ELECTRIC OSCILLATIONS AND ELECTRIC WAVES &.J PUBLISHERS OF BOOKS FOR^ Coal Age '* Electric Railway Journal Electrical World ^ Engineering News-Record American Machinist * Ingenieria Internacional Engineering 8 Mining Journal ^ p o we r Chemical 6 Metallurgical Engineering Electrical Merchandising ELECTEIC OSCILLATIONS AND ELECTKIC WAVES WITH APPLICATION TO RADIOTELEGRAPH Y AND INCIDENTAL APPLICATION TO TELEPHONY AND OPTICS BY GEORGE W. PIERCE, PH. D., PROFESSOR OF PHYSICS IN HARVARD UNIVERSITY FIRST !}DJ,TTQN , McGRAW-HILL BOOK COMPANY, INC. 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., LTD. JVERIE 1920 6 & 8 BOUVERIE ST., E. C. COPYRIGHT, 1920, BY THE McGRAw-HiLL BOOK COMPANY, INC. THK MAPX.EPRES8 YOHJC PA. PREFACE This book is designed to present a mathematical treatment of some of the fundamentals of the theory of electric oscillations and electric waves. Although the selection of material particularly applicable to radiotelegraphy has been the first consideration, yet, because the electromagnetic theory, which is fundamental to radioteleg- raphy, is fundamental also to optics, wire telephony and power transmission, it is hoped that the volume may be useful in these fields also. Book I on Electric Oscillations and Book II on Electric Waves are practically independent, so that a reader with a fair knowledge of mathematics may read the two books in either sequence. A student in optics might confine his attention almost entirely to Book II. A mature reader primarily interested in wire tele- phony or power transmission might begin at Chapter XVI of Book I, and continue through Chapter XVII, with such occa- sional references to the earlier chapters as are necessary for familiarity with the methods employed. He might then look into some of the earlier chapters in order to acquaint himself with the various transformer problems arising in connection with coupled circuits. It is suggested that students of radiotelegraphy begin at the beginning of Book I and read the various chapters consecutively, with the possible exception of Chapters IX, X, and XV, which may be omitted or postponed without rendering difficult the understanding of what follows. It is perhaps unnecessary to say that the theoretical work of this book should be supplemented by copious descriptive matter and laboratory work. Certain important subjects related to radiotelegraphy have been omitted particularly the matter of spark-gaps, arcs, va.cuum tubes, direction finders and the propagation of electric waves over the surface of the earth. These defects are to be partly remedied by including the omitted material in a revision vi PREFACE of the author's earlier book "The Principles of Wireless Teleg- raphy" and in other writings now in preparation. The writer takes pleasure in acknowledging his indebtedness to Mr. Yu Ching Wen for valuable assistance in reading the proofs; to Mr. H. E. Rawson for supplying a draftsman; and to the publishers for their accuracy and dispatch in the difficult com- position and manufacture of the book. G. W. P. CRUFT LABORATORY, HARVARD UNIVERSITY, CAMBRIDGE, MASS., U. S. A., December, 1919. CONTENTS Book I. Electric Oscillations CHAPTER I PAGE FUNDAMENTAL LAWS AND EQUATIONS . . : l CHAPTER II THE FLOW OF ELECTRICITY IN A CIRCUIT CONTAINING RESISTANCE, SELF-INDUCTANCE, AND CAPACITY. DISCHARGE, CHARGE, AND CURRENT INTERRUPTION ' 9 CHAPTER III ENERGY TRANSFORMATIONS DURING CHARGE OR DISCHARGE OF A CON- DENSER 32 CHAPTER IV THE GEOMETRY OF COMPLEX QUANTITIES . . 42 CHAPTER V CIRCUIT CONTAINING RESISTANCE, SELF-INDUCTANCE, CAPACITY AND A SINUSOIDAL ELECTROMOTIVE FORCE 51 CHAPTER VI ELECTRICAL RESONANCE IN A SIMPLE CIRCUIT 60 CHAPTER VII THE FREE OSCILLATION OF Two COUPLED RESISTANCELESS CIRCUITS. PERIOD AND WAVELENGTHS 73 CHAPTER VIII THE FREE OSCILLATION OF Two COUPLED RESISTANCELESS CIRCUITS. AMPLITUDES 86 CHAPTER IX THE FREE OSCILLATION OF Two INDUCTIVELY COUPLED CIRCUITS. PERIODS AND DECREMENTS. TREATMENT WITHOUT NEGLECTING RESISTANCES 94 vii viii CONTENTS CHAPTER X PAGE AMPLITUDE AND MEAN SQUARE CURRENT IN THE INDUCTIVELY COUPLED SYSTEM OF Two CIRCUITS 138 CHAPTER XI THEORY OF Two COUPLED CIRCUITS UNDER THE ACTION OF AN IM- PRESSED SINUSOIDAL ELECTROMOTIVE FORCE 156 CHAPTER XII RESONANCE RELATION IN RADIOTELEGRAPHIC RECEIVING STATIONS UNDER THE ACTION OF PERSISTENT INCIDENT WAVES 176 CHAPTER XIII A GENERAL RECIPROCITY THEOREM IN STEADY-STATE ALTERNATING- CURRENT THEORY WITH APPLICATION TO THE DETERMINATION OF RESONANCE RELATIONS ......... 204 CHAPTER XIV RESONANCE RELATIONS IN A CHAIN OF THREE CIRCUITS WITH CON- STANT PURE MUTUAL IMPEDANCES. STEADY STATE 226 CHAPTER XV RESONANCE RELATIONS IN A RADIOTELEGRAPHIC RECEIVING STATION HAVING A COUPLED SYSTEM OF CIRCUITS WITH THE DETECTOR IN SHUNT TO A SECONDARY CONDENSER 240 CHAPTER XVI ELECTRICAL SYSTEMS OF RECURRENT SIMILAR SECTIONS. ARTIFICIAL LINES. ELECTRICAL FILTERS 286 CHAPTER XVII ELECTRIC WAVES ON WIRES IN A STEADY STATE 324 Book II. Electric Waves CHAPTER I ELECTROSTATICS AND MAGNETOSTATICS 347 CHAPTER II MAXWELL'S EQUATIONS 358 CONTENTS ix CHAPTER III PAGE ENERGY OF THE ELECTROMAGNETIC FIELD. POYNTING'S VECTOR. . . 370 CHAPTER IV WAVE EQUATIONS. PLANE WAVE SOLUTION 377 CHAPTER V REFLECTION OF A PLANE WAVE FROM A PERFECT CONDUCTOR .... 391 CHAPTER VI VITREOUS REFLECTION AND REFRACTION 399 CHAPTER VII ELECTRIC WAVES IN AN IMPERFECTLY CONDUCTIVE MEDIUM 408 CHAPTER VIII ELECTRIC WAVES DUE TO AN OSCILLATING DOUBLET 421 CHAPTER IX THEORETICAL INVESTIGATIONS OF THE RADIATION CHARACTERISTICS OF AN ANTENNA 435 APPENDIX I 489 TABLE 1 RELATIONS OF CAPACITY INDUCTANCE PRODUCT TO UNDAMPED WAVELENGTH AND FREQUENCY OF A CIRCUIT, TOGETHER WITH SQUARES OF WAVELENGTHS . 502 TABLE II RADIATION RESISTANCE IN OHMS OF FLAT-TOP ANTENNA 509 TABLE III CONVERSION OF UNITS 510 INDEX . 511 BOOK I ELECTRIC OSCILLATIONS CHAPTER I FUNDAMENTAL LAWS AND EQUATIONS 1. Notation. I = Current (constant), Q = Quantity of electricity (constant), E = E.m.f., or difference of potential (constant), i = Instantaneous value of current at time t (variable), q = Instantaneous value of quantity at time t (variable), e = Instantaneous value of e.m.f. at time t (variable), R = Resistance, L = Self-inductance, C = Capacity. When several of these quantities enter into the same equation, they must all be measured in some common set of units. 2. Kirchhoff's Current Law 1 for a Steady State. When a conductor is traversed by a constant current and all the charges FIG. 1. Conductor with sections o, (5) As a generalization of this equation, if we consider current flowing into a given region bounded by a closed surface to be positive, and current flowing out to be negative, then Equation (6) may be stated as follows: The excess of the current flowing into a given region at a given time over the current flowing out at the same time is the time-rate of increase of quantity of electricity within the region at that time. TJiis is a statement of the Law of the Conservation of Elec- tricity, and applies to all cases of the flow of -electricity whether the flow is constant or variable. We shall call the equation Kirchhoff's Generalized Current Law, or Kirchhoff's Current Law. The terms employed in the statement and equations are explained in the next section. - 5. Explanation of Terms of Foregoing Statements and Equa- tions. Intrinsic Charge. The quantities q and ^ in equation (6) must be measured in the same set of units. If ii is in amperes, q must be in coulombs. If, on the other hand, i t is in absolute units of either the electrostatic or the electromagnetic system, q must be in absolute units of the same system. 1 The charge indicated by q is a charge that can be increased or diminished only by an actual transfer of electricity (free electrons) into the region containing q. Such a charge is known as an intrinsic charge, and is to be distinguished from certain induced charges to be considered later. The current i t must include any actual transfer of electricity into the region, whether of the ordinary conduction variety or whether the transfer is by an actual transfer of charged matter into the region; that is, ii must include conduction and convection currents of electricity. It is highly probable that all transfer of 1 For discussion of these units see PIERCE, "Principles of Wireless Teleg- raphy," Appendix I. CHAP. I] FUNDAMENTAL LAWS AND EQUATIONS 5 free electricity, even in metallic conduction, is accompanied by the flow of matter in the form of electrons, and is, therefore, essentially a convection current; but this subject may properly be deferred to later consideration. The current i i} however, in the present form of the equation does not include displacement currents to be treated in Book II. 6. Generalization of Kirchhoff's Electromotive -Force Law. If we have a circuit of the form of Fig. 5 in which an e.m.f. e is applied to a constant resistance R, a constant inductance L, and a constant capacity C in series, the instantaneous value of FIG. 5 Circuit containing e. m. f. e, resistance R, self-inductance L, and capacity C. the e.m.f. e at any time t is equal to the sum of the instantaneous values of the potential drops e R , e L , e c ; that is, e = e R + e L + e c , (7) in which C R = the fall of potential in the resistance R, e L = the fall of potential in the inductance L, e c = the fall of potential in the capacity C. Let us now adopt the following rule of signs: If e and i are in the direction of the arrows they are given a positive sign. If they are in the opposite direction they are given a negative sign. If the charge on the plate A (toward which positive i flows) is positive q is positive. If this charge is negative q is negative. Then by Ohm's law, 1 the fall of potential in the resistance R is e R = Ri, (8) 1 G. S. Ohm, "Die galvanische Kette mathematisch bearbeitet," Berlin, 1S27. 6 ELECTRIC OSCILLATIONS [CHAP. I where i is the instantaneous current through the resistance R. The fall of potential in the inductance L is e,=L| (9) where L is the self-inductance of the coil L, and i is the current through L. This current is the same as the current through R, since there is assumed to be no capacity and therefore no accumulation of charge within R and L. The fall of potential in the condenser C is ec-=|. (10) where + q is the charge on the plate A of the condenser, and C is the capacity of the condenser. It is to be noted that there is an equivalent charge of the opposite sign ( q) on the plate B; because, since there is no other capacity in the circuit, the current throughout the circuit at the time t is everywhere the same except within the dielectric of the condenser : and, there- fore, the current out of the condenser at B is always the same as the current into the condenser at A, and hence the deficit of charge (the negative charge) of B is always the same as the excess of charge (the positive charge) of A. By KirchhofTs Current Law (eq. 6) do T t =r ' therefore, q - fidt. (11) If now we substitute (8), (9), (10) and (11) in (7), we obtain ' (12) In this equation the applied e.m.f. e is usually called the im- pressed e.m.f. This impressed e.m.f. may be variable, constant or zero. If it is variable its instantaneous value at any time t is to be taken, and the current i is the instantaneous value of the current at the same time t. It may not be apparent just why the e.m.f. e, represented in Fig. 5 as produced by a dynamo, shall be considered as impressed e.m.f., while the other terms of the equation (12) are regarded as falls of potential. The reply is, that e is the terminal voltage CHAP. I] FUNDAMENTAL LAWS AND EQUATIONS 7 of the dynamo and is, therefore, impressed by a source of power external to the sequence of elements R, L and C. If e is not the terminal voltage of the dynamo, but the total e.m.f. generated in the dynamo, then equation (12) would still be true if we add the resistance of the dynamo to R and add the inductance of the dynamo to L, although in this case difficulty would arise because the equation presupposes a constant L, which would not be the case if the dynamo contained iron in its armature. As a further note on impressed e.m.f., if we regard e as the terminal voltage of the dynamo, it is evident that we may regard the quantity e L as the e.m.f. impressed on R; dt C for there is a terminal dynamo voltage e impressed on the circuit; this is opposed by the counter e.m.f. L -^ due to the magnetic field of the self inductance coil L and by the counter e.m.f. ^ (_/ due to the charge of the condenser, leaving e L ut C/ as the e.m.f. impressed on R. It is perhaps still more instructive to transpose also the term Ri to the left hand side of equation (12), giving We may now regard Ri as the counter e.m.f. of the resistance, and may interpret equation (13) as an algebraic statement of the fact that the impressed e.m.f. and the counter e.m.f. 