistance of two wires by, 269 , Murray's method, 260 , Varley's 263 , Phillips' method, 268 2 N 546 . HANDBOOK OF ELECTRICAL TESTING. Loss of current fault test, Kempe's, 256 Low resistance batteries, a method of measuring, 299 resistances, a method of measuring, 507 , measured by metre bridge, 213 , Carey Foster's method, 228 Thomson's bridge, 230 Lumsden's, or Lacoine's, method of measuring electromotive force, 155, 159 system of testing for faults in cables, 245 M. MANGE'S method of eliminating the effects of earth currents in conductivity tests, 237 testing for faults in cables, 249 resistance of battery test, 124 with slide wire bridge, 127 Manufacture of cables, specification for, 461 , tests during, 465, 478 Marine galvanometer, Thomson's, 63 Matthiessen's standard of copper resistance, 409 Maximum sensitiveness, angle of, in galvanometers, 23, 78 Merit, figure of, of galvanometers, 65 Metre bridge, 213 Mile, insulation per, of lines, 490 Milliampere, 495 Minotto battery, 283 Morning, or daily tests of land lines, 8, 494 Muirhead's battery resistance test, 297 standard cell, 141 Multiple arc, 70 Multiplying power of shunts, 69 Munro's battery resistance test, 298 Murray's loop test, 260 O. OBACH'S galvanometer, 37 Ohm, definition of, 1 , standard, 219 Ohm's law, 1 One cell, 283 , constant taken with, 364 P. PAEALLAX error in galvanometers, method of avoiding, 23 Parallel wires, joint resistance of, 70 INDEX. 547 Partial disconnection in cable, localisation of, 439 Pell's reversing key, 272 Phillips, S. E., method of measuring the individual resistance of two wires by loop test, 269 galvanometer resistance, 282 -making loop test, 268 Platinoid, use of, for resistance coils, 10 rheostat, 16 Poggendorff's method of measuring electromotive forces, 165 Polarisation in batteries, measurement of, 299 Pollard's theorem of a shunted battery, 505 Portable reflecting galvanometer, 54 Postal Telegraph Department, galvanometer used by, 498 , standard cell used by, 137 , standard of insulation adopted by, 6 , system of testing batteries, 515, 517 lines by received currents, 494, 500 -, Wheatstone bridge used by, 13 Potential, fall of, formulas for, 382 , measurement of, 284 = resistances by, 377 loss of, 380 , Gott's method, 381 -, Clark's test for fault by, 386 -, Siemens' equilibrium of, 393 equal, 30 Preece's fall of potential formula, 384 Proof condenser method of measuring resistances, Gott's, 381 Purity or conducting power of copper, effect of, on temperature corrections, 417 a. QUADRANT electrometer, Thomson's, 348 Quantity, unit of, 328 B. KECEIVED currents, table for calculating, 497 . -, testing telegraph lines by, 494, 500 Keflecting galvanometer (see Galvanometers) Keplenisher of electrometer, 351 Kesistance coils, 10 _ ., dial pattern, 14, 192 , , for cable testing, 12, 14, 192 . , Post Office pattern, 13 , slide, 15 548 HANDBOOK OF ELECTRICAL TESTING. Besistance, Current, and Electromotive force, between two points in a circuit, relation between, 292 , measurement of, by deflection, 3 half deflection, 5 fall of potential, 377 loss 380 , Gott's method, 381 substitution, 2 Wheatstone bridge (see Wheatstone bridge) -, unit of, 1 Besistances, combined, 520 , compensating, for galvanometer shunts, 71 : , high, measurement of, 5, 364 -, insulation, 368 -, joint, of several wires, 70 -, low, a method of measuring, 507 -, measurement by metre bridge, 213 -, Carey Foster's method, 228 Thomson's bridge, 231 Besultant fault, 265 Beversing keys, 271 , for electrometer, 354 , Pell's, 272 switches, 272 Eheostat, Thomson-Jolin, 16 Boberts, Martin, method of using metre bridge, 215 Bymer Jones' discharge key, 281 S. SCALE, and lamp, for Thomson's reflecting galvanometer, 54 , galvanometer, graduation or calibration of, 46, 47 , Jacob's transparent, for Thomson's reflecting galvanometer, 55 , Silvertown form of, ,, 58 -, skew, for tangent galvanometer, 30 Sealing up faults for testing, Gott's method, 456 Sections, two, of wires, insulation of, 234 Sensitiveness, angle of maximum, in galvanometers, 3, 78 , of galvanometers, 66 Short circuit keys, 270 Shunted battery, Pollard's theorem of, 505 Shunts, 67 , galvanometer, 59 , compensating resistance for, 71 , method of adjusting, 75 , multiplying power of, 69 INDEX. 549 Shunts, galvanometer, table of, 75 Siemens' battery resistance measurement, 118 electro-dynamometer, 318 , electrostatic capacity by loss of charge measurement, 327, 333, 344 , localisation of faults by potential, 390 -, or Lacoine's localisation of faults by potential, 393 , telegraph works, method of testing completed cable at, 375 transparent galvanometer scale in use at, 55 Silvertown compound key for cable testing, 509 galvanometer scale, 56 reflecting galvanometer, 52 Sine galvanometer, 19 Single wire cable, test during laying, 397, 399 . Skew scale for tangent galvanometer, 30 Slide resistance bridge, Varley's, 210 coils, 15 wire, or metre bridge, 213 , battery resistance by, 127 galvanometer resistance by, 98 Small resistances, a method ol measuring, 507 : , measurement by metre bridge, 213 , Carey Foster's method, 228 Thomson's bridge, 230 Smith, Willoughby, system of testing cables during laying, 399 Specific conductivity, 408 inductive or