The D. Van Nostrand Company intend this book to be sold to the Public at the advertised price, and supply it to the Trade on terms which will not allow of discount. ALTERNATING-CURRENT MACHINES BEING THE SECOND VOLUME OF DYNAMO ELECTRIC MACHINERY; ITS CONSTRUCTION, DESIGN, AND OPERATION BY SAMUEL SHELDON, A.M., PH.D. PROFESSOR OF PHYSICS AND ELECTRICAL ENGINEERING AT THE POLYTECHNIC INSTITUTE OF BROOKLYN, MEMBER OF THE AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS, FELLOW OF THE AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, AND FELLOW OF THE AMERICAN ELECTRO- THERAPEUTIC ASSOCIATION AND HOBART MASON, B.S., E.E. ASSISTANT IN ELECTRICAL ENGINEERING AT THE POLYTECHNIC INSTITUTE OF BROOKLYN, AND ASSOCIATE OF THE AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS FIFTH EDITION. NEW YORK: D. VAN NOSTRAND COMPANY 23 MURRAY AND 27 WARREN STS. LONDON: CROSBY LOCKWOOD & SON y STATIONERS' HALL COURT, LUDGATE HILL 1906 COPYRIGHT, 1902, BY D. VAN NOSTRAND COMPANY TYPOGRAPHY BY C. J. PETERS & SON. PRESSWORK BY THE F. H. GILSON COMPANY BOSTON, MASS., U. S. A. PREFACE. THIS book, like its companion volume on Direct Current Machines, is primarily intended as a text-book for use in technical educational institutions. It is hoped and be- lieved that it will also be of use to those electrical, civil, mechanical, and hydraulic engineers who are not perfectly familiar with the subject of Alternating Currents, but whose work leads them into this field. It is furthermore intended for use by those who are earnestly studying the subject by themselves, and who have previously acquired some proficiency in mathematics. There are several methods of treatment of alternating- current problems. Any point is susceptible of demonstra- tion by each of the methods. The use of all methods in connection with every point leads to complexity, and is undesirable in a book of this character. In each case that method has been chosen which was deemed clearest and most concise. No use has been made of the method of complex imaginary numbers. A thorough understanding of what takes place in an alternating-current circuit is not to be easily acquired. It is believed, however, that one who has mastered the first four chapters of this book will be able to solve any practi- cal problem concerning the relations which exist between power, electro-motive forces, currents, and their phases in iv PREFACE. series or multiple alternating-current circuits containing resistance, capacity, and inductance. The next four chapters are devoted to the construction, principle of operation, and behavior of the various types of alternating-current machines. Only American machines have been considered. A large amount of alternating-current apparatus is used in connection with plants for the long-distance transmission of power. This subject is treated in the ninth chapter. The last chapter gives directions for making a variety of tests on alternating-current circuits and apparatus. No apology is necessary for the introduction of cuts and material supplied by the various manufacturing companies. The information and ability of their engineers, and the taste and skill of their artists, are unsurpassed, and the informa- tion supplied by them is not available from other sources. For their courteous favors thanks is hereby given. CONTENTS. PAGE CHAPTER I. PROPERTIES OF ALTERNATING CURRENTS II. SELF-INDUCTION III. CAPACITY 29 IV. PROBLEMS ON ALTERNATING-CURRENT CIRCUITS. ... 44 V. ALTERNATORS ^7 VI. THE TRANSFORMER 9 2 VII. MOTORS I41 VIII. CONVERTERS IX. POWER TRANSMISSION J 2 X. TESTS . '9 8 ALTERNATING-CURRENT MACHINES. CHAPTER I. PROPERTIES OF ALTERNATING CURRENTS. 1. Definition of an Alternating Current. An alter- nating current of electricity is a current which changes its direction of flow at regularly recurring intervals. Between these intervals the value of the current may vary in any way. In usual practice, the value varies with some regularity from zero to a maximum, and decreases with the same regularity to zero, then to an equal max- imum in the other direction, and finally to zero again. In practice, too, the intervals of current flow are very short, ranging from jfa to ^^ second. 2. Frequency. When, as stated above, a current has passed from zero to a maximum in one direction, to zero, to a maximum in the other direction, and finally to zero again, it is said to have completed one cycle. That is to say, it has returned to the condition in which it was first considered, both as to value and as to direction, and is prepared to repeat the process described, making a second cycle. It should be noted that it takes two alternations to make one cycle. The tilde ( ~ ) is frequently used to denote cycles. A^E^NATING-CURRENT MACHINES. The term frequency is applied to the number of cycles completed in a unit time, i.e., in one second. Occasionally the word alternations is used, in which case, unless other- wise specified, the number of alternations per minute is meant. Thus the same current is spoken of as having a frequency of 25, or as having 3000 alternations. The use of the word alternations is condemned by good practice. In algebraic notation the letter / usually stands for the frequency. The frequency of a commercial alternating current depends upon the work expected of it. For power a low frequency is desirable, particularly for converters. The great Niagara power plant uses a frequency of 25. Lamps, however, are operated satisfactorily only on fre- quencies of 50 or more. Early machines had higher frequencies, 125 and 133 (16,000 alternations) being- usual, but these are almost entirely abandoned because of their increased losses and their unadaptability to the operation of motors and similar apparatus. In the Report of the Committee on Standardization of the American Institute of Electrical Engineers is the following : " In alternating-current circuits, the follow- ing approximate -frequencies are recommended as de- sirable : 25 or 30 40 60 120 "These frequencies are already in extensive use, and it is deemed advisable to adhere to them as closely as possible." The frequency of an alternating current is always that of the E.M.F. producing it. To find the frequency of the pressure or the current produced by any alternating-cur- PROPERTIES OF ALTERNATING CURRENTS. 3 rent generator, if V be the number of revolutions per minute, and / be the number of pairs of poles, then V 3. Wave-shape If, in an alternating current, the instantaneous values of current be taken as ordinates, and time be the abscissae, a curve, as in Fig. i, may be developed. The length of the abscissa for one com- plete cycle is seconds. Imagine a small cylinder, Fig. 2, carried on one end of a wire, and rotated uniformly about the other end in a vertical plane. Imagine a hori- zontal beam of parallel rays of light to be parallel to the plane of rotation, and to cast a shadow of the cylinder on Fig. i. Fig. 2. a plane screen perpendicular to the rays. The shadow will move up and down, passing from the top of its travel to the bottom in a half revolution, and from the bottom 4 ALTERNATING-CURRENT MACHINES. back to the top in another half revolution with a perfect harmonic motion. Now imagine the screen to be moved horizontally in its own plane with a uniform motion, and the positions of the shadow suitably recorded on it, as on sensitized paper or on a photographic film, a slotted screen protecting all but the desired portion from exposure. Then the trace of the shadow will be as in Fig. 3. The abscissae of this curve may be taken as time, as in the preceding curve, the ab- scissa of one complete cycle being the time in seconds of one revolution. , Or, with equal relevancy, the abscissae may be expressed in degrees. Consider the cylinder to be in a zero position when the radius to which it is attached is horizontal. Then the abscissa of any point is the angle which must be turned through in order that the cylinder may cast its shadow at that point. In this case the abscissa of a complete cycle will be 360, or 2?r. Consideration of the manner in which the curve has been formed shows that the ordinate of any point is proportional to the sine of the abscissa of that point, expressed in degrees. Hence this is called a sinusoid or sine curve. If the maximum ordinate of this curve be taken as E m , and time be considered to commence at the beginning of* any cycle, then the ordinate E' at any time / seconds later will be E' = E m sin 2 TT/?, which is equivalent to neglecting all those intervals of time corresponding to whole cycles, and considering only PROPERTIES OF ALTERNATING CURRENTS. $ the time elapsed since the end of the last completed cycle. As a numerical example : In an alternating-current circuit of 45 ~* and a maximum voltage of 100, what will be the pressure at 2! seconds after the beginning of a cycle ? E' = ioo sin (2 TT X 45 X 2.125) = sin 191.25 , = sin 1.25 , = - -J= , whence E' = 70.7 volts. Since the ordinates of the curve may represent either current or pressure, the expression /' = I m sin 27T/? is equally true. The ideal pressure curve from an alternator is sin- usoidal. Commercial alternators, however, do not gene- rate true sinusoidal pressures. But the sine curve can be treated with relative simplicity, and the curves of practice approximate so closely to the sine form, that mathematical deductions based on sine curves can with propriety be applied to those of practice. Two of these actual curves are shown in Fig. 4. The shape of the pressure curve is affected by irregular distribution of the magnetic flux. Also uneven angular velocity of the generator will distort the wave-shape, making it, relative to the true curve, lower in the slow spots and higher in the fast ones. Again, the magnetic reluctance of the armature may vary in different angular positions, particularly if the inductors are laid in a few large slots. This would cause a periodic variation in the ALTERNATING-CURRENT MACHINES. reluctance of the whole magnetic circuit and a correspond- ing pulsation of the total magnetic flux. All these influ- ences operate at open circuit as well as under load. E.M.F. CURVE 3 PHASE 40 POLE 2000 K.W. 25 ~ FULLY LOADED OX 2Q 40 60 80 100 120 140 160 1 E.M.F. CURVE SINGLE PHASE 8 POLE 500 WATTS 125 ^^ MOT LOAD.ED Fig. 4. There are two other causes which act to distort the wave-shape only when under load. For any separately excited generator, a change in the resistance or apparent resistance of the external circuit will cause a change in the PROPERTIES OF ALTERNATING CURRENTS. 7 terminal voltage of the machine. As is explained later, the apparent resistance (impedance) of a circuit to alter- nating currents depends upon the permeability of the iron adjacent to the circuit. Permeability changes with mag- netization. Now, because an alternating current is flow- ing, the magnetization changes with the changing values of current. This, by varying the permeability, sets up a pulsation in the impedance and affects the terminal volt- age of the machine, periodically distorting the wave of pressure from the true sine. There are cases of synchronously pulsating resistances. The most common is that of the alternating arc. With the same arc the apparent resistance of the arc varies in- versely as the current. So when operated by alternating currents, the resistance of a circuit of arc lamps varies syn- chronously, and distorts the pressure wave-shape in a manner analogous to the above. Summing up, the wave-shape of pressure may be dis- torted : At open circuit as well as under load ; by lack of uniformity of magnetic distribution, by pulsating of mag- netic field, by variation in angular velocity of armature ; and under load only ; by pulsation of impedance, by pulsa- tion of resistance. And the effects of any or all may be superimposed. 4. Effective Values of E.M.F. and of Current One ampere of alternating current is a current of such instan- taneous values as to have the same heating effect in a con- ductor as one ampere of direct current. This somewhat arbitrary definition probably arose from the fact that alter- nating currents were first commercially employed in light- ing circuits, where their utility was measured by the heat 8 ALTERNATING-CURRENT MACHINES. they produced in the filaments ; and further from the fact that the only means then at hand of measuring alternating currents were the hot-wire instruments and the electro- dynamometer, either of which gives the same indication for an ampere of direct current or for what is now called an ampere of alternating current. The heat produced in a conductor carrying a current is proportional to the square of the current. In an alternat- ing current, whose instantaneous current values vary, the instantaneous rate of heating is not proportional to the instantaneous value, nor yet to the square of the average of the current values, but to the square of the instantaneous cur- rent value. And so the average heating effect is proportional to ~~ the mean of the squares of the V instantaneous currents. Fi s- 5 - The average current of a sinu- soidal wave of alternating current, whose maximum value is 7 W , is equal to the area of one lobe of the curve, Fig. 5, divided by its base line TT. Thus r I m sin Odd T _ *J aV ~ ~ TT - J r - 7T 7T " 7T But the heating value of such a current varies, as f7 w 2 sinW0 7-2 J -*m \ " ! /i r T o 1 = = sm 20= 7. . 7T 7T 1_2 4 J 2 The square root of this quantity is called the effective value of the current, 7 = ^. This has the same heating V2 PROPERTIES OF ALTERNATING CURRENTS. 9 effect as a direct current /, and the effective values are always referred to unless expressly stated otherwise. Alternating-current ammeters are designed to read in effective amperes. Since current is dependent upon the pressure,, the resistance or apparent resistance of a circuit remain- ing constant, it is obvious that if / = -^ then does E 2 V2 also E = ~- Likewise if average / = - I m then does also V 2 T 2 average E = - E m . Or these may be demonstrated in a 7T manner analogous to the above. The maximum value of pressure is frequently referred to in designing alternator armatures, and in calculating dielectric strength of insulation. There have arisen vari- ous ways of indicating that effective values are meant, for instance, the expressions, sq. root of mean sq., "vV 2 , Vmean square. In England the initials R.M.S. are fre- quently used for root mean square. ~, . Effective E.M.F. . , . The ratio is called the form-factor, Average E.M.F. since its value depends upon the shape of the pressure wave. For the curve Fig. 6, the form- factor is unity. As a curve be- comes more peaked, its form- **& 6 - factor increases, due to the superior weight of the squares of the longer ordinates. In the sinusoid the values found above give i ~T-E m V2 Form-factor = - = i.u. 10 ALTERNATING-CURRENT MACHINES. IN PHASE Fig. 7. Probably no alternators give sine waves, but they ap- proach it so nearly that the value i.i I can be used in cal- culation without sensible error. 5. Phase The curves of the pressure and the current in a circuit can be plotted together, with their respective ordinates and common abscissae, as in Fig. 7. In some cases the zero and the maximum values of the current curve will occur at the same abscissae as do those values of the pressure curve, as in Fig. 7. In such a case the current is said to be in phase with the pressure. In other cases the current will reach a maximum or a zero value at a time later than the corresponding values of the pressure, and since the abscissae are indifferently time or degrees, the condition is represented in Fig. 8. In such a case, the current is said to be out of phase with, and to lag be- hind the pressure. In still other cases the / ^-Vx. LAGGING CURRENT curves are placed as in Fig. 9, and the current and pressure are again out of phase, but the current is said to lead Flg - 8< the pressure. The distance between the zero ordinate of one sine curve and the corresponding zero ordinate of another, may be measured in degrees, and is called the angular displacement or phase difference. This angle of lag or, of lead is usually represented by <. When one PROPERTIES OF ALTERNATING CURRENTS. II LEADING CURRENT Fig. 9- RIGHT ANGLES Vox. curve has its zero ordinate coincident with the maximum ordinate of the other, as in Fig. 10, there is a displacement of a quarter cycle ( = 90), and the curves are said to be at right angles. This term owes its origin to the fact that the radii whose projections will trace these curves, as in 3, are at right angles to each other. If the zero ordinates of the two curves coincide, but the positive maximum of one coincides with the negative maxi- mum of the other, as in Fig. n, then < = 180, and the curves are in op- posite pJiase. An alternator arranged to give a single pressure wave to a two- wire circuit is said to be a single phaser, and the current in the circuit a single-phase current. Some machines are arranged to give pressure to two dis- tinct circuits each of which, considered alone, is a single-phase circuit but the time of maxi- mum pressure in one is the time of zero pres- sure in the other, so that simultaneous pres- sure curves from the two circuits take the form of Fig. 1 2. Such is said to be a two-phase or quarter-pJiase OPPOSITE PHASE 12 ALTERNATING-CURRENT MACHINES. system, and the generator is a two-phaser. A three-phase system theoretically has three circuits of two wires each. The maximum positive pressure on any circuit is displaced from that of either of the other circuits by 120. As the algebraic sum of the cur- rents in all these circuits (if balanced) is at every in- stant equal to zero, the three return wires, one on TWO PHASE Fig. 12. each circuit, may be dis- pensed with, leaving but three wires. The three sim- ultaneous curves of E.M.F. are shown in Fig. 1 3 . The term polyphase applies to any system of two or more phases. An ^-phase system has n circuits and n pressures with successive phase differences , 360 . of - degrees. n 6. Power in Alternating-Current Circuits With a direct- current circuit, the power in the circuit is equal to the product of the pressure in volts by the current strength in amperes. In an alternating- current circuit, the instan- taneous power is the product of the instantaneous values of current strength and pressure. If the current and pressure are out of phase there will be some instants when the pressure will have a positive value and the current a negative value or vice versa. At such times the instantaneous power will be a negative quantity, i.e., PROPERTIES OF ALTERNATING CURRENTS. 13 power is being returned to the generator by the disappear- ing magnetic field which had been previously produced by the current. This condition is shown in Fig. 14, where the power curve has for its ordinates the product of the corresponding ordinates of pressure and current. These are reduced by multiplying by a constant so as to make them of convenient size. The circuit, therefore, receives power from the generator and gives power back again in alternating pulsations having twice the frequency of the gen- erator. It is clear that the relative magnitudes Flg * I4 * of the negative and positive lobes of the power curve will vary for. different values of , even though the original curves maintain the same size and shape. So it follows that the power in an alternating-current circuit is not merely a function of E and /, as in direct-current circuits, but is a function of E, /, and , and the relation is deduced as follows : Let the accent (') denote instantaneous values. If the current lag by the angle , then from 3, E' = E m sin a, where, for convenience, a = 2 IT ft, and /' = I m sin (a ). Remembering that "P r E = -, and / = ( 4) the instantaneous power, V2 V2 P r E' I' = 2EI sin a sin (a <). 14 ALTERNATING-CURRENT MACHINES. But sin (a <) = sin a cos < cos a sin <, so P' = 2 j5"/(sin 2 a cos (f> sin a cos a sin <). Remembering that is a constant, the average power over 1 80, 2 vS'/cos C n . 2J5 /sin cos a -- sin 2 a 4 Q 7T = El cos <. /* . I Sin a ,7o ri . 1^ -sin 2 a . \_2 J Should the current /?#*/ the pressure by ^>, then the leading equation would be P'=2 fsm a sin (a + ), which gives the same expression, P = El cos <, which is the general expression for power in an alternating- current circuit. Since, to get the true power in the circuit, the apparent power, or volt -amperes, must be multiplied by cos <, this quantity is called \htpowcr factor of the circuit. If the pressure and current are in phase, < = o, and the power factor is unity. SELF-INDUCTION. 1 5 CHAPTER II. SELF-INDUCTION. 7. Self -Inductance. The subject of inductance was briefly treated of in 15, vol. i., of this work ; but, since it is an essential part of alternating-current phenomena, it will be discussed more fully in this chapter. When lines of force are cut by a conductor an E.M.F. is generated in that conductor ( 13, vol. i.). A conductor carrying cur- rent is encircled by lines of force. When the current is first started in such a conductor, these lines of force must be established. In establishing itself, each line is con- sidered as having cut the conductor, or, what is equivalent, been cut by the conductor. This notion of lines of force is a convenient fiction, designed to render an understand- ing of the subject more easy. To account for the E.M.F. of self-induction, the encircling lines must be considered as cutting the conductor which carries the current that establishes them, during their establishment. It may be considered that they start from the axis of the conductor at the moment of starting the current in the circuit ; that they grow in diameter while the current is increasing ; that they shrink in diameter when the current is decreasing; and that all their diameters reduce to zero upon stopping the current. At any given current strength the conductor is surrounded by many circular lines, the circles having various diameters. Upon decreasing the strength those of V 16 ALTERNATING-CURRENT MACHINES. smaller diameter cut the conductor and disappear into a point on the axis of the conductor previous to the cutting by those of larger diameter. The number of lines accom- panying a large current is greater than the number accom- panying a smaller current. The E.M.F. of self-induction is always a counter E.M.F. By this is meant that its direction is such as to tend to prevent the change of current which causes it. When the current is started the self -induced pressure tends to oppose the flow of the current and prevents its reaching its full value immediately. When the circuit is interrupted the E.M.F. of self-induction tends to keep the current flowing in the same direction that it had originally. 8. Unit of Self -Inductance. The self -inductance, or the coefficient of self-induction of a circuit is generally rep- resented by L or /, and is that constant by which the time rate of change of the current in a circuit must be multi- plied in order to give the E.M.F. induced in that circuit. Its absolute value is numerically equal to the number of lines of force linked with the circuit, per absolute unit of current in the circuit, as is shown below. By linkages, or number of lines linked with a circuit, is meant the sum of the number of lines surrounding each portion of the circuit. For instance, a coil of wire consisting of ten turns, and threaded completely through by twelve lines of force, is said to have 1 20 linkages. The absolute unit of self-inductance is too small for ordinary purposes, and a practical unit, the henry, is used. This is io 9 times as large as the c. G. s. or absolute unit. The Paris electrical congress of 1900 adopted as the unit of magnetic flux the maxwell, and of flux density the SELF-INDUCTION. 17 gauss. A maxwell is one line of force. A gauss is one line of force per square centimeter. If a core of an electro- magnet has a transverse cross-section of 30 sq. cm., and is uniformly permeated with 60,000 lines of force, such a core may be said to have a flux of 60,000 maxwells and a flux density of 2000 gausses. In 13, vol. i., it has been shown that the pressure gene- rated in a coil of wire when it is cut by lines of force is where ;/ is the number of turns in a coil, and where e is measured in c. G. s. units, 3> in maxwells, and / in seconds. In a simple case of self-induction the maxwells set up are due solely to the current in the conductor. Now let K be a constant, dependent upon the permeability of the mag- netic circuit, such that it represents the number of max- wells set up per unit current in the electric circuit ; then, indicating instantaneous values by prime accents, & = Ki r , and d = Kdi. The E.M.F. of self-induction may then be written w di e -=- Kn n; By the definition of the coefficient of self-induction, whose c. G. s. value is represented by /, /''- '" l ~dt From the last two equations, it is seen that / = Kn. Kn is evidently the number of linkages per absolute unit current. The negative sign indicates that the pressure is counter E.M.F, 18 ALTERNATING-CURRENT MACHINES. In practical units, E--L dl '. *- L ~di A circuit having an inductance of one henry will have a pressure of one volt induced in it by a uniform change of current of one ampere per second. 9. Practical Values of Inductances. To give the stu- dent an idea of the values of self-inductance met with in practice, a number of examples are here cited. A pair of copper line wires, say a telephone pole line, will have from two to four milhenrys (.002 to .004 henrys) per mile, according to the distance between them, the larger value being for the greater distance. The secondary of an induction coil giving a 2" spark has a resistance of about 6000 ohms and 50 henrys. The secondary of a much larger coil has 30,000 ohms and about 2000 henrys. A telephone call bell with about 75 ohms has 1.5 henrys. A coil found very useful in illustrative and quantitive experiments in the alternating-current laboratory is of the following dimensions. It is wound on a pasteboard cylinder with wooden ends, making a spool 8.5 inches long and 2 inches internal diameter. This is wound to a depth of 1.5 inch with No. 16 B. and S. double cotton-covered copper wire, there being about 3000 turns in all. A bundle of iron wires, 16 inches long, fits loosely in the hole of the spool. The resistance of the coil is 10 ohms, and its in- ductance without the core is 0.2 henry. With the iron core in place and a current of about 0.2 ampere, the induc- tance is about 1.75 henrys. This coil is referred to again in S ii. SELF-INDUCTION. 19 The inductance of a spool on the field frame of a gene- rator is numerically Jw _- where < is the total flux from one pole, ;/ the number of turns per spool, and I f the field current of the machine. It is evident that the value of L may vary through a wide range with different machines. 10. Things Which Influence the Magnitude of L. If all the conditions remain constant, save those under considera- tion, then the self-inductance of a coil will vary : directly as the square of the number of turns ; directly as the linear dimension if the coil changes its size without changing its shape ; and inversely as the reluctance of the magnetic circuit. Any of the above relations is apparent from the follow- ing equations. The numerical value of the self-induc- tance is / = # - . i As shown in Chapter 2, vol. i., M.M.F. 4 irni reluctance c where c is the mean length in centimeters of the magnetic circuit, A its mean cross-sectional area in square centi- meters, and /A is permeability. Then, if 24 ALTERNATING-CURRENT MACHINES. = I'di, I' being also a variable ; but an average of the val- ues of /' throughout the range may be called /, and the formula for energy stored in the field holds true. Since iron has always a hysteretic loss, some of the energy is consumed, and the work given back at the dis- appearance of the field is less than that used to establish the field by the amount consumed in hysteresis. 14. Current Produced by a Harmonic E.M.F. in a Cir- cuit Having Resistance and Inductance. Given a circuit of resistance R and inductance L upon which is impressed a harmonic E.M.F. E of frequency /, to find the current / in that circuit. Represent by o> the quantity 2Trf. At any instant of time, /, let the instantaneous value of the current be I' . To maintain this current requires an E.M.F. whose value at this instant is I'R. Represent this by E' r . From 3, in a harmonic current, /' = I m sin o>/, hence, E r ' = RI m sin /. Evidently E r ' has its maximum value RI m = E rm at tat = go or 270, and its effective value is E r = RI. The counter E.M.F. of self-induction at the same instant of time, /, is * T d1 ' E *=- L ~dt' But as before, /'= f m sin /, so dl' = v7 m cos o)/ dt, and .' = - <*LI m cos o>/. SELF-INDUCTION. 