's constitute a system in equilibrium. If we have several dynamos or batteries of terminal voltages ei, 62, 6 3 . . ., these e.m.f. 's being estimated positive when tend- ing to send currents in the direction of the arrows and negative when tending to send currents in the opposite direction, and if we have several capacityless resistances 1 Ri, R 2) Rs . . ., several capacityless inductances LI, L 2 , L 3 - . , and several condensers of capacities Ci, C 2 , C 3 . . . , all in series, we shall have ei -f 02 -M+ . . . -- (Ri + R* + # 3 + . . .)* - 1 Some or all of the resistances may be in whole or part the resistances of the inductance coils. 8 ELECTRIC OSCILLATIONS [CHAP. I or Ze - (2R)i - (SL) j t - (s^) y^d* = 0. (14) Equation (14) presupposes that the L's, R's, and C's are in- dependent of the time t. It may readily be seen how the equation is to be modified to make it applicable to cases in which these coefficients are variables. We shall, however, have occasion to discuss chiefly those problems in which R, L, and C are constants independent of current I and independent of time t, and shall at present limit ourselves to these conditions. The group of results constituting Kirchhoff's Generalized Electromotive Force Law, or Kirchhoff's Second Law, may be summarized as follows:' 7. Summary of Kirchhoff's E.M.F. Law: 1. When there is an instantaneous current i flowing in a constant capacityless and inductanceless resistance R at the time t, there is impressed at the same time at the terminals of the resistance by some source of power external to the resistance a difference of potential e R equal to Ri and in the direction of i\ 2. When there is at the time t an instantaneous current i flowing in a constant capacityless inductance L of resistance R L , there is impressed at the same time at the terminals of the inductance by some source of power external to the inductance a difference of potential e L equal to R L i -}- L-r and in the direction oft; 3. When there is at the time t an instantaneous current i flowing into the positively charged plate of a condenser of con- stant capacity C, there is an equal current i flowing away from the other plate 1 of the condenser, and there is impressed upon the condenser from some source of power external to the con- denser a difference of potential between the plates of the value e c equal to ^^ and in the direction of i; C 4. When several of these elements (resistance, inductance and condensers) are in series the total instantaneous impressed e.m.f. is equal to the sum of the instantaneous e.m.f.'s impressed on the elements. 1 Care must be exercised in determining what is the other plate of the condenser. It is the aggregate of all bodies on which terminate lines of static force from the first plate. CHAPTER II THE FLOW OF ELECTRICITY IN A CIRCUIT CONTAIN- ING RESISTANCE, SELF-INDUCTANCE, AND CAPACITY. DISCHARGE, CHARGE, AND CURRENT INTERRUPTION 8. Notation. R = Resistance, L = Self inductance, C = Capacity, I = Initial constant current, E = Constant impressed e.m.f., E Q = Initial difference of potential between the plates of a condenser, Qo = Initial charge on one plate of a condenser prechosen as positive, Q = Final charges on this plate, q = Charge at the time t on the condenser plates, A (Fig. 1), prechosen as positive, i = Instantaneous current flowing toward the plate A at the time t, e = Impressed e.m.f. at the time t. Let the positive direction of e be toward that plate of the condenser designated as positive. As we proceed we shall need also the following additional abbreviations : r\ / R ~ fcl= - r . s R I~R~ 2 * 2= * 2L ,...x W4= : (iv) - (v) a = R/2L. 9 10 ELECTRIC OSCILLATIONS [CHAP. II = a Among these expressions the following algebraic relations are seen to exist: (vi) ki = a + co* = a + j'w, (vii) (viii) J-J\s (ix) ki - kz = 2a h = 2ju, As these relations occur in the text, we shall refer to them by their respective Roman Numerals. 9. Differential Equation of Current and Quantity. If in a circuit of the form of Fig. 1, we equate the impressed e.m.f. e to the sum of the counter e.m.f. 's (that is, the counter e.m.f. of FIG. 1. Circuit containing R, L, C and impressed e.m.f. c. self inductance + the counter e.m.f. of resistance + the counter e.m.f. of capacity) we have, by (7), (8), (9), and (10) of Chapter I, We have also the following relation of i to q (6), Chapter I, 1 = ~dV or q = fidt. Differentiating (1) and introducing (2), we obtain e& = cfc* dt C (2) (3) (4) CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 11 Likewise, if we replace i in equation (1) by its value in terms of q from equation (2), we obtain Equations (4) and (5) are the differential equations for the current in the circuit and for the charge in the condenser at any time t in terms of the e.m.f. impressed upon the circuit. 10. General Solution. A general solution of equations of the form of (4) and (5) is given in Appendix I, Note 6. Instead of making direct use of the solution there given, it is instructive to solve (4) and (5) by elementary methods for specific values of e such as arise in important practical cases. 11. Important Special Problems. By assigning different values to the impressed e.m.f., e, various special problems arise in connection with the flow of current in condenser circuits. The following of these problems are highly important and interesting: I. To find i and q during the discharging of a condenser initially charged. II. To find i and q during the charging of a condenser under a constant impressed e.m.f. III. To find i and q produced by interrupting a current flowing in an inductance which is shunted by a condenser. IV. To find i and q under the action of a sinusoidal impressed e.m.f. These problems will be treated in order (the first three in this chapter, and the fourth in Chapter V). Each problem will be examined in detail, partly on account of the interest that it pre- sents in itself, and partly as introductory to other matter. I. THE DISCHARGING OF A CONDENSER* 12. Differential Equation for Current and Quantity During Discharge. Suppose a condenser of Capacity C, Fig. 2, to be initially charged with a quantity of electricity + Q on one plate and QQ on the other plate, and at the time t = 0, let the gap G be closed in such a way that there is no spark 2 at G, then we have the initial conditions. 1 This problem was first solved by Sir Wm. Thomson, Phil. Mag., 5, p. 393, 1853. 2 Because a spark has a resistance that is a function of the current through the spark. 12 ELECTRIC OSCILLATIONS [CHAP. II When t = 0, q = Q = CE , (6) where E is the initial difference of potential between the plates of the condenser. We have also: When I = 0, i = 0. (7) In addition to these initial conditions, we have the fact that the e.m.f. impressed upon the circuit is zero; whence the dif- ferential equations (4) and (5) take the respective forms d 2 i . ^di , i dt (8) (9) It is seen that (8) and (9) are identical in form. They are the differential equations for the current i and the quantity q during the condenser discharge. t-t FIG. 2. Illustrating condenser discharge. Left-hand diagram is the condition at t = o; right-hand, at i = t. 13. General Solution of Equations (8) and (9). Let us fix our attention upon equation (8). This equation is a homoge- neous linear differential equation of the second order, with con- stant coefficients. This terminology, which is used generally in the theory of differential equations, has the following significance. (* di d^i \ i, -=-> -j- v . . . ] as the elements of the equation, the equation is linear in these elements, since products or squares or higher powers of these elements do not enter. It is homogeneous, since every significant term of the equation contains one of the elements to the same power; namely, the first power. It has the constant coefficients L, R, and 1/C. CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 13 The equation is of the second order, by which is meant that the highest order of any derivative is the second order. The following general propositions in the theory of differential equations are applicable to the problem: /. The sum of two or more solutions of a linear homogeneous equation is a solution of the equation; that is, the solutions are additive. II. If we can in any way find a solution of a linear, homogeneous equation of the nth order, the solution, if it contains n independent arbitrary constants, is the most general solution, or the complete integral of the equation. The proofs of these two propositions are found in Appendix I, Notes 1 and 4. We shall employ the propositions in obtaining the solution of equations (8) and (9). In the beginning let us attempt to find by inspection a par- ticular solution of (8). We may try anything we like in the search for a solution; for example, let us try i = A, a constant. This substituted in (8) yields = + + A/C, and, therefore, A = 0, and i = 0. Such a value will not contribute anything by addition to any other solution that may be found. We might make various other random attempts to find a particular solution of (8), but we shall make greater progress by basing our attempts upon some rational experience, particularly upon experience with the use of exponential functions. Let us try i = Ae kt , (10) where A and k are constants and e is the base of the natural logarithms. This value of i substituted in (8) gives, = {Lk 2 + Rk + l/C}Ae kt . (11) It is seen that we may divide out Ae kt from (11), obtaining = L/c 2 + Rk + 1/C. (12) We have thus the result that (10) is a solution of (8) provided k satisfies the quadratic 'equation (12"). Solving this quadratic equation (12) for k, we find the two roots " W = kl by (i); Art * 8 ' (13) and - =L = k 2 by (ii), Art, 8. (14) 14 ELECTRIC OSCILLATIONS [CHAP. II Equations (13) and (14) give two specific values of k either of which will make the exponential value of i given in (10) satisfy the differential equation (8). The coefficient A of equation (10) is entirely arbitrary and may have any values whatsoever so far as may be determined by the given differential equation. The constant k is determined by (13) and (14). ^ In the attempt to find one particular solution of (8) we have really found two particular solutions, namely, either i = Aie**; .(15) or . (16) In these equations A\ and A z are arbitrary constants, which are in general independent of each other. Now by Proposition I, Art. 13, the sum of these two solutions is a solution. That is, i = A l klt + A**" (17) is a solution of equation (8). In fact, this is the most general solution, or complete integral, of (8), provided ki and & 2 are dif- ferent quantities; for then A\ and A 2 are two independent arbi- trary constants; and by Proposition II, Art. 13, such a solution is general. If on the other hand ki = k 2) the solution (17) reduces to i = (A 1 + A 2 ) * 1 ', (18) and, therefore, possesses only one arbitrary constant; for the sum of A i and A 2 is no more arbitrary than AI alone. The exceptional case with ki equal to fc 2 arises when R* = 4L/C, or CO A = CO = 0, as may be seen by reference to (13) and (14) and to (iii) and (iv), Art. 8. This is called the Critical Case. The critical case re- quires a special treatment, which is given in Appendix I, Note 7, where the result is obtained in the form of _ R i = (A, + A 2 l) e 2L (19) CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 15 If the reader does not care to follow the reasoning of the Note 7 in Appendix I, he can satisfy himself that (19) is a solution of (8) in the critical case by substituting (19) directly in (8) and intro- ducing also the condition R 2 = 4L/C. Since (19) contains two independent arbitrary constants, i 1 is the complete integral for the critical case. To sum up, we have found the general solution of (8) to be i = A** 1 ' + A 2 e k2t , provided R 2 ^ 4L/C, (20) _Rt_ i = (Ai + A 2 t) e 2L , provided R 2 = AL/C. (21) Now to obtain the value of q we may solve directly equation (9), just as we have solved (8). We shall, however, adopt the alternative method of obtaining q by integrating (20) and (21), employing the relation q = fidt. (22) Equation (20) gives A kit A kzt q = ! + ^- 2 e , provided R 2 ^ 4L/C, (23) KI KZ and equation (21) gives q = f (Ai.