electrostatic capacity, 413 insulation, 411 measurements, 408 Specification for manufacture of cables, 461 Standard cell, Clark's, 140 , De la Kue's, 143 , Fleming's, 139 , Muirhead's, 141 , Post Office, 137 , Wheatstone's, 137 of copper resistance, Matthiessen's, 409 of insulation for land lines, 6 ohm, 219 Substitution method of measuring resistances, 2 Switches, reversing, 272 T. TABLE for calculating battery resistances, 516 electromotive forces, 514 insulation resistances, 8 550 HANDBOOK OF ELECTEICAL TESTING. Table for calculating insulation resistances and strengths of received currents, 497 Tangent galvanometer, 7, 19, 498 , angle of maximum resistance of, 23 ? best conditions for using, 28 , corrections for, 21, 34 , insulation resistance by, 8 , principle of, 20 , skew scale for, 30 Taylor, Herbert, galvanometer shunt tables, 375 Temperature corrections for conductor resistance, 414, 422, 425 insulation resistance, 419, 424, 425 -, effect on electrification, 369 -of cable determined by conductor resistance, 421 Theorem, Pollard's, of a shunted battery, 505 Thomson's bridge, 230 electrostatic capacity test, 335 method of measuring battery resistance, 144 galvanometer resistance, 93, 98 quadrant electrometer, 348 reflecting galvanometer, 46 , dead-beat form of, 59 , Gray and March Webb's arrangement of, 52 , lamp and scale for, 54 -, marine, 63 -, portable form of, 54 Thomson-Jolin rheostat, 16 Three wires, individual resistance of, 231 Two by loop test, 269 Transparent scale, Jacob's, for reflecting galvanometer, 55 TJ. UNITS, electrical, 1 V. VARLEY'S loop test, 263 slide resistance bridge, 210 Volt, definition of, 1 W. WARREN'S test for small faults in insulated wires, 436 Webb, F. 0., discharge key, 276 , March, arrangement of reflecting galvanometer, 52 Wheatstone bridge, 188 INDEX. 551 Wheatstone bridge, conditions for accurate measurements by, 192 , conductivity resistance by, 231 , insulation 233 , measurement by, when exact equilibrium cannot be obtained, 209 , of wires traversed by earth currents, 235 -, method of connecting up, 191 -, slide wire or metre, 213 -, used by Postal Telegraph Department, 13 -, Varley's slide resistance, 210 Wheatstone' s method of measuring electromotive force, 152 standard cell, 137 "Wiedemann's method of measuring electromotive force, 146 Willoughby Smith's system of testing cables during laying, 399 Wires, copper, specific conductivity of, 408 , temperature corrections for, 414, 422, 425 , individual resistance of three, 231 two, by loop test, 269 -, joint resistance of, 70 ZERO, false, 238, 265 , inferred, 65, 362 , skew, of tangent galvanometer", 30 LONDOX : PRINTED BY WILLIAM CLOWES AND SONS, LIMITED, STAMFORD STREET AND CHARING CROSS. 1887. 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Long-Span Railway Bridges, comprising Investiga- tions of the Comparative Theoretical and Practical Advantages of the various adopted or proposed Type Systems of Construction, with numerous Formulae and Tables giving the weight of Iron or Steel required in Bridges from 300 feet to the limiting Spans ; to which are added similar Investigations and Tables relating to Short-span Railway Bridges. Second and revised edition. By B. BAKER, Assoc. Inst. C.E. Plates, crown 8vo, cloth, 5^. Elementary Theory and Calcidation of Iron Bridges and Roofs. By AUGUST RITTER, Ph.D., Professor at the Polytechnic School at Aix-la-Chapelle. Translated from the third German edition, by H. R. SANKEY, Capt. R.E. With 500 illustrations, 8vo, cloth, 15^. The Builders Clerk : a Guide to the Management of a Builder's Business. By THOMAS BALES. Fcap. 8vo, cloth, i s. 6J, The Elementary Principles of Carpentry. By THOMAS TREDGOLD. Revised from the original edition, and partly re-written, by JOHN THOMAS HURST. Contained in 517 pages of letter- press, and illustrated with 48 plates' and 150 wood engravings. Fourth edition, reprinted from the third, crown 8vo, cloth, 12s. 6d. Section I. On the Equality and Distribution of Forces Section II. Resistance of Timber Section III. Construction of Floors Section IV. Construction of Roofs Sec- tion V. Construction of Domes and Cupolas Section VI. Construction of Partitions Section VII. Scaffolds, Staging, and Gantries Section VIII. Construction of Centres for Bridges Section IX. Coffer-dams, Shoring, and Strutting Section X. Wooden Bridges and Viaducts Section XI. Joints, Straps, and other Fastenings Section XII. Timber. Our Factories, Workshops, and Warehouses: their Sanitary and Fire-Resisting Arrangements. By B. H. THWAITE, Assoc. Mem. Inst. C.E. With 183 wood engravings, crown 8vo, cloth, gs. Gold : Its Occurrence and Extraction, embracing the Geographical and Geological Distribution and the Mineralogical Charac- ters of Gold-bearing rocks ; the peculiar features and modes of working B 2 CATALOGUE OF SCIENTIFIC BOOKS Shallow Placers, Rivers, and Deep Leads ; Hydraulicing ; the Reduction and Separation of Auriferous Quartz ; the treatment of complex Auriferous ores containing other metals ; a Bibliography of the subject and a Glossary of Technical and Foreign Terms. By ALFRED G. LOCK, F.R.G.S. With numerous illustrations and maps, 1250 pp., super-royal 8vo, cloth, 2l. I2s. 6d. A Practical Treatise on Coal Mining. By GEORGE G. ANDRE, F.G.S., Assoc. Inst. C.E., Member of the Society of Engineers. With 82 lithographic plates. 