2$ Evidently E^ has a maximum value of - <*>LI m = E sm at &t = o or 1 80, and its effective value E, = - <*>LL It is clear that the impressed E.M.F. must be of such a value as to neutralize E s and also supply E r . But these two pressures cannot be simply added, since the maximum value of one occurs at the zero value of the other ; that is, they are at right angles to each other, as denned in sf - 5. Reference to Fig. 17 will Fig * I7 ' make it clear that combining these at right angles will give as a resultant the pressure V/f r 2 + E* ; and it is this pressure that the impressed E.M.F. E must equal and oppose. So E= ^(IRJ- + from which 77 7 = This is a formula which must be used in place of Ohm's Law when treating inductive circuits carrying harmonic currents. It is evident that, if the inductance or the fre- quency be negligibly small (direct current has f = o), the formula reduces to Ohm's Law ; but for any sensible val- ues of o> and L the current in the circuit will be less than that called for by Ohm's Law. The expression ^R' 2 4- w 2 Z 2 is called the impedance of the circuit, and also the apparent resistance. The term R is of course called resistance, while the term , may be a considerable factor in determining the impedance of a circuit. Having recourse once again to the harmonic shadow- graph described in 3, the phase relation between im- pressed E.M.F. and current may be made plain. It has already been shown that E r and E, are at right angles to SELF-INDUCTION. each other. Since the pressure E r is the part of the im- pressed E.M.F. which sends the current, the current must be in phase with it. Therefore there is always a phase displacement of 90 between / and E s . This relation is also evident from a consideration of the fact that when / reaches its maximum value it has, for the instant, no rate of change ; hence the flux, which is in phase with the cur- rent, is not changing, and consequently the E.M.F. of self- induction must be, for the instant, zero. That is, / is maxi- mum when E s is zero, which means a displacement of 90. In Fig. 1 8 the triangle of E.M.F.' 's of Fig. 17 is altered to the corresponding parallelogram of E.M.F. ,'s, and the maximum values substituted for the, effective. It now the parallelogram re- volve about the center o, the traces of the har- monic shadows of the extremities of E m , E rm and will develop as shown. It is evident that the curve EJ and so also the curve of current lags behind the curve E' by the angle <. It is clear that the magnitude of depends upon the relative values of L and R in the circuit, the exact relation being derived from the triangle of forces. Fig. 18. E 8 27T/Z Furthermore COS < = EI -~ that is, the cosine of the angle of lag is equal to the ratio 28 ALTERNATING-CURRENT MACHINES. of the volts actually engaged in sending current to the volts impressed in the circuit, and this ratio is again equal to the power-factor as stated in 6. 15. Choke Coils. The term choke coil is applied to any device designed to utilize counter electromotive force of self-induction to cut down the, (flow of current in an alternating-current circuit. Disregarding losses by hyster- esis, a choke coil does not absorb any power, except that which is due to the current passing through its resistance. It can therefore be more economically used than a rheostat which would perform the same functions. These coils are often used on .alternating-current cir- cuits in such places as resistances are used on direct- current circuits. For instance, in the starting devices employed in connection with alternating-current motors, the counter E.M.F. of inductance is made to cut down the pressure applied at the motor terminals. The starter for direct-current motors employs resistance. Since a lightning discharge is oscillatory in character and of enormous frequency, a coil which would offer a negligible impedance to an ordinary alternating current will offer a high impedance to a lightning discharge. This fact is recognized in the construction of lightning arresters. A choke coil of but few turns will offer so great an impe- dance to a lightning discharge that the high-tension, high- frequency current will find an easier path to the ground through an air gap suitably provided than through the machinery, and the latter is thus protected. Choke coils are also used in connection with alternating- current incandescent lamps, to vary the current passing through them, and in consequence to vary the brilliancy. CAPACITY. 29 CHAPTER III. CAPACITY. 16. Condensers Any two conductors separated by a dielectric constitute a condenser. In practice the word is generally applied to a collection of thin sheets of metal separated by thin sheets of dielectric, every alternate metal plate being connected to one terminal of the instru- ment, the intervening plates to the other terminal. The Leyden jar is also a common form of condenser. The office of a condenser is to store electrical energy by utilizing the principle of electrostatic induction. If a con- tinuous E.M.F. be applied to the terminals of a condenser, a current will flow, large at first and gradually diminish- ing, till the plates of the condenser have been charged to an electrostatic difference of potential that equals and opposes the electrodynamic pressure applied. Then there is a balance of E.M.F.'s, and no current will flow if there be no leakage. A frequent misconception as to the capacity of a con- denser is that it is equal to the quantity of electricity it will hold. The quantity of electricity a given condenser will hold is directly proportional to the tension of the charge, and a consideration of this fact leads to the follow- ing definition : The capacity of a condenser is numerically equal to the quantity of electricity with which it must be charged in 30 ALTERNATING-CURRENT MACHINES. order to raise the potential difference between its terminals from zero to unity. If the quantity and potential be measured in c. G. s. units, the capacity, c, will be in c. G. s. units. If practical units be employed, the capacity, c, is expressed in farads. The farad is the practical unit of capacity. A condenser whose potential is raised one volt by a charge of one cou- lomb has one farad capacity. The farad is icr 9 times the absolute unit, and even then is too large to conveniently express the magnitudes encountered in practice. The term microfarad (y^oToro" ^ arac O ' 1S m most general use. In electrostatics, both air and glass are used as dielec- trics in condensers ; but the mechanical difficulties of con- struction necessitate a low capacity per unit volume, and therefore render these substances impracticable in electro- dynamic engineering. Mica, although it is expensive and difficult of manipulation, is generally used as the dielectric in standard condensers and in those which are intended to withstand high voltages. Many commercial condensers are made from sheets of tinfoil, alternating with slightly larger sheets of paraffined paper. Though not so good as mica, paraffin will make a good dielectric if properly treated. It is essential that all the moisture be expelled from the paraffin when employed in a condenser. If it is not, the water particles are alternately attracted and repelled by the changes of potential on the contiguous plates, till, by a purely mechanical action, a hole is worn completely through the dielectric, and the whole condenser rendered useless by short-circuit. Ordinary paper almost invariably contains small particles of metal, which become detached from the calendar rolls used in manufacture. CAPACITY. 31 These occasion short-circuits even when the paper is doubled. A distinctly different form of condenser is the electro- lytic condenser. It consists of two electrodes dipping into an electrolyte, as, for instance, two carbon electrodes in zinc sulphate. A charge of electricity will deposit zinc upon one electrode and set up an E.M.F. of polariza- tion. Such condensers should not be subjected to volt- ages in excess of their E.M.F. of polarization. Electro- lytic condensers have about the same volume as other condensers of the same volt-ampere capacity. The maximum voltage that may be applied to a con- denser is limited by the dielectric strength of the material employed. If this limit be exceeded, the dielectric will be ruptured, which renders the condenser useless. The ohmic resistance of condenser dielectrics is not infinite. There is always a leakage from one charged plate to the other through the insulation and over its surface. Poor insulation may occasion a considerable loss, which appears as heat in the apparatus when in use. There is also a dielectric hysteresis which is analogous to magnetic hystere- sis in iron. A dielectric with a high hysteretic constant may consume considerable power when in use, which will also appear as heat. The capacity of a condenser may be calculated by using the following formula : C = .000225 An / ' where C capacity in microfarads, A = area of dielectric between two conducting plates * in square inches, ALTERNATING-CURRENT MACHINES. n = number of sheets of dielectric, / = thickness of dielectric in mils, k specific inductive capacity of dielectric as obtained from the following table. TABLE. Glass 3 to 7 Ebonite 2.2 to 3 Gutta-percha 2.5 Paraffin 2 to 2.3 Shellac 2.75 Mica 6.6 Beeswax 1.8 Kerosene 2 to 2.5 17. Connection of Condensers in Parallel and in Series. -Condensers may be connected in parallel as in Fig. 19. If the capacities of the individual condensers be respectively C^ C 2 , C s , etc., the capacity C of the combina- tion will be The parallel arrangement of sev- Fig * l9 ' eral condensers is equivalent to in- creasing the number of plates in one condenser. An increase in the number of plates results in an increase in the quantity of electricity necessary to raise the potential difference between the terminals of the condenser one volt ; that is, an increase in the capacity results. If the condensers be connected in series, as in Fig. 20, the capacity of the combination will be C= " '+i + i+ : .. v Cj C 8 C 9 CAPACITY. 33 For, if a quantity of positive electricity, Q, flow into the left side of C lt it will induce and keep bound an equal neg- ative quantity on the right side of C v and will repel an equal positive quantity. This last quantity will constitute the charge for the left side of The operation is repeated in the case of each of Fi - 20 - the condensers. It is thus clear that the quantity of charge in each condenser is Q. The impressed E.M.F. must consist of the sum of the potential differences on the separate condensers. Let these differences be respectively E lt Ep E S) etc. Then the impressed E.M.F. But E, = E 2 = E z = etc., <-i c 2 c s and also, E = -, o therefore =+ or 63 63 As an example, consider three condensers of respective capacities of i, 2, and 5 microfarads. Since the factor to reduce to farads will appear on both sides of the equations, it may here be omitted. With the three in multiple (Fig. 19), the capacity of the combination will be C = i + 2 + 5 = 8 mf . 34 ALTERNATING-CURRENT MACHINES. With the three in series (Fig. 20), C = = .588 mf. With the two smaller in parallel and in series with the larger (Fig. 21), = 1.875 mf- 1 + 2 5 Fig. ax. With the two smaller in series and in parallel with the larger (Fig. 22), Si + 5 = 5-666 mf. If with any condensers d = C 2 = C 3 = then, with n in multiple, = C., and with n in series, C = C\. It is interesting to note that the formulas for capacities in parallel and in series respectively are just the reverse of those for resistances in parallel and in series respectively. 18. Growth of Current in a Condensive Circuit. The opposition to a flow of current which is caused by a con- CAPACITY. 35 denser is quite different from that which is caused by a resistance. To be sure, there is some resistance in the leads and condenser plates, but this is generally so small as to be negligible. The practically infinite resistance of the condenser dielectric does not obstruct the current as an ordinary resistance is generally considered to do. The dielectric is the seat of a polarization E.M.F. which is de- veloped by the condenser charge and which grows with it. It is a counter E.M.F. ; and when it reaches a value equal to that of the impressed voltage, the charging current is forced to cease. To find the current at any instant of time, /, in a circuit (Fig. 23) containing a resistance R and a capacity C, the constant impressed pressure E must be considered as consisting of two variable parts, one E r , being active in sending current through the re- sistance, and the other part, E c , being required to balance the po- f igm 23> tential of the condenser. Then at all times E = E; + E:. Let time be reckoned from the instant the pressure E is applied ; when, therefore, / = o and 7 = . Consider the JK. current at any instant of time to be 7'. Then if it flow for dt seconds it will cause dQ coulombs to traverse the circuit, and from which e-fr*. 36 ALTERNATING-CURRENT MACHINES. By definition, _ Q' therefore, And by Ohm's Law, so at this instant of time E = E! ~C~ whence = RCI'+ f /' in the positive direction from the negative maximum to the positive maximum of pressure, and in a negative direction from the positive maximum to the negative maximum, as described above. This necessitates that the zero values of current occur at the maximum values of pressure ; and since the curves are both sinusoids, their relation may be plotted as in Fig. 25. It is immediately seen that these curves are at right angles, as described in 5, and that the current leads the pressure by 90. Reference again to the hydraulic analogy will show that the condenser is completely charged at the instant of maximum positive pressure, discharged at the instant of zero pressure, charged in the opposite direction at the in- CAPACITY. 39 \3 Fig. 26. stant of maximum negative pressure, and finally dis- charged at the instant of the next zero pressure. Thus the charge is zero at 3 the maximum current flow, and at a maximum at zero current, that is, when the current turns and starts to flow out. These points are marked in Fig. 26. 21. Current and Voltage Relations. If a sinusoidal pressure E of frequency/ be impressed upon a condenser, the latter is charged in - . seconds, discharged in the 4 / next - seconds, and charged and discharged in the oppo- 4/ site direction in the equal succeeding intervals. The maximum voltage E m = ^2E ( 4), hence the quantity at full charge is The quantity flowing through the circuit per second is This number therefore represents the average current, or = 4 From 4, the effective current whence and 7 = - ^7 av , 2 V2 7=27T/C^, E = 27T/C 7. 40 ALTERNATING-CURRENT MACHINES. The last is an expression for the volts necessary to send the capacity current through a circuit. The expression is called the capacity reactance of the circuit. It is analogous to 2 irfL, the inductance reactance of an induc- tive circuit. If the circuit contain both a resistance R and a capa- city C, the voltage E impressed upon it must be considered as made up of two parts, E r , which sends current through the resistance and is therefore in phase with the current, and E & which balances the counter pressure of the conden- ser and is therefore 90 behind the current in phase. By Ohm's Law E r = RI, and from above c -. 27T/C The impressed E must overcome the resultant of these two E.M.F.'s ; and since they are at right angles E = or i I 7 = ,.. The relation of the E.M.F.'s is shown graphically in Fig. 27, where the current, which is in phase with the pressure E r , is seen to lead the impressed pressure by the angle <. 22. Resistance, Inductance, and Capacity in an Alter- nating-Current Circuit. The general case of an alter- CAPACITY. nat ing-current circuit is one that contains resistance, inductance, and capacity. To derive the expression for current flow in such a circuit, it is but necessary to com- bine the results already found ; and this is most readily done graphically. In 14 it was shown that the counter 2TTfL 27T/C Pig. E.M.F. due to the inductance reactance of a circuit is 2?r/Z, and leads the current by 90. In 21 it was shown that the E.M.F. of capacity reactance of a circuit is -- -~ 2 7T/C and lags behind the current by 90. These two E.M.F. 's are, then, in exactly opposite phases, or 180 apart, and Fig. 29. the resultant reactance is merely their numerical difference. These relations are shown in Fig. 28, where the reactance of inductance is greater than that of condensance, and in Fig. 29, where the latter exceeds the former, the resistance being the same in either case. Clearly the impedance resulting from the three factors R, Z, and C is represented 42 ALTERNATING-CURRENT MACHINES. in direction and in magnitude by the hypothenuse as shown, and the impressed pressure is proportional to this quantity. The general expression for the flow of an alternating current through any kind of circuit is therefore Z7 / = /= , I the quantity within the brackets indicating an angle of lag of current if positive, and an angle of lead if negative. 23. Resonance. If in a circuit containing inductance and capacity as well as resistance, the two former are proportioned so that the expression reduces to the capacity being of a proper magnitude to balance induc- tance. At one instant energy is being stored in the field at the same rate as it is being given to the circuit by the condenser, and at another instant energy is being released from the field at the same rate as it is being stored in the condenser. When this condition prevails, resonance is said to be attained, or the circuit is said to be in tune. If the capacity and inductance be in parallel, enormous CAPACITY. 43 currents may flow between the two. This is because the two are balanced, and the one is at any time ready to receive the energy given up by the other ; and a surging once started between them receives periodical increments of energy from the line. This is analogous to the well- known mechanical phenomena that a number of gentle, but well-timed, mechanical impulses can set a very heavy suspended body into violent motion. The frequency of these impulses must correspond exactly to the natural period of oscillation of the body. If the capacity and inductance be in series, the differ- ence of potential between the terminals of either may be far greater than the E.M.F. impressed upon the circuit. In the first case damage is likely to result from the overloading of the conductors between the inductance and the capacity, even to burning them out, while in the second case the pressure may rise to such a point as to puncture the insulation of all the apparatus in the circuit of the generator itself. even that L 44 ALTERNATING-CURRENT MACHINES. CHAPTER IV. PROBLEMS ON ALTERNATING-CURRENT CIRCUITS. 24. Definitions of Terms. In considering the flow of alternating currents through series circuits and through parallel circuits, continual use must be made of various expressions, some of which have been denned during the development of the previous chapters. For convenience the names of all the expressions connected with the general equation E will be given and defined. / is the current flowing in the circuit. It is expressed in amperes, and lags behind or leads the pressure, by an angle whose value is , 2 v " = tan- 1 - E is the harmonic pressure, of maximum value V2 E, which is applied to the circuit, and has a frequency/. It is expressed in volts. R is the resistance of the circuit, and is expressed in ohms. It is numerically equal to the product of the im- pedance by the cosine of <. PROBLEMS. 45 L is the inductance of the circuit, and is expressed in henrys. C is the localized capacity of the circuit, and is expressed in farads. 2 TT/L is the inductive reactance of the circuit, and is expressed in ohms. 2TT/C is the capacity reactance, or capacitance, of the circuit, and is expressed in ohms. is the reactance of the circuit, and is expressed in ohms. It is numerically equal to the product of the impedance by the sine of <. is the impedance or apparent resistance of the circuit, and is expressed in ohms. the 'reciprocal of the impedance, is the admittance of the circuit. It is expressed in terms of a unit that has never CONDUCTANCE Fig. 30. been officially named, but which has sometimes been called the mho. There are two components of the admittance, as shown in Fig. 30. 46 ALTERNATING-CURRENT MACHINES. The conductance of a circuit is that quantity by which E must be multiplied to give the component of / parallel to E. It is measured in the same units as the admittance, and is numerically equal to cos impedance and also to 27T/C, The susceptance of a circuit is that quantity by which E must be multiplied to give the component of / perpen- dicular to E. It is measured in the same units as the admittance, and is numerically equal to sn impedance ' and also to It should be noticed that while admittance is the recip- rocal of impedance, conductance is not the reciprocal of resistance, nor is susceptance the reciprocal of reactance. This becomes evident, upon considering numerical values in connection with the impedance right-angled triangle, e.g., 3, 4, and 5 for the sides. 25. E.M.F.'s in Series Alternating E.M.F.'s that may be put in series may differ in magnitude, in frequency, in phase relation, and in form or shape of wave. Forms other than that of the sinusoid need not be discussed. PROBLEMS. 47 E.M.F.'s of different frequencies in series will give an irregular wave-form whose maximum values will recur at intervals. The duration of these intervals is the least common multiple of the durations of the component half -cycles. If two harmonic E.M.F.'s of the same frequency and phase be in series, the re- sulting E.M.F. is merely the sum of the separate E.M.F.'s. This condition is shown in Fig. 31, in which the two E.M.F.'s are plotted Fig- 31- together, and . the resulting E.M.F. plotted by making its instantaneous values equal to the sum of the correspond- ing instantaneous values of the component E.M.F.'s. The maximum of the resultant E.M.F. is evidently = E lm and since and = V2 V2 as was stated. If two E.M.F. 1 s of the same frequency, but exactly opposite in phase, be placed in series, it may be similarly shown that the resultant E.M.F. is the numerical differ- ence of the component E.M.F.'s. This case may occur in the operation of motors. The most general case that occurs is that of a number of alternating E.M.F.'s of the same frequency, but of 4 8 ALTERNATING-CURRENT MACHINES. different magnitudes and phase displacements. The changes in magnitude and phase and the ~oh ise . elation of the resulting curve of E.M.F. are shown in Fig. j2, where recourse is had once again to the harmonic shadowgraph, But two components, E l and E 2 , are treated, whose phase displacement is < r The radii vectors E im and E tm are laid off from o with the proper angle L between them, and the shadows traced by their extremities are shown in the dotted curves. The instantaneous value of the result- ant E.M.F. is the algebraic sum of the corresponding in- Fig. 32- stantaneous values of the component E.M.F.'s, and the resultant curve of E.M.F. is traced in the figure by the solid line. But this solid curve is also the trace of the ex- tremity of the line E m , which is the vector sum (the result- ant of the force polygon) of the component pressures, E im and E 2m . This is evident from the fact that any instan- taneous value of the resultant pressure curve is the sum of the corresponding instantaneous values of the component curves, or ( 3) E'= E lm sin / + E 2m sin (to/ + ---E- 2 - - E- 3 E = and ] 2 sn sn cos cos .,-70 Fig. 34- As a numerical example, suppose three alternators, Fig. 33, to be connected in series. Suppose these to give sine waves of pressure of values E l = 70, E, z = 60, and E 3 = 40 ALTERNATING-CURRENT MACHINES. R 2 L 2 volts respectively. Considering the phase of l to be the datum phase, let the phase displacements be <^ = o, < 2 = 4O, and ^ = 75, respectively. It is required to find E and . Completing the parallelograms or completing the force polygon as shown in Fig. 34, it is found that E = 148.7 volts and < = 32.1. 26. Polygon of Impedances. Consider a circuit having a number of pieces of apparatus in series, each of which may or may not possess resistance, inductance, and capacity. There can be but one current in that circuit when a pressure is applied, and that current must have the same phase throughout the circuit. The pressure at the terminals of the various pieces of apparatus, necessary to maintain through them this current, may, of course, be of different magnitude and in the same or different phases, being dependent upon the values of R, L, and C. Therefore to de- termine the pressure necessary to send a certain alternating cur- rent through such a series circuit, it is but necessary to add vec- torially the pressures needed to send such a current through the separate parts of the circuit. This is readily done graphically. Fig. 35 shows the pressures (according to 22) neces- sary to send the current / through several pieces of ap- 35- PROBLEMS. 51 paratus, and the combination of these pressures into a polygon giving the resultant pressure E necessary to send the current / through the several pieces in series. In these diagrams, impedance is represented by the letter Z. 6~ and C 3 are localized, not distributed capacities. For practical purposes, the quantity /, which is common to each side of the triangle, may be omitted ; and merely the impedances may be added vectorially in a " polygon of impedances," giving an equivalent impedance, which, when multiplied by /, gives E. Inspection of the figure shows that the analytical ex- pression for the required E is The pressure at the terminals of any single part of the circuit is It is evident that E l + E v + ....> E, and it is found by experiment that the sum of the potential differences, as measured by a voltmeter, in the various parts of the circuit, is greater than the impressed pressure. 27. A Numerical Example Applying to the Arrangement Shown in Fig. 35. Suppose the pieces of apparatus to have the following constants : 2 ALTERNATING-CURRENT MACHINES. ? i = 85 ohms, Z x = .25 henry, C = .000018 farad (18 mf.) ? 2 = 40 ohms, Z 2 = .3 henry, C s = .000025 farad, {4 = 100 ohms. With a frequency of 60 cycles whence (0 = 377 it is required to find the pressure necessary to be applied to the circuit to send 10 amperes through it. Fig. 36. The completion of the successive parallelograms in Fig. 36, is equivalent to completing the impedance poly- gon, and the parts are so marked as to require no explana- tion. The solution shows that the equivalent impedance, ^=229.5 ohms, that the equivalent resistance (= actual resistance in series), ^ = 225 ohms, that the equivalent re- actance is condensive and equals 46. 2 ohms, and that = PROBLEMS. 53 11.55 of lead. Hence the pressure required to send 10 amperes through the circuit is E = 10 x 229.5 = 22 95 v lts. To obtain the same results analytically E = 10 [85 + 4o-|-ioo] 2 4-[(i47.3 94- 2 ) 113.1 + io6.2] 2 , E = 2295 volts. The voltages at the terminals of the various pieces of ap- paratus are : E 1 = 10 V85 2 + (147.3 94- 2 ) 2 I001 volts, 1 13. i 2 =1200 E z = 10 Vo 2 + io6.2 2 =1062 " 4 = 10 Vioo 2 -f- o 2 = 1000 " Et + Et + Et + Ei =-4263 " which is greater than ^ = 2195 volts; showing that the numerical sum of the pressures is greater than the im- pressed pressure ; while the vectorial sum of the separate pressures is equal to the impressed pressure. 28. Polygon of Admittances. If a group of several impedances, Z } , Z^ etc., be connected in parallel to a common source of harmonic E.M.F. of E volts, their equivalent impedance is most easily determined by con- sidering their admittances FJ, F 2 , etc. The currents in these circuits wouid be The total current, supplied by the source, would be the vector sum of these currents, due consideration being given to their phase relations. Calling this current 7, the equation I=EY can be written, where Fis the equivalent admit- 54 ALTERNATING-CURRENT MACHINES. tance of the group. To determine F, a geometrical addition of F 1? F 2 , etc., must be made, the angular relations being the same as the phase relations of f lt 1^ etc., respectively. The value of the equivalent admittance may therefore be represented by the closing side of a polygon, whose other sides are represented in magnitude by the several admit- tances F x , F 2 , etc., and whose directions are determined by the phase angles of the currents I lt 7 2 , etc., flowing through the admittances respectively. The equivalent im- pedance then is equal to the reciprocal of F. The sum of the instantaneous values of the currents in the branch circuits is equal to the corresponding instantaneous values Fig. 37. in the supply main. As, however, the maximums occur at various times, the sum of the effective currents in the branches is generally greater than the main supply current. Fig. 37 is a polygon of admittances, showing the method of obtaining the admittance F and its phase angle referred to a datum line, which is equivalent to a number of parallel admittances, F 1? F 2 , and F 3 , with angles lt <.,, and < 3 , respectively. By taking its reciprocal, the equivalent admittance can be transformed into the equivalent impedance. A convenient dimensional scale should be employed. The impedance may be resolved into its equivalent reactance Fig. 38. PROBLEMS. and its equivalent resistance. The equi- valent resistance is not the resistance of the parallel arrangement as measured by direct -current methods. As a numerical example, consider the same apparatus as was used in the pre- ceding example, 27, to be arranged in parallel, as in Fig. 38. All the other conditions and values are as stated before. It is re- quired to find the current that will flow through the mains when ten volts are impressed Y on the circuit. The ** . r<~>^ diagram, Fig. 39, is self-explanatory. The solution shows that the equivalent From this Fig. 39- admittance Y .0224 and that 16.1. the equivalent impedance Z = = 44.6 ohms, .0224 the equivalent reactance o>Z I = Zsm = 12.4 ohms, L < C J and the equivalent resistance R = Z cos = 42.9 ohms. The current that will flow under a pressure of 10 volts is = = 10 x .0224 = .224 amperes. 56 ALTERNATING-CURRENT MACHINES. If a circuit have some impedances in series and some in parallel, or in any series parallel combination, the equivalent impedance can always be found by finding the equivalent impedances of the several groups, and then combining these equivalent impedances tc get the total equivalent impedance sought. ALTERNATORS. 57 CHAPTER V. ALTERNATORS. 