+ A^e-o'dt, where by (v) a = R/2L', whence When the last term of this equation is integrated by parts, we obtain q = - + - ZT ', provided R 2 = 4L/C. (24) Equations (20), (21), (23), and (24) are the general solutions of the differential equations (8) and (9). In these equations AI and A 2 are arbitrary constants; while ki, & 2 and a are constants of the circuits defined in equations (i), (ii), and (v), Art. 8, respectively. 14. Determination of the Arbitrary Constants AI and A 2 Subject to the Initial Conditions. We may now determine the arbitrary constants subject to the initial conditions written above as (7) and (6). These initial conditions are: 16 ELECTRIC OSCILLATIONS [CHAP. II When t = 0, i = 0, and when t = 0, q = Q = CE . In the non-critical case (R 2 ^ 4L/C), these initial conditions substituted in (20) and (23) give = A! + A 2, (25) Qo = ~ + ^ (26) whence iS4) This last equation, by (viii) and (ix), Art. 8, gives Therefore (27) ^OJfcjL/O 64L/C); and (35) whenever co is real (that is, when R 2 <4L/C). In the critical case (where R 2 = 4L/C) the solution is equation (36). The values of a, co^ and w are given in Art. 8. 16. Complete Solution of Quantity Subject to the Initial Conditions. -The value of q may be obtained by substitution of the values of the constants AI and A 2 into the equations for q (23) and (24), but we shall adopt the alternative method ef integrating i with respect to time. 18 ELECTRIC OSCILLATIONS [CHAP. II In the non-critical case, by taking the time integral of (35) we obtain q = fidt - ~ ye- ' sin J sm tan J- \ Therefore by (viii), Art. 1, En i - ? = f 5 - ViC -" sin j at + tan- 1 (-) 1 (37) .Leo I \ft/ j The corresponding integration of (34) gives 5 = \/LC e-< sinh *< + tanh- 1 (38) In the critical case, in which R 2 = 4L/C, we may obtain q simply by substituting the values of AI and A 2 from (28) and (31) into (24), obtaining but by (v) and by the fact that in the critical case R z = 4L/C, we have a 2 = R 2 /4L 2 = 1/LC, and therefore, q = E C(l + at) - = Qo(l + at) e-'- (39) Equation (37) or (38) g^es the value of q in the non-critical case. Either of these equations may be used, but it is simpler and more direct to use (38) when CO A is real (that is, when R 2 >4L/C ) and (37) when w is real (that is, when R 2 <4L/C ). In the critical case (where R 2 = 4L/C ) the solution is (39). The values of a, co^ and co are given in Art. 8. 17. Identity of the Critical Case Results with the Non-critical. It is to be noted that, although the form of the expression derived for i and q in the critical case is different from the form obtained in the non-critical case, in reality the non-critical results reduce to the critical results if we make R 2 = 4L/C. CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. jf 19 This may be shown as follows : If R* = 4L/C equation (iv) gives Now lim co = co = 0. lim co = CO = *; whence by (35) the current, as co approaches 0, approaches E t E Q t -A which is the current in the critical case, as given in (36). If next we concern ourselves with the limit approached by the charge q of equation (37) as co approaches 0, and note that we may expand tan - 1 (co/a) for small values of co/a in the form tan- 1 (-) = - -I (-) V . . [B. O. Peirce's Tables, No. 779] \a/ a 6 \a/ we find that _ lim w = This substituted in (37) gives Now by the fact that in the critical case R 2 = 4L/C, we have VLC therefore * + CE, (44) by Appendix I, Note 8, or as follows: The result (44) may be obtained by adding the particular integral of (42) (namely, CE) to the complementary function which is the solution of (42) with the constant E replaced by zero. This complementary function is Bie* 1 ' + B 2 e k *. Having now the values of i and q in (43) and (44), we are now to observe that to make i equal to the time derivative of q, we must require that BI = Ai/ki and J5 2 so that we may write q in the form Let us now insert the initial conditions that the condenser shall start uncharged and that the initial current shall be zero; that is, when t =-0, q = 0, and i = 0. These conditions give = Ai + A 2 , and whence \ 2 ** lj and fcifc, A 1 = CE E by (viii) and (ix). Now by comparison it will be seen that AI and A 2 are the negatives of the values obtained for these quantities in equations (25) and (27) (which give the current and quantity during the discharge) , except that the E which appears in the present prob- lem is the e.m.f. impressed on the circuit, while in the discharge 22 ELECTRIC OSCILLATIONS [CHAP. II problem E is the potential difference to which the condenser was initially charged. In the event that the condenser is first charged under the impressed e.m.f. E and then discharged, these two values of E are the same. If t is measured from the beginning of the charging in the one case and from the beginning of the discharg- ing in the other case, it will be seen that the current in the two cases differs only in sign, and that the quantity during charge is a constant CE minus the quantity during discharge. Expressed mathematically, these results are contained in the following table: 19. Comparison of Discharge with Charge. During discharge t = Time from beginning of dis- charge, E = Difference of potential be- tween the plates of the con- denser at t = 0, Q = Charge on positive plate when t = 0, i = Current toward the positive plate at time t, q = Charge on the positive plate at time t. then .= j e~ at sin co/, Leo (46) During charge i = Time from beginning of charge, E = Difference of potential be- tween the plates of the con denser at t = o ; Q = Charge on positive plate when t = co, i = Current toward the positive plate at time t, q = Charge on the positive plate at time '. then E Loo =r- e~ at sin q = CE - Yorf + tan- 1 ^ (47) sin co/, E Leo (co/ -f tan- 1 W \ ct (48) sn (49) Note that in the case in which co is not real, these quantities are the same as here given, but may be more conveniently used with hyperbolic sines and hyperbolic antitangents in place of sin and tan, and with CO A substituted for co. Also the result for the critical case is comprised in the above equations. We may, however, simplify the result in the critical case, by taking the limits of i and q above as w approaches zero. This process gives for the critical case During discharge During charge Ed Et i = 7- e~ at , (50) i = + r ~ e , q = Qo(l (51) CE - Q(l + at)e- at CE - CE(l + at)- at (52) (53) CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 23 III. PERIOD, DAMPING FACTOR, AND LOGARITHMIC DECREMENT 20. Determination of Period During Discharge. We come now to a discussion of the results obtained in the case of the condenser discharge. We have found for the current and quantity during discharge the equations .E q = VLC~e- at sm (ut + tan" 1 -), L/CO CL (54) (55) in which i is the current flowing toward the plate that was ini- tially positively charged, and q is the quantity of electricity on this plate at the time t. FIG. 4. FIGS. 3 and 4. Giving respectively current i and quantity q (on positive plate of condenser) plotted against time. If co is real (that is, if R 2 <4L/C) both of these quantities are seen to be periodic and to have a factor that is a sinusoidal function of the time. A diagram of i plotted against t is given in Fig. 3. A similar diagram for q is given in Fig. 4. The period of oscillation of the current in Fig. 3 may be defined as the time between alternate zero values of the current; that is, the time between the points ai and a 2 , #2 and 3 , etc. These points are the values of t for which i becomes zero after successive 24 ELECTRIC OSCILLATIONS [CHAP. II complete cycles, and by (54) they occur at values of t for which sin cot = 0. Since only alternative points are considered, this relation gives ut = 0, 27r, 47r, etc.; whence, giving subscripts to different values of t satisfying the relation, we have ti = 0, C 2 = 27T/CO, 3 = 47r/co, etc. and, therefore, the period T is T = t 2 ~ti = t 3 ~t 2 = . . . = 27T/0). Putting in the value of w from (iv), we obtain 27T =jj[ (Thomson's Formula). (56) \LC 4L 2 Equation (56) gives T the period of oscillation of the current during the discharge of a condenser. Similar reasoning gives the same period of oscillation of the quantity q. It is seen that this period is real, only provided R* < 4L/C. (56a) 21. Approximate Value of the Period of Discharge. Assuming that the inequality (56a) is satisfied, it is seen that & < 4L 2 - LC } then the equation (54) may be expanded by the binomial theorem into where * R Now if we note by (viii) that -"'+*' CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 25 we shall see that equation (57) reduces to T = 2irVXC, (58) (Thomson's approximation formula) provided | 2 ^,or^^ (680) where the symbol < < means "is negligible in comparison with." The approximate period T calculated by the formula (58) has an important role in some of the work of the later chapters and is called the Undamped Period of the Circuit and will often be desig- nated by S to distinguish it from the true period T. The Equation (58) gives the Undamped Period of the oscilla- tions of current during the discharge of the condenser. This is sensibly the actual free period of the oscillations if a 2 /2 is negligible in comparison with co 2 . The oscillations of q have the same period as the oscillations of i. 22. The Time between Successive Positive or Negative Maxima is the Same as the Time between Alternate Zero Values. Let us next find the time between successive positive or negative maxima of current and show that this gives the same period as the time between alternate zero values of current. Equation (54) for the current is Differentiating this with respect to the time and setting di/dt 0, we have = 'sin (4 whence at = tan - 1 , tan - 1 -- + 2ir, tan- 1 + 4ir, a a a If we let the successive values of t obtained from this equation be ti, t 2 , it, etc. and let