2 vols., royal 4to, cloth, 3/. I2J. Iron Roofs : Examples of Design, Description. Illus- trated with 64 Working Drawings of Executed Roofs. By ARTHUR T. WALMISLEY, Assoc. Mem. Inst. C.E. Second edition, revised, imp. 4to, half-morocco, 3/. A History of Electric Telegraphy, to the Year 1837. Chiefly compiled from Original Sources, and hitherto Unpublished Docu- ments, by J. J. FAHIE, Mem. Soc. of Tel. Engineers, and of the Inter- national Society of Electricians, Paris. Crown 8vo, cloth, gs. Scons' Information for Colonial Engineers. Edited by J. T. HURST. Demy 8vo, sewed. No. i, Ceylon. By ABRAHAM DEANE, C.E. 2s. 6d. CONTENTS : Introductory Remarks Natural Productions Architecture and Engineering Topo- graphy, Trade, and Natural History Principal Stations Weights and Measures, etc., etc. No. 2. Southern Africa, including the Cape Colony, Natal, and the Dutch Republics. By HENRY HALL, F.R.G.S., F.R.C.I. With Map. 3-r. 6d. CONTENTS : General Description of South Africa Physical Geography with reference to Engineering Operations Notes on Labour and Material in Cape Colony Geological Notes on Rock Formation in South Africa Engineering Instruments for Use in South Africa Principal Public Works in Cape Colony : Railways, Mountain Roads and Passes, Harbour Works, Bridges, Gas Works, Irrigation and Water Supply, Lighthouses, Drainage and Sanitary Engineering, Public Buildings, Mines Table of Woods in South Africa Animals used for Draught Purposes Statistical Notes Table of Distances Rates of Carriage, etc. No. 3. India. By F. C. DANVERS, Assoc. Inst. C.E. With Map. 4*. 6d. CONTENTS : Physical Geography of India Building Materials Roads Railways Bridges Irriga- tion River Works Harbours Lighthouse Buildings Native Labour The Principal Trees of India Money Weights and Measures Glossary of Indian Terms, etc. A Practical Treatise on Casting and Founding, including descriptions of the modern machinery employed in the art. By N. E. SPRETSON, Engineer. Third edition, with 82 plates drawn to scale, 412 pp., demy 8vo, cloth, 18*. PUBLISHED BY E. & F. N. SPON. 5 The Depreciation of Factories and their Valuation. By EWING MATHESON, M. Inst. C.E. 8vo, cloth, 6s. A Handbook of Electrical Testing. By H. R. KEMPE, M.S.T.E. Fifth edit-on, revised and enlarged, crown 8vo, cloth, i6j. Gas Works : their Arrangement, Construction, Plant, and Machinery. By F. COLYER, M. Inst. C.E. With 31 folding plates, 8vo, cloth, 24*. The Clerk of Works: a Vade-Mecum for all engaged in the Superintendence of Building Operations. By G. G. HOSKINS, F.R.I.B.A. Third edition, fcap. 8vo, cloth, is. 6d. American Foundry Practice: Treating of Loam, Dry Sand, and Green Sand Moulding, and containing a Practical Treatise upon the Management of Cupolas, and the Melting of Iron. By T. D- WEST. Practical Iron Moulder and Foundry Foreman.' Second edition, with numerous illustrations, crown 8vo, cloth, I CM. 6d. The Maintenance of Macadamised Roads. By T. CODRINGTON, M.I.C.E, F.G.S., General Superintendent of County Roads for South Wales. 8vo, cloth, 6s. Hydraulic Steam and Hand Power Lifting and Pressing Machinery. By FREDERICK COLYER, M. Inst. C.E., M. Inst. M.E. With 73 plates, 8vo, cloth, iSs. Pumps and Pumping Machinery. By F. COLYER, M.I.C.E., M.I.M.E. With 23 folding plates, 8vo, cloth, I2s. 6d. Pumps and Pumping Machinery. By F. COLYER. Second Part. With II large plates, 8vo, cloth, 12s. 6d. The Municipal and Sanitary Engineer s Handbook. By H. PERCY BOULNOIS, Mem. Inst. C.E., Borough Engineer, Ports- mouth. With numerous illustrations, demy 8vo, cloth, 12s. 6d. CONTENTS : The Appointment and Duties of the Town Surveyor Traffic Macadamised Roadways Steam Rolling Road Metal and Breaking Pitched Pavements Asphalte Wood Pavements Footpaths Kerbs and Gutters Street Naming and Numbering -Street Lighting Sewer- age Ventilation of Sewers Disposal of Sewage House Drainage Disinfection Gas and Water Companies, &c., Breaking up Streets Improvement of Private Streets Borrowing Powers Artizans' and Labourers' Dwellings Public Conveniences Scavenging, including Street Cleansing Watering and the Removing of Snow Planting Street Trees Deposit of pj ans Dangerous Buildings Hoardings Obstructions Improving Street Lines Cellar Openings Public Pleasure Grounds Cemeteries Mortuaries Cattle and Ordinary Markets Public Slaughter-houses, etc. Giving numerous Forms of Notices, Specifications, and General Information upon these and other subjects of great importance to Municipal Engi- neers and others engaged in Sanitary Work. CATALOGUE OF SCIENTIFIC BOOKS Tables of the Principal Speeds occurring in Mechanical Engineering, expressed in metres in a second. By P. KEERAYEFF, Chief Mechanic of the Obouchoff Steel Works, St. Petersburg ; translated by SERGIUS KERN, M.E. Fcap. 8vo, sewed, 6d. A Treatise on the Origin, Progress, Prevention, and Cure of Dry Rot in Timber; with Remarks on the Means of Preserving Wood from Destruction by Sea- Worms, Beetles, Ants, etc. By THOMAS ALLEN BRITTON, late Surveyor to the Metropolitan Board of Works, etc., etc. With 10 plates, crown 8vo, cloth, "js. 6d. Metrical Tables. By G. L. MOLESWORTH, M.I.C.E. 32010, cloth, is. 6d. CONTENTS. General Linear Measures Square Measures Cubic Measures Measures of Capacity Weights Combinations Thermometers. Elements of Construction for Electro- Magnets. By Count TH. Du MONCEL, Mem. de 1'Institut de France. Translated from the French by C. J. WHARTON. Crown 8vo, cloth, 4^. 