29. Single-phase Alternators As is the case with direct-current machines, alternators have a field and an armature. The direct-current machine's commutator is replaced, in the single-phaser, by a pair of slip-rings ; and the current, instead of being rectified, is lead out as alternating current by brushes playing on the rings, as described in 29, vol. i. Revolving field and inductor alternators differ from this arrangement, as will be shown hereafter. It is necessary that all but the very smallest alternators should be multipolar to fit them to commercial require- ments. For alternators must have in general a frequency between 25 and 125 cycles per second; the armature must be large enough to dissipate the heat generated at full load without its temperature rising high enough to injure the insulation ; and finally, the peripheral speed of the armature cannot safely be made to greatly exceed a mile a minute. With these restrictions in mind, and knowing that a point on the armature must pass under two poles for each cycle, it becomes evident that alterna- tors of anything but the smallest capacity must be multi- polar. In practice it is quite as common to have the field of an alternator revolve inside the armature as to have the 58 ALTERNATING-CURRENT MACHINES. armature revolve. In a few instances, notably at Niagara, the fields revolve outside the armature. The chief advan- tage of the revolving field is that it avoids the collection of high-tension currents through brushes, since the arma- ture may be permanently connected up, and only low- tension direct current need be fed through the rings to the field. Other advantages are increased room for arma- ture insulation, and, in polyphasers, the necessity for only two instead of three or more slip-rings. 30. Polyphase Alternators Single-phase currents are satisfactory for lights, but not, as yet, for power. As polyphase currents are equally well adapted to both pur- poses, and since they are generally more economical of transmission than the single-phase, they are much more generally employed. If a motor be operated on a single- phase circuit, the supply of power to it is pulsating. These pulsations occur with great rapidity, there being in the case of unit power factor two for each cycle. A single-phase motor must be larger for the same capacity than a polyphase motor. Windings for any number of circuits or phases may be placed on a single-armature core, and these may each be separately connected to an outside circuit through slip- rings, or they may be connected together in the armature according to some scheme whereby one slip-ring will be common to two phases. These windings can be placed so that the E.M.F.'s generated therein will have any desired phase relations with each other. It is customary to place them so that the E.M.F.'s of a two-phase or four-phase system are 90 apart, of a three-phase system are 120 apart, of a six-phase system are 60 apart. ALTERNATORS. In the following diagrams the curled lines are supposed to represent armature windings, which revolve in a bipolar field. In some cases they are supposed to be wound on cores so as to form pole armatures and in the other cases to form ring armatures. The dots at the terminals repre- sent points of transition between slip-rings and brushes, which are in connection with line wires. It is desirable to consider the relations between the E.M.F.'s generated in the armature coils and the pressure between the line-wires, as well as between the currents in the armature coils and the currents in the line-wires. The assumption is made that the different phases are equally loaded, both as to current and as to its phase. The system is then said to be balanced. It is further assumed that the effective E.M.F. in each armature coil is E volts, and the effective current / amperes. 31. Two-phase Systems In the case of two coils and four wires, the pressure is E volts between the wires attached respectively to each coil. There is no connection between the two coils and their wires. In case three wires be employed, as shown in Fig. 40, the pressure between m and n or between / and n is E volts, and ^fzE volts between / and m. in / and m and V2/ in n. Fig. 40. / amperes flows 32. Four-phase or Quarter-phase Systems When con- nected, as in Fig. 410, i.e., star connected, the pressure between / and m or n and / is 2E volts ; between n or / and / or m it is *^2,E volts. The current in each line-wire is / amperes. If connected as in Fig 41^, i.e., ring con- 6o ALTERNATING-CURRENT MACHINES. nected, the pressure is E volts between / and n, n and m, m and /, or p and /, and ^2E volts between / and m, or n and/. The current in each line-wire is V2/ amperes. d b 02J Fig. 41. 33. Three-phase Systems The pressure and current relations in three-phase apparatus are often puzzling to the student. Consider three similar coils, x> y, and z, on a ring armature, each covering 120, as in Fig. 42^. The E.M.F's generated in these coils, when they are rotated in a bipolar field, will have the same maximum values, but they will differ in phase from each other by 120. If two of the coils, x and y, be connected as in b, then the pres- sure between the free terminals would be the result of adding the two E.M.F.'s at 120 with each other. If, instead of this connection, the one shown in c be made, known as the star connection or Y connection, the pres- sure between the free terminals would be the result of ALTERNATORS. 6l subtracting the E.M.F. of coil y from that ofxat 120. Subtraction is necessary because the connections of coil y to the circuit have been reversed. To subtract one quantity from another it is but necessary to change its sign and add. Therefore the pressure between the free terminals is that which results from adding the E.M.F '.'s of x and y at 300 (= 120 + 1 80) as shown in Fig. 43. It is ^I^E volts. The star connection is generally represented as in Fig. 44, where the pressure between any two line-wires is \T$E volts, and the current in each line-wire is / amperes. If the three coils be connected as in Fig. 45, the result is termed a delta (A) or mesh-connection. The pressure be- tween any two of the line-wires is E volts. Each line-wire is supplied with current from two coils, connection being made at the junction between the beginning of one coil and the ending of the other. The value of the current in each wire is VJ/ amperes. This results from subtracting the current in one coil from that in the other at 120, which, as before, is the same as adding the currents at 300. The power which is delivered by a three-phase machine is not altered by changing the method of connection. In one case each phase is supplied with / amperes at V3 E volts, in the other case with V^ / amperes at E volts. Fig. 44. Fig. 45- 62 ALTERNATING-CURRENT MACHINES. At any instant the current in one wire of a three-phase system is equal and opposite to the algebraic sum of the currents in the other two wires. This is clearly shown in Fig. 46, where the curve found by adding at each instant the ordi- nates of two of the three-phase currents is similar, exactly equal, and opposite to the third current. 34. Electromotive Force Generated In 13, vol. i., it was shown that the pressure generated in an armature is E av = where and p = number of pairs of poles, < = maxwells of flux per pole, V revolutions per minute, S = number of inductors. In an alternating current E k^E av , where k l is the form-factor, i.e., the ratio of the effective to the average E.M.F. Hence in an alternator yielding a sine wave E.M.F., v E = 2.22 v I0 60 Inasmuch as/ g- represents the frequency,/, E= 2.22&S/IO- 8 . An alternator armature winding may be either concen- trated or distributed. If, considering but a single phase, there is but one slot per pole, and all the inductors that are intended to be under one pole are laid in one slot, then ALTERNATORS. 63 the winding is said to be concentrated, and if the inductors are all in series the above formula for E is applicable. If now the inductors be not all laid in one slot, but be dis- tributed in n more or less closely adjacent slots, the E.M.F. generated in the inductors of any one slot will be - of that generated in the first case, and the pressures in the differ- ent slots will differ slightly in phase from each other, since they come under the center of a given pole at different times. The phase difference between the E.M.F. gener- ated in two conductors which are placed in two successive armature slots, depends upon the ratio of the peripheral distance between the centers of the slots to the peripheral distance between two successive north poles considered as 360. This phase difference angle _ width slot -h width tooth ~~ circumference armature * no. pairs poles If the inductors of four adjacent slots be in series, and if the angle of phase difference between the pressures generated in the successive ones be , then letting lf E^ Eg and E represent the respective pressures, which are fcv. Fig. 47- supposed to be harmonic, the total pressure, E, generated in them is equal to the closing side of the polygon as shown in Fig. 47. Obviously E < E^ -\- y + 8 4- E f If the winding had been concentrated, with all the indue- 6 4 ALTERNATING-CURRENT MACHINES. .97 .96 .94 .90 ction of Pole Distar Occupied by jSlotsj \\\ Slot. tors in one slot, the total pressure generated would have been equal to the algebraic sum. The ratio of the vector sum to the algebraic sum of the pressures generated per pole and per phase is called the distribution constant. Not only may the number of slots under the pole vary, but they may be spaced so as to occupy the whole surface of the armature between succes- sive pole centers (the peripheral distance between two poles is termed the pole distance), or they may be crowded together so as to occupy only one- half, one-fourth, or any other fraction of this space. Both the number of slots and the fractional part of the pole dis- tance which they occupy affect the value of the distri- bution constant. A set of curves, Fig. 48, has been drawn, showing the values of this constant for various conditions. Curves are drawn for one slot (concentrated winding), 2, 3, 4 slots in a group, and many slots (i.e., smooth core with wires in close contact on the surface). The ordinates are the distribution constants, and the abscissae the frac- tional part of the pole distance occupied by the slots. The distribution constant, k.^ must be introduced into the formula for the E.M.F. giving >L -8 60 ' ' .3 Fig. 48. 2 Slots. 3 Slots. 4 Slots. Many Slots. ALTERNATORS. or, for sine waves, E = 2.22 35. Armature Windings -- Some simple diagrams of the windings of multipolar alternators are given in Fig. 49 et seq. The first is a single-phase concentrated winding, with the winding which is necessary to make it two-phase in dotted lines. If the two windings be electrically con- nected where they cross at point P the machine becomes /' 4- POLE. / SINGLE PHASE. CONCENTRATED. ADDITION OF DOTTED WINDING MAKES IT TWO PHASE 4- POLE. \ 3- PHASE. A CONCENTRATED. Fig. 49. Fig. 50. a star-connected four-phaser. Fig. 50 is a three-phase, A connected, concentrated winding. Fig. 5 i is the same but y connected. The common junction of the windings would have to be provided with a slip-ring if it were desired to operate a three-phase, four-wire system with the fourth wire connected to the machine. Fig. 52 is a three-phase, A connected winding distributed over two slots. In all these diagrams the radial lines represent the inductors ; other lines the connecting wires. The induc- tors of different phases are drawn differently for clearness. 66 ALTERNATING-CURRENT MACHINES. Where but one inductor is shown, in practice there would be a number wound into a coil and placed in the one slot. For simplicity all the inductors of one phase are shown in series. In concentrated windings, all inductors of one \ 4- POLE. 3- PHASE. Y. CONCENTRATED. \ Fig. 51. 4- POLE. 3- PHASE. A. DISTRIBUTED. Fig. 52. phase carrying current in the same direction could be connected in multiple if desired ; but with distributed wind- ings, the coils cannot all be placed in multiple, because the small phase differences between them would set up local currents and give rise to undue heating. To determine the interior connections for a three-phase A winding, place the inductors of a coil of one phase under the centers of the poles, then a maximum pressure in a given direction is generated therein. Since the algebraic sum of the pressures around the A must be zero, the other two phases must be connected so that their pressures oppose the first. To determine the y connection, place the inductors of one phase under the centers of the poles. The E.M.F. of this phase will now be at a maximum, say, away from the common center. The other two phases ALTERNATORS. 6/ must be so connected as to have F.M.F.'s toward the com- mon center at this instant. 36. Armature Reaction The armature reaction of an alternator consists of two parts, distortion and magnetiza- tion or demagnetization. These depend upon the arma- ture ampere-turns and upon the lag or lead of the armature current. The maximum pres- sure is generated in a coil when its opposite inductors are re- spectively under the centers of north and south poles. This con- dition is represented in Fig. 53. If the armature current be in phase with the pressure, /, in the coils coincides with E m and poles on the armature are formed as shown. It is seen that the M.M.F.'s both of the field and of the armature conspire to concentrate the flux in the trailing pole tips. So with / in phase with E the armature M.M.F. chiefly effects a distortion of the lines, entailing a greater flux density, hence a lower per- meability, and also a greater length of air-gap path. This slightly decreases the flux, and affects the regulation of the alternator. If, now, the current be lagging, the armature will have reached a position in advance, at the instant of maximum current. Therefore, like poles of the field and of the armature will be more directly opposite to each other. The distorting influence will be present in a degree ; and there will be considerable demagnetization of the field, due to the opposing M.M.F.'s of the armature and field ampere- turns. If the current be leading, then, at the instant of maximum current, a south armature pole will be more 68 ALTERNATING-CURRENT MACHINES. nearly opposite to a north field pole, and their M.M.F.'s will be cumulative. The field will be strengthened if the magnetizing reaction exceeds in effect the skewing reac- tion. Alternators have a much better regulation on non- inductive loads than on inductive loads. 37. Armature Inductance The impedance of an alter- nator armature is made up of its ohmic resistance, R, combined at right angles with its reactance, 2 -nfL. In practice the inductance, L, is likely to be so great that R becomes negligible, and the impedance equals the reac- tance. The armature reactance may or may not be an appreciable part of the impedance offered by the completed circuit. If it is appreciable, then the current in the circuit will lag even with a non-inductive load. In any case there will be loss of voltage due to armature impedance which (when R is negligible) is equal to 2 nfLI. This is at right angles to the current, and must be properly combined with / times the equivalent impedance of the external cir- cuit to determine the pressure actually generated in the machine. In special cases the armature reactance is the predominant feature of the circuit ; for instance, alternators for series arc lighting are made with so great a reactance that the impedance of the external circuit within the limits of operation is negligible in comparison. The altera- tion in the value of this impedance does not, then, appre- ciably alter the total impedance of the circuit, and the alternator therefore operates as a constant-current gen- erator. Many commercial alternators have sufficient arma- ture reactance to prevent their injuring themselves on dead short circuit for a limited time. It is necessary that armatures should have some considerable inductance ALTERNATORS. 69 when alternators are to be operated satisfactorily in parallel. 38. Synchronous Reactance When an alternator is op- erating on a load, the pressure, which would be generated on open circuit at the same speed and excitation, is made up of the following parts, and might be found by adding them together in their proper phase relations : (a) terminal voltage, E, (b) ohmic drop in armature, 77?, in phase with the cur- rent, (c) armature inductance drop, 90 with the current, (d) deficit of actually generated volts due to increase of magnetic reluctance accompanying distortion, (e) deficit or increment of actually generated volts due to the demagnetization of a lagging current or the mag- netization of a leading current. All the parts, except the first mentioned, can be grouped together, and be dealt with collectively by the use of a quantity called the synchronous impedance. It is that im- pedance, which, if connected in series with the outside cir- cuit and an impressed voltage of the same value as the open-circuit voltage at the given speed and excitation, would permit a current of the same value to flow as does flow. This quantity for any load can be determined experimen- tally with ease. The synchronous impedance has two fac- tors, namely, the armature resistance and a quantity termed the synchronous reactance. The two, when combined at right angles, give the synchronous impedance. Since the synchronous impedance takes account of all the diverse causes of voltage drop above enumerated, it is clear that it has not a physical existence, but is merely a ALTERNATING-CURRENT MACHINES. fiction. It is of great use in determining the performance of a machine. Its value is the same for all excitations of the field, but is somewhat different for various loads. These two facts afford a very convenient means of deter- mining its value. Run the alternator at its proper speed. Short-circuit the armature through an ammeter. Excite the field until the ammeter indicates the desired load. Then open the load circuit and read the terminal voltage. The quotient of the volts by the amperes is the synchron- ous impedance. It may happen that the resistance of the armature is negligibly small, in which case the synchron- ous reactance equals the synchronous impedance. 39. Saturation Coefficient A no-load saturation curve of an alternator may be obtained by measuring the termi- nal voltage corresponding to various strengths of field cur- rent, when the machine is running at its proper speed and without load. Laying off E.M.F.'s, E as ordinates and exciting currents, I f) as abscissae, a curve is found as in Fig. 54. Field Excitation The ratio is called dE the no-load saturation co- efficient of the machine. Another curve, called the load-saturation curve can be obtained by using a variable non-inductive resistance for maintaining the constant full load. The terminal volts corresponding to various field excitations are read on a 54. ALTERNATORS. 71 voltmeter. This curve will approximately parallel the no- load saturation curve. It will have a zero voltage value for that excitation which causes sufficient voltage to send the full-load current through the synchronous impedance of the armature. A full-load saturation coefficient curve might be drawn from the full-load saturation curve. It will nearly coincide with the other coefficient curve. These saturation curves have forms similar to magneti- zation curves for iron. The knee, however, is less abrupt than is general in an iron curve, because of the unvarying permeability of air, and because the different magnetic parts of the generator do not reach saturation at the same time. If the alternator is normally excited to above the knee of the saturation curve, it will require a considerable increase of field current to maintain the terminal voltage when the load is thrown on, while if normally excited below the knee, a slight increase of excitation will suffice. The regulation is, however, better when the magnetization is above the knee ; that is, with unaltered field strength, the voltage rise upon throwing off the load is less than if the excitation were below the knee. 40. Leakage Coefficient As in direct-current machines, the leakage coefficient of an alternator may be defined as the number of maxwells set up by the field divided by the number of maxwells passing through the armature. It is always greater than unity. Its value depends upon the design of the machine, upon the permeability of the various parts making up the magnetic circuit, upon the load on the machine, and upon the degree of saturation in the fields. In modern commercial machines of size its values lie between i.i and 1.5. 72 ALTERNATING-CURRENT MACHINES. 41. Efficiency The following is abstracted from the Report of the Committee on Standardization of the Ameri- can Institute of Electrical Engineers. Only those por- tions are given which bear upon the efficiency of alternators. They will, however, apply equally well to synchronous motors. The " efficiency" of an apparatus is the ratio of its net power output to its gross power input. Electric power should be measured at the terminals of the apparatus. In determining the efficiency of alternating-current apparatus, the electric power should be measured when the current is in phase with the E.M.F. unless otherwise specified, except when a definite phase difference is in- herent in the apparatus, as in induction motors, etc. Where a machine has auxiliary apparatus, such as an exciter, the power lost in the auxiliary apparatus should not be charged to the machine, but to the plant consisting of the machine and auxiliary apparatus taken together. The plant efficiency in such cases should be distinguished from the machine efficiency. The efficiency may be determined by measuring all the losses individually, and adding their sum to the output to derive the input, or subtracting their sum from the input to derive the output. All losses should be measured at, or reduced to, the temperature assumed in continuous operation, or in operation under conditions specified. In synchronous machines the output or input should be measured with the current in phase with the terminal E.M.F. except when otherwise expressly specified. Owing to the uncertainty necessarily involved in the approximation of load losses, it is preferable, whenever ALTERNATORS. 73 possible, to determine the efficiency of synchronous ma- chines by input and output tests. The losses in synchronous machines are : a. Bearing friction and windage. b. Molecular magnetic friction and eddy currents in iron, copper, and other metallic parts. These losses should be determined at open circuit of the machine at the rated speed and at the rated voltage, + IR in a synchronous generator, IR in a synchronous motor, where / = cur- rent in armature, R = armature resistance. It is undesir- able to compute these losses from observations made at other speeds or voltages. These losses may be determined by either driving the machine by a motor, or by running it as a synchronous motor, and adjusting its fields so as to get minimum cur- rent input, and measuring the input by wattmeter. The former is the preferable method, and in polyphase ma- chines the latter method is liable to give erroneous results in consequence of unequal distribution of currents in the different circuits caused by inequalities of the impedance of connecting leads, etc. c. Armature-resistance loss, which may be expressed by/ I^R ; where R = resistance of one armature circuit or branch, 7 = the current in such armature circuit or branch, and p = the number of armature circuits or branches. d. Load losses. While these losses cannot well be determined individually, they may be considerable, and, therefore, their joint influence should be determined by observation. This can be done by operating the machine on short circuit and at full-load current, that is, by deter- mining what may be called the " short-circuit core loss." 74 ALTERNATING-CURRENT MACHINES. With the low field intensity and great lag of current existing in this case, the load losses are usually greatly exaggerated. One-third of the short-circuit core loss may, as an approximation, and in the absence of more accurate infor- mation, be assumed as the load loss. e. Collector-ring friction and contact resistance. These are generally negligible, except in machines of extremely low voltage. f. Field excitation. In separately excited machines, the I^R of the field coils proper should be used. In self- exciting machines, however, the loss in the field rheostat should be included. 42. Regulation for Constant Potential. Alternators feeding light circuits must be closely regulated to give satisfactory service. The pressure can be maintained constant in a circuit by a series boosting transformer, but it is generally considered better to regulate the dynamo by suitable alteration of the field strength. The simplest method of regulating the potential is to have a hand-operated rheostat in the field circuit of the alternator, when the latter is to be excited from a com- mon source of direct current, or in the field circuit of the exciter, if the alternator is provided with one. The latter method is generally employed in large machines, since the exciter field current is small, while the alternator field current may be of considerable magnitude, and would give a large I^R loss if passed through a rheostat. A second method of regulation employs a composite winding, analogous to the compound windings of direct- current generators. This consists of a set of coils ; one ALTERNATORS. 75 on each pole. These are connected in series, and carry a portion of the armature current which has been rectified. The rectifier consists of a commutator, having as many segments as there are field poles. The alternate segments are connected together, forming two groups. The groups are connected respectively with the two ends of a resis- tance forming part of the armature circuit. Brushes, >S 6-60-900 For J1S d-90-900For JflS CH20-900 For Commutator-Collector ; manner of placing spool* The observer f supposed to be loohtng at race* or pole piece* marKea A and B. The ser-Iea field winding should be nearest the armature -that ift. toward the observer, The arrovws correaporx> t.o t-hose on spool flange*, tne spools being so placed that the arrows point in opposite TO ; then the flux at any time, t, is 3> OT cos w/, and the counter E.M.F., which is equal and opposed to the im- pressed primary pressure E p , may be written (13, vol. i.) . cos CD/) . '"lo 8 " dt and since & m and are constant EP = i o~ 8 # p ci)$ m sin from which and n o> This equation is used in designing transformers and choke coils. The values of <& m for 60 cycle transformers of different capacities, as determined by experiment and use, are shown in the curve, Fig. 75. It is usual in such designs to also assume a maximum flux density, (B m . While the value assumed differs much with different man- ufacturers, it is safe to say that for 25 cycles (B m varies between 9 and 1 4 kilogausses ; for 60 cycles between 6 and 9 kilogausses; and for 125 cycles between 5 and 7 9 6 ALTERNATING-CURRENT MACHINES. kilogausses. The necessary cross-section, A, of iron, neces- sary to give the desired counter E.M.F., as well as the number of turns of wire in the primary, is then found from the above, as 3> m =& m A. (b) With secondary closed through an outside impedance. The flux, which is linked with the primary, is also linked with the secondary. Its variations produce in the secon- Lighting Transformers 10 12 14 16 13 20 Capacity in Kilowatts Fig. 75- dary an E.M.F. T times as great as the counter E.M.F. in the primary, since there are T times as many turns in the secondary coil as in the primary, or If this secondary be closed through an external impedance, a current I s will flow through this circuit. In the secon- dary coil the ampere turns, nj# will be opposed to the ampere turns of the primary, and will thus tend to demag- netize the core. This tendency is opposed by a read- justment of the conditions in the primary circuit. Any demagnetization tends to lessen the counter E.M.F. in the primary coil, which immediately allows more current to THE TRANSFORMER. 97 flow in the primary, and thus restores the magnetization to a value but slightly less than the value on open-circuited secondary. Thus the core flux remains practically con- stant whether the secondary be loaded or not, the ampere turns of the secondary being opposed by a but slightly greater number of ampere turns in the primary. So nJ 8 = n p f p , very nearly, and f s =^f p = -f p . n T The counter E.M.F. in the primary of a transformer accommodates itself to variations of load on the secondary in a manner similar to the variation of the counter E.M.F. of a shunt wound motor under varying mechanical loads. If the secondary load be inductive or condensive, then I s will lag or lead E s by the same angle that I p lags or leads E p) still neglecting R p , R s , (R, and hysteresis. In such case I p is 180 from, or opposite to, 7 S , and E p is oppo- site to E f For a more exact statement than the above, see 54. 47. Equivalent Resistance and Reactance of a Trans- former. If a current of definite magnitude and lag be taken from the secondary of a transformer, a current of the same lag and r times that magnitude will flow in the primary, neglecting resistance, reluctance, and hysteresis. An impedance which, placed across the primary mains, would allow an exactly similar current to flow as this primary current, is called an equivalent impedance, and its components are called equivalent resistance and equivalent reactance. If the whole secondary circuit of a transformer with its load have a resistance R 8 and a reactance X 3 , and if the 98 ALTERNATING-CURRENT MACHINES. primary pressure be E p and the secondary total pressure E s , then the current that will flow in the secondary circuit is X s and it lags behind E t by an angle , whose tangent is Therefore "W?. 2 + X? = ^ . * If the equivalent impedance have a resistance R and a X X reactance X then the ratios and ~ must be equal, since R R * the angle of current lag is the same in both primary and secondary. And since the current in the equivalent im- pedance has the same magnitude as that in the primary and But and t T i E i ,-- therefore, V^? 