\ sin ( ut + tan 1 j- ) (64) V a / IV. EXCITATION BY CURRENT INTERRUPTION 25. The Production of Oscillations by Buzzer Excitation. In many of the experiments employed in high-frequency measure- ments electrical oscillations are produced by excitation of the condenser circuit by the use of a buzzer, 1 which acts by making and breaking a current flowing in an inductance. The accompanying figure (Fig. 5) represents a battery B supplying current to an inductance L through an interrupter J. The inductance L has resistance R } and is shunted by a condenser of capacity C. The interrupter J is here represented as a buzzer with its field coil Z/o also shunted by a condenser C . The mathematical theory which follows applies to the heavy line circuit LRC, which is a circuit of frequency high in compari- son with that of the circuit Z/ C . 1 This form of buzzer excitation is due to Zenneck, Leitfaden der drahtlosen Telegraphic, p. 3. 28 ELECTRIC OSCILLATIONS [CHAP. II Let us measure time from the instant of interruption of current at /. Let the current flowing in L at any time t seconds after the interruption be i, which is a function of t, and let the charge in the condenser C at the same time be q. Then * = dq/dt. (65) From the time t = 0, when the circuit is broken at J, there is no external impressed e.m.f., so the differential equations for current and quantity are and (66) (67) 1 1 1 i 1 1 D i i r S3 ** + '/ C (^ q i n L, 1 iR FIG. 5. Diagram of circuits for buzzer excitation. The complete solution of (66) and (67) gives i = Aie*" + A 2 e**, q = ^ e*" + ^ e*", (68) where ki and k 2 have the values given in (i) and (ii), Art. 8. The initial conditions are and When t = 0, i = I, when t = 0, q = - CRI, (69) (70) where 7 is the current flowing in the coil L at the time of inter- ruption at /. Equation (70) is obtained on the assumption that the current is in practically steady state immediately before interruption, so that the counter e.m.f. in the coil is RI. This is the potential of the lower plate of the condenser in excess of the CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 29 upper plate. The capacity C times this potential gives the charge on the lower plate as CRI; but the upper plate is regarded as positive, whence the negative sign in (70). The charge is - CRI. If now we introduce the initial conditions (69) and (70) into the pair of equations (68), we obtain 7 = Ai + A 2 , (71) i 9 _ 2l i 2 . , = 17 " T 2 ~ IbST" A determination of the A '& from these equations may be made as follows: From (72), by the relations (v), (vii), and (viii), we obtain -CRI = LC{(Ai + A*) (-a) + j(A 2 -A,)}. Now a = R/2L, whence T - Al + Az _l_ M^l ~ A 2) 2 2a This equation, combined with (71), gives Ai - A 2 = al/jw, which combined with (71) gives al Aa= '2^ Substitution of these values of AI and A 2 into the equation (68) for i gives, in view of (vi) and (vii) , i = e-<{ A l ^ + A***} (73) = - e~ at (w cos co + a sin co By (viii) this last equation gives = - -;= e ~ at sin [ VLC \ + tan" 1 - ) 30 ELECTRIC OSCILLATIONS [CHAP. II Equation (74) gives the current in the coil L in the direction of the original current 7, at a lime t seconds after the interruption of latJ. 26. On the E.M.F. Induced in a Very Loosely Coupled Sec- ondary Circuit by Buzzer Excitation. In the preceding sections there has been discussed the oscillations produced in a circuit by a method known as buzzer excitation. Oscillations produced in this way are often employed to impress an e.m.f . on a secondary circuit for the purpose of making measurements in the secondary circuit. A diagram of this arrangement of apparatus is shown in Fig. 6. The oscillations occurring in the circuit LC impress an e.m.f. on the circuit L 2 C 2 . Let us now specify that the circuit Z/ 2 C 2 , I 1 Primary o L _= ii L J- ex > J, j Secondary M u FIG. 6. Buzzer excitation of primary circuit inducing e.m.f. in secondary circuit. which we shall call the secondary circuit, shall be so far away from the primary circuit LC that the current induced in the secondary does not materially influence the current flowing in the primary, and let us determine the e.m.f. induced in the secondary circuit. The induced e.m.f. has the instantaneous value determined by the mutual inductance M between the two circuits and by the time rate of change of the primary current. The relation is dt (75) Substituting in (75) the value of t from (74) and performing the differentiation, we have e 2 = -. e~ at { ~ a sin (coi + (p) + co cos (co + ^) } , where CHAP. II] CONDENSER DISCHARGE, CHARGE, ETC. 31 Expanding the sine and cosine terms by the formulas for sines and cosines of a sum and noting that sin

The energy Wi 2 supplied to the inductance during the time from ti to tz is seen by (2) and (8) to be where / = J '<2 L d(i 2 ) , 1 . /QX O 77-^ = o^W - /I 2 ) W 41 2 t + tan- 1 + sin 2 cot [ (13) 2Lco 2 o CHAP. Ill] ENERGY TRANSFORMATIONS 37 This is the electrical energy resident in the system at any time t. If now we take any two times t and t + T 7 , where T is one whole period of oscillation, the sine terms will be identical at t and t + T 7 , and we shall have as the ratio of the energies in the system at the two times W t , T e -2*+r> whence log Wt - log W t+T = 2aT = 2d, (14) where d = aT, and is the logarithmic decrement of current or quantity per cycle. Equation (14) gives the logarithmic decrement of electrical energy in a condenser and inductance during discharge, and shows by comparison with (61) of Art. 24 that the log. dec. of energy per cycle is twice the log. dec. of current or quantity per cycle. 34. Energy Expended in Resistance During Condenser Dis- charge. In the preceding paragraph, we have examined the energy resident in the condenser and inductance during the dis- charge of a condenser. We shall now attack the complementary problem of determining how much energy has been dissipated in the resistance from the beginning of the discharge to a time t seconds thereafter. If the time extends from t = to t = t, the expended energy by (11) is W R = R *dt. (15) w ~ cos Letting the initial value of condenser potential be E, we have, equation (46), Art. 19, E 2 i 2 = j^-~v e ~ 2at sin 2 o>Z L 2 co 2 Substituting this value of i 2 into (15) we obtain L 2 w 2 f < - ~ cos " dt Jo 2 ~ -4a -ivV+a) 2 -a 38 ELECTRIC OSCILLATIONS [CHAP. Ill This expression can be simplified by noting the trigonometric relation cos(# + y) cos (x y} = 1 sin 2 # - sin 2 ?/, whence . , 1 sin 2 # sin 2 ?/ . <* + > cost* - y) (16a) Now - cos (2at - tan- 1 - 60 -) ^cos (2wt + tan" 1 -) (17) \ a/ \ a I If now we let x = a)t + tan -1 (co/a) y = 2 cos 2co< tan" 1 -- ) = - \ -a) a 1 - sin 2 ( ut + tan- 1 - J - sin 2 ut (18) This result introduced into (16) gives, on replacing a 2 + co 2 and a by their values from (viii) and (v), Art. 8, W = [f - Sin2 (' + tan ~ X a) - Sln2 ^ 1 e " 20( ]l ' - 2 [sin 2 (co* + tan- 1 ~) + sin'orf] je" 2 ^. (19) Equation (19) gr^es i/i6 energy W R expended in the resistance R during the discharge of the condenser in the interval of time from the beginning of the discharge to t seconds thereafter, the condenser being originally charged to a potential difference E. It is to be noticed that this energy expanded in the resistance + the energy left in the circuit (13) is E 2 C/2, which is the energy origi- nally in the condenser. It is also to be noticed that if we make t infinite, the terms in- volving t in (19) become zero, and the total energy expended in the resistance becomes W R = E*C/2, (20) so that the energy lost in the resistance is the total energy of the condenser charge. 35. Average Current and Mean-square Current During N Complete Condenser Discharges per Second. If we suppose that the condenser is charged N times per second, and after each CHAP. Ill] ENERGY TRANSFORMATIONS 39 charging, the charging source is removed and the condenser is discharged through a current-measuring instrument whose deflections are proportional to the average current, we should have a measure of the average current of the N discharges per second. If we assume that each discharge is practically complete, we can easily calculate this average current from fundamental considerations, as follows. The quantity of electricity flowing from the condenser at each discharge is its original charge = CE. . Per second the quantity is N times this, so that The average current for N complete discharges per second = NCE = NQ. (21) On the other hand, certain types of current-measuring in- struments read the square root of the mean-square current (R.M.S. current). This is true of hot-wire ammeters, thermal- junctions, dynamometers, etc. We shall, therefore, calculate from elementary considerations the R.M.S. current during N complete condenser discharges per second. If there are N discharges per second, the energy dissipated in the resistance R of the circuit per second (that is, the average power P dissipated) is by (20) . P = NE*C/2. (22) This average power divided by R gives the mean-square current, hence * - TOT where 7 2 with a dash over it means the mean-square current. Taking the square root of the mean-square current (23), we obtain R.M.S. current for N complete discharges per second Equations (21) and (24) give respectively the mean current and the R.M.S. current obtained from N condenser discharges per second. The condenser is charged each time to a potential differ- ence E } the charging source is removed, and the condenser is then 40 ELECTRIC OSCILLATIONS [CHAP. Ill discharged through any inductance L and resistance R. The in- ductance of the circuit is found to be immaterial, provided the dis- charge is complete. In the case of the average current, both inductance and resistance are immaterial. The number of discharges N per second is sup- posed to be sufficiently smaU to permit practically complete discharges. 36. Energy Lost in the Resistance of the Circuit During the Charging of a Condenser. We shall next prove a very interesting fact concerning loss of energy when a condenser is charged by applying a constant e.m.f. E. During the process of charging a condenser through any re- sistance and inductance under the action of a constant im- pressed e.m.f. E, the energy lost in the resistance of the circuit from time to t is W R = fl' iVt, (25) where t is measured from the beginning of the charge. It is to be noticed that i 2 during the charge is of the same form as i 2 during discharge (equations (48) and (46), Art 19) so that (25) when integrated gives the same result as (19), and when t is made infinite (see (20)) W K - ** (26) Equation (26) gives the energy dissipated in the resistance of the circuit when a condenser C is charged by the application of a constant e.m.f. E. This amount of energy dissipated is inde- pendent of the inductance and resistance through which the con- denser is charged. This energy dissipated is equal in amount to the energy finally delivered to the condenser, equation (7), so that the efficiency of the process of charging a condenser from a constant e.m.f. applied through any inductance and resistance to the con- denser is J^; which means that in order to deliver any given amount of energy to a condenser by applying a constant e.m.f. an equal amount of energy must be dissipated in the resistance of the circuit , however small we make that resistance. 37. Energy and Power Supplied to a Condenser Circuit Excited by Current Interruption. Reference is made to Fig. 5, Chapter II, which shows a circuit LRC excited by sending a practically steady current 7 through L and interrupting the current in the feed line. CHAP. Ill] ENERGY TRANSFORMATIONS 41 After each interruption of the feed circuit at J, if the oscilla- tions in the LRC circuit have time to die practically to zero before a new make of the interrupter, the energy expended in the resistance R is T 72 ^#272 ^ = 4+ Y ' (27) as may be seen from the principle of the conservation of energy, since the first of these terms is the energy in the inductance and the second is the energy in the condenser at the beginning of the discharge. If there are N makes and breaks of current at / each second, the energy per second (mean power P) dissipated in this circuit is P =N(- ^- 2 - -). (28) Equation (28) gives the average power P delivered to the oscil- latory circuit LRC and expended in that circuit, provided the circuit is actuated by making and breaking a current I, N times per second, at J (Fig. 5, Chapter II), and provided the interrup- tions are sufficiently infrequent to allow a practically complete dis- charge of the inductance between interruptions, and provided the feed current I has time to come to a steady state in L. CHAPTER IV THE GEOMETRY OF COMPLEX QUANTITIES 38. Utility. In the mathematical treatment of periodic phe- nomena a considerable simplification is made by the use of imagi- nary and complex quantities. As aids to the memory, the complex quantities may be represented geometrically by simple diagrams, which are easier to remember than the algebraic formulas. By the use of a simple set of rules for the geometrical representation of algebraic quantities and algebraic operations (rules due to Argand and Demoivre) many of the algebraic manipulations may be performed by the aid of geometrical constructions; and the final results obtained may be reinterpreted, if necessary, into algebraic symbols for the purposes of calculations. 