6d. Electro -Telegraphy. By FREDERICK S. BEECHEY, Telegraph Engineer. A Book for Beginners. Illustrated. Fcap. 8vo, sewed, 6d. H andr ailing : by the Square Cut. By JOHN JONES, Staircase Builder. Part Second, with eight plates, 8vo, cloth, 3^. 6d. Practical Electrical Units Popularly Explained, with numerous illustrations and Remarks. By JAMES SWINBURNE, late of J. W. Swan and Co., Paris, late of Brush-Swan Electric Light Company, U.S.A. i8mo, cloth, u-. 6d. Philipp Reis, Inventor of the Telephone : A Biographical Sketch. With Documentary Testimony, Translations of the Original Papers of the Inventor, &c. By SILVANUS P. THOMPSON, B.A., Dr. Sc., Professor of Experimental Physics in University College, Bristol. With illustrations, 8vo, cloth, "js. 6d. A Treatise on the Use of Belting for the Transmis- sion of Power. By J. H. COOPER. Second edition, illustrated, 8vo, cloth, 15^. Hints on Architectural Draughtsmanship. By G. W. TUXFORD HALLATT. Fcap. 8vo, cloth, is. 6d. PUBLISHED BY E. & F. N. SPON. A Pocket-Book of Useful Formula and Memoranda for Civil and Mechanical Engineers. By GuiLFORD L. MoLESWORTH, Mem. Inst. C.E., Consulting Engineer to the Government of India for State Railways. With numerous illustrations, 744 pp. Twenty-first edition, revised and eularged, 32mo, roan, 6s. SYNOPSIS OF CONTENTS: Surveying, Levelling, etc. Strength and Weight of Materials Earthwork, Brickwork, Masonry, Arches, etc. Struts, Columns, Beams, and Trusses Flooring, Roofing, and Roof Trusses Girders, Bridges, etc. Railways and Roads Hydraulic Formulae Canals. Sewers, Waterworks, Docks Irrigation and Breakwaters Gas, Ventilation, and Warming Heat, Light, Colour, and Sound Gravity : Centres, Forces, and Powers Millwork, Teeth of Wheels, Shafting, etc. Workshop Recipes Sundry Machinery Animal Power Steam and the Steam Engine Water-power, Water-wheels, Turbines, etc. Wind and Windmills- Steam Navigation, Ship Building, Tonnage, etc. Gunnery, Projectiles, etc. Weights, Measures, and Money Trigonometry, Conic Sections, and Curves Telegraphy Mensura- tion Tables of Areas and Circumference, and Arcs of Circles Logarithms, Square and Cube Roots, Powers Reciprocals, etc. Useful Numbers Differential and Integral Calcu- lus Algebraic Signs Telegraphic Construction and Formulae. Spons Tables and Memoranda for Engineers; selected and arranged by J. T. HURST, C.E., Author of 'Architectural Surveyors' Handbook,' ' Hurst's Tredgold's Carpentry,' etc. Seventh edition, 64010, roan, gilt edges, is. ; or in cloth case, I j. 6d. This work is printed in a pearl type, and is so small, measuring only 2t in. by if in. by i in. thick, that it may be easily carried in the waistcoat pocket. " It is certainly an extremely rare thing for a reviewer to be called upon to notice a volume measuring but 2^ in. by if in., yet these dimensions faithfully represent the size of the handy little book before us. The volume which contains 118 printed pages, besides a few blank pages for memoranda is, in fact, a true pocket-book, adapted for being carried in the waist- coat pocket, and containing a far greater amount and variety of information than most people would imagine could be compressed into so small a space The little volume has been compiled with considerable care and judgment, and we can cordially recommend it to our readers as a useful little pocket companion." Engineering. A Practical Treatise on Natural and Artificial Concrete, its Varieties and Constructive Adaptations. By HENRY REID, Author of the ' Science and Art of the Manufacture of Portland Cement.' New Edition, with 59 woodcuts and 5 plates, 8vo, cloth, 1 5 s. Notes on Concrete and Works in Concrete; especially written to assist those engaged upon Public Works. By JOHN NEWMAN, Assoc. Mem. Inst. C.E., crown 8vo, cloth, $s. Hydrodynamics : Treatise relative to the Testing of Water- Wheels and Machinery, with various other matters pertaining to Hydrodynamics. By JAMES EMERSON. With numerous illustrations, 360 pp. Third edition, crown 8vo, cloth, 4J 1 . 6d. Electricity as a Motive Power. By Count TH. Du MONCEL, Membre de 1'Institut de France, and FRANK GERALDY, Inge- nieur des Pontset Chaussees. Translated and Edited, with Additions, by C. J. WHARTON, Assoc. Soc. TeL Eng. and Elec. With 113 engravings and diagrams, crown 8vo, cloth, 7*. 6d. CATALOGUE OF SCIENTIFIC BOOKS Treatise on Valve-Gears, with special consideration of the Link-Motions of Locomotive Engines. By Dr. GUSTAV ZEUNER, Professor of Applied Mechanics at the Confederated Polytechnikum of Zurich. Translated from the Fourth German Edition, by Professor J. F. KLEIN, Lehigh University, Bethlehem, Pa. Illustrated, 8vo, cloth, I2s. 6d. The French- Polisher s Manual. By a French- Polisher; containing Timber Staining, Washing, Matching, Improving, Painting, Imitations, Directions for Staining, Sizing, Embodying, Smoothing, Spirit Varnishing, French-Polishing, Directions for Re- polishing. Third edition, royal 32mo, sewed, 6d. Hops, their Cultivation, Commerce, and Uses in various Countries. By P. L. SIMMONDS. Crown 8vo, cloth, 4^. 