2 + X z = - = - = - V^ s 2 -h ^ 2 - ry 8 r^ /, r j But I=S- Solving ft = - 2 ^ g , which are the values of the equivalent resistance and re- actance respectively. THE TRANSFORMER. 99 48. Transformer Losses. The transformer as thus far discussed would have 100% efficiency, no power whatever being consumed in the apparatus. The efficiencies of loaded commercial transformers are very high, being gen- erally above 95% and frequently above 98%. The losses in the apparatus are due to (a) the resistance of the elec- tric circuits, (b) reluctance of the magnetic circuit, (c) hysteresis, and (d) eddy currents. These losses may be divided into core losses and copper losses, according as to whether they occur in the iron or the wire of the trans- former. 49. Core Losses. (a) Eddy current loss. If the core of a transformer were made of solid iron, strong eddy cur- rents would be induced in it. These currents would not only cause excessive heating of the core, but would tend to demagnetize it, and would require excessive currents to flow in the primary winding in order to set up sufficient counter E.M.F. To a great extent these troubles are prevented by mak- ing the core of laminated iron, the laminae being trans- verse to the direction of flow of the eddy currents but longitudinal with the magnetic flux. Each lamina is more or less thoroughly insulated from its neighbors by the natural oxide on the surface or by Japan lacquer. The eddy current loss is practically independent of the load. The E.M.F. producing these eddy currents is in phase with the counter E.M.F. of the primary coil, both being produced by the same flux. Its value E e is expressed by p the fraction , where P e is the power loss in watts due A to eddy currents, and ^ is the exciting or no-load primary current. The value of P, is calculated from the following 100 ALTERNATING-CURRENT MACHINES. empirical formula, in which perfect insulation between the laminae is assumed : P e = kvftp&J*, where k = a constant depending upon the reluctivity and resistivity of the iron. v = volume of iron in cm. 3 , / = thickness of one lamina in cm., / = frequency, and ($> m = maximum flux density ( m per cm. 2 ). In practice k has a value of about 1.6 x io~ u . (b) Hysteresis loss. A certain amount of power, P ht due to the presence of hysteresis, is required to carry the iron through its cyclic changes. The value of P h can be calculated from the formula expressing Steinmetz's Law, where v = volume of iron in cm. 3 , f = frequency, (B OT = the maximum flux density, and rj = the hysteretic constant (.002 to .003). The portion of the impressed E.M.F. which must be expended in the primary circuit to balance the hysteretic loss is E - -- h ~ A This is in phase with / 1 . Closely associated with E h is another portion of the impressed E.M.F. which is consumed in producing the cyclical and sinusoidal variations of magnetic flux. This is not easily considered distinct from E h . Consider, however, the primary current. There is but one primary current. THE TRANSFORMER^, //; At any instant of time a portion of it is balanced and its magnetic effect is neutralized by the demagnetizing cur- rent in the secondary ; another portion is balanced by the demagnetizing action of the eddy currents ; and the rem- nant is useful in producing the cyclical variations of the magnetic flux. If the flux be sinusoidal this portion of the current cannot be sinusoidal. This is due to the change in permeability with saturation of the iron core. Neither is the rising current curve the reverse of the fall- ing current curve. This is due to the fact that, owing to hysteresis, the permeability on rising flux is smaller than on falling under a given magnetomotive force. This last portion of the primary current is therefore not sinusoidal. As it is but a small percentage of the total current, it is, however, for convenience generally considered as sinusoi- dal. To send this distorted portion of the primary current requires a portion of the impressed E.M.F., and this is made up of two components, E h in phase with the pri- mary current and discussed above, and E mag at right angles with the primary current. This E mag may be considered as sending that portion of the current sufficient to overcome the magnetic reluctance of the core. Being at right angles with I p it represents no loss of power. During half of the time I p and E mag have the same direction and during the other half they are in opposite directions. The core there- fore alternately receives energy from the circuit and gives it back to the circuit. To determine the value of E mag consider that it must be of such a magnitude as will send through the primary coil, of resistance R p , that portion, 7^, of the main current which produces the flux, i>tb2^\ .^ ALTERNATING-CURRENT MACHINES. Representing the reluctance of the core by 61, and the magnetomotive force necessary to produce the flux < m by 3C, from 21 and 25, vol. i., oc whence I = . , 4 V2 7TH p and E mag = * m - 4 Y2 im p I mag is called the magnetizing current of a transformer. The primary counter E.M.F., E, is less than the primary line voltage by the slight pressure necessary to send this current through the primary resistance, thus, The value of 61 is calculated ( 24, vol. i.) from where / is the length of magnetic circuit, A its cross-sec- tion and p the reluctivity of the iron (-i- permeability. In modern commercial transformers the core loss at 60^ may be about 70% hysteresis and 30% eddy current loss. At 1 25^ it may be about 55% hysteresis and 45% eddy current loss. This might be expected, since it was shown that the first power of /enters into the formula for hysteresis loss, while the second power off enters into the formula for eddy current loss, THE TRANSFORMER. 103 The core loss is also dependent upon the wave-form of the impressed E.M.F., a peaked wave giving a somewhat lower core loss than a flat wave. It is not uncommon to find alternators giving waves so peaked that transformers tested by current from them show from 5% to 10% less core loss than they would if tested by a true sine wave. On the other hand generators sometimes give waves so flat that the core loss will be greater than that obtained by the use of the sine wave. The magnitude of the core loss depends also upon the temperature of the iron. Both the hysteresis and eddy cur- rent losses decrease slightly as the temperature of the iron increases. In commercial transformers, a rise in tempera- ture of 40 C. will decrease the core loss from 5% to 10%. An accurate statement of the core loss thus requires that the conditions of temperature and wave-shape be specified. The core loss is practically constant at all loads, and is the same whether measured from the high-tension or the low-tension side, the exciting current in either case being the same percentage of the corresponding full-load current. The exciting current varies in magnitude with the design of the transformer. In general it will not exceed 5 % of the full-load current, and in standard lighting transformers it may be as low as i%. In transformers designed with joints in the magnetic circuit the exciting current is largely influenced by the character of the joints, increasing if the joint is poorly constructed. In the measurement of core loss, if the product of the impressed volts by the exciting current is less than twice the measured watts (i.e., if cos >.5 or < < 60) there is reason to suspect poorly constructed magnetic joints or higher densities in the iron than good practice allows. 104 ALTERNATING-CURRENT MACHINES. 50. Copper Losses. The copper losses in a transformer are almost solely due to the regular current flowing through the coils. Eddy currents in the conductor are either negligible or considered together with the eddy cur- rents in the core. When the transformer has its secondary open-circuited the copper loss is merely that due to the exciting current in the primary coil, P mag R p . This is very small, much smaller than the core loss, for both I mag and R p are small quantities. When the transformer is regularly loaded the copper loss in watts may be expressed At full load this loss will considerably exceed the core loss. While the core loss is constant at all loads, the copper loss varies as the square of the load. 51. Efficiency. Since the efficiency of induction appa- ratus depends upon the wave-shape of E.M.F., it should be referred to a sine wave of E.M.F., except where expressly specified otherwise. The efficiency should be measured with non-inductive load, and at rated frequency, except where expressly specified otherwise. The efficiency of a transformer is expressed by the ratio of the net power output to the gross power input or by the ratio of the power output to the power output plus all the losses. The efficiency, c, may then be written, V.I. where V % is the difference of potential at the secondary terminals. If the transformer be artificially cooled, as many of the THE TRANSFORMER. IO5 larger ones are, then to this denominator must be added the power required by the cooling device, as power con- sumed by the blower in air-blast transformers, and power consumed by the motor-driven pumps in oil or water cooled transformers. Where the same cooling apparatus supplies a number of transformers or is installed to supply future additions, allowance should be made therefor. Inasmuch as the losses in a transformer are affected by the temperature, the efficiency can be accurately specified only by reference to some definite temperature, such as 2 S C. The all-day efficiency of a transformer is the ratio of energy output to the energy input during the twenty-four hours. The usual conditions of practice will be met if the calculation is based on the assumption of five hours full- load and nineteen hours no-load in transformers used for ordinary lighting service. With a given limit to the first cost, the losses should be so adjusted as to give a maximum all-day efficiency. For instance, a transformer supplying a private residence with light will be loaded but a few hours each night. It should have relatively much copper and little iron. This will make the core losses, which con- tinue through the twenty-four hours, small, and the copper losses, which last but a few hours, comparatively large. Too much copper in a transformer, however, results in bad regulation. In the case of a transformer working all the time under load, there should be a greater proportion of iron, thus requiring less copper and giving less copper loss. This is desirable in that a loaded transformer has usually a much greater copper loss than core loss, and a halving of the former is profitably purchased even at the expense of doubling the latter. 106 ALTERNATING-CURRENT MACHINES. 52. Regulation The definition of the regulation of a transformer as authorized by the American Institute of Electrical Engineers is as follows : " In transformers the regulation is the ratio of the rise of secondary terminal voltage from full-load to no-load (at constant impressed primary terminal voltage) to the secondary full-load volt- age." Further conditions are that the frequency be kept constant, that the wave of impressed E.M.F. be sinusoidal, and that the load be non-inductive. Not the whole primary impressed pressure is operative in producing secondary pressure, for I p R p volts are lost in overcoming the resistance of the primary coil. Besides this there is a flux linked with the primary that does not link the secondary. This induces a counter pressure in the primary which neutralizes a part of the impressed pressure. Such flux, linking one coil but not the other, is called leakage flux. Furthermore, not all of the E.M.F. induced in the secondary is utilizable at the terminals. There is a drop of I s R s volts due to the resistance of the secondary coil, and another drop due to a leakage flux which links the secondary but not the primary. All these drops increase with load, and therefore, neglecting core loss effects, at no load E s =rE pt but on load, E s , then the value of the inductance, Z, is n& ** = s~7 I0 8 / THE TRANSFORMER. IO7 where n is the number of turns in the coil. A method of calculating Z, the equivalent inductance, is given in the next article. The resistance of the secondary causes a drop /, R s . The same effect on the regulation would be caused if the secondary resistance were zero and another resistance rj whose value is R s -^ were inserted in the primary cir- cuit. The imaginary primary drop, resulting from this insertion, has to be but - as great as the actual secondary drop to be as great a percentage of the impressed E, and there is r times as much current to cause it, hence rj R s = --* . The power lost in this imaginary resistance is I*R# and this equals the power really lost in the secondary I?R S , since I p = T/., and s =-;-- In order to calculate the regulation, consider this equiv- alent of secondary drop to be accounted for in the primary. Then for a given impressed E.M.F. on the primary, E p , the terminal voltage on the secondary will be at no load E t rE p , at primary load f p , V g =r [_E p - I p (R p + .#,) cos $ - LI P sin <], where E t = secondary pressure generated, V. = difference of potential at secondary terminals, Z = L p -\- Z 3 as calculated in the next section, and = angle of lag of I p behind JS P . Then from the definition of regulation, when / in the above is made equal to the full-load current, rE V, Regulation == ? - ^ io8 ALTERNATING-CURRENT MACHINES. 53. Calculation of Equivalent Leakage Inductance. The arrangement of one of the most usual kinds of core type transformers called the " type H," is shown in Fig. 76. The coarse wire is wound inside the fine wire, and as these are more generally used as step-down transformers the latter will be called the primary. Fig. 77 shows one leg of the transformer, giving the paths of leakage flux Fig- 76. and the system of nota- tion employed. The discussion is carried on entirely in c. G. s. units. Consider the secondary (coarse wire) coil first. ^ Primary. Secondary i. p --il I ! B - II 1 i\i c 1 '! * < 2 ! g Ji ii i 1 | H ]j H Fig. 77- The M.M.F. tending to send flux through the elemen- sy~ tary portion dx and back through the iron is of the J\. whole M.M.F. of the secondary, so for any element, 00 M.M.F. = 4 irn 8 t' 8 - - * THE TRANSFORMER. IOQ Since the permeability of iron is roughly 1000 times that of air, no appreciable error is introduced by consider- ing the whole reluctance of the circuit of the leakage flux to be in the air portion of that circuit. If it be assumed that the lines of force follow a circular path from the end of the coil to the iron, the length of the air portion of the magnetic circuit for any element is C -f- irx. The use of this value will result in an integral expression, simple enough in theory, but too unwieldy to be introduced on these pages. Since the portion of the air path outside the coil (the curved portion) is a small part of the whole path, no serious error will be introduced by assuming that the leakage flux from any element follows a path V whose length is the average length C + TT The cross- section area of the air part of the magnetic circuit for any element is (2A-{-2B + %x)dx. Therefore the reluctance* of any element is C + -X 61 = 2 (A -f B + 4 x) dx The elementary leakage flux, d, is then _ M.M.F. _ 4 irn,i s x 2 (A -f- B + 4 x) dx X Since this flux links with of the secondary turns, the number of linkages is 8 Trn s i 8 {A H- B -\- 4 x) xdx xn s _ 8 TT n? i g (A + B -f- 4 x) xrdx X -y ' -~ " -- * V"2 A HO ALTERNATING-CURRENT MACHINES. By definition ( 8) the coefficient of self-induction, /, is numerically equal to the number of linkages per unit current. Therefore _ linkages _ 8 irn t *(A -f- B + 4 x) xdx aig : - The limits of the variable x are o and X, therefore 8 r* 8 \ ~V A + **)$**** + ^^l (c+=jr)j This applies to one leg of the transformer. For the two legs, upon reverting to practical units, 16 Z.= , all the terms of which are either absolute numbers or linear dimensions in centimeters. It cannot be objected that this analysis does not take account of the leakage flux that does not travel the whole length of the coil, C. It is a true statement for any length, and therefore might be applied to the elementary length dC, which when integrated would give the result stated above. The value of L p is determined in the same way, and the THE TRANSFORMER. Ill expression therefore is quite similar. There can be no iron in the path of the leakage flux from the outside coil, so the reluctance will be twice as great. The value that is represented by A for the inner coil becomes A 4- 2X -h 2g for the outer. Likewise B is replaced by B -j- 2X + 2g, g being the space occupied by insulation between the coils. Then r 8 (A + 2X+2g) + (B + 2X+2g)+ Z Y L P ~ o * n p * 3(C + 7r y) If the secondary circuit is open the secondary coil is idle, equivalent to so much air, and all the flux set up by the primary is leakage flux. As the secondary resistance can be replaced by an r> equivalent primary resistance, R 3 = ~ , for purposes of calculation, so also the secondary inductance can be re- placed by an equivalent inductance in the primary, Z 3 = '. These values, L p and L^ are to be used in the formula at the end of the last section for determining the regulation of a transformer. 54. Exact Solution of a Transformer In the treat- ment of regulation, efficiency, etc., heretofore, certain small errors have been allowed, due to neglecting the effects of the core, eddy currents, and hysteresis losses. The following graphic solution, adapted from Steinmetz, takes account of all these effects, and is general in all respects. It must first be understood that there are three fluxes to be considered: (i) The useful flux that links both coils. It is not in any definite phase with either I p or I v It is, however, always. at right angles to the E.M.F. it induces, 112 ALTERNATING-CURRENT MACHINES. the direct in the secondary, and the counter in the primary. (2) The leakage flux of the primary coil. This links the primary only, and being independent of F a is always in phase with 7 P . (3) The leakage flux of the secondary coil. This is similarly in phase with /.. Let E s = E.M.F. induced in secondary, V s = difference of potential at secondary ter- minals, E p = impressed primary pressure, E = operative part of E p (E = - ) , I p and I s = primary and secondary currents respectively, 4> p and s = lag of primary and of secondary currents respectively behind E p and E s , y = angle of lag of M.M.F. behind useful flux. The problem is : Given the necessary data of the trans- former, to determine its behavior with any specified load on the secondary. As in Fig. 78, draw the line 3>, representing the direc- tion of the flux, vertically for convenience. In this analy- sis, the no-load exciting current is separated into two com- ponents. One is used in neutralizing the demagnetizing effect of the eddy currents. The other, I h) is the magne- tizing current and is also made up of two components, one in phase with the primary pressure E p and the other at right angles with it. The relative magnitudes of these two components are dependent upon the shape of the hysteresis curve of the iron. Once determined they may be represented as I h cos (3 and I h sin ft where ft is termed the angle of hysteretic lag. When multiplied by E p the first represents the power lost in hysteresis ; the second the power passing backward and forward between the THE TRANSFORMER. magnetic field and the circuit. If to the former the power lost in eddy currents, W e , be added and the two be com- bined with the latter as in Fig. 79 an angle y results, which Fig. 78. represents the lag of the magnetomotive force. Determine the angle y in this manner. Draw the line M.M.F. (Fig. 78) y ahead of < indicating in direction and magnitude the ampere turns which must exist to set up the flux $>. Its value is determined dur- ing the transformer design. Draw from the center the line E , 90 ahead of the flux, representing the operative primary Its length is - E a and as it T We I E pi,, cos/? Fig. 79- pressure opposes the counter primary pressure, it is set ahead of the flux. Draw the line , 90 behind , representing the pressure induced in the secondary. Its length is proportional to the no-load sec- ondary terminal pressure, 114 ALTERNATING-CURRENT MACHINES. The angle s , the lag due to the whole secondary cir- cuit, is known. Draw I 8 at 8 behind E s , and extend it till its length is proportional to the secondary ampere turns, 7 8 ;/.. This line represents one component of the magnetizing force. From this component line and the resultant line M.M.F. determine the other component I p n p . Divide this by n p and the primary current is dis- covered in magnitude and phase. There is a drop of I p R p volts in the primary. The im- pressed pressure that compensates for. this is in phase with 7 p . A counter voltage 90 behind I p will be set up clue to the primary leakage flux. Its value is &L P I p . To over- come this an impressed pressure must be supplied opposite it in phase or 90 ahead of the current. In a side figure vectorially add I p R p in the phase of I p and <*L V I P 90 ahead of this phase. This gives the direction and magni- tude of the drop d p in the primary. Properly add d p to the operative pressure j5" and the necessary impressed pressure E p is the resultant. The angle between I p and E p is the angle of lag p of the primary current. It slightly exceeds < g . The pressure E 8 is generated in the secondary coil. There is a drop of 7 g R a volts in this coil in phase with I t . A counter voltage 90 behind I s will be set up due to the secondary leakage flux. Its value is L S 7 s . To overcome this 7 8 7 8 at 90 ahead of this pha"se. This gives the drop d s in the secondary coil. This drop must be subtracted from the pressure generated to give the secondary terminal volts. To subtract a vector, revolve it 180 and proceed as in addition, Properly sub- THE TRANSFORMER. tract d t from E t and the resultant, V# is the potential dif- ference at the terminals of the secondary coil of the transformer. By constructing this diagram for full load I s and then for I t = o, the regulation of a transformer can be found by the ratio of the difference between the values of V % in each case to the full load V t . The efficiency at any load can be determined from the diagram for that load, by /,, cos s E p l p cos Fig. 78 is not the true diagram of a commercial trans- former. For clearness a ratio of i to I has been portrayed and the losses greatly exaggerated. In practice it will be found impossible to complete the solution graphically because of the extreme flatness of the triangles. The better way is to draw an exaggerated but clear diagram, and obtain the true values of the sides by the algebra of complex imaginary quantities, or if the student is unfa- miliar with this method, by the more laborious methods of trigonometry and geometry. 55. Methods of Connecting Transformers -- There are numerous methods of connecting transformers to distribut- ing circuits. The simplest case is that of a single transformer in a single-phase circuit. Fig. 80 shows such an arrangement. This and the succeeding figures have the pressure and current values of the different parts marked on them, as- suming in each case a I K.W., i to 10 step-down transformer. As in Fig. 81, two or more trans- Fig, so. n6 ALTERNATING-CURRENT MACHINES. formers may have their primaries in parallel on the same circuit, and have their secondaries independent. If the two secondaries of this case are connected properly in series a secondary system of double the potential will result, or by adding a third wire to the point of juncture, as shown by the dotted line of Fig. 82, a three-wire system of dis- tribution can be secured. The secondaries must be con- nected cumulatively ; that is, their instantaneous E.M.F.'s must be in the same direction. If connected differentially, there would be no pressure between the two outside sec- Fig. 81. Fig. 82. ondary wires, the instantaneous pressures of the two coils being equal and opposed throughout the cycle. Again, with the same condition of primaries, the secondaries can be connected in multiple as in Fig. 83. Here the connec- tions must be such that at any instant the E.M.FJs of the secondaries are toward the same distributing wire. The connection of more than two secondaries in series is not common, but where a complex network of secondary dis- tributing mains is fed at various points from a high-tension system, secondaries are necessarily put in multiple. In many types of modern transformers it is usual to THE TRANSFORMER. 117 wind the secondaries (low-tension) in two separate and similar coils, all four ends being brought outside of the case. This allows of connections to two-wire systems of either of two pressures, or for a three-wire system accord- ing to Figs. 82 and 83, to be made with the one transformer, this being more economical than using two transformers of half the size, both in first cost and in cost of operation. In many transformers the primary coils are also wound in two parts. In these, however, the four ter- minals are not always brought outside, but in some cases are Fig ' 8s * led to a porcelain block on which are four screw-connectors and a pair of brass links, allowing the coils to be arranged in series or in multiple according to the pressure of the line to which they are to be connected. From this block two wires run through suitably bushed holes outside the case. A two-phase four-wire system can be considered as two inde- pendent single-phase systems, transformation being accom- plished by putting similar single- ts- 8 4. phase transformers in the circuit, one on each phase. If it is desired to tap a two-phase circuit to supply a two-phase three-wire circuit, the arrangement of Fig. 84 is employed. By the reverse connections two-phase three-wire can be transformed to two-phase four-wire, An interesting transformer connec- n8 ALTERNATING-CURRENT MACHINES. tion is that devised by Scott, which permits of transfor- mation from two-phase four-wire to three-phase three-wire. Fig. 85 shows the connections of the two transformers. If one of the transformers has a ratio of 10 to i with a tap at the middle point of its secondary coil, the other must have a ratio of 10 to .867 (io to One ter- minal of the secondary of the latter is connected to the Fig. 85. 100 Pig. 86. middle of the former, the remaining three free terminals being connected respectively to the three-phase wires. In Fig. 86, considering the secondary coils only, let mn rep- resent the pressure generated in the first transformer. The pressure in the second transformer is at right angles ( 5) to that in the first, and because of the manner of connection, proceeds from the center of mn. Therefore the line 0/ represents in position, direction, and magnitude the pressure generated in the second. From the geo- metric conditions mnp is an equilaterial triangle, and the pressures represented by the three sides are equal and at 60 with the others. This is suitable for supplying a three-phase system. In power transmission plants it is not uncommon to find the generators wound two-phase, and the step-up transformers arranged to feed a three- phase line, THE TRANSFORMER. In America it is common to use one transformer for each phase of a three-phase circuit. The three transform- ers may be connected either Y or A. They may be Y on the primary and A on the secondary, or vice versa. Fig. 87 shows both primary and secondary connected A. The pressure on each pri- mary is 1000 volts, and as a I-K.W. transformer was assumed, i.e., i K.W. per phase, there will be one ampere in each, calling for J -7 (^3) amperes in each Fig. 87. primary main ( 33). This arrangement is most desirable where continuity of service is requisite, for one of the transformers may be cut out and the system still be operative, the remaining transformers each taking up the difference between 1 and \ the full load ; that is, if the system was running at full load, and one trans- former was cut out, the other two would be over- loaded i6| per cent. Even if two of them were cut out, service over the remaining phase could be maintained. It is not uncommon to reg- ularly supply motors from three-phase mains by two some- what larger transformers rather than by three smaller ones. Fig. 88 shows the connections for both primaries and secondaries in Y- If in this arrangement one trans- Fig. 88. I2O ALTERNATING-CURRENT MACHINES. former be cut out, one wire of the system becomes idle, and only a reduced pressure can be maintained on the re- maining phase. The advantage of the star connection lies in the fact that each transformer need be wound for only 57.7 per cent of the line voltage. In high-tension trans- mission this admits of building the transformers much smaller than would be necessary if they were A connected. Fig. 89 shows the connections for primaries in A, second- aries in Y ; and Fig. 90 those for primaries in Y and sec- ondaries in A. By taking advantage of these last two arrangements, it is possible to raise or lower the voltage Fig. 89. Fig. 90. with i to i transformers. With three I to I transform- ers, arranged as in Fig. 89, 100 volts can be transformed to 173 volts; while if connected as in Fig. 90, 100 volts can be transformed into 58 volts. Fig.