39. Representation of Real Quantities. Real quantities are represented along a horizontal axis. This axis is called the axis of reals. As in analytical geometry, the numerical magnitudes of the real quantities are represented by lengths proportional to these -magnitudes. Positive values of real quantities are rep- resented by lengths drawn to the right along the axis of reals, from some arbitrary origin; negative values are represented by lengths drawn to the left from the origin. A negative quantity may be looked upon as making an angle of 180, or 180 + n 360 with the positive axis of reals; while a positive quantity makes an angle + n 360 with this axis, where n is an integer. Let us examine the result obtained by multiplying -\-a by b. The result is ab, a quantity having a magnitude equal to the product of the magnitude of the factors, and an angle (180) equal to the sum of the angle of the factors. Likewise, the product of a by 6 is -\-ab, a quantity as be- fore having a magnitude equal to the product of the magnitudes of the factors and an angle (360) equal to the sum of the angles of the factors, since a line making an angle of 360 with the positive axis is coincident with a line making with this axis. As a third example, the multiplication of a quantity a by 1 42 CHAP. IV] GEOMETRY OF COMPLEX QUANTITIES 43 reverses it, and a double multiplication of a by 1 is equivalent to a double reversal, or rotation through 180 + 180. 40. Representation of Imaginary Quantities. Argand's Method. The quantity \/ 1 is a number that multiplied by itself gives 1. Also the double application of A/ 1 to a quantity a as a multiplier gives a: this result is equivalent to the result obtained by rotating a through an angle of 180. Consistent with this and with the fact that with real quantities double multiplication resulted in the addition of angles, let us postulate that the single operation of multiplying by \/~ 1 amounts to a changing of a into a position it would have if rotated through 90. That is, we shall represent geometrically \/ 1 X a by a length a along an axis perpendicular to the axis of reals. This vertical axis is called the axis of imaginaries. The + and sign before imaginary quantities, as before real quantities, shows opposition in direction; that is, -\-\/ 1 a and \/l a have, opposite directions along the axis of imaginaries as shown in Fig. 1. A detailed consideration of this method of represent- ing real and imaginary quantities along two mutually perpen- dicular axes in the same plane shows that the system is entirely self-consistent. In order to avoid repeatedly writing \/ 1, we shall follow the prevailing custom in electrical engineering and adopt the symbol j for this quantity; that is j = V^ (1) j' 2 = -1. 41. Representation of Complex Quantities. The complex quantity a + bj shall be represented by the directed sect, or vector, OP, with a component a along the axis of reals and a component bj along the axis of imaginaries, Fig. 2. The directed sect, or vector, OP may be called the vectorial representation of the complex quantity, or briefly the vector OP may be called the vector a + bj. A vector has magnitude and direction. The magnitude of the vector OP is the length of OP, which is Va 2 + & 2 = r (say). (2) The direction of OP is determined by the angle

= b/a, (6) sin

) is /ie trigonometric polar co- ordinate expression for the complex quantity a + bj, or for the vector OP, Fig. 2. 43. Exponential Expression for the Vector OP. Demoivre's Formula. Another form of expression for the vector OP in polar coordinates may be obtained by examining the series expansions of cos = 1 - + - (10) CHAP. IVJ GEOMETRY OF COMPLEX QUANTITIES 45 (ID sin

+ j sin

in a positive direction, as indicated by the arrow. The function y* (16) therefore represents a uniform circular motion in which the angle co radians is described in a unit time. The angle co described per unit time is called the angular velocity of the revolution. 46 ELECTRIC OSCILLATIONS [CHAP. IV For the radius OP to move through an angle 2ir radians (i.e., once around) requires a time T such that or 2T = coT 7 , (17) (18) T given by (18) is the period of revolution. 45. The Addition of Complex Quantities, and the Summation of Vectors. Returning now to general elementary considera- tions, let us suppose that we have two complex quantities = a>i + and Zz = 2 -f- bz By direct algebraic addition their sum z is seen to be z = zi + Z' 2 = ai + a 2 + (&i + b 2 )j. (19) (20) From this it is seen that the geometrical representation of , which is the sum of the complex quantities z\ and z z , is ob- FIG. 5. Addition of vectors. tained by laying off ai + #2 on the axis of reals, giving the point x, Fig. 5, then at x a length 61 + b z must be laid off in the direc- tion of the axis of imaginaries. This brings us to the point P. The vector OP is z, the sum of Zi and z 2 . If now through the points M and TV respectively we draw the vertical line MS and the horizontal line NS, and jpin the inter- section point S with and P, we see that OS and SP are in magnitude and direction equivalent to z\ and z% respectively. Therefore, the geometrical sum of two vectors Zi and 2 2 is obtained by putting 2 2 on the end of z\ y and joining the beginning CHAP. IV] GEOMETRY OF COMPLEX QUANTITIES 47 of Zi with the end of z z . The same result is obtained if^i is put on end of z 2 , as shown by dotted lines in Fig. 5. The sum is again the vector OP and is now obtained by joining the beginning of the dotted z\ to the end of the dotted z z . In like manner, the vector z is the sum of the vectors Zi, z z , 23, 24, 2 8 , in Fig. 6. The vector sum of z\, 22, ... 25 is independ- ent of the order of the addition of terms. For example, if the order Zi, z 5 , z z , 23, z be taken the construction in Fig. 7 is obtained, which has the same sum as that obtained in Fig. 6. FIG. 6. Addition of five vectors. FIG. 7. Addition of same vectors in different order. 46. The Multiplication of Complex Quantities, and the Geo- metrical Representation of the Product. Given 21 = 2 = Let + &1 2 , n = r 2 = \a 2 2 +1 j sin a>o) } 6 -a {(^[ 1 -f A 2 )cos a> + ./(Ai A z ) sin co Z}. (18) If now in (18) we let ,. ,. - A 2 ) cos i/'o = - A 2 )} 2 ' sin \!/Q = and /o = we obtain 1*2 = /o e~ aoi sin In equation (19) 7 and \J/ are new arbitrary constants which are to be determined by the initial conditions of the problem. Equation (19) is a perfect equivalent of (3), and after the deter- mination of the arbitrary constants gives correct results whether R 2 is equal to, less than, or greater than 4L/C; that is, whether co is zero, real, or imaginary. Only, however, when the angular velocity co of free oscillation of the circuit is real does the solution remain periodic. If o> is zero or imaginary (19) goes over into the exponential, or hyperbolic, form, which is non-periodic. If we add equation (19) to (12) we obtain the complete ex- pression in the transformed aspect; to wit, ^ + W. (20) Equation (20) is the complete expression for the current in the circuit containing resistance, self-inductance and capacity, and an impressed sinusoidal e.m.f. This equation is alternative to (13). The impressed e.m.f. has angular velocity w, while o> is CHAP. V] SINUSOIDAL IMPRESSED E.M.F. 55 the angular velocity of free oscillation of the circuit, and a = R/2L. IQ and i/'o are arbitrary constants to be determined by the initial conditions. 57. The Quantity Constituting the Charge of the Condenser. In equation (20) is given an expression for the current flowing in the circuit under the action of an impressed sinusoidal e.m.f . To obtain q the quantity of electricity constituting the charge of the condenser at any time t, it is only necessary to form the integral q = fidt - E/a I X cos " I 4 ,X\ ( ut tan" 1 \ RI + x 2 -Io\/LC e- B " sin (co * + *<> + tan- 1 ) (21) \ flo / 58. Determination of the Arbitrary Constants when the E. M.F. is Impressed on a Circuit without Current or Charge. The reader who is not immediately interested in the determination of these constants may omit this and the next section and resume the reading at the section on the Steady-state Solution (Art. 60). In equations (20) and (21) two arbitrary constants 7 and \l/ occur. These are to be determined for each specific problem by the use of the initial conditions. We cannot in general impose the condition that t = when the initial current and charge are zero, for this implies that the dynamo impressing the e.m.f. (E sin coO is thrown into the circuit containing no current and no charge when the dynamo e.m.f. is itself just zero. Now if the dynamo is thrown into the circuit at a random time this will not be the case. Our problem, in case the initial charge and current are zero, imposes the condi- tions t = t 1} i = 0, q = 0, (22) where ti is the random time determining the phase of the e.m.f. at the time of impressing it. If now, for abbreviations, we let tan - 1 ti and X !^l 6 -o(-fa) sin {u Q (t - ti) + sin-^P}]- (30) Equation (30) gives the complete value of the current i when the e.m.f. is impressed at a time ti upon a circuit without current or charge. In this equation (pi and P have the values given in (23) and (29). In the expression for i, t is greater than ti, which is the time at which the e.m.f. E sin ut was thrown into the circuit. 59. Condition That Makes the Transient Term in (30) Zero. The term involving the exponential in (30) is called the transient term. One method of making this transient term zero is to let t be infinite. We shall consider this in the next section. Another method of making the transient term zero is to make S -^ = 0. (31) Let it be noted that, if sin ! (32) L \ I COCOo 00 i Setting the radical equal to zero and expanding it, we obtain cto 2 ~h ^o 2 i , &o 2 ~h fc>o 2 2do . > ^ s ismVi H s cosVi sin 4L/C). 60. Results in the Steady State. Apart from the method outlined in the preceding section for making the transient term in the current equation zero, it is seen that this transient term in each case is multiplied by an exponential factor with an expo- nent that approaches minus infinity with increase of time. If the time is sufficiently long after the application of the sinusoidal e.m.f., the transient term becomes negligible. The state of things after the transient term has become prac- tically zero is called the steady state, and the solution for the steady state is called the steady state solution. In the steady state, after the transient term has become practically zero, it is seen from (20) and (21) that the current and quantity are given by the equations (36) _ pj I / ~y\ r- cos ( ut tan- 1 -- ) (37) VR Z + X* \ R/ J CHAP. V] SINUSOIDAL IMPRESSED E.M.F. 59 in which E sin wt the impressed e.m.f., and X = Leo 1/Cw = the reactance of the circuit. (38) R, L, and C = the resistance, inductance, and capacity of the circuit. Equations (36) and (37) give the values of the current i and the quantity of electricity q constituting the charge of the condenser at the time t, under the action of a sinusoidal e.m.f. E sin c*t which has been in application sufficiently long to permit the establishment of a steady state. Some of the characteristics of the steady-state flow of current will be discussed in the next Chapter on Electrical Resonance in Simple Circuits. CHAPTER VI ELECTRICAL RESONANCE IN A SIMPLE CIRCUIT 61. Wave Length, Actual and Conventional. We have seen in Chapter II that an electrical circuit containing capacity and self -inductance, if the resistance is not too great, has a characteris- tic period of oscillation. We shall show in subsequent chapters, treating Maxwell's Electromagnetic Theory that, with certain forms of these circuits, energy is radiated into surrounding space in the form of electromagnetic waves. If a circuit of period T radiates waves, the wave length X of the waves radiated is related to the period T by the equation \=cT, (1) where X = wave length, and c = velocity of propagation of the waves. This relation follows from the elementary consideration that of two successive positive wave crests one is emitted at a time T seconds later than the other. The first, in the time T, travels a distance cT, so that the first crest is a distance cT ahead of the second; hence the distance between these two successive positive v/ave crests, which is the wave length, is X = cT. In free space, we shall show from Maxwell's Theory, that c, the velocity of the waves in free space is the velocity of light; that is, c = 3 X 10 10 centimeters per second. If it is required to obtain the wave length in meters, as is usual in radiotelegraphic practice, and if T is in seconds, the velocity of propagation must be expressed in meters per second ; that is ^^^^ c = 3 X 10 8 meters per second. (2) In the case of an actual radiation of electric waves into space, the wave length X is the actual distance between adjacent posi- tions of similar phase in the emitted wave system. It has become customary in radiotelegraphic practice to specify 60 CHAP. VI] RESONANCE IN SIMPLE CIRCUIT 61 the period of all periodic electric circuits in terms of the wave lengths corresponding to the periods of the circuits, even when the circuits happen to be of such form as actually to radiate only an insignificant amount of energy as characteristic waves. We thus attribute conventionally to every oscillatory circuit a wave length X satisfying the relation (1). Although we have not yet taken up the matter of electro- magnetic radiation, it is often an advantage to express results in terms of wave lengths as well as in terms of periods, and to use, in experimental investigations with these circuits, apparatus cali- brated in wave lengths. 