6d. A Practical Treatise on the Manufacture and Distri- bution of Coal Gas. By WILLIAM RICHARDS. Demy 4to, with numerous wood engravings and 29 plates, cloth, 2%s, SYNOPSIS OF CONTENTS : Introduction History of Gas Lighting Chemistry of Gas Manufacture, by Lewis Thompson, Esq., M.R.C.S. Coal, with Analyses, by J. Paterson, Lewis Thompson, and G. R. Hislop, Esqrs. Retorts, Iron and Clay Retort Setting Hydraulic Main Con- densersExhausters Washers and Scrubbers Purifiers Purification History of Gas Holder Tanks, Brick and Stone, Composite, Concrete, Cast-iron, Compound Annular Wrought-iron Specifications Gas Holders Station Meter Governor Distribution Mains Gas Mathematics, or Formula: for the Distribution of Gas, by Lewis Thompson, Esq. Services Consumers' Meters Regulators Burners Fittings Photometer Carburization of Gas Air Gas and Water Gas Composition of Coal Gas, by Lewis Thompson, Esq. Analyses of Gas Influence of Atmospheric Pressure and Temperature on Gas Residual Products Appendix Description of Retort Settings, Buildings, etc., etc. Practical Geometry, Perspective, and Engineering Drawing', a Course of Descriptive Geometry adapted to the Require- ments of the Engineering Draughtsman, including the determination of cast shadows and Isometric Projection, each chapter being followed by numerous examples ; to which are added rules for Shading, Shade-lining, etc., together with practical instructions as to the Lining, Colouring, Printing, and general treatment of Engineering Drawings, with a chapter on drawing Instruments. By GEORGE S. CLARKE, Capt. R.E. Second edition, with 21 plates. 2 vols., cloth, los. 6d. The Elements of Graphic Statics. By Professor KARL VON OTT, translated from the German by G. S. CLARKE, Capt. R.E., Instructor in Mechanical Drawing, Royal Indian Engineering College. With 93 illustrations, crown 8vo, cloth, $s. The Principles of Graphic Statics. By GEORGE SYDENHAM CLARKE, Capt. Royal Engineers. With 112 illustrations. 4to, cloth, \2s. 6d. Dynamo-Electric Machinery : A Manual for Students of Electro-technics. By SILVANUS P. THOMPSON, B.A., D.Sc., Professor of Experimental Physics in University College, Bristol, etc., etc. Second edition, illustrated^ 8vo, cloth, I2s. 6d. PUBLISHED BY E. & F. N. SPON. The New Formula for Mean Velocity of Discharge of Rivers and Canals. By W. R. K UTTER. Translated from articles in the 'Cultur-Ingenieur,' by Lowis D'A. JACKSON, Assoc. Inst. C.E. 8vo, cloth, I2s. 6d. Practical Hydraulics ; a Series of Rules and Tables for the use of Engineers, etc., etc. By THOMAS Box. Fifth edition, numerous plates, post 8vo, cloth, $s. A Practical Treatise on the Construction of Hori- zontal and Vertical Waterwheels, specially designed for the use of opera- tive mechanics. By WILLIAM CULLEN, Millwright and Engineer. With n plates. Second edition, revised and enlarged, small 4to, cloth, \2s. 6d. Tin: Describing the Chief Methods of Mining, Dressing and Smelting it abroad ; with Notes upon Arsenic, Bismuth and Wolfram. By ARTHUR G. CHARLETON, Mem. American Inst. of Mining Engineers. With plates, 8vo, cloth, I2s. 6d. Perspective, Explained and Illustrated. By G. S. CLARKE, Capt. R.E. With illustrations, 8vo, cloth, 3J-. 6d. The Essential Elements of Practical Mechanics; based on the Principle of Work, designed for Engineering Students. By OLIVER BYRNE, formerly Professor of Mathematics, College for Civil Engineers. Third edition, -with 148 wood engravings, post 8vo, cloth, -JS. (xt. CONTENTS : Chap. I. How Work is Measured by a Unit, both with and without reference to a Unit of Time Chap. 2. The Work of Living Agents, the Influence of Friction, and introduces one of the most beautiful Laws of Motion Chap. }. The principles expounded in the first and second chapters are applied to the Motion of Bodies Chap. 4. The Transmission of Work by simple Machines Chap. 5. Useful Propositions and Rules. The Practical Millwright and Engineers Ready Reckoner; or Tables for finding the diameter and power of cog-wheels, diameter, weight, and power of shafts, diameter and strength of bolts, etc. By THOMAS DIXON. Fourth edition, I2mo, cloth, %s. Breweries and Mailings : their Arrangement, Con- struction, Machinery, and Plant. By G. SCAMELL, F.R.I.B.A. Second edition, revised, enlarged, and partly rewritten. By F. COLYER, M.I.C.E., M.I.M.E. With 20 plates, 8vo, cloth, i8j. A Practical Treatise on tJie Manufacture of Starch, Glucose, Starch-Sugar, and Dextrine, based on the German of L. Von Wagner, Professor in the Royal Technical School, Buda Pesth, and other authorities. By JULIUS FRANKEL ; edited by ROBERT HUTTER, proprietor of the Philadelphia Starch Works. With 58 illustrations, 344 pp., 8vo, cloth, i8j. io CATALOGUE OF SCIENTIFIC BOOKS A Practical Treatise on Mill-gearing, Wheels, Shafts, Riggers, etc. ; for the use of Engineers. By THOMAS Box. Third edition, with 1 1 plates. Crown 8vo, cloth, 'js. 6d. Mining' Machinery: a Descriptive Treatise on the Machinery, Tools, and other Appliances used in Mining. By G. G. ANDRE, F.G.S., Assoc. Inst. C.E., Mem. of