- 91 shows a transformer and another one connected as an autotransformer doing the same work. Since the required ratio of transformation is i to 2, the autotrans- former does the work of the regular transformer with one- half the first cost, one-half the losses, and one-half the drop in potential (regulation). The only objection to this method of transformation is that the primary and second- THE TRANSFORMER. 121 ary circuits are not separate. With the circuits grounded at certain points, there is danger that the insulation of the low-tension circuit may be subjected to the voltage of the high-tension circuit. One coil of an autotransformer must be wound for the lower voltage, and the other coil for the One 100 Kw. Transformer Ratio 1 to 2 Losses not considered < 2000-V: >j| One 50 Kw. Transformer a Ratio 1 to 1 Losses not considered. Fig. 91. difference between the two voltages of transformation. The capacity of an autotransformer is found by multiply- ing the high-tension current by the difference between the two operative voltages. Autotransformers are often called compensators. Compensators are advantageously used Losses not considered Fig. 92. where it is desired to raise the potential by a small amount, as in boosting pressure for very long feeders. Fig. 92 shows three I to 2 transformers connected in A on a three-phase system, and three i to I compensators con- nected in Y to do the same work. From a two-phase circuit, a single-phase E.M.F. of any 122 ALTERNATING-CURRENT MACHINES. desired magnitude and any desired phase-angle may be secured by means of suitable transformers, as shown in Fig- 93- Suppose the two phases X and Fof a two-phase system be of 100 volts pressure, and it is desired to obtain a single-phase E.M.F. of 1000 volts and leading the phase X by 30. As in Fig. 94, draw a line representing the DIRECTION OF PHASE X. Fig. 94. direction of phase X. At right angles thereto, draw a line representing the direction of phase Y. From their inter- section draw a line 1000 units long, making an angle of 30 with X. It represents in direction and in length the phase and the pressure of the required E.M.F. Resolve this line into components along X and F, and it becomes evident that the secondary of the transformer connected to X must supply the secondary circuit with 866 volts and that the secondary of the other must supply 500 volts. Therefore the transformer connected to X must step-up i to 8.66 and that connected to Fmust step-up i to 5. If 10 amperes be the full load on the secondary circuit, the first transformer must have a capacity of 8.66 K.W., and the second a capacity of 5 K.W. The load on X and Y is not balanced. 56. Lighting Transformers. Because of their extensive use on lighting distributing systems, the various manufac- turers have to a great extent standardized their lines of lighting transformers. Power transformers are not as yet THE TRANSFORMER. well standardized, probably because they are generally used in such large units as to warrant a special design for each case. The Wagner Electric Mfg. Co.'s "type M " transformer is illustrated in Fig. 95. It is of the shell type of con- struction, makers using this type claiming for it superiority of regulation and cool running. In the shell type the iron Fig. 95- is cooler than the rest of the transformer, in the core type it is hotter. As the "ageing" of the iron, or the increase of hysteretic coefficient with time, is believed to be aggra- vated by heat, this is claimed as a point of superiority of the shell type. However, the prime object in keeping a transformer cool is not to save the iron, but to protect the insulation ; and as the core type has less iron and generally less iron loss, the advantages do not seem to be remarkably 124 ALTERNATING-CURRENT MACHINES. in favor of either. In the Wagner " type M " transformers the usual practice of having two sets of primaries and sec- ondaries is followed. Fig. 96 shows the three coils composing one set. A low-tension coil is situated between two high-tensioned coils, this arrangement being conducive to good regulation. Th*> ideal method would be to have the coils still more subdivided and interspersed, but prac- tical reasons prohibit this. Fig. 97 shows the arrangement of the coils in the shell. The space between the coils and the iron is left to facilitate the circulation of the oil in which they are submerged. The laminae for the shell are stamped each in two parts and assembled with joints staggered. As can be seen from the first cut, all the terminals of the two primary and the two secon- Fi fi- 97. dary coils are brought outside the case. The smaller sizes of this line of transformers, those under 1.5 K.W., have Fig. 96. THE TRANSFORMER. 125 sufficient area to allow their running without oil, so the manufacturers are enabled to fill the retaining case with an insulating compound which hardens on cooling. The General Electric Co.'s "H" transformers are of the core type. In Fig. 76 was shown a sectional view giv- ing a good idea of the arrangement of parts in this type. Fig. 71 is also one of this line of transformers. In it is shown the tablet board of porcelain on which the connec- tions of the two high-tension coils may be changed from series to parallel or vice versa, so that only two high-tension wires are brought through the case. Fig. 98 shows the ar- rangement of the various parts in the assembled apparatus. The makers claim for this type that the coils run cooler because of their being more thoroughly surrounded with oil than those of the shell type. Another point brought forward is that copper is a better conductor of heat than iron ; the heat from the inner portions of the apparatus is more readily dissipated than in the shell type. The core has the advantage of being made up of simple rectangular punchings, and the disadvantage of having four instead of two joints in the magnetic circuit. A particular advantage of the " type H " transformer is the ease and certainty with which the primary windings can be sepa- rated from the secondary windings. A properly formed seamless cylinder of fiber can be slipped over the inner winding and the outer one wound over it. This is much Fig. 98. 126 ALTERNATING-CURRENT MACHINES. more secure than tape or other material that has to be wound on the coils. Fig. 99. The Westinghouse " O. D." transformers are of the shell type. The construction of the separate parts is shown in Fig. 99. The coils are wound narrow and to the full depth, and high-tension and low-tension coils alternate side by side instead of from the center out. Fig. 100 shows a 2 K.W. O. D. transformer with- out the case. A tablet board is used for the terminals of the high-tension coils, but the low-tension wires are all run out of the case. Fig. 101 shows one of the coils. Type O. D. transformers are built from i to 25 K.W. for lighting and to 50 K.W. for power. Those of 10 K.W. or less are Fig> I00< THE TRANSFORMER. 127 Fig. 101. in cast-iron cases, those above 10 K.W. in corrugated iron cases with cast tops and bottoms. The corrugations quite Fig. 102. materially increase the radiating surface. The windings are submerged in oil. An example of the Stanley Electric Manufacturing Co/s 128 ALTERNATING-CURRENT MACHINES. standard line of "type A. O." transformers is given in Fig. 1 02. These are also of the shell type, with divided primaries and secondaries, four of the eight which belong to a single transformer being shown in Fig. 103. Fig. 103. 57. Cooling of Transformers The use of oil to assist in the dissipation of the heat produced during the opera- tions of transformers is almost universal in sizes of less than about 100 K.W., especially if designed for outdoor use. Some small transformers are designed to be self- ventilating, taking air in at the bottom, which goes out at top as a result of being heated. They are not well pro- tected from the weather, and are liable to have the natural draft cut off by the building of insects' nests. Larger transformers that are air cooled and that supply their own draft are used to some extent in central stations and other places where they can be properly protected and attended to. A forced draft is, however, the more common. Where such transformers are employed, there are usually a number THE TRANSFORMER. 129 of them ; and they are all set up over a large chamber into which air is forced by a blower, as indicated in Fig. 104. Fig. 104. Dampers regulate the flow of air through the transformers, They can be adjusted so that each transformer gets its proper share. Fig. 105 shows a General Electric Company's air-blast transformer in process of construction. The iron core is built up with spaces between the laminae at intervals ; and the coils, which are wound very thin, are assembled in small intermixed groups with air spaces maintained by pieces of insulation between them. The assembled struc- ture is subjected to heavy pressure, and is bound together to prevent the possibility of vibration in the coils due to the periodic tendency to repulsion between the primary and the secondary. These transformers are made in sizes from 100 K.W. to 1000 K.W. and for pressures up to 35,000 volts. Another method of cooling a large oil transformer is to circulate the oil by means of a pump, passing it through a radiator where it can dissipate its heat. Again cold water is forced through coils of pipe in the transformer case, and it takes up the heat from the oil. There is the slight dan- ger in this method that the pipes may leak and the water may injure the insulation. Water-cooled transformers have been built up to 2000 K.W. capacity. 130 ALTERNATING-CURRENT MACHINES. In those cases where the transformer requires some outside power for the operation of a blower or a pump, the power thus used must be charged against the trans- Fig. 105. former when calculating its efficiency. In general this power will be considerably less than I % of the trans- former capacity. 58. Constant-Current Transformers. For operating series arc-light circuits from constant potential alternating- current mains, a device called a constant-current trans- former is frequently employed. A sketch showing the principle of operation is given in Fig. 106. A primary coil is fixed relative to the core, while a secondary coil is THE TRANSFORMER. allowed room to move from a close contact with the primary to a considerable distance from it. This secon- dary coil is nearly but not entirely counter-balanced. If no current is taken off the secondary that coil rests upon the primary. When, however, a current flows in the two coils there is a repul- sion between them. The counter- poise is so adjusted that there is an equilibrium when the current Flg ' Io6 * is at the proper value. If the current rises above this value the coil moves farther away, and there is an increased amount of leakage flux. This lowers the E.M.F. induced Fig. 107, 132 ALTERNATING-CURRENT MACHINES. in the secondary, and the current falls to its normal value. Thus the transformer automatically delivers a constant current from its secondary when a constant potential is impressed on its primary. Fig. 107 shows the mechanism of such an apparatus as made by the General Electric Company. The cut is self- explanatory. Care is taken to have the leads to the mov- Fig. 108. ing coil very flexible. Transformers for 50 lamps or more are made with two sets of coils, one primary coil being at the bottom, the other at the top. The moving coils are balanced one against the other, avoiding the necessity of a very heavy counterweight. Fig. 108 shows a 5o-light constant-current transformer without its case. Fig. 109 shows a complete 25-lamp apparatus. The tank THE TRANSFORMER. 133 is filled with oil, the same as an ordinary transformer. Great care must be taken to keep these transformers level, and to assist in this the larger sizes have spirit-levels built Fig. 109. into the case. A pair of these transformers can be spe- cially wound and connected to supply a series arc-light circuit from a three-phase line, keeping a balanced load on the latter. 59. Design of a Transformer The method of design- ing a transformer depends upon the specifications as to con- struction and operation, and upon various values which the designer is forced or sees fit to assume. The following is one method : Specifications. These usually give the capacity in watts, the frequency, the primary voltage, the secondary 134 ALTERNATING-CURRENT MACHINES. voltage, and the conditions of operation, place of installa- tion, whether loaded all day or not, etc. Assumptions. The assumption of the following quanti- ties is usually preliminary to any calculation, the shape of transformer, -the current density in the primary, the current density in the secondary, the turns in the primary coil, and the maximum flux density in the iron. The method of design is one of cut and try. A number of values of flux density and various numbers of primary turns are assumed. Efficiency curves are calculated for the various arrangements. The most efficient is ultimately selected ; or if none are satisfactory, the course of the design will have brought out the proper direction to take in making new assumptions. The following design refers to a core-type, step-down, lighting transformer of about 5 K. w. capacity. The as- sumptions are: 1000 circular mils per ampere in the primary, 1500 circular mils per ampere in the secondary (because this is inside, and has less opportunity of dissi- f r Fig. no. pating its heat), 500, 700, and 1000 turns primary successively, and 2000, 3000, and 4000 gausses maximum flux density. The transformer will have the shape shown in Fig. no. Because of the general use of the English units of measure by most practical mechanics, the dimen- THE TRANSFORMER. 135 sions indicated are all expressed in inches. The ratio = ;;/ may be conveniently assumed as m = 1.5, and the b ratio - = n is likewise generally made n = i. a I. To obtain tJie area, A, of the core in square centi- meters. Let E = impressed primary E.M.F., > m = assumed maximum flux density, T p = assumed number of turns in primary, and f = frequency. The instantaneous value of the counter E.M.F. of self- induction will be ( 13, vol. i., 3) ,_ , _ p sn ~~dt~ e' = T p & m a> cos 2 (t p + + insulation and clearance). Assume * = fte + i + g THE TRANSFORMER. 137 Now a = mb, and /, = rt p t-j \*9l so a = 2 But also, t f = so substituting and transposing, All the terms of the right-hand member are known, so it may be reduced to a simple number, and set equal to K. Then and m V. To obtain the volume v of iron in cubic centimeters. About 90% of a volume occupied by laminated iron is metal. v=2(a + b+2c) xrx^/X I754 3 x 0.9. VI. To obtain the watts P h lost in hysteresis. Accord- ing to Steinmetz's Law, using 77 = .003, Hysteresis loss = .003 v& m lf * ergs per cycle. 138 ALTERNATING-CURRENT MACHINES. VII. To obtain the resistance of the secondary R a in ohms. Although surrounding a rectangular core, the coils are usually approximately circular in section, for con- venience in winding and in insulating. If the section of the core varies considerably from the square, allowance can be made in estimating the length of a mean turn. Considering the coil as truly cylindrical, and allowing \ inch insulation between it and the core, the length of a mean turn The total length of secondary wire (both limbs) is then rT p l, and its resistance can be found directly in a wire table giving the hot resistances of wires ; or, it may be assumed that the transformer will operate at such a tem- perature that one mil foot has 1 1 ohms resistance, then 12 x circular mils VIII. To obtain the resistance of the primary R p in ohms. Similarly to the above, the length of a mean turn ^8^ r 76^ allowing T 3 g- inch insulation between the two coils, and the total length of primary wire is T p l. The resistance can be found in a table, or calculated from !*--'" "I I A I v * I x- 10 12 X circular mils THE TRANSFORMER. 139 IX. To obtain the foucault current loss P f in watts. Steinmetz has given the empirical formula P f = m suggested, the efficiency curve of each transformer should then be drawn, taking points at -^,\, J, f , and full load. After having selected the most suitable, determine the following values. XI. To determine the all-day efficiency in per cent. The average lighting transformer is found to be loaded equivalent to full load for 5 hours, and no load for 19 hours, per day. The all-day efficiency is watt hours output watt hours input per day. 5 . with non-inductive load, / being the full-load secondary current. XII. To determine the regulation in percent. In 53 was shown the method of calculating the magnetic leakage 140 ALTERNATING-CURRENT MACHINES. of this type of transformer. Call the flux linking only the primary coils 3> p (this is twice that which links the coil of one limb of the transformer). Call that which links only the secondary coils 3> g . There is practically no voltage drop at no load, so E t = rE p . At full load there is a drop in the primary and in the secondary, due (a) to IR drop, (b) to self-induction caused by leakage flux. Knowing this leakage flux, by the formula of paragraph I., this sec- tion, calculate the voltage drop in primary and in secondary coils, thus, JE pi = lo-'VJ and E gd = I0 - 8 A/2 The regulation, expressed in per cent, is Regulation = E > " ^+ + * where I p = rl st and is the full-load current. Regulation as stated refers to a non-inductive load. MOTORS. 141 CHAPTER VII. MOTORS. 60. Rotating Field. Suppose an iron frame, as in Fig. in, to be provided with inwardly projecting poles, and that these be divided into three groups, arranged as in the dia- gram, poles of the same group being marked by the same letter. If the poles of each group be alternately wound in opposite directions, and be connected to a single source of E.M.F., then the resulting current would magnetize the interior faces al- ternately north and south. If the im- pressed E.M.F. were alternating, then the polarity of each pole would change Fi - IXI - with each half cycle. If the three groups of windings be connected respectively with the three terminals of a three-phase supply circuit, any three successive poles will assume successively a maximum polarity of the same sign, the interval required to pass from one pole to its neighbor being one-third of the duration of a half cycle. The maximum intensity of either polarity is therefore passed from one pole to the next, and the result is a rotat- ing field. If the frequency of the supply E.M.F. be/, and if there be / pairs of poles per phase, then the field will 142 ALTERNATING-CURRENT MACHINES. make one complete revolution in - seconds. It will there- / V * fore make - = complete revolutions per second. A rotating field can be obtained from any polyphase supply- circuit by making use of appropriate windings. 61. The Induction Motor If a suitably mounted hollow conducting cylinder be placed inside a rotating field, it will have currents induced in it, due to the relative motion be- tween it and the field whose flux cuts the surface of the cylinder. The currents in combination with the flux will react, and produce a rotation of the cylinder. As the cur- rent is not restrained as to the direction of its path, all of the force exerted between it and the field will not be in a tangential direction so as to be useful in producing rota- tion. This difficulty can be overcome by slotting the cylinder in a direction parallel with the axis of revolution. Nor will the torque exerted be as great as it would be if the cylinder were mounted upon a laminated iron core. Such a core would furnish a path of low reluctance for the flux between poles of opposite sign. The flux for a given magnetomotive force would thereby be greater, and the torque would be increased. Induction motors operate according to these principles. The stationary part of an induction motor is called the stator, and the moving part is called the rotor. It is common practice to produce the rotating field by impressing E.M.F. upon the windings of the stator. There are, however, motors whose rotating fields are produced by the currents in the rotor windings. Fig. 1 1 2 shows the stator core and frame of a Westing- house induction motor, and Fig. 1 1 3 shows the same with MOTORS. 143 the windings in place. Each projection of the core does not necessarily mean a pole ; for it is customary to employ a distributed winding, there being several slots per pole Fig. 112. Fig. 113. per phase. Fig. 1 14 shows the rotor. The inductors are copper bars embedded in slots in the laminated steel core. They are all connected, in parallel, to copper collars or short-circuiting rings, one at each end of the rotor. They offer but a very small resist- ance, and the currents induced in them are forced to flow in a direction parallel with the axis. The reaction against the field flux is therefore in a Fi s- "4- proper direction to be most efficient in producing rotation. A rotor or armature of this type is called a squirrel cage. 62. Principle of Operation of the Induction Motor. If the speed of rotation of the field be V R. P. M. and that of 144 ALTERNATING-CURRENT MACHINES. the rotor be V R. P. M., then the relative speed between a given inductor on the rotor and the rotating field will be V V R. P. M. The ratio of this speed to that of the field, V V f viz., = s, is termed the slip, and is generally ex- Y pressed as a per cent of the synchronous speed. If the flux from a single north pole of the stator be < maxwells, then the effective E.M.F. induced in a single rotor inductor Y is 2.22 p =*?-. The power jP l which the generator gives to the circuit is and the power P 2 which the motor gives to the circuit is Now, if in either of the above expressions for power, the cosine has any other value than unity, then the power will consist of energy pulsations, there being four pulsa- tions per cycle. The energy is alternately given to and received from the circuit by the machine. If the cosine be positive, the amount of energy in one pulsation, which 160 ALTERNATING-CURRENT MACHINES. is given to the circuit, will exceed the amount in one of the received pulsations. The machine is then acting as a generator. If the cosine be negative the opposite takes place, and the machine operates as a motor. As a and /? are but functions of E lt E 2 , and 0, and as these latter are the quantities to be considered in operation, it is desir- able to eliminate the former. By a somewhat complicated analytical transformation it can be shown that cos (6 + <) and P 2 = -.* cos (0 - If there were no losses due to resistance, etc., P l would be numerically exactly equal to P 2 . Neglecting any losses in the machines, except that due to resistance, the alge- braic sum of P l and P 2 is equal to RI*. In order to determine the behavior of a synchronous motor when on a given circuit, use is made of the above formula for power, and each case must be considered by itself. The method of procedure is shown in the next article. 73. Special Case. Suppose a single-phase synchronous motor, excited so as to generate 2100 volts, to be con- nected to a generator giving 2200 volts, the total resis- tance of the circuit being 2 ohms and the reactance I ohm. Then the angle < of current lag behind the resultant E.M.F. has a value tan < = = 0.5, whence = 26 34'. A preliminary calculation, using the formulas of the pre- vious article, shows that both machines act as generators for values of between o and 1 20, and between 240 and 360 approximately. MOTORS. 161 Calculations of P l and P 2 for various values of between 120 and 240 have been made, and are embodied in the form of curves in Fig. 127. From an inspection of these Fig. 127. curves, and a consideration of the equations from which the curves are derived, the following conclusions may be drawn : (a) The motor will operate as such for values of be- tween 175 and 238. The difference between these angles may be termed the operative range. (b) The generator would operate as a motor for values of B between 133 and 174, providing the motor were mechanically driven so as to supply the current and power ; i.e., what was previously the motor must now operate as a generator. (c) The motor, within its operative range, can absorb any amount of power between zero and a certain maxi- mum. To vary the amount of received power, the motor has to but slightly shift the phase of its E.M.F. in respect to the impressed E.M.F., and then to resume running in 162 ALTERNATING-CURRENT MACHINES. synchronism. The sudden shift of phase under change of load is the fundamental means of power adjustment in the synchronous motor. It corresponds to change of slip in the induction motor, to change of speed in the shunt motor, and to change of magnetomotive force in the transformer. (d) For all values of the received power, except the maximum, there are two values of phase difference 0. At one of these phase differences more current is required for the same power than at the other. The value of the current in either case can be calculated as follows : Since /> + P 2 = RI Z /= V R The values of /are plotted in the diagram. The efficiency p of transmission e = _ ? is also different for the two values of . It is also represented by a curve. If the phase alteration, produced by an added mechan- ical load on the motor, results in an increase of power received by the motor, the running is said to be stable. If on the other hand, the increase of load produces a decrease of absorbed power, the running is unstable. (e] If for any reason the phase difference 0, between the E.M.F.'s of the motor and generator, be changed to a value without the operative range for the motor, the motor will cease to receive as much energy from the circuit as it gives back, and it will, therefore, fall out of step. Among the causes which may produce this result are sudden variations in the frequency of the generator, variations in the angular velocity of the generator, or excessive me- MOTORS. 163 chanical load applied to the motor. In slowing down, all possible values of will be successively assumed ; and it may happen that the motor armature may receive suffi- cient energy at some value of to check its fall in speed, and restore it to synchronism, or it may come to a stand- still. (/) Under varying loads the inertia of the motor armature plays an important part. The shifting from one value of 9 to another, which corresponds to a new mechan- ical load, does not take place instantly. The new value is overreached, and there is an oscillation on both sides of its mean value. This oscillation about the synchronous speed is termed hunting. If the armature required no energy to accelerate or retard it, this would not take place. (g) The maximum negative value of P 2 that is, the maximum load that the motor can carry is evidently when cos ( <) = i or when = 180. The formula for the power absorbed by the motor then reduces to . E.E* (h) The operative range of the motor can be determined by making P 2 equal to zero. By transformation the for- mula then becomes E? cos COS (0 - ) = ^-^r- ' Two values of (0 <) result, one on each side of 1 80. In the case under consideration cos (0 ) = .851 and - < = 211 40' or 148 20'. Since <6 = 26 34', 0= 238 14' or 174 54'. 1 64 ALTERNATING-CURRENT MACHINES. 74. The Motor E.M.F. To determine what value of E 2 will give the maximum value of power to be absorbed by a motor, consider E 2 as a variable in the equation given in (g) above. Differentiating cos <> and setting this equal to zero and solving, = 1230 volts. 2 COS CURRENT LAGGING E, Fig. 128. At this voltage the maximum possible intake of the motor is 6 1 1 K. w. If the voltage of the motor be above this or below it, its maximum intake will be smaller. Remembering that the current lags behind the resultant pressure of the generator and motor pressures by an angle , which is solely dependent upon w, I,, and R, it will be easily seen, from an inspection of Figs. 128, 129, and 1 30, that the current may be made to lag behind, lead, or be in phase with E lf by simply altering the value of E This may be done by vary- ing the motor's field excitation. A proper excitation can produce a unit power factor in the transmitting line. The over-excited synchronous motor, therefore, acts like a con- denser in producing a leading cur- rent, and can be made to neutralize the effect of induct- ance. The current which is consumed by the motor for a given load accordingly varies with the excitation. The' CURRENT LEADING Ej Fig. 129. CURRENT IN PHASE WITH E, Fig. I 3 0. MOTORS. I6 5 relations between motor voltage and absorbed current for various loads are shown in Fig. 131. Synchronous motors are sometimes used for the purpose of regulating the phase relations of transmission lines. MOTOR VOLTAGE Fig. 131. The excitation is varied to suit the conditions, and the motor is run without load. Under such circumstances the machines are termed synchronous compensators. The capacity of a synchronous motor is limited by its heating. If it is made to take a leading current in order to adjust the phase of a line current, it cannot carry its full motor load in addition without heating. 75. Polyphase Synchronous Motor. The discussion which has just been given applies to the single-phase motor. The facts brought out are equally applicable to the polyphase motor. In the latter case each leg or phase is to be considered as a single-phase circuit. The total power is that of each phase multiplied by the number of phases. 