62. Mean Square Current and Amplitude of Current in a Circuit Containing Resistance, Self -inductance, and Capacity, and a Sinusoidal E.M.F. The circuit upon which the e.m.f. is FIG. 1. Circuit containing impressed sinusoidal e.m.f. impressed we shall designate as Circuit II, or as the Receiving Circuit. The e.m.f. may be impressed by a generator in the circuit (see Fig. 1), or it may be impressed by induction from a Circuit I (Fig. 2), containing persistent sinusoidal oscillations, provided the Circuit I be so far from the Circuit II that the reaction of Circuit II in changing the current in Circuit I is negligible. The subject of these reactions will be taken up in Chapters VII and VIII, but the reactions will here be considered zero. Let the e.m.f. impressed on II be e = E sin <*it. (3) Let the resistance, inductance and capacity of the receiving circuit (Circuit II) be R,L, and C, and, as in the previous chapters, let the capacity be disposed in one or more discrete condensers so that there is no distributed capacity. 62 ELECTRIC OSCILLATIONS [CHAP. VI Then, after a steady state is reached, the current in II, desig- nated by i, is, by (36), Chapter V, E 1 = where sin [ uit tan" 1 -= X 1/Ccoi. (5) Since many types of measuring instruments, when placed at A in series in Circuit II, indicate the average square of the current or else the square root of the mean square current (R.M.S. current), let us obtain the value of these quantities. First let y = sin (cat + i are numerically equal to each other and opposite in sign and are sometimes said to neutralize each other. CHAP. VI] RESONANCE IN SIMPLE CIRCUIT 65 65. Ratio of Current in the General Case to Current at Current - resonance. Let us now divide (11) by the square of (15) obtaining ^ = 2/(say) = (17) I max. 1 + A */tt* This equation is equally true whether I 2 and / 2 max - are the squares of the amplitudes of current or the mean-square values, since the ratio of amplitudes squared and the ratio of the time average of the squares of instantaneous values are the same. If, in (17), we replace X by its value from (5), we obtain = 1 V i+^ 2 li _ " ! I 2 ( 18 ) ' 7?2 where V = /V/ 2 ma X . = /V/ 2 ma X . (19) Equation (18) is the equation to a resonance curve of current square against the circuit adjustments. We can apply (18) to specific cases in which different elements of the system are variable. We shall discuss two such cases. 66. Resonance Curve of Relative Current Square with a Fixed Impressed E.M.F. and Variation of Capacity in the Receiv- ing Circuit. Referring to Fig. 1 or Fig. 2, we have called the circuit II, with constants L, R, and C, the receiving circuit. Impressed upon Circuit II is a sinusoidal e.m.f. of value e = E sin coitf, in which coi is the angular velocity of impressed e.m.f. We shall now suppose that coi and E are kept constant, and we shall compute the relative current square in the receiving circuit when the condenser C of the receiving circuit is given various values. The fundamental equation of the result is given in (18), and we shall merely transform this equation into a form involving wavelengths and decrements instead of inductances, capacities, resistances, and angular velocities. Regarding the decrement, we have denned in (62) of Chapter II a quantity in which d is the logarithmic decrement per cycle of a circuit 5 66 ELECTRIC OSCILLATIONS [CHAP. VI whose period of free oscillation is T, and whose resistance and inductance are R and L. Now the period of free oscillation of a circuit is exactly given in (56), of Chapter II. This period is given approximately in (58), Chapter II; namely T = 2v\LC (21) Although (21) is only an approximate value of the free period of oscillation of the circuit, it is the exact value of the Undamped Period of the Circuit. We shall, accordingly define a new logarithmic decrement, indicated by 5, with the exact equation ilHf (22) and shall designate this decrement d as the logarithmic decrement per undamped period of the circuit. Since we are going to vary C in the present article, 6 as defined in (22) is a variable. Let us fix our attention on one particular value of 5, namely the value of 5 when C has the value to give a maximum value of y, and designate this value of d as 5o. Now by (18) for a maximum of y, it is seen that LC = 1/wi 2 (23) where Co = value of C that makes y a maximum. From (22) and (23), we have = j^-> (24) where 5 = logarithmic decrement per undamped period at current-resonance. Let us next examine the question of wavelengths. The period of the impressed e.m.f. T l (say) is related to coi by the equation Ti = 27T/CO! (25) According to equation (1) the wavelength Xi of the impressed e.m.f. is Xi = cT l = 2nrc/ui (26) CHAP. VI] RESONANCE IN SIMPLE CIRCUIT 67 If we call the period of the circuit T, the wavelength of the circuit X is X = cT (27) where c = velocity of propagation of the waves. T = free period of oscillation of the circuit. Since T is not exactly given by (21), while the undamped period of the circuit is exactly given oy (21), let us define the undamped wavelength of the circuit as the wavelength of the .002 1.002 1.004 1.006 1.008 1.01 FIG. 3. Curves of relative current vs. relative wavelengths for various values of decrement. undamped period, and indicate this undamped wavelength by a Greek capital Lambda A, then A = (28) In general when the circuits have small decrements A does not differ appreciably from X, but when the decrements are large, we should find it inaccurate to replace A by X. If now we substitute (24), (26), and (28) into (18), we obtain 1 (29) Equation (29) gives the value of relative square-current y, as defined in (19), in terms of the undamped wavelength A of the re- 68 ELECTRIC OSCILLATIONS [CHAP. VI ceiving circuit, for a fixed value of the wavelength \\of the impressed e.m.f. 67. Sample Curves of Relative Current for Fixed Impressed E.M.F. and Variation of the Capacity of the Receiving Circuit. If we extract the square root of (29) we have (30) Equation (30) is true whether I and I max . are amplitude values or R.M.S. Values. CHAP. VI] RESONANCE IN SIMPLE CIRCUIT 69 Figures 3, 4, 5, and 6 contain plots of equation (30) for differ- ent values of 5o- These curves were traced from blue prints FIG. 5. Same as Fig. 3, but with different values of 5 and different horizontal scale. 1.0 .9 .8 ..7 3.6 !: .3 .2 .1 .0 X \ 4.0 2.0 1.0 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 FIG. 6. Same as Fig. 5, but with different values of 5 . kindly supplied to me by Mr. J. Martin, Expert Radio Aid of the U. S. Navy. 70 ELECTRIC OSCILLATIONS [CHAP. VI 68. Determination of Decrement from Relative Current Square with Fixed E.M.F. and Variation of C. From (29) we may obtain 7TJ1 -XiVA 2 ) 50 = TT (3D Vy ~ in which y = / 2 /7 2 max = ^ 2 /^ 2 max Xi = wavelength of impressed e.m.f., and A = undamped wavelength of the circuit. By plotting a curve of y vs. Xi 2 /A 2 , we may compute a value of 6 for every value of A. All of the values of So so obtained should agree within the limits of accuracy of the measurements. It is apparent that this accuracy is not very great, but fortunately it is not generally of importance to know S with great accuracy. 69. Approximate Method of Rapidly Determining 5 . As an approximate method of determining So, 'let A' and A" = the two values of A at which y has the value J, and let A" > A', then by (31) S = 7r(l-Xi 2 /A" 2 ) and Adding these equations and dividing by 2, we may take the following steps 5o = "TlA 7 " 2 ~ A 77 " 2 ! { A "2 _ A /2J 2A' 2 A" 2 2A /2 A" 5 I X] We may now introduce approximations as follows: Let A'A" = Xi 2 , (32) and A' + A" = 2Xi approximately, (33) then So = Tr{- - ; approximately. (34) To the same degree of approximation 6 = d. We may state this result in the following rule. CHAP. VI] RESONANCE IN SIMPLE CIRCUIT 71 70. Rule for Approximate Determination of Logarithmic Decrement d of a Circuit with Variable Capacity. To obtain the logarithmic decrement of a circuit, impress upon it an un- damped e.m.f. of constant amplitude and frequency, take the dif- ference of the two wavelength adjustments of the circuit that give a mean square current equal to half the maximum mean square current, divide this difference by the wavelength adjustment of the circuit that gives a maximum mean square current, and multiply the quotient by TT. This gives 60 which is approximately d. 71. Problem. For practice it is recommended that the reader apply this rule to the curves of Figs. 3, 4, 5, and 6, noting that the ordinates of these curves are the square roots of y, and that for y to fall to a half value, the square root of y falls to .707 times the maximum value. 78. Determination of Decrement by Impressing an Undamped E.M.F. of Fixed Amplitude and Variable Frequency on a Circuit of Fixed Inductance, Capacity and Resistance. The starting point for this paragraph is the general equation (18). The decrement per undamped period of the fixed circuit is given in (22), from which we obtain This substituted into (18) gives 1 - Now introducing the wavelength values given in (26) and (28) we obtain 1 + __ i _ ili- h a 2 X! 2 ! A 2 i : i + i 2 *L 2 /A! 7T 2 h ~5 2 A 2 ^ 2 l I This last equation, solved for d gives 5 = ^A^ vTT" "t" ^ in which the sign is to be chosen so as to make 6 positive. 72 ELECTRIC OSCILLATIONS [CHAP. VI Equation (38) gives the decrement d per undamped period of the fixed circuit upon which is impressed an e.m.f. of constant amplitude and of wavelength Xi. The undamped wavelength of the fixed circuit is A defined by (28). 79. Approximate Method for Rapidly Determining 5 with Fixed Circuit and Variable Impressed Angular Velocity. Analogously to the case of fixed e.m.f. and variable circuit, as approximately treated in Art. 69, we may treat approximately the case of fixed circuit with a variable frequency of impressed e.m.f. Let X'i and X"i = impressed wavelengths at which y has half the maximum value, then by (38), if X"i > X'i, we have A \X'i 2 / ' and Adding these two equations, dividing by 2, and factoring, we obtain (41) Equation (41) is exact. Now X"i is greater than A and X'i is less than A, so that if X"i and X'i are not too far apart, their product is approximately equal to A 2 , so that (41) reduces to .