the Society of Engineers. Royal 4to, uniform with the Author's Treatise on Coal Mining, con- taining 182 plates , accurately drawn to scale, with descriptive text, in 2 vols., cloth, 3/. I2s. CONTENTS : Machinery for Prospecting, Excavating, Hauling, and Hoisting Ventilation Pumping Treatment of Mineral Products, including Gold and Silver, Copper, Tin, and Lead, Iron Coal, Sulphur, China Clay, Brick Earth, etc. Tables for Setting out Curves for Railways, Canals, Roads, etc., varying from a radius of five chains to three miles. By A. KENNEDY and R. W. HACKWOOD. Illustrated, 32mo, cloth, 2s. 6d. The Science and Art of the Manufacture of Portland Cement, with observations on some of its constructive applications. With 66 illustrations. By HENRY REID, C.E., Author of 'A Practical Treatise on Concrete,' etc., etc. 8vo, cloth, i$s. The Draughtsman s Handbook of Plan and Map Drawing '; including instructions for the preparation of Engineering, Architectural, and Mechanical Drawings. With numerous illustrations in the text, and 33 plates (15 printed in colours']. By G. G. ANDRE, F.G.S., Assoc. Inst. C.E. 4to, cloth, 9^. CONTENTS : The Drawing Office and its Furnishings Geometrical Problems Lines, Dots, and their Combinations Colours, Shading, Lettering, Bordering, and North Points Scales Plotting Civil Engineers' and Surveyors' Plans Map Drawing Mechanical and Architectural Drawing Copying and Reducing Trigonometrical Formulae, etc., etc. The B oiler-maker s andiron Ship-builder s Companion, comprising a series of original and carefully calculated tables, of the utmost utility to persons interested in the iron trades. By JAMES FODEN, author of ' Mechanical Tables,' etc. Second edition revised, with illustra- tions, crown 8vo, cloth, $j. Rock Blasting: a Practical Treatise on the means employed in Blasting Rocks for Industrial Purposes. By G. G. ANDRE, F.G.S., Assoc. Inst. C.E. With 56 illustrations and 12 plates, 8vo, cloth, ioj. 6d. Painting and Painters Manual: a Book of Facts for Painters and those who Use or Deal in Paint Materials. By C. L. CONDIT and J. SCHELLER. Illustrated, 8vo, cloth, IQJ. 6d. PUBLISHED BY E. & F. N. SPON. n A Treatise on Ropemaking as practised in piiblic and private Rope-yards, with a Description of the Manufacture, Rules, Tables of Weights, etc., adapted to the Trade, Shipping, Mining, Railways, Builders, etc. By R. CHAPMAN, formerly foreman to Messrs. Huddart and Co., Limehouse. and late Master Ropemaker to H.M. Dockyard, Deptford. Second edition, I2mo, cloth, 3*. Laxtons Builders and Contractors Tables ; for the use of Engineers, Architects, Surveyors, Builders, Land Agents, and others. Bricklayer, containing 22 tables, with nearly 30,000 calculations. 4to, cloth, 5-r. Laxtons Builders and Contractors Tables. Ex- cavator, Earth, Land, Water, and Gas, containing 53 tables, with nearly 24,000 calculations. 4to, cloth, 5^. Sanitary Engineering: a Guide to the Construction of Works of Sewerage and House Drainage, with Tables for facilitating the calculations of the Engineer. By BALDWIN LATHAM, C.E., M. Inst. C.E., F.G.S., F.M.S., Past-President of the Society of Engineers. Second edition, with numerous plates and woodcuts, 8vo, cloth, I/, icxr. Screw Cutting Tables for Engineers and Machinists, giving the values of the different trains of Wheels required to produce Screws of any pitch, calculated by Lord Lindsay, M.P., F.R.S., F.R.A.S., etc. Cloth, oblong, 2s. Screw Cutting Tables, for the use of Mechanical Engineers, showing the proper arrangement of Wheels for cutting the Threads of Screws of any required pitch, with a Table for making the Universal Gas-pipe Threads and Taps. By W. A. MARTIN, Engineer. Second edition, oblong, cloth, is., or sewed, 6d. A Treatise on a Practical Method of Designing Slide- Valve Gears by Simple Geometrical Construction^ based upon the principles enunciated in Euclid's Elements, and comprising the various forms of Plain Slide- Valve and Expansion Gearing ; together with Stephenson's, Gooch's, and Allan's Link-Motions, as applied either to reversing or to variable expansion combinations. By EDWARD J. COWLING WELCH, Memb. Inst. Mechanical Engineers. Crown 8vo, cloth, 6s. Cleaning and Scouring : a Manual for Dyers, Laun- dresses, and for Domestic Use. By S. CHRISTOPHER. i8mo, sewed, 6d. A Handbook of House Sanitation ; for the use of all persons seeking a Healthy Home. A reprint of those portions of Mr. Bailey-Denton's Lectures on Sanitary Engineering, given before the School of Military Engineering, which related to the "Dwelling," enlarged and revised by his Son, E. F. BAILEY -DENTON, C.E., B.A. IVith 140 illustrations, 8vo, cloth, 4^. 6d. 4 12 CATALOGUE OF SCIENTIFIC BOOKS A Glossary of Terms used in Coal Mining. By WILLIAM STUKELEY GRESLEY, Assoc. Mem. Inst. C.E., F.G.S., Member of the North of England Institute of Mining Engineers. Illustrated "with numerous woodcuts and diagrams, crown 8vo, cloth, 5^. A Pocket- Book for Boiler Makers and Steam Users, comprising a variety of useful information for Employer and Workman, Government Inspectors, Board of Trade Surveyors, Engineers in charge of Works and Slips, Foremen of Manufactories, and the general Steam- using Public. By MAURICE JOHN SEXTON. Second edition, royal 32mo, roan, gilt edges, 5-r. Electrolysis: a Practical Treatise on Nickeling, Coppering, Gilding, Silvering, the Refining of Metals, and the treatment of Ores by means of Electricity. By HIPPOLYTE FONTAINE, translated from the French by J. A. BERLY, C.E., Assoc. S.T.E. With engravings. 