76. Starting Synchronous Motors. These motors do not have sufficient torque at starting to satisfactorily come up 1 66 ALTERNATING-CURRENT MACHINES. to speed under load. They are, therefore, preferably brought up to synchronous speed by some auxiliary source of power. In the case of polyphase systems an induction motor is very satisfactory. Its capacity need be but ^ that of the large motor. Fig. 132 shows a 750 K. w. quarter- phase General Electric motor with a small induction motor Fig. 132. geared to the shaft for this purpose. This motor may be mechanically disconnected after synchronism is reached. Before connection of the synchronous motor to the mains it is necessary that the motor should not only be in syn- chronism, but should have its electromotive force at a difference of phase of about 180 with the impressed pressure. To determine both these points a simple device, MOTORS. 167 known as a synchronizer, is employed. It consists of an incandescent lamp connected in series with the sec- ondaries of two transformers, whose primaries are con- nected respectively with the line and with the motor terminals. The brightness with which the lamp glows is a measure of the phase difference between the two E.M.FJs. It is customary to so connect the transformers that when the motor E.M.F, is at 180 with the line pressure, the lamp will have its greatest brilliancy. As the motor is coming up to speed, the lamp will be alternately bright and dark. The alternations will grow slower as synchronism is approached, and will finally be so slow as to permit the closing of the main switch at the proper instant. Synchronous motors may be brought up to speed with- out any auxiliary source of power. The field circuits are left open ; and the armature is connected either to the full pressure of the supply, or to this pressure reduced by means of a starting compensator, such as was described in 66. The magnetizing effect of the armature ampere turns sets up a flux in the poles sufficient to supply a small starting torque. When, after running a sufficient time as an induction motor, synchronism is nearly attained, the fields may be excited and the motor will come into step. The load is afterwards applied to the motor through friction clutches or other devices. There is great danger of per- forating the insulation of the field coils when starting in this manner. This is because of the high voltage produced in them by the varying flux. In such cases each field spool is customarily open-circuited on starting. Switches which are designed to accomplish this purpose are called break-up switches. 168 ALTERNATING-CURRENT MACHINES. 77. Parallel Running of Alternators Any two alter- nators adjusted to have the same E.M.F., and the same frequency, may be synchronized and run in parallel. Ma- chines of low armature reaction have large synchronizing power, but may give rise to heavy cross currents, if thrown out of step by accident. The contrary is true of machines having large armature reaction. Cross currents due to differences of wave-form are also reduced by large arma- ture reaction. The electrical load is distributed between the two machines according to the power which is being furnished by the prime movers. This is accomplished, as in the case of the synchronous motor, by a slight shift of phase between the E.M.F.'s of the two machines. The difficulties which have been experienced in the parallel running of alternators have almost invariably been due to bad regulation of the speed of the prime mover. Trouble may arise from the electrical side, if the alternators are designed with a large number of poles. Composite wound alternators should have their series compounding coils con- nected to equalizing bus bars, the same as compound wound direct-current generators. CONVERTERS. 169 CHAPTER VIII. CONVERTERS. 78. The Converter. The converter is a machine hav- ing one field, and one armature, the latter being supplied with both a direct-current commutator and alternating- current slip-rings. When Brushes, which rub upon the slip-rings, are connected with a source of alternating current of proper voltage, the armature will rotate syn- chronously, acting the same as the armature of a synchronous motor. While so revolving, direct current can be taken from brushes rubbing upon the commu- tator. The intake of cur- rent from the alternating- current mains is sufficient to supply the direct-current circuit, and to overcome the losses due to resistance, friction, windage, hyster- esis, and eddy currents. The windings of a converter armature are closed, and simply those of a direct-current dynamo armature with properly located taps leading to the slip-rings. Each ring must be connected to the armature winding by as many taps as there are pairs of poles in the field. These taps are equidistant from each other. Fig. 133- I/O ALTERNATING-CURRENT MACHINES. There may be any number of rings greater than one. A converter having ;/ rings is called an //-ring converter. The taps to successive rings are -th of the distance be- n tween the centers of two successive north poles from each other. Fig. 133 shows the points of tapping for a 3-ring multipolar converter. A converter may also be supplied with direct current Fig. 134. through its commutator, while alternating current is taken from the slip-rings. Under these circumstances the machine is termed an inverted converter. Converters are much used in lighting and in power plants, sometimes receiving alternating current, and at other times direct current. In large city distributing systems they are often used in connection with storage batteries to charge them CONVERTERS. I?! from alternating-current mains during periods of light load, and to give back the energy during the heavy load. They are also used in transforming alternating into direct currents for electrolytic purposes. A three-phase machine for this purpose is shown in Fig. 1 34. A converter is sometimes called a rotary converter or simply a rotary. 79. E.M.F. Relations. In order to determine the re- lations which exist between the pressures available at the various brushes of a converter, Let E d = the voltage between successive direct-current brushes. E n = the effective voltage between successive rings of an ;2-ring converter. a the maximum E.M.F. in volts generated in a single armature inductor. This will exist when the conductor is under the center of a pole. b = the number of armature inductors in a unit electrical angle of the periphery. The electrical angle subtended by the centers of two successive poles of the same polarity is considered as 2 IT The E.M.F. generated in a conductor may be considered as varying as the cosine of the angle of its position relative to a point directly under the center of any north pole, the angles being measured in electrical degrees. At an angle ft, Fig. 135, the E.M.F. generated in a single inductor G is a cos ft volts. In an element dft of the periphery of the armature there are bd$ inductors, each with this E.M.F. If connected in series they will yield an E.M.F. ALTERNATING-CURRENT MACHINES. of ab cos ft dp volts. The value of ab can be determined if an expression for the E.M.F. between two successive direct-current brushes be determined by integration, and be set equal to this value E d as follows : r + l E d = I ab cos (3dj3 = 2 ab. Fig. 135- 2ir successive rings is n In an ^-ring converter, the electrical angular distance between the taps for two The maximum E.M.F. will be generated in the coils between the two taps for the succes- sive rings, when the taps are at an equal angular distance from the center of a pole, one on each side of it, as shown in the figure. This maximum E.M.F. is ab cos BdB = 2 ab sin - r ' = ^si%. The effective voltage between the successive rings is therefore By substituting numerical values in this formula, it is found that the coefficient by which the voltage between CONVERTERS. 173 the direct -current brushes must be multiplied in order to get the effective voltage between successive rings is for 2 rings 0.707 3 rings 0.612 4 rings 0.500 6 rings 0.354 In practice there is a slight variation from these co-effi- cients due to the fact that the air-gap flux is not sinusoid- ally distributed. 80. Current Relations. In the following discussion it is assumed that a converter has its field excited so as to cause the alternating currents in the armature inductors to lag 1 80 behind the alternating E.M.F. generated in them. The armature coils carry currents which vary cyclically with the same frequency as that of the alternating-current supply. They differ widely in wave-form from sine curves. This is be- cause they consist of two currents superposed upon each other. Consider a coil B, Fig. 136. It car- ries a direct current whose value is half that car- ried by one direct-current brush, and it reverses its direction every time that Fig - 136 ' the coil passes under a brush. The coil, as well as all others between two taps for successive slip-rings, also carries an alternating current. This current has its zero 174 ALTERNATING-CURRENT MACHINES. value when the point A, which is midway between the successive taps, passes under the brush. The coil being \(/ electrical degrees ahead of the point A, the alternating current will pass through zero of a cycle later than 2 7T the direct current. The time relations of the two currents are shown in Fig. 137. To determine the maximum value of the alternating current consider that, after subtracting the machine losses, Fig. 137- the alternating-current power intake is equal to the direct- current power output. Neglecting these losses for the present, if E n represents the pressure and / the effective alternating current in the armature coils between the suc- cessive slip-rings, then for the parts of the armature wind- ings covered by each pair of poles E d l d = nEJ n Therefore, the maximum value of the alternating current is . 7T n sin The time variation of current in the particular coil B is obtained by taking the algebraic sum of the ordinates of CONVERTERS. 175 the two curves. This yields the curve shown in Fig. 138. Each inductor has its own wave-shape of current, depend- ing upon its angular distance \j/ from the point A. \ Fig. 81. Heating of the Armature Coils The heating ef- fect in an armature coil due to a current of such peculiar wave-shape as that shown in Fig. 138 can be determined either graphically or analytically. The graphic determina- tion requires that a new curve be plotted, whose ordinates shall be equal to the squares of the corresponding current values. The area contained between this new curve and the time axis is then determined by means of a planimeter. The area of one lobe is proportional to the heating value of the current. This value may be determined for each of the coils between two successive taps. An average of these values will give the average heating effect of the currents in all the armature coils. The heating is different in the different coils. It is a maximum for coils at the points of tap to the slip-rings and is a minimum for coils midway between the taps. The analytical determination is made as follows : For a coil which is ^ electrical degrees from a point midway 176 ALTERNATING-CURRENT MACHINES. between successive slip-ring taps, at the time / seconds after passing a direct-current positive brush, the instanta- neous value of the current is r _I d { 4 sin (2 y j 2 { . n sin I The effective value of the current in this coil is, therefore, i . 7T n sin - n 1 6 cos if/ . 7T 7T 7T/2 sm - ;r sin- - n n where 2 represents for simplicity the value of the radical. The heating, due to the current in this coil, is propor- / 2 O 2 tional to - l -^-, and the average heating over the whole 4 armature is proportional to Inasmuch as the heating of this armature when run as a simple direct-current generator is proportional to , it 4 can, with the same heating, when operating as a conver- ter, put out r== ' as much direct current. ^ ^ ri* sin 2 - n CONVERTERS. 82. Capacity of a Converter By inserting numerical values in the above equation it is found that a machine has different capacities, based upon the same temperature rise, according to the number of slip-rings, as shown in the fol- lowing table. The armature is supposed to have a closed- coil winding : CONVERTER CAPACITIES. USED AS A KILOWATT CAPACITY Direct-current generator 100 Single-phase converter 85 Three-phase converter 134 Four-phase converter 164 Six-phase converter 196 Twelve-phase converter 227 The overload capacity of a converter is limited by com- mutator performance and not by heating. As there is but small armature reaction, the limit is much higher than is the case with a direct-current generator. 83. Starting a Converter. Converters may be started and be brought up to synchronism by the same methods which are employed in the case of synchronous motors. It is preferable, however, that they be started from the direct-current side by the use of storage batteries or other sources of direct current. They may be brought to a little above synchronous speed by means of a starting resistance as in the case of a direct-current shunt motor, and then, after disconnecting and after opening the field circuit, the connections with the alternating-current mains may be made. This will bring it into step. 84. Armature Reaction. The converter armature cur- rents give rise to reactions which consist of direct-current ALTERNATING-CURRENT MACHINES. generator armature reactions superposed upon synchronous motor armature reactions. It proves best in practice to set the direct-current brushes so as to commutate the cur- rent in coils when they are midway between two succes- Fig. 139- sive poles. The direct-current armature reaction, then, con- sists in a cross-magnetization which tends to twist the field flux in the direction of rotation. When the alternating currents are in phase with the impressed E.M.F. they also exert a cross-magnetizing effect which tends to twist the CONVERTERS. 179 field flux in the opposite direction. The result of this neu- tralization is a fairly constant distribution of flux at all loads. Within limits even an unbalanced polyphase con- verter operates satisfactorily. There is no change of field excitation necessary with changes of load. The converter is subject to hunting the same as the synchronous motor. As its speed oscillates above and below synchronism, the phase of the armature current, in reference to the impressed E.M.F., changes. This results in a distortion of the field flux, of varying magnitude. This hunting is much reduced by placing heavy copper circuits near the pole horns so as to be cut by the oscillat- ing flux from the two horns of the pole. The shifting of flux induces heavy currents in these circuits which oppose the shifting. Fig. 139 shows copper bridges placed be- tween the poles of a converter for this purpose. When running as an inverted converter from a direct- current circuit, anything which tends to cause a lag of the alternating current behind its E.M.F, is to be avoided. The demagnetization of the field by the lagging current causes the armature to race the same as in the case of an unloaded shunt motor with weakened fields. Converters have been raced to destruction because of the enormous lagging currents due to a short circuit on the alternating- current system. 85. Regulation of Converters The field current of a converter is generally taken from the direct-current brushes. By varying this current the power factor of the alternating-current system may be changed. This may vary, through a limited range, the voltage impressed between the slip-rings. As the direct-current voltage i8o ALTERNATING-CURRENT MACHINES. Step- down Transformer. Regulator. bears to the latter a constant ratio it may also be varied. This is, however, an uneconomical method of regulation. Converters are usually fed through step-down transform- ers. In such cases there are two com- mon methods of regu- lation, which vary the voltage supplied to the converter's slip- of Stillwell, which is Fig. 140. rings. The first is the method shown in the diagram, Fig. 140. The regulator consists of a transformer with a sectional Fig. 141. CONVERTERS. l8l secondary. Its ratio of transformation can be altered by moving a contact-arm over blocks connected with the various sections, as shown in the diagram. The primary of the regulator is connected with the secondary terminals of the step-down transformer. The sections of the second- ary, which are in use, are connected in series with the step- down secondary and the converter windings. The second method of regulation is that employed by the General Electric Co. The ratio of transformation of a regulating transformer, which is connected in circuit in the same manner as the Stillwell regulator, is altered by shifting the axes of the primary and secondary coils in respect to each other. Fig. 141 shows such a transformer, the shifting being accomplished by means of a small, direct-current motor mounted upon the regulator. The primary windings are placed in slots on the interior of a laminated iron frame, which has the appearance of the stator of an induction motor. The secondary windings are placed in what corresponds to the slots of the rotor core. The winding is polar ; and if the secondary core be rotated by an angle corresponding to the distance between two successive poles, the action of the regulator will change from that of booster to that of crusher. 182 ALTERNATING-CURRENT MACHINES. CHAPTER IX. POWER TRANSMISSION. 86. Superiority of Alternating Currents. The two great sources of energy for use in manufacturing establish- ments and in land transportation systems are the coal mines and the water powers. While coal can be trans- ported to the point of utilization of the energy, the energy of the waterfall cannot be commercially transmitted to a long distance without the use of electricity. In many cases it is uncertain whether it is not cheaper to transmit the energy of the coal in the form of electrical energy than to transport the coal itself. There is gen- erally greater convenience and greater flexibility in the application and utilization of the transmitted electrical energy. The electrical transmission can be accomplished by means of direct currents or by means of alternating cur- rents. For transmission over anything but quite short distances the alternating current is preferable to the direct current. Even for short distances, when these pass through densely populated districts, the alternating cur- rent is adopted for pure transmission purposes. The direct current has its points of superiority. Its use is not attended by inductive disturbances with the ac- companying drop and sometimes low power factor ; it is attended by no appreciable capacity effects; it is not POWER TRANSMISSION. 183 subject to electric surgings, which sometimes cause insu- lation perforations, short circuits, and arcing. It permits the use of direct-current motors with their very satisfac- tory operation as to efficiency, small starting current, overload capacity, and speed control. Its use on trans- mission lines of over a few miles' length is prohibited by the cost of the line which it necessitates. As will be seen later, long distance electrical power transmission, to be economical or even commercially possible, must be ef- fected by high voltages. Direct-current sparkless com- mutation is limited to 1000 volts. This limit is dependent upon the economical and mechanical limits of armature peripheral velocity, current density, gap-flux density, and temperature elevation. Furthermore service conditions demand other voltages than those of the transmission line. The direct -current transformer or dynamotor is expensive and not very efficient. The use of alternating currents is attended by the evil effects of inductance and capacity ; the operation of alter- nating-current motors can be called only fairly satisfactory ; but the employment of the very satisfactory, highly effi- cient, and moderately priced static transformer, makes possible the transmission at high voltages with its accom- panying small currents, small line wires, and cheap pole line construction. The use of the synchronous converter for distribution purposes in connection with alternating-current transmis- sion, constitutes a very satisfactory system, and seems to best meet all the engineering requirements. 87. Frequency. It is customary to call frequencies above 60 high, and those below 60 low. The proper fre- 1 84 ALTERNATING-CURRENT MACHINES. quency for a transmission and distributing system is dependent upon a number of variables as follows : - a. High-frequency transformers are smaller and cost less than those of lower frequency. This is seen by inspec- tion of the formula in article 59, I. For the same volt- age and flux density, the product of the iron cross-section and the number of turns varies inversely as the frequency. The cross-section of copper would be the same for the same capacity, irrespective of the frequency. b. High-frequency generators may be constructed cheaper than those of low frequency. For the same field multipolarity a high frequency is associated with high arm- ature speed, and, therefore, greater output. On the other hand, if an armature be run at the greatest peripheral velo- city mechanically permissible, a high frequency necessitates a greater field multipolarity, and, therefore, a greater cost and complexity of construction. c. High frequencies permit of the satisfactory oper- ation of both arc and incandescent lamps. Arcs do not operate well on any frequencies below 40. The satisfac- tory operation of incandescent lamps depends upon their voltage and candle-power. Low-voltage lamps have fat filaments of large heat capacity which do not drop in tem- perature so rapidly as high-voltage thin filaments. The same is true of high candlepower filaments. These lamps may be operated satisfactorily at 25 cycles per second. Standard i ic-volt, 16 candle-power lamps, however, fatigue the eye at frequencies under 30 cycles. d. The inductive line drop, 2 -n-fL, varies directly as the frequency. Its value will be considered later. Being greater for high frequencies, it is then more liable to pro- duce poor regulation at points of distribution. POWER TRANSMISSION. 185 e. The capacity charging current also varies directly as the frequency. /. The wattless currents due to inductance and capacity, therefore, increase with the frequency, and thereby lower the operative capacity of the generator, the transformers, and the line. They also lower the efficiency of operation. g. High frequencies may necessitate so high a field multipolarity that the angular speed variation of the prime mover will prevent the satisfactory paralleling of the gen- erators. For the same reason, the running of synchronous motors and of synchronous converters may be unsatisfac- tory. h. Induction motors are best suited for operation on low- frequency circuits. At high frequencies the speed must be high or the motor must be large to avoid running on a low-power factor. The speed could be lowered by increas- ing the number of poles ; i.e., by placing the poles nearer to each other. If the diameter remained the same, this would result in an increase of stator flux leakage, which would reduce the power factor. 88. Voltage If the frequency, the amount of trans- mitted power, and the percentage of power lost in the line, remain constant, the weight of line wire will vary in- versely as the square of the voltage impressed upon the line. This depends upon the fact that the cross-section of the wire is not determined by the current density and the limit of temperature elevation, but by the permissible voltage drop. If the impressed voltage on a line be multiplied by n, the drop in the line may be increased n times without altering the line loss. For the line loss is to the total power given to the line as the drop in volts is to 1 86 ALTERNATING-CURRENT MACHINES. the impressed voltage. To transmit the same power, but - th the previous current is necessary ; and this current, to produce n times the drop, must, therefore, transverse a resistance ^ 2 times as great as previously. In transmitting power electrically over long distances, the line cost constitutes a large part of the total invest- '12.5 20 Fig. 142. ment. In such cases it is desirable to employ as high volt- ages as possible. There is, however, a limit to the voltage which may be employed. Mr. Charles F. Scott has given some interesting results of experiments carried out on vari- ous pole lines. He found that the power lost through the air between wires increased with the impressed voltage, and after a certain voltage was reached, increased very rapidly ; that, with a given impressed voltage, the loss decreased as POWER TRANSMISSION. I8 7 the distance between the wires was increased ; that atmos- pheric conditions, such as snow, rain, and humidity, had no appreciable effect on the loss ; that peaked wave-shaped E.M.F.'s gave a greater loss than flat -topped ones ; and that the loss decreased as the diameter of the wires was increased. The relations between the distance between wires, the impressed voltage, and the power loss, is shown in Fig. 142. 10, C. U 6 Q- to fj o i-_ t^. ^2 o ft O 1 0~ Jo ^^"N M-OO S vO vD t^OO O^ O N -*OO CJ t^. * t^ 1-1 t^ IO 1O O*OO N 1O xD t^ t^oo a^ q N 1000 \OCONiON t -?>o. * N t ~vo'oo > * ^ M moo co co ii S lUl.ssSS^&^ll 1000 O N N M $>) xO 00 00 Q xOt^O^Oi-Nco^t iOsO t^ C> N oo s N PO\> s N m i q i q i q i q ) q ) q > q 5 q 1 q ) qq i 88888888 8 8 8 ^ t^. M HHXHWVIQ; ocoooooooooo OOOOOt^cOTj-oioN" \O_OO_ >-_ OxO i cosO, tx -_ c^oo^ 10 h- tC co 10 co^O N ** co\D O sD vO coOOOvD in-tcoN M M O NOO IOM OOO-O -tN t^ rfaS 1 coS^-S'N 2"cov co\D_ i-^vO co O^OO vD 10 < co 335? -a izigl ^ II + Corresponding to Current in line P.F. of .90 or .80 Inductance factor of .44 or .602. current of line: 1=2 X ^.i^id/CE icr 6 . E.M.F. f= Frequency. in Microfarads. Zrrlnductancein henrys. es of X, Z, A per mile of wire. es of C and / per mile of circuit. ance between wires, 18". u U ^-2. volt POWER TRANSMISSION. CURRENT IN MAIN CONDUCTOR. VALUES OF T. 195 SYSTEM. PER CENT POWER FACTOR. 100 95 .90 .85 .80 Sin le base 1. 000 .500 .576 1.052 .526 .607 i. in 555 .642 1.172 .588 .679 1.250 .625 .729 Two-phase (4 wire) .... Three-phase (3 wire) . . . Current in main conductors Output in Watts E is quite appreciable. The capacity of a two-w microfarads per mile of line is approximately two-wire line in C = 0.04 . where d is the distance between centers of wires, and R is the radius of the wire, both being measured in the same units. Because of its capacity, a line which is unloaded takes a current when an alternating E.M.F. is impressed upon it. If the capacity be C microfarads, then E volts at a frequency/ would send a charging current (21) /= 2 7rfCio~ G amperes. 94. Line Constants The various constants of a trans- mission line are given in the table on the preceding page. In calculating the sizes of lines, transformers, and generators of a transmission system, allowance has to be made for the various power factors of the load drawn off at various points. Induction motors, arc lights, and synchronous motors under some excitations have other than unit power-factor. Therefore transformers which supply them must have an excess of capacity sufficient to 196 ALTERNATING-CURRENT MACHINES. Line loss in per cent, of power delivered 28 27 2625 2423 22 21 2019 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 28.272.62.52423222120191817161514.131211109 87654 3.2 1 Line loss in per cent, of power delivered Fig. 148. POWER TRANSMISSION. 197 carry the extra current. The line, the step-up trans- formers, and the generator which supplies the electrical energy, must all have increased capacity. The prime mover, which drives the generator, however, does not need to have this extra capacity. The actual current in all the apparatus being larger than it would be if the power- factor were unity, is accompanied by increased heat losses at every point. The excess of capacity is needed to get rid of this heat, without undue elevation of temperature in the apparatus. The equivalent impedance of the loads and their equivalent power-factor as affecting the line can be determined as shown in the problems of Chapter IV. 95. Weight of Copper In the curves of Fig. 148 are shown the relations which exist between the transmission loss of power in per cent, the impressed volts per mile, and the weight of copper per K.W. delivered. The loss is expressed as a percentage of the power delivered. The curves apply to a three-phase transmission at unit power- factor. Five per cent has been allowed for sag of lines be- tween poles. To determine the values for aluminium wire, correct by the constants given in 90. 198 ALTERNATING-CURRENT MACHINES. CHAPTER X. TESTS. 96. Apparatus In the following pages are given direc- tions for a series of experiments designed to give the stu- dent dexterity in handling apparatus, a firmer grasp of the phenomena connected with alternating currents, and a knowledge of the methods employed in testing alternating- current apparatus. This course was laid out for use in a laboratory with but a moderate amount of apparatus, and all this apparatus will be here described to avoid the necessity of introducing such descriptions in the directions for the experiments. The laboratory is supplied with power from an Edison direct-current three-wire system with 117 volts on a side. The largest machine is a 7.5 K.W. double-current gener- ator, which is run as an inverted converter from the Edi- son current. This is a four-pole machine whose speed can be regulated from 1200 to 1800 R.P.M. This gives a range on the alternating end of 40 ~* to 60 ^. There are six slip-rings on the armature, so connected that single- phase current can be had from rings I and 4, quarter-phase from 1-4 and 2-6, and three-phase from 1-3-5. The volt- age, of course, cannot be altered. A laboratory not sup- plied with current from a street service could use such a machine, running it as a double-current generator by a steam or gas engine. This would be more desirable than running a regular alternator ; as frequently direct current, TESTS. 