(* yf 1 ' , approximately, = d, approximately. (42) This result may be stated in the following rule. 80. Rule for Approximate Determination of Logarithmic Dec- rement d, with Circuit Fixed and Frequency of Impressed E.M.F. Varied. To obtain the logarithmic decrement of a circuit of fixed constants, impress upon it an e.m.f. of fixed amplitude variable as to frequency. Take the difference of the two impressed wavelengths that produce a mean-square current equal to half the maximum mean-square current, divide this difference by the wavelength that gives a maximum mean-square current, and multiply the quotient byjr. CHAPTER VII THE FREE OSCILLATION OF TWO COUPLED RESIST- ANCELESS CIRCUITS. PERIODS AND WAVELENGTHS 1 81. Differential Equations for Inductively Coupled System of Two Circuits. If we have two circuits, as in Fig. 1, with the inductances of the two circuits near enough together to permit currents flowing in one of the circuits to induce electro- motive forces of appreciable values in the other circuit, the circuits are said to be coupled. FIG. 1. Two circuits I and II, coupled by a transformer. In the illustration the coupling is by mutual induction and is said to be inductive coupling. In setting up the differential equations both circuits will be assumed to have inductance, capacity, and resistance. The electromotive forces impressed upon the system from without is supposed to be zero. 1 The following is a partial list of references to theoretical works on the free oscillation of two coupled circuits: Lord Rayleigh, "Theory of Sound;" J. von Geitler, Sitz. d. k. Akad. d. Wiss. z. Wien, February and October, 1905; B. Galizine, Petersb. Ber., May and June, 1895; V. Bjerkness, Wied. Ann., 55, p. 120, 1895; Oberbeck, Wied. Ann., 55, p. 625, 1895; Domalip and Kolac'ek, Wied. Ann., 57, p. 731, 1896; M. Wien, Wied. Ann., 61, p. 151, 1897, and Ann. d. Phys., 8, p. 686, 1902; Drude, Ann. d. Phys., 13, p. 512, 1904; B. Macku, Jahrb. d. drahtlos. Teleg., 2, p. 251, 1909; Cohen, Bui Bu. of Standards, 5, p. 5UJF 1909. ^T 73 74 ELECTRIC OSCILLATIONS [CHAP. VII Independent of the method of setting up the currents in the system, the current i\ flowing in the Circuit I induces an electro- motive force M ~ in Circuit II, and likewise the current i z flowing in Circuit II induces an electromotive force M~ in Circuit I, where M mutual inductance between the two circuits. Consequently the differential equations for the currents in the two circuits are T d ^ _i p v i_ fi ldt K/T diz . , N LI-J- + Kill -\ ~ = M -y-, (1) = M ^, (2) where LI, Ri, Ci = respectively inductance, resistance and ca- pacity of Circuit I, L 2 , R 2 , C z = corresponding values for Circuit II. 82. * Differential Equations for a Direct Coupled System of Two Circuits. In the inductively coupled system described R' FIG. 2. Two circuits I and II coupled by an auto-transformer. above the two coils LI and L 2 , which acted mutually upon each other, had no part of their metallic circuits in common. The mutual action between them was by means of the transformer with separate and distinct primary and secondary coils. Circuits are also often connected by an auto-transformer, as in Fig. 2, where the two circuits have a metallic part Z/ in common. This connection is called a direct connection or direct 1 This article is somewhat confusing and may be omitted at first reading. CHAP. VII] RESISTANCELESS CIRCUITS 75 coupling. It will now be shown that this system leads to a set of differential equations that under certain conditions are the same as the equations for the inductively coupled system. For the sake of generality we may suppose that certain coils of the system, as L' and L", have no mutual action upon each other or upon other parts of the system, while other coils, as Z/o and L"o do have mutual induction. Let M = the mutual inductance between these two coils^ Z/o and L" , M r = the mutual inductance between Z/ and L' " , where L' "o is the part of the coil Z/' which is not common to Z/o. Let L! = L' + Z/o, Rl = R f ~f~ RQ, L 2 = L" + L" = L" + Z/o + L'" + 2M f , RZ = R ~T~ RO ~\~ R o- Then as before LI and L 2 are the total self-inductances of the Circuits I and II respectively, and Ri and R% are the total resistances, and M the total mutual inductance. Now taking the counter e.m.f.'s around the two circuits, noting that the coil Z/o is traversed by a current ii i 2) we have (3) (4) j ii * v,,. , \T> (I ; - L -TT + K * 2 H -- 7^ -- + R 0^2 + AJo(^2 * + L' (t s - z\) + M' 2 + M' (t s - t'O = (5) Equations (1) an^ (2) are tffte differential equations for the currents i\ and i% in the two circuits respectively when the two circuits are connected by having mutual inductance, and part of a coil in common. Introducing the values of LI, L 2 , Ri and R z from (3) , we obtain from (4) and (5) Lxf- 1 + Bin + ^ = (M' + L'o)J 2 + KOH (6) (7) 76 ELECTRIC OSCILLATIONS [CHAP. VII Now M' + Z/o = M (8) as may be seen by the following considerations. M is the mag- netic flux linkage common to Z/o and L" for a unit current in Z/o, which is the linkage with itself (=Z/o) plus the linkage withL " ( = M'). Substituting (8) in (6) and (7) we have L& + *, + ^ = M % + BA, (9) at O i at ' ; Equations (9) cmd (10) are ^e differential equations for the currents i\ and i z in the two circuits respectively, when the two cir- cuits ar$ direct coupled. R is the resistance of the element common to the two circuits. It is seen that these two equations are identical with those (1) and (2) for the inductively coupled circuits, provided the resistance of that part of the coil common to the two circuits is negligible. It is evident that various other methods of coupling 1 the cir- cuits together may be employed; for example, they may be connected together by having a condenser in common, but we shall at present confine our attention to the two types of coupling here illustrated, and shall proceed to treat the special case in which all the resistances of the two circuits are negligible. We shall describe both types of circuits here illustrated as magnetically coupled. 83. Differential Equations for Two Magnetically Coupled Circuits of Negligible Resistances. If all of the resistances of the two circuits are negligible, the equations (1) and (2) for the inductively coupled circuits and the equations (9) and (10) for the direct coupled circuits reduce to the form '* + =TJT - * <'+ = "'- These are the differential equations in the resistanceless case of two magnetically coupled circuits. 1 See subsequent chapters. CHAP. VII] RESISTANCELESS CIRCUITS 77 84. Steps toward a Solution of (11) and (12). The two equations (11) and (12) are to be solved as simultaneous. The elimination of one of the i's from those two equations will give a homogeneous linear differential equation of the fourth order 1 in the other i and its derivatives. The solutions are, therefore, additive, and the complete solution must contain four and only four arbitrary constants. Instead of performing the elimination it is simpler and more instructive to solve by inspection by assuming ii = Ae kt , (13) i z = B. (14) That these values are solutions is seen by a direct substitution of them in equations (11) and (12), giving A l ,* + = MBk, (15) + ~ = MAk. (16) J The product of these two equations gives which is independent of A and B. Equation (17) is a relation that must be satisfied by k, in order that (13) and (14) may be a simultaneous system of values satis- fying (11) and (12) . 85. Expression of (17) in Terms of Angular Velocities of the Separate Circuits. Let us now write o>! 2 = I/Lid, (18) o> 2 2 = 1/L 2 C 2 . (19) It is seen that, since the resistances are negligible, o?i and co 2 are the angular velocities of free oscillation of the two circuits of the system respectively, when each is alone and uninfluenced by the other. (Cf Arts. 8 and 15). If now we divide (17) by LikL 2 k, we obtain 1 The steps of this process are given in Art. 98 below. 78 ELECTRIC OSCILLATIONS [CHAP. VII where *-& The quantity r is called the coefficient of coupling of the circuits. Equation (20) may be solved as a quadratic in k 2 . It is some- what more direct to our purpose to solve (20) for the reciprocal of k rather than for k itself. For this purpose, let us perform the indicated multiplication in (20) and divide the result, by o>i 2 co 2 2 , obtaining - - -=- (22) fc 4 n W T cc 2 2 /fc coi 2 co 2 2 ' Completing the square and solving we obtain i 1/1 \ i / i\ ,4 , fc = *\ aw + 5 :t 2V W 5*) + ^w' Since r, by the physics of the problem, is less than unity, it is seen that the quantity under the main radical is negative whether the plus or the minus be used before the second radical, since the original circuits are oscillatory. Whence, k is a pure imaginary quantity, and there are seen to be four different values of k consistent with (23). These four values may be written X (say), fc 3 = ja>"(say), (24) where co' and w" are given by following relation, somewhat simplified from (23), 1, = + Jl/l i+ _L f )+l /(J. -J- i co \ 2\oji 2 co 2 / 2 \ Vcoi 2 co 2 2 , (26) Taking the products of (25) and (26) and taking the recip- rocal of the result, we find that V' = -^ (27) CHAP. VII] RESISTANCELESS CIRCUITS 79 which used as a multiplier of (25) and (26) gives - C0 2 2 ) 2 + ^) + \ '-7== - (29) VI - T 2 In seeking for a solution of our original differential equations (11) and (12) we have now found four solutions, one correspond- ing to each value of k. These solutions are of the form of (13) and (14), and for each of the four solutions for i\ we have a different arbitrary constant. Similarly for each of the four solutions for i 2 we have a separate arbitrary constant, but there are some relations among these constants. Taking the sum of the four solutions for i\ and likewise for i z , we obtain 1\ == A. i |~ A.% ~| A- 3^ "i ^J- 46 (oUJ D kit I D kit I D k3t [ D ktt fQI \ Equations (30) and (3 1) are the complete solutions of the differ- ential equations (11) and (12). In these solutions the several k's are given by (24) taken in connection with (28) and (29). The four A's and the four B's are arbitrary, except that each B is related to the corresponding A by a relation of the form of (14) and (15). The two relations (14) and (15) are not, however, independent since their product was used in determining the k's. 86. Determination of the Periods of the Magnetically Coupled Pair of Resistanceless Circuits. Let us leave for the present the question of the values of the arbitrary constants A and B, which are to be obtained from the initial conditions, and return to an examination of the k's, which may be used to give us the period or periods of the resulting oscillations that occur in the coupled system. Since the k's are all imaginary quantities with the values given in (24), we may transform 1 the equations for i\ and i% (namely, (30) and (31)) into the trigonometric forms ii = I' i sin (o/ + 's are constants derivable from the A' s and B's or from the initial conditions. Fixing our attention upon the w' and co", it is to be seen that both currents are doubly periodic, and that the two periods of the current ii in Circuit I are the same as the two periods of the current i 2 in the Circuit II. These two periods may be obtained from the corre- sponding angular velocities ' and co". Let these two periods be 7" and T", which are related to the corresponding angular velocities by the equations T' = 27T/"' (34) T" = 27T/co" (35) Therefore, if we multiply equations (25) and (26) through by 27T, and recall that the periods TI and T z of the two circuits when alone are T 1 = 27T/ui (36) T 2 = 27T/C02 (37) we obtain T 1 = +^l(Ti 2 + TV) + \ J (T 7 ! 2 - TV) 2 + 4r 2 2 7 1 2 T 2 2 (38) T" = + TV + TV - Ti 2 - TV 2 + 4r 2 7V7Y (39) These two equations may be written in a different form as follows : V 2 (41) That (40) and (41) are respectively identical with (38) and (39) may be shown by squaring and extracting the square root of (40) and (41), by which operation we arrive at (38) and (39). The equations (38) and (39), or the alternative equations (40) and (41), give the two periods T' and T" of the doubly periodic CHAP. VII] RESISTANCELESS CIRCUITS 81 oscillation that occurs in the primary circuit of the coupled system. The same two periods occur also in the secondary circuit of the coupled system. These equations are exact only provided the resistances are negligible in their effects on the periods. 