8vo, cloth, 9-f. A Practical Treatise on the Steam Engine, con- taining Plans and Arrangements of Detail's for Fixed Steam Engines, with Essays on the Principles involved in Design and Construction. By ARTHUR RIGG, Engineer, Member of the Society of Engineers and of the Royal Institution of Great Britain. Demy 4to, copiously illustrated with woodcuts and 96 plates, in one Volume, half-bound morocco, 2/. 2s. ; or cheaper edition, cloth, 25^. This work is not, in any sense, an elementary treatise, or history of the steam engine, but is intended to describe examples of Fixed Steam Engines without entering into the wide domain of locomotive or marine practice. To this end illustrations will be given of the most recent arrangements of Horizontal, Vertical, Beam, Pumping, Winding, Portable, Semi- portable, Corliss, Allen, Compound, and other similar Engines, by the most eminent Firms in Great Britain and America. The laws relating to the action and precautions to be observed in the construction of the various details, such as Cylinders, Pistons, Piston-rods, Connecting- rods, Cross-heads, Motion-blocks, Eccentrics, Simple, Expansion, Balanced, and Equilibrium Slide-valves, and Valve-gearing will be minutely dealt with. In this connection will be found articles upon the Velocity of Reciprocating Parts and the Mode of Applying the Indicator, Heat and Expansion of Steam Governors, and the like. It is the writer's desire to draw illustrations from every possible source, and give only those rules that present practice deems correct. Barlow s Tables of Squares, Cubes, Square Roots, Cube Roots, Reciprocals of all Integer Numbers up to 10,000. Post 8vo, cloth, 6s. Camus (M.) Treatise on the Teeth of Wheels, demon- strating the best forms which can be given to them for the purposes of Machinery, such as Mill-work and Clock-work, and the art of finding their numbers. Translated from the French, with details of the present practice of Millwrights, Engine Makers, and other Machinists, by ISAAC HAWKINS. Third edition, -with 18 plates, 8vo, cloth. $s. PUBLISHED BY E. & F. N. SPON. 13 A Practical Treatise on the Science of Land and Engineering Surveying, Levelling, Estimating Quantities, etc., with a general description of the several Instruments required for Surveying, Levelling, Plotting, etc. By H. S. MERRETT. Fourth edition, revised by G. W. USILL, Assoc. Mem. Inst. C.E. 41 plates, with illustrations and tables, royal 8vo, cloth, I2J. 6d. PRINCIPAL CONTENTS : Part i. Introduction and the Principles of Geometry. Part 2. Land Surveying ; com- prising General Observations The Chain Offsets Surveying by the Chain only Surveying Hilly Ground To Survey an Estate or Parish by the Chain only Surveying with the Theodolite Mining and Town Surveying Railroad Surveying Mapping Division and Laying out of Land Observations on Enclosures Plane Trigonometry. Part 3. Levelling Simple and Compound Levelling The Level Book Parliamentary Plan and Section- Levelling with a Theodolite Gradients Wooden Curves To Lay out a Railway Curve- Setting out Widths. Part 4. Calculating Quantities generally for Estimates Cuttings and Embankments Tunnels Brickwork Ironwork Timber Measuring. Part 5. Description and Use of Instruments in Surveying and Plotting The Improved Dumpy Level Troughton's Level The Prismatic Compass Proportional Compass Box Sextant Vernier Panta- graph Merrett's Improved Quadrant Improved Computation Scale The Diagonal Scale Straight Edge and Sector. Part 6. Logarithms of Numbers Logarithmic Sines and Co-Sines, Tangents and Co-TangentsNatural Sines and Co-SinesTables for Earthwork, for Setting out Curves, and for various Calculations, etc., etc., etc. Saws : the History, Development, Action, Classifica- tion, and Comparison of Saws of all kinds. By ROBERT GRIMSHAW. With 220 illustrations, 4to, cloth, Ms. 6d. A Siipplement to the above ; containing additional practical matter, more especially relating to the forms of Saw Teeth for special material and conditions, and to the behaviour of Saws under particular conditions. With 120 illustrations, cloth, gs. A Guide for the Electric Testing of Telegraph Cables. By Capt. V. HOSKICER, Royal Danish Engineers. With illustrations, second edition, crown 8vo, cloth, 4^. 6d. Laying and Repairing Electric Telegraph Cables. By Capt. V. HOSKICER, Royal Danish Engineers. Crown 8vo, cloth, 3*. 6d. The Assayers Manual: an Abridged Treatise on the Docimastic Examination of Ores and Furnace and other Artificial Products. By BRUNO KERL. Translated by W. T. BRANNT. With 65 illustrations, 8vo, cloth, I2s. 6d. The Steam Engine considered as a Heat Engine : a Treatise on the Theory of the Steam Engine, illustrated by Diagrams, Tables, and Examples from Practice. By JAS. H. 