199 as well as alternating, is called for in the experiments. In such case, both frequency and voltage could be regulated. Besides this, there is a 500- watt, 8-pole, 125^ alternator belt-driven by a direct-current motor. The wave-shape of this machine was given in Fig. 4. The machine on which most of the tests are run is a double-current generator of about i K.W. capacity. This is a bipolar machine fitted with four slip-rings on one end and a commutator on the other. The rings are arranged so that three-phase cur- rent is obtained from rings 1-2-3, an d single-phase from rings 1-4. This machine serves a multitude of purposes. It can be run as a direct-current motor ; as a synchronous motor, either single-phase or three-phase ; as a converter, either direct or inverted ; and when driven by a belt, as an alternator, single-phase or three-phase ; or as a direct-cur- rent generator, either shunt wound or separately excited. Its speed can be varied from 1500 to 2400, giving fre- quencies of 25 ~ to 40^. It may be run in parallel with the larger converter when that is slowed down to 40 /-*. The equipment of rotating apparatus is completed by two induction motors, one of one-horse power, the other of a half-horse power capacity. They are both wound for three-phase ; but the smaller is equipped with a condenser compensator, as described in 67, and can be run when desired on a single-phase circuit. The transformer equipment consists of three I-K.W. I to I oil-cooled transformers, a half-K.w. I to 2 air- cooled transformer, and an old ring-wound armature ar- ranged with taps so that it can serve to transform from I to i, 2, 3, or 4. For inductive circuits three coils are used. The first, known as Coil i, was described in 9. It has about 3000 200 ALTERNATING-CURRENT MACHINES. turns of No. 16 B. & S. wire, 10 ohms resistance, and 0.2 henrys inductance without iron. A bundle of iron wires 1 6" long and i?" diameter can be inserted in either of the three coils. Coil 2 is in the shape of a hollow cylinder, whose external diameter is 3i", internal diameter 2f", and length 3J". It consists of about 6500 turns of No. 26 B. & S. wire, with an inductance of o.i henry and a resist- ance of 60 ohms. Coil 3 is of the same external appear- ance as Coil 2, but is made of about 7600 turns of No. 25 B. & S. wire, giving an inductance of 0.141 henry and a resistance of 60 ohms. It will be noticed that Coil 2 and Coil 3 have the same resistance, and that their inductances are as i to V2. Six paraffined paper condensers of about two microfarads each, are used when condensive circuits are desired. The instruments used are as follows : Four hot-wire ammeters, with ranges of i, 3, 15, and 20 amperes respec- tively. All but the first work across shunts, the small one, however, taking the whole current through its hot wire. These, of course, are used for either alternating or direct currents. Two inclined coil ammeters have ranges respec- tively of 5 amperes and 50 amperes. There are three voltmeters, an inclined coil instrument reading to 65 volts ; a Cardew hot-wire instrument, read- ing to i 50 volts; and a Weston standard portable voltmeter with two scales, one up to 100 volts, the other up to 200. Any of these may be used on either alternating- or direct- current circuits. For all the larger measurements a 2.5 K.W. indicating wattmeter is used. For the finer measurements a Weston standard wattmeter, having two scales, is used. The lower scale, for use with pressures of 75 volts or less, reads up TESTS. 201 to 75 watts ; the upper scale, for use with pressures of 150 volts, reads up to 150 watts. For this instrument a shunt has been constructed, having a coil similar to the current coil of wattmeter, so as to have the same resistance and the same time constant as the latter. This is placed in parallel with the current coil, a small resistance for ballast having first been placed in series with each. The watt- meter then reads up to 300 watts, and is as accurate on inductive as on non-inductive loads. Certain direct-current instruments are occasionally used, principally a Weston standard portable i5O-volt direct- current voltmeter, and a similar five-ampere ammeter. These instruments are used for convenience, and could be dispensed with if necessary. A means of measuring the rate of rotation of the various machines is essential, and a portable tachometer is by far the best instrument for the purpose. Of course, a greater accuracy can be obtained by using a revolution counter, and noting the number of revolutions in a con- siderable length of time ; but this method is too slow to be satisfactory, and is useless if the speed be fluctuating. To load a machine electrically, two lamp boards are used. These have each ten key sockets arranged in two rows between three wires. Thus, three wires of a three- wire system may be connected thereto, or the outside wires may be connected together, and all the lamps be put in multiple ; and finally, by using the two outside wires only, all the lamps turned on in one row can be put in series with all those on in the other row. Thus a wide range of resistances can be obtained by very small steps, if a few each of 8, 16, 32, 50, and 100 candle-power lamps are in the sockets. 202 ALTERNATING-CURRENT MACHINES. In the following descriptions of experiments, for the sake of brevity, the apparatus needed will not be named ; but such notation will be used in the figures, showing the arrangement of apparatus, that the particular apparatus will be indicated. All measuring instruments will be marked with a letter indicating their kind, and a number indicating their capacity ; thus A s is a three-ampere am- meter, W 2m is a 2.5 K.W. wattmeter. In many cases, the manner of drawing will indicate the apparatus, thus : Q is an alternating-current ammeter or voltmeter. <^jft is a direct-current ammeter or voltmeter, ra is a wattmeter, the binding posts of the current coil .** being conspicuously large to avoid confusion. \-^ is a switch designed to shift one ammeter out of cir- cuit and another in without interrupting the con- tinuity of the circuit. ^ is a contact-maker, giving a short contact at any desired point in a revolution. .^ ;,. is a commutator designed to change the direction of * * current flow in a circuit. %%S% is a lamp board as described above. _OOOQ is an inductive coil, is a condenser, is a transformer, the numbers indicating the relative number of turns. --- is the armature and field coils of a direct-current * *- machine or the direct-current end of a converter. jar- is the armature and field coils of an alternating-cur- rent machine or the alternating-current end of a converter. is to represent a belt-drive between two armatures, is to represent a direct connection, or, in the case of a converter, the two ends of the same armature. A.C._ Al TESTS. 203 97. Exp. i. Peculiarities of Alternating-Current Circuits. -This experiment consists of some merely qualitative observations calculated to illustrate to the student the dif- ference between alternating currents and the direct cur- rents he has hitherto used. First Part. Arrange the apparatus as in Fig. 149, the lamp being by way of protection, in the case of accidental short circuit. Let;r be first, 100 . p. the inductive coil known as yj) Coil i, second, the same with 60 ^-' the iron core inserted in it, third a condenser of about 10 M.F. capacity, and fourth a 5o-candle power lamp. Apply to these circuits a Fig- MQ- uniform potential of about 60 volts. Let the frequency be successively 125 ^, 40 / ', and o ^, i.e., direct current. With each change note the ammeter reading. It will be observed that with an inductive circuit the current in- creases as the frequency decreases, and that the maximum current possible flows in the form of direct current. With a condensive circuit the current decreases as the frequency decreases, and is zero with direct current. With a non- reactive circuit, such as 117V.' r~l R*o eo v the 50 c. p. lamp, the current flow is indepen- 'j' 16 ' dent of the frequency. Fig - I5 - Direct current at 60 volts for this experiment can be secured, of course, by running a small dynamo at suitable excitation, but more easily from the 1 1 /-volt street service by the arrangement shown in Fig. 150. The lamps can be adjusted to give 204 ALTERNATING-CURRENT MACHINES. 60 volts, and the rheostat can take up the difference. This adjustment will be, of course, somewhat different for different loads. Second Part. The following solution is one of the many used for blue-prints, and has, besides, the property of turning blue, at the anode only, when a current is passed through it, if the anode be of iron. Mix 25 parts (by weight) of ammonium nitrate, NH 4 NO 3 , and 12.5 parts of ammonium muriate, NH 4 C1. Dissolve 1.3 parts of ferri- cyanide of potassium, K 3 Fe (CN) (; , (red prussiate of potash) in 1000 parts of water. Add the ammonium salts. The chemicals should be pure and the water distilled. Keep in a dark place, and use within twenty-four hours. Prepare an insulating handle, Fig. 151, with three piano- wire projections long enough to be elastic, and whose points may touch a plane surface in a right line and 16 c. p. 75 v - 3 Phase ' 40 Fig. 151. near together. Let these wires be connected through 16 c. p. lamps respectively to the terminals of a three-phase system, the pressure being 100 volts or less, and the fre- quency 40 or less. Lay an uncalendered paper well moistened, but not soaked, in the blue-print solution upon a metal plate, and draw the marking-points quickly across its surface. Blue marks will be left when the current is passing in one of its directions ; and these, by their inter- rupted nature, will show the change of direction in the alternating current. Also the relative displacement or TESTS. 205 120- 90- 240- 210- " stagger " of the rows of marks will show the phase displacements of a three-phase current, as in Fig. 152. Third Part. Excite a i6-candle power lamp with alternating current at its rated voltage and a low frequency, say 40 ^. Hold one end of a bar magnet against the bulb and in various positions. The fila- ment will vibrate synchronously with the alternations, due to the regularly recurring attraction and repulsion between the per- manent magnetic field of the magnet and the alternating field of the filament. If this experiment fails at first, try varying the frequency, the strength, and polarity of the magnet, and even try other lamps. Often the filament can be made to so vi- brate as to touch the glass, and finally rupture itself. 98. Exp. 2. Shape of E.M.F. Wave of Alternator. To perform this experiment use is made of a balance, as shown in Fig. 153. It consists of a hard graphite rod, C, of high resistance, through which current is passed from a direct-current con- stant potential source, two 16 c. p. lamps being in series to guard against accident in case of accidental short cir- cuit. A rolling contact bears upon this rod, and allows of a nice adjustment of the pressure applied to the testing circuit. This pressure can be accurately measured by the standard direct -current voltmeter V. In one branch of o-l Fig. 152. 206 ALTERNATING-CURRENT MACHINES. the test circuit is placed the telephone receiver, T. The operation is as follows : The test circuit, consisting of the armature of the alternator, a lamp or other non-inductive resistance for protection, a contact-maker, and the E.M.F. balance just described, is closed for an instant at some point of the revolution which corresponds to some point of the curve of instantaneous pressures. At such instants current will flow through the test circuit, causing the telephone receiver to click sharply ; and this click comes with a rapidity corresponding to the rate of revolution of the contact -maker, say 1800 per minute. The sliding con- tact on the graphite rod is then operated until the con- tinuous direct E.M.F. is just equal and opposite to the in- stantaneous E.M.F. put forth by the alternator. Then there will be no flow of current whether the contact-maker be Fig * I53 ' opened or closed, and the re- ceiver will cease to click. The voltage can be read di- rectly on the voltmeter, thus obviating the use of any reduction constants. This method, due to Mershon, is very delicate, since a telephone receiver is sensitive to very small currents. To obtain an E.M.F. curve from an alternator, arrange the apparatus as in Fig. 154. The contact-maker is con- nected directly to the shaft of the generator, and is obliged to revolve in unison therewith. Run the alternator at its rated speed and voltage. Set the brush of contact-maker TESTS. 207 at the desired beginning point, and balance the instanta- neous E.M.F. by sliding the balance until no clicking is heard in the receiver. Note the setting of the contact- Balance 1 wvwvi Fig. 154. maker and the reading of the voltmeter in the balance. Set the contact-maker ahead by five electrical degrees (i mechanical degree =/ electrical degrees, where/ is the number of pairs of poles), and repeat as before. Take readings thus by steps of five degrees throughout one complete cycle, i.e., under two poles. Since the instan- taneous E.M.F. will be in one direction during half a cycle, but in the opposite direction during the other half, and balancing E.M.F. is always in the same direction, a commutator must be introduced in the test circuit, as shown. When the commutator is in one position, the voltage readings should be marked -f, when in the other position, they should be marked . Plot a curve with volts as ordinates and degrees as abscissae. Indicate on the margin of the curve-sheet the effective value of the curve as obtained from the alternat- ing-current voltmeter. By means of a planimeter meas- ure the area of one lobe of the curve, and find its average ordinate, by dividing the area by the base line, i.e., the 208 ALTERNATING-CURRENT MACHINES. length corresponding to 1 80. This may be done in inch and square inch units, if the planimeter be so calibrated, without reducing. Lay this average ordinate off on the margin also. Divide the effective value by the average value to obtain the form-factor ( 4) of the pressure wave. If this value be about 1 . 1 1 , the curve is practically a sinusoid. 99. Exp. 3. Shape of Current Wave of Alternator with Inductive Load. Arrange the apparatus as shown in Fig. 155. The method of procedure is that of Exp. 2. The instantaneous drop of potential along a non-induc- tive resistance is proportional to, and in phase with, the current in that resistance. Measure the resistance of Coil 1 I WWVWW\R To Balance the 50 c. p. lamp under the conditions of ?/se, since the resistance of a carbon filament varies widely with the temperature ; and from this, and the values of instanta- neous pressures observed, calculate the instantaneous cur- rents according to Ohm's Law. Plot a curve with amperes as ordinates, and degrees as abscissae, TESTS. 209 ioo. Exp. 4. Simultaneous Pressure, Current, and Power Curves from Alternator with an Inductive Load. Arrange apparatus as in Fig. 156. It will be seen that either a point on the pressure curve (Exp. 2), or a point on the current curve (Exp. 3), can be taken by suitably placing the two-throw switch. Putting the commutator in the main circuit instead of the test circuit is possible when the load is light, does not affect the validity of the observations, and eliminates a possible source of trouble from bad contacts in the test circuit. Readings are to be Coil 1 Fig. 156. taken every five degrees through 400, a little over one cycle. Take readings for both pressure and current curves each time before moving the contact-maker. This is better than taking a complete current curve, then going back and taking a pressure curve, since there is more liability to distortion due to change in conditions in the latter case. The voltmeter, ammeter, and wattmeter readings should not vary during the test, and occasional observations should be made to see that this condition is complied with. If it cannot be, readings at stated intervals should be taken, and their averages used in the subsequent calculations. 210 ALTERNATING-CURRENT MACHINES. Plot three curves on one sheet, having degrees as their common abscissae, and volts, amperes, and watts as their respective ordinates. The instantaneous watts at any abscissa equal the product of the instantaneous volts and amperes for that same abscissa. In general, a separate scale of ordinates will be required for each curve. The curves will have the general relations shown in Fig. 14. Note the number of degrees intercepted on the axis, between the pressure curve and the current curve. This is the angle of lag, <, the cosine of which is the power- factor of the circuit if the pressure wave is sinusoidal. By the method given in Exp. 2, find the form-factor of the pressure curve. Divide this by the form-factor of a true sinusoid, i.e., i.i i, and call the quotient K. Then K cos < is the power-factor of the curve, whether sinusoidal or not. By means of a planimeter, measure the area of the lobes of the power curve, being careful to go around the nega- tive part in a counter-clockwise direction. Find the mean ordinate of this curve by dividing the area by the base line, and determine its value in watts by laying off on the scale of ordinates for the power curve. Fill out the following table, putting in the last column the percentage variation of the individual values from the average. How DETERMINED. WATTS. % By Wattmeter By Planimeter By E x I X K cos . . . . Average O TESTS. 211 The variations should be within the limits of errors of instruments and observations, say 2%. 101. Exp. 5. Measurement of Self -inductance. There are various methods of measuring the coefficient of self- induction, two of which are given here. The first is applicable to any series circuit, and consists in the deter- mination of the quantities in the general expression E -K27T/Z) 2 and solution for Z. If E, /, and R be measured respec- tively in volts, amperes, and ohms, L will be expressed in henrys. (a) Arrange apparatus as shown in Fig. 157, all the lamps being turned off. Insert at x successively Coil i, Coil 2, and Coil 3. Turn on lamps until a good ammeter deflection is ob- tained, and note readings of ammeter and volt- meter. The ohmic re- I I ToX - sistance must in each Fig ' I57 ' case be independently determined if not already known. Take four sets of observations with each coil; with and without iron core, at 40 /-*, and 60 ~. Solve for the inductance in each case from =41 -J? Without iron in the magnetic circuit, L is a constant of the circuit, independent of / and/; but when iron is pres- 212 ALTERNATING-CURRENT MACHINES. ent, it varies considerably with / and slightly with f. The variation of inductance with load is the subject of Exp. 7. Caution must be used in this experiment, that the am- meter be not injured. For instance, the careless removal of an iron core with closed circuit may cause a destructive increase of current. (b) The above method of measuring the inductance is not applicable to branched or parallel circuits with different time constants, for the reason that the resistance of the whole circuit, as measured by direct current, is not the equivalent resistance of the circuit, as explained in 28. A method using a voltmeter, ammeter, and wattmeter is entirely general, is equally accurate, and does not require 120V. A.C M C/l v V 200 ToX. _? Y- W 150 Fig. 158. the independent determination of the resistance. Arrange the apparatus the same as in the first part of this experi- ment, with the addition of a wattmeter, as shown in Fig. 158. The method of procedure is the same as be- fore, save that the wattmeter reading is also noted in each case. If /, E, and P be the instrument readings in am- peres, volts, and watts respectively, then the inductance in henrys is E . / P z= 7 s:n l CC El **/ TESTS. 213 This equation results from a consideration of the fol- lowing : P cos < = - Z sin < (see Fig. 30). E P 102. Exp. 6. Measurement of Capacity -- When there is no resistance and no inductance in a circuit as is the case with a condenser the general formula reduces to / = hence _ C- == 27T/E Arrange the apparatus as in Fig. 159. Let x be the six condensers taken, first two at a time, then three at a time, then all together, always arranging them in parallel. Note the current and pressure in each case, and 125^ solve for C by the above formula. The capacity of any parallel com- bination of condensers is the sum ^s- *& of the capacities of the component parts, and should so be shown by this experiment. If E and / be in volts and To X. 214 ALTERNATING-CURRENT MACHINES. amperes respectively, then C will be in farads. In the report reduce these results to microfarads by multiplying by io 6 . This method is not open to the objections to the similar method of measuring inductance, since here the resistance is practically zero. Yet the second method, using the wattmeter, could be employed. The formula would be - ^ = Z sin 0>C But the wattmeter will read zero, since little power is lost in a condenser, so < = 90, and - = Z, or "=-4 (*) which is the same as deduced from the general formula. 103. Exp. 7. Variation of Coefficient of Self-Induction Under Load. This experiment may be performed in two parts ; (a) by varying the magnitude of the measuring current, (b) by using a constant measuring current, and varying the saturation of the magnetic circuit by a separate current in a separate winding. (a) Measure the coefficient of self-induction of the fine wire coil of a i to 2 transformer by either of the methods of Exp. 5 . This current must be made to vary by suitable steps, and this can most easily be done by applying differ- ent pressures to the coil. A wide assortment of pressures can be obtained by using different brushes of the converter supplying the energy, and the different steps of the i, 2, 3, 4 transformer. Determine the value of for each TESTS. 215 of the conditions, and plot a curve having these values as ordinates, and the corresponding currents used in measur- ing as abscissae. A curve, such as Fig. 160, will result .3*1 .30 It -.26 8.22 Q 1 1! 1 ^ MAGNETIZING CURRENT. AMPERES Fig. 160. 3.5 with certain irons if the current be started low enough. The sharp rise of the curve at first is due to the fact that at very low densities the permeability increases with density, as is shown in the curves on page 24, Vol. i. A 10 'XAAAA/V Reo Fig. 161. (b) Arrange the apparatus as in Fig. 161. The meas- urements of L are made on the fine-wire side of the I : 2 2l6 ALTERNATING-CURRENT MACHINES. transformer, while the permeability is altered by direct current in the low-pressure side. The measuring current should be kept constant ; and as it has a tendency to rise as L decreases, resistance will have to be inserted in the alternating-current circuit by adjusting R^. Take read- ings at suitable steps from zero amperes direct current to the maximum safe temporary ampere capacity of the coil in question, say 15 amperes for a i K. w. 55-volt coil. Calculate the value of L for each of the steps, and plot a curve, using these values as ordinates and the direct- current magnetizing amperes corresponding thereto as abscissae. 104. Exp. 8. Measurement of Mutual Induction Arrange the apparatus as in Fig. 162, the requisite pres- Iron core through both Fig. 162. sure being secured by stepping up in the I : 2 transformer. The experiment consists of three parts : - (a) Determine the coefficient of mutual induction be- tween the two coils from the formula Transpose Coil 2 and Coil 3, and determine M again, the formula changing, of course, to The results should be alike if the- same current flows in each case. TESTS. 217 Finally calculate the theoretical value of M on the as- sumption of no magnetic leakage from M= VZ 2 Z 3 . Z 2 and Z 3 were determined in Exp. 5. If the same meas- uring currents be used throughout, this last value of M will be somewhat above the others, since there is some leakage flux. (b) With the arrangement of Fig. 162 place the iron core with its end flush with the outside of Coil 2, and pro- jecting clear through Coil 3. Move Coil 3 by steps of 2 cm. each from o to 24 cm. from Coil 2, and measure the value of M for each step. Be careful that the iron core be not moved relatively to Coil 2. Plot a curve with centimeters as abscissae and values of M as ordinates. (c) Repeat the last, keeping the iron flush with Coil 3 however, and moving Coil 2. In this case the current in Coil 2 will vary, and the curve will be distorted by the effects of load and saturation as investigated in Exp. 7. Plot curve of (b) and that of (c) on the same sheet and to the same scale. Be careful not to remove the iron core entirely from the coil that is carrying the current, or the current will exceed the capacity of the ammeter. 105. Exp. 9. Measurement of Power in a Single-phase Inductive Circuit. --There are various ways of measuring power in alternating-current circuits besides using a watt- meter, but none are as satisfactory. In the following it is desired to measure the power in Coil i : (a) By the three-voltmeter method. Arrange the appa- 2l8 ALTERNATING-CURRENT MACHINES. At sov. Coil 1 ! LgJ 1 1 |r ^ K E 1 h F _ Fig - l63 - sures indicated. The power, P, in the coil is ratus as in Fig. 163, the non-inductive lamp resistance, R, having been previ- ously determined. With a loo-volt al- ternating-current volt- meter note the pres- . (b) By the three-ammeter method. Arrange the appa- ratus as in Fig. 164. If / be the reading of A e>) 7 1 that of A z , and 7 2 that of A lt the power in the circuit is where R is the non-inductive resistance of the lamps, which must be independently determined for the condi- tions of operation. Fig. 164. Fig. 165. (<:) By the combined method. Arrange apparatus as in Fig. 165. If /be the reading of A & 7 2 the reading of A^ and E the reading of the voltmeter, then the power in the TESTS. 219 If it be desired to compare the results of (a), (d), and (V), arrangements must be made so that the same difference of potential may be applied to the terminals of Coil I in each case. These methods are rather impractical, and open to the two serious objections that a small error of observation may lead to a serious error in the result, and that the maximum accuracy can only be obtained when about as much power is consumed in the auxiliary devices as in the circuit under test. 106. Exp. 10. Measurement of Power in Polyphase Circuits by Indicating Wattmeters In any two-phase cir- cuit of four wires the load can be measured by two watt- meters, one connected regularly in each phase. The sum of their readings is the power in the circuit. In a two- phase four-wire system with a balanced load, one of the wattmeters may be dispensed with, and the reading of the other multiplied by two. In any two-phase, three-wire system the power can be measured by two wattmeters connected as in Fig. 166. Load. Fig. 166. The sum of the instrument readings is the whole power. In a two-phase, three-wire system, where all the load is con- 220 ALTERNATING-CURRENT MACHINES. 167. nected between the outside wires and the common wire, and none between the outside wires themselves, and where the load is balanced, then one wattmeter can be used to measure the whole power by connecting its current coil in the common wire nd its pressure coil between the com- mon wire and one outside wire first, then shifting this connection to the other outside wire, as indicated in Fig. 167. The sum of the instrument readings in the two positions is the whole power. A wattmeter made with two pres- sure coils could have one connected each way, and the instru- ment would automat- ically add the readings, giving the whole power directly. Or, again, a high non-reactive resistance could be placed between the two outside wires and the pressure coil of the wattmeter connected between the common wire and the center point of this resistance. This requires that the wattmeter be recalibrated with half of this high resistance in series with its pressure coil. With the exception of the two-phase systems, the power in any balanced polyphase system may be measured by one wattmeter whose current coil is placed in one wire, and whose pressure coil is connected between that wire and the neutral point. The instrument reading multiplied by the number of phases gives the whole power. The neutral point may be on an extra wire, as in a three-phase, four-wire system ; or may be artificially constructed by con- necting the ends of equal non-reactive resistances together, and connecting f:he free ends one to each of the phase wires. TESTS. 221 With the exception of the two-phase systems, the power in any //-phase, //-wire system, irrespective of bal- ance, may be determined by the use of ni wattmeters. The current coils are connected, one each, in ni of the wires, and the pressure coils have one of their ends con- nected to the respective phase wires, and their free ends all connected to the nth wire. The algebraic sum of the readings is the power in the whole circuit. Depending upon the power factor of the circuit, some of the watt- meters will read negatively, hence care must be taken that all connections are made in the same sense ; then those instruments which require that their connections be changed, to make them deflect properly, are the ones to whose readings a negative sign must be affixed. Some specific connections for indicating wattmeters in three-phase circuits are shown in the following figures. Fig. 1 68 shows the connection of three wattmeters to meas- Fig. 168. ure the power in an unbalanced three-phase system. All the readings will be in the positive direction, and their sum is the total power. If a fourth, or neutral wire be present, it should be used, instead of creating an artificial neutral, as shown. The magnitude of the equal non- reactive resistances, used to secure this neutral point, 222 ALTERNATING-CURRENT MACHINES. 