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Translated from the French of E. HOSPITALIER, Editor of " L'Electricien," by C. J. WHARTON, Assoc. Soc. Tel. Eng. Numerous illustrations. Demy 8vo, cloth, gs. CONTENTS : i. Production of the Electric Current 2. Electric Bells 3. Automatic Alarms 4. Domestic Telephones 5. Electric Clocks 6. Electric Lighters 7. Domestic Electric Lighting 8. Domestic Application of the Electric Light 9. Electric Motors 10. Electrical Locomo- tion XI< Electrotyping, Plating, and Gilding 12. Electric Recreations 13. Various appli- cationsWorkshop of the Electrician. Wrinkles in Electric Lighting. By VINCENT STEPHEN. With illustrations. i8mo, cloth, 2s. 6d. CONTENTS : i The Electric Current and its production by Chemical means 2. Production of Electric Currents by Mechanical means 3. Dynamo- Electric Machines 4. Electric Lamps 5. Lead 6. Ship Lighting. The Practical Flax Spinner ; being a Description of the Growth, Manipulation, and Spinning of Flax and Tow. By LESLIE C.MARSHALL, of Belfast. 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Second edition, with additions. Royal 8vo, sewed, 4^. A Treatise on the Manufacture of Soap and Candles, Ltibricants and Glycerin. By W. LANT CARPENTER, B.A,, B.Sc. (late of Messrs. C. Thomas and Brothers, Bristol). With illustrations. Crown 8vo, cloth, icxr. 6d. PUBLISHED BY E. & F. N. SPON. 17 The Stability of Ships explained simply, and calculated by a new Graphic method. By J. C. SPENCE, M.I.N.A. 410, sewed, Steam Making, or Boiler Practice. By CHARLES A. SMITH, C.E. 8vo, cloth, los. 6d. CONTENTS : i. The Nature of Heat and the Properties of Steam 2. Combustion. 3. Externally Fired Stationary Boilers 4. Internally Fired Stationary Boilers 5. Internally Fired Portable Locomotive and Marine Boilers 6. Design, Construction, and Strength of Boilers 7. Pro- portions of Heating Surface, Economic Evaporation, Explosions 8. Miscellaneous Boilers, Choice of Boiler Fittings and Appurtenances. The Fireman s Guide ; a Handbook on the Care of Boilers. By TEKNOLOG, fdreningen T. I. Stockholm. Translated from the third edition, and revised by KARL P. DAHLSTROM, M.E. Second edition. Fcap. 8vo, cloth, 2s. A Treatise on Modern Steam Engines and Boilers, including Land Locomotive, and Marine Engines and Boilers, for the use of Students. By FREDERICK COLYER, M. Inst. C.E., Mem. Inst. M.E. With 36 plates. 4to, cloth, 25 s. CONTENTS : z. Introduction 2. Original Engines 3. Boilers 4. High-Pressure Beam Engines 5. Cornish Beam Engines 6. Horizontal Engines 7. Oscillating Engines 8. yertical High- Pressure Engines 9. Special Engines 10. Portable Engines n. Locomotive Engines 12. Marine Engines. Steam Engine Management; a Treatise on the Working and Management of Steam Boilers. By F. COLYER, M. Inst. C.E., Mem. Inst. M.E. i8mo, cloth, 2s. Land Surveying on the Meridian and Perpendicular System. By WILLIAM PENMAN, C.E. 8vo, cloth, Ss. 6d. 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London : E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. JTTST PUBLISHED. Crown 8vo, cloth, 480 pages, with 183 illustrations, 5-r. WORKSHOP RECEIPTS, THIRD SERIES. BY C. G. WARNFORD LOCK. Uniform with the First and Second Series. SYNOPSIS OF CONTENTS. Alloys. Indium. Rubidium. Aluminium. Iridium. Ruthenium. Antimony. Iron and Steel. Selenium. Barium. Lacquers and Lacquering. Silver. Beryllium. Lanthanum. Slag. Bismuth. Lead. Sodium. Cadmium. Lithium. Strontium. Caesium. Lubricants. Tantalum. Calcium. Magnesium. Terbium. Cerium. Manganese. Thallium. Chromium. Mercury. Thorium. Cobalt. Mica. Tin. Copper. Molybdenum. Titanium. Didymium. Nickel. Tungsten. Electrics. Niobium. Uranium. Enamels and Glazes. Osmium. Vanadium. Erbium. Palladium. Yttrium. Gallium. Platinum. Zinc. Glass. Potassium. Zirconium. Gold. Rhodium. London : E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. WORKSHOP RECEIPTS, FOURTH SERIES, DEVOTED MAINLY TO HANDICRAFTS & MECHANICAL SUBJECTS, BY C. 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Mechanical Drawing Casting and Founding in Iron, Brass, Bronze, and other Alloys Forging and Finishing Iron Sheetmetal Working Soldering, Brazing, and Burning Carpentry and Joinery, embracing descriptions of some 400 Woods, over 200 Illustrations of Tools and their uses, Explanations (with Diagrams) of 116 joints and hinges, and Details of Construction of Workshop appliances, rough furniture, Garden and Yard Erections, and House Building Cabinet-Making and Veneering Carving and Fretcutting Upholstery Painting, Graining, and Marbling Staining Furniture, Woods, Floors, and Fittings Gilding, dead and bright, on various grounds Polishing Marble, Metals, and Wood Varnishing Mechanical movements, illustrating contrivances for transmitting motion Turning in Wood and Metals Masonry, embracing Stonework, Brickwork, Terracotta, and Concrete Roofing with Thatch, Tiles, Slates, Felt, Zinc, &c. Glazing with and without putty, and lead glazing Plastering and Whitewashing Paper-hanging Gas-fitting Bell-hanging, ordinary and electric Systems Lighting Warming Ventilating Roads, Pavements, and Bridges Hedges, Ditches, and Drains Water Supply and Sanitation Hints on House Construction suited to new countries. London: E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ENGINEERING LIBMAHV wuv 5 1948 LD 21-100m-9,'47(A5702sl6)476 YB 23986 80*73 Library THE UNIVERSITY OF CALIFORNIA LIBRARY