4 * Balanced Load. t 1 ( xy J Fig. 169. must be so chosen that the resistances of the pressure-coils of the wattmeters will be so large, compared thereto, as not to disturb the po- tential of the artificial neutral point. Fig. 169 shows the connection of one watt- meter, so as to read one-third of the whole power in a balanced, three-phase, four-wire system. If the system be three-wire, a neutral point may be created as in Fig. 168. Fig. 170 shows the connections of two wattmeters for the determination of the power in balanced or unbalanced three-phase systems, avoiding the necessity of a neutral point. The algebraic sum of the instru- ment indications is the whole power. If the power-factor be greater than .5, both instruments will give positive readings ; if it be less, one instru- ment will give a negative reading. With low power-factors, such as given by a partially loaded induction motor, it is sometimes difficult to determine whether the smaller read- ings are negative or not. If in doubt, give the wattmeters a separate load of lamps (power-factor = i) and make the connections such that both instruments deflect properly. Then connect them to the load to be measured. If the Fig. 170. TESTS. 223 terminals of one instrument have to be exchanged, then to the readings of that instrument must be affixed the nega- tive sign. Fig. 171 shows the connections for one watt- meter in a balanced three-phase circuit, independent of a neutral point. The free end of the pressure coil is con- nected first to one of the wires opposite that in which the current coil is con- nected, then to the other. The alge- braic sum of the readings in the two positions is the total *~~[ power. Both read- Fig- 171. ings will be positive if the power-factor is greater than .5 ; but one of them will be negative if it is less. Hence care must be used to avoid confusion of signs at low power-factors. This method, requiring a two-throw switch to change the connection, two readings of the instrument, and, if used on a load varying from high to low power-factor, a commutator, to change the pressure coil connections, has little advantage over the method of Fig. 169, save that it dispenses with the necessity for a neutral point. Six-phase circuits are used generally only between the step-down transformers of three-phase transmission sys- tems and the alternating-current ends of rotary conver- ters ; hence they are always balanced. They can then be measured by the method of Fig. 169, where a neutral is employed, or the three alternate wires may be considered a three-phase system, the method of Fig. 171 employed, and the three-phase power thus determined multiplied by 2 to give the total power. If the circuit should be unbal. 224 ALTERNATING-CURRENT MACHINES. anced, five instruments would be necessary, as stated earlier in this section. The student is expected to construct circuits according to the various figures just given, and convince himself that the wattmeters do give the true power. If the load be of lamps, the power in each may be measured by a volt- meter and ammeter used at their terminals ; then by con- necting in star and in delta, balanced and unbalanced, the accuracy of the wattmeter indications can be checked. In following Fig. 166 or Fig. 167, it should be remem- bered that a two-phase current cannot be secured from an armature with a mesh winding, such as a rotary converter must have ; and that any attempt to make a two-phase, three-wire system out of a quarter-phase system will be disastrous. To get two-phase current from such a ma- chine, the quarter-phase current must be passed through the primaries of two similar transformers, two opposite wires going to one, the other two to the other. The transformer secondaries will then deliver true two-phase current, and the circuits may be united in a three-wire system. 107. Exp. ii. Calculation and Measurement of the Re- sulting Impedance of a Number of Impedances in Series. Arrange apparatus as shown in Pig. 172. Determine the impedance Z of the whole circuit from the readings of the voltmeter and ammeter. Independently determine the ohmic resistance of the cir- cuit with the same current flowing. The magnitude of the current affects the resistance of the lamp. TESTS. 225 Solve for the reactance, X = 2 -nfL, from the equation, Z = \IJ? + ... 226 ALTERNATING-CURRENT MACHINES. A t lOOc. p. 120V. Coil 1. Co;i2. Fig. 172. 108. Exp. 12. Calculation and Measurement of the Re- sulting Impedance of a Number of Impedances in Parallel. - Use the same impedances as in the last experiment, but arranged as in Fig. 173. As before stated, the voltmeter- ammeter-resistance method of solving inductive circuits is inapplicable to branched circuits ; so the wattmeter must be used as shown. Determine the equiva- / ^=^v 100 lent impedance from E Determine the angle of lag in the main circuit bv 60 v ' u y 4o~ Determine the equiva- lent resistance, R (which Fig. 173. is not the actual resistance of the parallel arrangement), from R = Z cos <. Determine the equivalent reactance from X = Z sin <>. TESTS. 227 All the constants of Coil I and Coil 2 are known ; but the resistance of the lamp had better be redetermined for the particular current used in it. Combine the admittances of these parts of the branched circuit into a polygon of admittances according to 28. Take the reciprocal of the resulting admittance, that is, the equivalent impedance, and resolve it into its com- ponent parts of equivalent reactance and equivalent resist- ance. The actual plotting may be done on 25" x 30" drawing-board to the scales 2 ohms = I cm. and I unit of admittance = 1000 cm. Make a report in the form of a table such as is used in the last experiment. The variation should not exceed 3%. 109. Exp. 13. Calculation and Measurement of Result- ing Impedance of any Series-Parallel or Parallel-Series Arrangement of a Number of Impedances. Arrange the apparatus as in Fig. 174, or according to any other scheme if it be desired to vary the experiment. Fig. 174. Determine the values of the resulting or equivalent I X, Z) and , as in Exp. 12, 228 ALTERNATING-CURRENT MACHINES. Also determine the same quantities for the individual parts of the circuit under the conditions of use if they be not already known. In the graphic determination pursue the following steps : 1. Find the equivalent impedance of Coil 3, and the 100 C.P. lamp, calling it M. 2. Find the equivalent impedance of the 50 c. P. lamp, and J/", calling it W. 3. Find the equivalent impedance of Coil i, and Coil 2, calling it P. 4. Find the equivalent impedance of P and N. This will be the required impedance of the whole circuit, and should be resolved into its com- ponent parts of equivalent resistance and equivalent reactance. Measure <, the angle between the impedance and the resistance. Make a report in the form of a table as in the two pre- ceding experiments. The variation of the determinations by the two method should not exceed 3 % . no. Exp. 14. Efficiency and Regulation of a Trans- former. Arrange the apparatus as in Fig. 175. A two-throw switch allows the same voltmeter to read either primary or secondary pressure. The ammeter A 3 may be used on the lower readings. The transformer used is the J K. w. i to 2, stepping up from about 58 volts to 116, its rated range. It is operated at its rated frequency, 60^. Increase the load from o to i K. w. (100% overload) by suitable steps. At each step take readings of the primary volts, primary watts ; secondary volts and secondary am- TESTS. 229 peres. Since the load is non-inductive, the product of the secondary volts and amperes gives the secondary watts. Determine the efficiency and the regulation, both in per cent, for each set of readings from watts secondary % efficiency = - X 100. watts primary ] regulation = T volts prim. volts sec. full-load sec. volts X 100. 1 to 2 Fig. 175- Plot two curves on the same sheet, having as their common abscissae both watts and per cent of full-load secondary, and as their respective ordinates per cent effi- ciency and 1 00% per cent regulation. in. Exp. 15. Determination of Load Losses in a Transformer The core losses are usually considered independent of the load, while all those that vary with the load are called the load losses. Their chief component is, of course, the PR loss in the copper, but there may be some eddy current and local hysteresis losses that vary with the load, and a determination of them all is made as follows : Arrange the apparatus as in Fig. 176. The I to 2 230 ALTERNATING-CURRENT MACHINES. Fig. 176. 0.5 K. w. transformer is used with its low-tension side short-circuited. There will be but a small pressure gener- ated therein, and its current will demagnetize the core almost entirely ; hence AIK all the losses measured may be considered as load, not core losses. Care must, of course, be taken to control the amount of current pass- ing through the trans- former. Adjust the lamps so that about 100% overload current, 10 amperes, is shown by A 15 . Read the ammeter and the wattmeter. Reduce the current by a suitable amount, and read again. So continue down to zero amperes, substituting A 3 for A 15 when the readings on the latter become unsatisfac- tory. Plot a curve with load in amperes as abscissae and load loss in watts as ordinates. Take care that the wire short-circuiting the low- pressure coil is of low resistance, 'and has good contacts. Note also that the pressure leads from the wattmeter should go direct to the terminals of the transformer, as, in general, the resistance of the wires leading to it is not negligible in comparison with the resistance of the coil itself. If the current exceeds the ampere capacity of the watt- meter, it is advisable to put in a single-pole switch to short-circuit the current coil at all times save when a reading is being taken, TESTS. 231 112. Exp. 16. Determination of Core Losses of a Transformer, and Construction of an Efficiency Curve. - The core losses, hysteresis chiefly, are constant at all loads. Hence, the energy supplied to a transformer when its secondary is open-circuited, is practically a measure of these losses. Connect a wattmeter in the primary circuit of the i K.W. transformer when its primary is supplied with pressure at its rated voltage and frequency, and its sec- ondary is open-circuited. The wattmeter reading is the core loss. From a knowledge of the core loss and the load losses at various loads, construct an efficiency curve for various loads from o to 100% overload (secondary), the efficiency in per cent at any load P l being where P r and P c are the load and core losses respectively at the load P r This curve should be similar to the efficiency curve found in Exp. 14. 113. Ex. 17 and 18. Simultaneous Pressure and Cur- rent Curves from Primary and Secondary of a Trans- former. It is desired to get these curves for two con- ditions. First, Exp. 17, with a full non-inductive load, and second, Exp. 18, with an equal (in amperes) very inductive load. Arrange apparatus as in Fig. 177. For the non- inductive load, lamps are suitable, for the inductive load the primaries of unloaded transformers are good ; and to get a nice adjustment Coil I can be put into circuit, and 232 ALTERNATING-CURRENT MACHINES. the current 'n it adjusted by moving its iron core in or out. It might be here remarked that if the transformer is supplied with current from a rotary converter, the E.M.F. balance described in Exp. 2 cannot use direct current from the same source as that which runs the converter, even though they be put on opposite sides of a three-wire system. A separate source of direct E.M.F. must in such case be supplied for the balance, either from a separate direct-current generator or from a sufficient number of To Load and Balance Fig. 177. cells of storage battery. If, however, the alternating cur- rent be passed through another transformer before being applied to the one under test, this trouble does not arise ; but the introduction of the second transformer has a dis- turbing effect on the wave-shape. It will be seen that the apparatus is merely an elabora- tion of that in Exp. 4, a four-way double-pole switch being used instead of the two-throw switch of the former experi- ment. This switch is conveniently made of mercury cups in a block of wood. For the i K. w. transformer at 58 to 1 1 6 volts, the reading on A 15 should be about 4.4 amperes. TESTS. 233 Suitable values for the non-inductive resistances are, R p = 2.2 ohms, R a = 4.4 ohms. These must be able to carry the currents without overheating, and must not be allowed to change their resistance due to change of temperature. Proceed as directed in Exp. 4, taking readings every twelve electrical degrees throughout half a cycle. Take all four readings before moving the contact-maker. Plot the four curves of Exp. 1 7 on one paper, and those of Exp. 1 8 on another. In each case degrees will be the common abscissae. Both pressure curves of either experiment must be plotted to one scale of ordinates, both current curves to another. Careful work will show a phase difference slightly less than 180 between the primary and secondary pressures and currents, particularly in Exp. 1 8. 114. Exp. 19. Calculation and Measurement of the Mutual-inductance of Transformer Coils at No Load. - (a) Measure the self-inductance of the primary and of the secondary coils by the method of Ex. 5, first part, the coil not under test being left open-circuited. (b) Do the same by the method of Exp. 5, second part. The results in the two cases should be alike if attention is paid to the following point. In measuring the primary inductance, apply the rated voltage so that it will send the charging current. In measuring the secondary, adjust the impressed voltage so that just such a current will flow as will give the same ampere-turns in the secondary as there were in the primary when it was being measured. If this precaution be not taken, the results will be changed by the effects of varying load, according to Exp. 7. (c) Using the average of the values of L p and L t as 234 ALTERNATING-CURRENT MACHINES. found in (a) and (b), calculate the value of M on the sup- position of no magnetic leakage, from M ' = VZ 8 Z P . (d) Measure the mutual induction by the method of Exp. 8 a, taking care that the ampere turns are the same in each case, and the same as were used in (a) and (b). This last result may be slightly less than that arrived at in (c) because of magnetic leakage. Care should be taken throughout that the frequency be kept constant. 115. Exp 20. Practice in Three-Phase Transformer Connections Three similar I to I transformers may be conveniently used for this experiment. The student may have to exercise some ingenuity in determining the direc- tion of winding in the coils. When each of the following connections has been made, excite the primaries by a three- phase current, and measure the pressure between each of the secondary wires, seeing that all three sides have the same and the expected voltage. (a) Connect both primaries and secondaries in Y> as shown in Fig. 88. See that E s E p . Then make the secondary a three-phase, four-wire system, and see that the voltage between any outside wire and the middle wire is . VI (b) Connect both primaries and secondaries in A, Fig. 87, and see that E 8 E p . Disconnect one transformer from the circuits, and observe that the three-phase pressure is still maintained in the secondary. (c) Connect the primaries in Y> and the secondaries in E A, as in Fig. 90. Observe that E t = *=. TESTS. 235 (d) Connect the primaries in A, and the secondaries in Y, as in Fig. 89, with a four-wire secondary system. Ob- serve that, with reference to the outside wires, E a = V3 E p , while considering any outside wire and the middle wire, E. = E, 116. Exp. 21. External Characteristic of an Alter- nator Run the alternator at normal speed and field excitation, both being kept constant during the experiment. Arrange a variable non-inductive load lamps preferably - so that readings can be taken from o load to 50% overload at suitable intervals. At each step note the armature current and the terminal pressure. Plot a curve with currents as abscissae, and pressures as ordinates. 117. Exp. 22. Field Compounding Curve of an Alter- nator Run the alternator at constant rated speed. Arrange a variable non-inductive load of lamps, ranging by suitable steps from o load to 50% overload ; at each step adjust the field current, so that the rated terminal voltage is maintained. Take simultaneous readings of field- current and armature current. Plot a curve with armature currents as ordinates, and field currents as abscissae. Note that the speed must be kept constant, that the terminal pressure must be kept constant, and that read- ings should be taken only with ascending values of field currents, as magnetic retentivity will distort the curve somewhat if the field current is run too high, and then brought down to the required point. 118. Exp. 23. No-Load Saturation Curve of an Alter- nator. - Run the alternator at constant rated speed, and ALTERNATING-CURRENT MACHINES. excite the fields from zero up to full excitation, taking, at suitable intervals, readings of the field current, and the no-load armature voltage. Repeat, carrying the excitation from full excitation down to zero. Plot the two curves on one sheet, using field currents as abscissae, and terminal pressures as ordinates. The two curves will not exactly coincide, because of the magnetic retentivity of the iron. Care must be taken always to adjust the field current by increasing from a lower value to a higher when taking the ascending curve ; and by decreasing from a higher value to a lower when taking the descending curve. 119. Exp. 24. Full-Load Saturation Curve of an Al- ternator Arrange apparatus as in Fig. 1 78. The alter- nator is a IK. w. 80 volt, single-phase machine. The machine is given a non-inductive load of lamps, and a heavy current rheostat which has zero resistance on the Fig. 178. last point. Run the alternator at its rated speed. Make the resistance of the external circuit zero i.e., short-circuit it through the ammeter. Adjust the field rheostat to its maximum resistance, and close the field switch. Increase the excitation by manipulating the field rheostat until the rated full-load current is flowing in the external circuit, as TESTS. 237 shown on^( 15 . Take readings of the field amperes, and the terminal volts, the latter being zero at this step. Increase the resistance of the armature circuit by a suitable amount, and readjust the excitation till the rated full-load current is again flowing in the armature, and take readings of field current and terminal voltage. Repeat at suitable steps until full field excitation is obtained. Plot a curve on the same paper, and to the same scale as that of Exp. 23, using field currents as abscissae, and terminal volts as ordinates. Take heed that readings are always taken with ascend- ing values of field current, and when the ammeter in the armature circuit shows rated full-load current. The speed must be kept constant. 120. Exp. 25. Synchronous Impedance of an Alterna- tor As stated in 38, the synchronous impedance of an alternator varies somewhat with the load, but is practically constant at all excitations ; hence its determination is easily accomplished in the following manner : Arrange the apparatus as shown in Fig. 179. Run the alternator at its rated speed. By means of the field rheo- stat cut the excita- tion down to a mini- mum. Short-circuit the armature through an am- meter and switch as shown. Adjust . . Fig. 179. excitation so that the ammeter shows about |- full-load current, and note the ammeter reading. Open the switch in the armature cir- 238 ALTERNATING-CURRENT MACHINES. cuit and note the terminal volts. Close the switch, read- just excitation so that the load is increased by a suitable amount, and repeat the readings. Repeat until a limit is reached, either because full field excitation has been ob- tained, or because the machine is being too severely over- loaded. Which of these two conditions arises first depends upon the synchronous impedance of the machine. Calculate the synchronous impedance for each set of readings from open circuit voltage Syn. Imp. = - , short circuit current when the readings are for the same excitation and speed. Plot a curve with armature currents as abscissae, and the values of the synchronous impedance as ordinates. Particular attention should be paid that the speed be kept constant, as it is liable to rise on throwing off the load. 121. Exp. 26. Core loss of an Alternator The core loss of any armature is determined by measuring the dif- ference in power required to run it with and without field excitation. With an alternator this is most easily done by running the armature by a rated motor and observing the power input thereto. It is desirable to have the quantity sought as large as possible in comparison with the quanti- ties observed ; hence the rated motor used should be as small as is practicable. The alternator must be driven at its rated speed, and the pulleys so proportioned that the motor will run at its rated speed also ; or else a special efficiency curve of the motor must be obtained for the speed at which it will be required to run. TESTS. 239 A wattmeter placed in the motor circuit will indicate the power input thereto ; or, if it be a direct-current motor, a voltmeter and ammeter can be used. Let A = watts input to motor when the alternator field is not excited. and m = efficiency of motor at this input. Let J?= watts input to motor when the alternator fields are fully excited, and n = efficiency of motor at this input. Then the core loss in watts is P c = Bn - Am. It is well to repeat the measurements a number of times and average the results. Since the losses in shafting and belting are practically the same at all loads, these do not affect the accuracy of the results. 122. Exp. 27. Complete Test of a i-H.P. Three-Phaso Induction Motor As a test of the motor performance solely, the voltage at the motor terminals should be kept constant throughout the test. This may easily be accom- plished if the motor is run from a separate alternator. If, however, it is run from an inverted converter, and particu- larly if the desired voltage has to be obtained by transfor- mation, there will be a slight drop of voltage as the load increases. Since the power-factor in this test will run from very low to about 80%, the method of measuring three-phase power shown in Fig. 169 will be used, as it requires but one instrument reading, and leads to no uncertainty as to 240 ALTERNATING-CURRENT MACHINES. algebraic signs. The apparatus used is simply an amme- ter, a voltmeter, and a wattmeter connected into the motor circuit. Fig. 180 shows the arrangement, the two watt- meters not being used at once, but being alternative, one for high readings, the other for low, thus securing a greater accuracy over a wide range. The motor when stalled takes about 18 amperes; so this is its momentary Fig. 180. starting current. Care must be taken in starting up that the measuring instruments are not injured by such a flow of current. The larger wattmeter has a capacity of 2.5 K. w., and a 25-ampere limit ; the smaller a capacity for 300 watts, and a 5-ampere limit. Either of these, as well as the voltmeter, will stand the rated motor pressure, no volts. The power output of the motor is absorbed in a strap TESTS. 241 brake, as shown in Fig. 181. With a 4.5" pulley at 1800 revolutions the spring balances should have ranges of about 30 Ibs. and 4 Ibs. respectively. The motor must be supplied with current at its rated voltage and frequency, and the frequency must be kept constant throughout the experiment. Observations. Take readings at suitable intervals, say steps of 4 Ibs. each on the larger scale, from no load to the stalling of the motor. Do not leave the motor stalled, as it overloads the instruments. At each step take readings of the watt- meter, ammeter, voltmeter, both spring balances, and the speed of the motor. Repeat the experiment three times with fifteen-minute intervals between the repeti- tions. The scale readings, P, will be the Flg> l8l> same, at any one step, for all three trials, and the other values can be averaged to partially eliminate errors of observation. Calculations. -- Using the average values of the three readings at any one step, fill out the following table : ' 2 3 4 5 6 7 8 9 H ^ H h a $ * WATTS OUTPU WATTS INPUT. VOLTS AT TERMINA AMPERES PER PHA VOLT-AMPERES INP Po\VER-FACTOI EFFICIENCY %. APPARENT EFFICIENCY * a. 3 CO 242 ALTERNATING-CURRENT MACHINES. Kir (*-/") (1) Watts output = -- 746, 33,000 where d = diameter of pulley in inches. V = revolutions per minute. (P P f ) = difference in scale readings in pounds. (2) Watts input = 3 X wattmeter reading. (3) Volts at terminals = voltmeter reading. (4) Amperes per phase = ammeter reading. (5) Volt-amperes input = V3 X volts at terminals X amperes per phase. Watts input (6) Power-factor % = ==-r - X 100. Volt-amperes input / \ -i-rc / Watts output (7) Efficiency % = - - X 100. Watts input ,_. . Watts output (8) Apparent efficiency = - : - X 100. Volt-amperes input (9) P where V = revolutions per minute, f = frequency, p = number of pairs of poles. Plotting of Curves. Plot eight curves on one paper. All the curves will have watts output as abscissae. The points of 25%, 50%, 75%, 100%, and 125% of full load should also be indicated. The ordinates for the first seven curves are taken from columns 2 to 8 respectively, TESTS. 243 The ordinates for the last curve are found by subtract- ing the per cent slip from 100%. Curves should be marked with the names appearing at the heads of the columns from which their ordinates were taken. The curves from columns 2 and 5 should be to the same scale of ordinates. Those from columns 6, 7, 8, and 9 will all have the same scale of ordinates, which will be per cents, and should run from o to 100%. There will thus be four scales of ordinates, and they should be marked respectively, " Watts or Volt Amperes," "Volts," "Amperes," and "Per Cent." On the margin state the name and size of the machine, and the date of test. 123. Exp. 28. Complete Test of a 1 H.P., Three-Phase Induction Motor, run on a Single-phase Circuit Through a Condenser-Compensator. The function of the condenser- compensator was discussed in 67. The arrangement of apparatus is shown in Fig. 182. Another wattmeter and Motor Fig. i8a. another ammeter may be used to alternate with those shown to secure greater accuracy in the lower ranges if considered advisable. The same absorption dynamometer is used as in Exp. 27 ; and the directions there given for taking observations, and for calculating and plotting results, should be followed with the following exceptions : take readings at 2-lb. steps, 244 ALTERNATING-CURRENT MACHINES. since the motor is half the size of the other ; since this is single-phase, the wattmeter readings go direct in column 2, and the products of the volts by the amperes go direct in column 5. It may be found that the capacity of the condenser- compensator has been so proportioned that in plotting the results, curves 2 and 5, and also 7 and 8, will be nearly coincident, and that curve 6 is practically a straight hori- zontal line. 124. Exp. 29. Methods of Synchronizing. Synchro- nous motors and also converters must be synchronized before being connected to the mains from which they receive their power. There are a number of ways of doing this, of which the best depends upon attendant cir- cumstances, (a) The motor and generator may be elec- trically connected while at rest, and the latter started up slowly, the motor not loaded then starting up and running synchronously, (ft) The field circuit of the motor may be left open, and the armature started up without load as an induction motor until near synchronism, and the field switch then closed. In large machines this endangers the insulation of the field coils, (c) The arma- ture may be brought to speed mechanically, either by a small direct connected induction motor or by a belt from some moving pulley, (d) In converters the machine can be started and brought to speed from the direct-current end like a direct-current motor, if there be direct cur- rent available. This requires a starting-box and a field rheostat. The two convenient methods for synchronizing the I K, w. three-phase converter are (d) and (d) ; the former TESTS. 245 will be practiced in Exp. 30, the latter is the subject for the present experiment. Arrange the apparatus as shown in Fig. 183. At starting, the field coils of the converter must be excited from the source of direct current, but when running as a converter the machine must be excited from its own brushes. This necessitates the two switches, a and b. These switches must not be both open at once, at least while the machine is running from the direct-current end ; and if they are not rightly connected the direct-current Main Switch 3 Phase Fig. 183. source will be short-circuited when they are both closed at once. It is best, after the set-up is made, to test across the switches with a voltmeter. The switch must not be closed if any pressure shows across the gap it is intended closing. When the connections have been properly made, open the main switch and switch