87. Determination of the Wavelengths of the Magnetically Coupled Pair of Resistanceless Circuits. To obtain the result- ing wavelengths in the coupled system, it is only necessary to multiply the periods by the velocity of light, and employ the relations \' rT f \" rT" } A - Cl A - Cl \ . Xi = cT l X 2 = cT 2 I These values substituted into (38), (39), (40) and (41) give V = (Xi 2 + X 2 2 ) + (X! 2 - X 2 2 ) 2 + 4r 2 X! 2 X 2 2 (43) X " = \2 (>l2 + V) ~ \ V(Xi 2 - X 2 2 ) 2 + 4r 2 X! 2 X 2 2 (44) or the alternative results V = >X! 2 + X 2 2 + 2XiX 2 X 2 2 - 2XA 2 2 (45) Xx 2 + X 2 2 - 2XiX 2 \ 2 (46) Equations (43) and (44), or the alternative equations (45) and (46), give the two wavelengths \ r and \" of the doubly periodic oscillation that occurs in the primary circuit and also in the secondary circuit of the coupled system,* provided the resistances are negligible in their effects on the resulting wavelengths. 1 88. Graphical Method of Finding X' and X". The equations given in the preceding section permit the calculation of X' and X" when Xi, X2, and r are given. When great accuracy is not required, the following graphical method may be employed. In Fig. 3, lay off AB equal to Xi 1 For experimental tests of these equations see PIERCE, Physical Review, 24, p. 152, 1907; also "Principles of Wireless Telegraphy," p. 228, McGraw- Hill, 1910. 82 ELECTRIC OSCILLATIONS [CHAP. VII and BD also equal to Xi and in the same straight line with AB. At the point B draw the line BC making with BD an angle whose sine is T. Make the length of BC equal to X 2 , then draw A C and DC. Call the lengths of A C and BC, b and a respectively. Then half the sum of b and a is the required wavelength X', and half their difference is the required wavelength X". This may be readily proved as follows : Since sin e = T, cos 6 = \/l r 2 . By the geometrical proposition concerning the square of the side of a triangle opposite to an obtuse or an acute angle A B D FIG. 3. Showing geometrical construction for obtaining resultant wavelengths. = Xl 2 -f- iX 2 COS 6 = iX 2 cos e = whence, from (45) and (46) X' = b + a b - a (47) (48) (49) Exactly similar construction may be employed to give 7" and T") if all the X's are replaced by the corresponding T's. 89. Simple Relations Among Wavelengths or Periods in a Magnetically Coupled Pair of Resistanceless Circuits. By taking the sum of the squares of (38) and (39) and likewise the sum of the squares of (43) and (44), we obtain T' 2 + T 7 " 2 = TV + T 2 2 , X'2 + x" 2 = Xi 2 + X 2 2 . (50) (51) CHAP. VII] RESISTANCELESS CIRCUITS 83 Also, by taking the products of the same two pairs of equations, we obtain T'T" = T^r^T 2 (52) XV' = XiXsVT^ 2 (53) 90. Special Cases of the Coupled System of Negligible Resistances. Case I. Isochronism. If the two circuits have the same period when each is alone, X x = \ 2 = \ (say) (54) and Tl = T 2 = T (say) (55) then equations (43), (44), (38), and (39) give T = T^/Y+r T" = Tf^r (56) X' = X-vlT^ V = xr (57) Case II. Negligible Coupling. Whether the circuits are iso- chronous or not, if T is sufficiently small so that terms involving it in (38) to (43) are negligible, these equations give T' = T l T" = T 2 (58) X' = Xi X" = X 2 (59) As to how small r must be in order to be negligible depends upon the relative values of Xi and X 2 . If Xi = X 2 , then by (57), to be negligible r/2 < < 1 (60) where < < means "is negligible in comparison with." If, on the other hand, Xi and X 2 are sufficiently different to make 4r 2 X 1 2 X 2 2 2 2 (7) in which the last term is obtained by replacing l/L^C^ by co 2 2 . In using this equation we must give B } A, and k the same sub- script. Doing this and replacing the subscripted k by its value from (6) we obtain the system of equations Bi B* M co /2 AT /9 2 xJ2 w AT 2 . .//2 4 JLt.vt (8) in which X and Y are abbreviations for the quantities set im- mediately before them in (8). Now introducing our initial conditions (1) into (2), (3), (4) and (5) and making use of the equations (6) and (8), we obtain = A. + A 2 -f A 3 + A 4 from (2) (9) 88 ELECTRIC OSCILLATIONS [CHAP. VIII = X(A l -f A 2 ) + F-(A 3 + A 4 ) from (3) and (8) (10) Q = Al .~/ 2 + A3 ~/ 4 from (4) and (6) (11) - X(A }~> A * } + ' from (5), (6) and (8) (12) Equations (9) and (10) give by elimination (X - F) (A 1 + A 2 ) = (13) (X - F) (A, + A,) = (14) while equations (11) and (12) give by elimination -QY = (X - F) ~ = (Z - F) ^A 4 (16) Unless Z = F, (13) and (14) give A, = -A 2 , A, = -A, (17) and these values substituted into (15) and (16) give and ' This derivation of the constants AI, A 2 , A 3 and A 4 is valid unless X = F. By a comparison of (11) with (12) it is seen that X cannot equal F unless both are zero. If both are zero, (8) shows that M is zero. If M is zero the Circuit II will have no current in it, and the Circuit I will be a single circuit with a condenser discharge in it satisfying the conditions given in Chapter IT. 93. Periodic Equations for the Currents. With these values of the A's and with proper values of the fc's from (6) introduced into (2) we obtain . = ~X~-^Y ~2~ H X~-^Y 2 If we introduce j as a factor in the denominators of the ex- ponential factors they become sines, and we have fi = (^F sin 'J - o"X sin u"t] (21) CHAP. VIII] RESISTANCELESS CIRCUITS 89 In like manner using the values of the B's as given in (8), we obtain sin w * ~ w sin w (22) As a step toward replacing X and F by their values, let us note from (8) that 1 M *- I - /.,2/,./2 whence co 2 2 /o/ M T, 2 In like manner from (8) (23) (24) (25) X Y ~ y'2 rr 1 " 2 (^v) Further, if we replace T f and T" by their values from Chapter VII equations (38) and (39) we obtain Y 1 Y = From these values of X and Y, we obtain y \r /TT/ 2 _ X -Y 2 X 1 X - Y 2 + + 1 - 1 v- 2 rp 2 \ 2 Introducing these values into (21) we obtain 1 (27) (28) ^l = + 1 co' sin u't -1 + 1 + to sm (29) Let us next determine z' 2 , which can be done by multiplying the equation (23) by (25) obtaining XY M T, 2 X - Y L 2 T' 2 - T"' (30) 90 ELECTRIC OSCILLATIONS [CHAP. VIII which by (38) and (39) Chapter VII 2 \/(7Y- TSY This introduced into (22) gives /* - co"sih"i}(31) Equations (29) and (31) grwe 2/ie complete expressions for the currents in the two circuits of the coupled system having negligible resistances and excited by discharging at the time t = the con- denser Ci with an initial charge Q. 94. Relative Amplitudes of Current in the Coupled System of Negligible Resistances Excited by a Condenser Discharge. If we write the equations for i\ and i% respectively in the form *i = 7'i sin w't + /"i sin w" (32) it = I't sin w't + 7" 2 sin w"t (33) it is seen that the ratios of amplitudes for the same frequency in the different circuits may be written. [See (21) and (22).] T, - T' 2 (35) T 2 2 - T" 2 Also it is seen that the ratio of amplitudes of the two different frequencies in the same circuit are for primary and secondary respectively r\ = -<*"x -rx -T f (Tz 2 - T" 2 } /'i " w'y T"Y T"(T Z 2 - T' 2 ) Equation (34) gives the ratio of amplitude of current in the sec- ondary circuit to that in the primary circuit for the frequency T'. Equation (35) gives a similar ratio of amplitudes for the frequency T". Equation (36) gives the ratio of amplitude of current of fre- quency T" to the amplitude of current of frequency T' in the same (primary) circuit. Equation (37) is a similar ratio for the sec- CHAP. VIII] RESISTANCELESS CIRCUITS 91 ondary circuit. These equations hold for excitation by the discharge of the primary condenser with resistanceless circuits. H. DISCHARGE OF AN INDUCTANCE 95. Determination of Amplitudes when the Coupled System of Negligible Resistances is Excited by the Discharge of the Primary Inductance. Let us now determine the solution of the resistanceless coupled circuit problem when the excitation is produced by sending a steady current through the inductance of Circuit I, and then isolating it as was done in the buzzer ex- citation process of Chapter II. The differential equations are the same as in the problem already treated and give therefore the same frequencies as before. The amplitudes, however, which are determined by the initial conditions will now be different from those of the previous sections. If we measure time from the instant of isolating the current in the primary inductance, the initial conditions are as follows: When t = 0, ii = 7, i 2 = 0, qi = 0, q 2 = (38) By comparison with the equations (9) to (12) it will be seen that these initial conditions require / = A! + A 2 + A* + A 4 (39) = X(At + A t ) + Y(A. + A,), (40) - At) Y(A 3 - A,) ~ ' Elimination among these equations gives XI The several B's have the same ratio to the corresponding A' s as in the condenser discharge problem. 92 ELECTRIC OSCILLATIONS [CHAP. VIII These constants substituted into equations (2) and (3) give, after trigonometric transformation as before, the results f i = Y^Y { Y COS "'* ~ Z COS w "' ! -JZF TF .A l //!> COS CO t COS CO "t\ 145) (46) By comparison of these equations for current in this case of inductance excitation with the corresponding equations for cur- rent in the previous problem of capacity excitation, it will be seen that equations (45) and (46) take the form 1 1 + - T 2 2 ) cos "t] (48) MI Equations (47) and (48) give respectively the primary and secondary current in a coupled system of two circuits of negligible resistances, excited by sending a steady current I through the in- ductance of the primary circuit and isolating it at a time t = 0. 96. Relative Amplitudes of Current in the Resistanceless Coupled System Excited by Isolating a Current in the Primary Circuit. If now in this case we write the expressions for the currents in the abbreviated forms ii = J'i cos a)'/ + J"i cos u"t iz = J'z cos u't + J"z cos co"# and compare the amplitudes we have Jf ^ /7T T, T n "2 -v V ^2 -L 1-L 2 <^j _ v _ Vc* ~T t t "" -^ 9 / J'l J': - X Y TV - T" z T 2 2 - T" 2 (49) (50) (51) (52) (53) (54) CHAP. VIII] RESISTANCELESS CIRCUITS 93 Equations (51) to (54) give the relative amplitudes of current in the resistanceless coupled system of two circuits excited by the discharge of an inductance in the primary circuit. The discharge is produced by isolating a constant current I in the primary in- ductance at t = 0. It is to be noted that two of the ratios (51) and (52) are the same as in the case of the condenser excitation, and two of the ratios (53) and (54) are different from the case of condenser excitation. It is also to be noted that cosines appear in the present case, where sines appeared in the case of the other form of excitation. CHAPTER IX THE FREE OSCILLATION OF TWO INDUCTIVELY COUPLED CIRCUITS. PERIODS AND DECRE- MENTS. TREATMENT WITHOUT NEG- LECTING RESISTANCES 1 97. Differential Equations. It is proposed to treat in the present chapter the theory of the free oscillation of two coupled circuits such as are shown diagrammatically in Fig. 1. The method is similar to that employed in Chapters VII and VIII, except that now the resistances are to be retained wherever their values are significant. FIG. 1. Diagram of circuits. The differential equations are those given in equations (1) and (2) of Chapter VII, which are here rewritten with all the terms transposed to the left-hand side; namely, (2) 98. Elimination to Show that the Resulting Equations are of the Fourth Order. Let us eliminate iz from the two equations and show that the resulting equation in i\ is a differential equa- tion of the fourth order. 1 See references at beginning of Chapters VII and VIII. The present treatment is more complete than the treatment in the references. 94 CHAP. IX] THE FREE OSCILLATION 95 First add M times equation (2) to L 2 times equation (1), and differentiate, obtaining (T T M2\ d2il j T pd?i /Vi pM-^2,^2 n /o\ *** M } ^ + L2jRl W " ~cT " h ^ 2 "5" + ~c7 = Add Rz times (1) to (3) and differentiate, obtaining Add 1/Cz times (1) to (4), and differentiate, obtaining A ,,, In the same way the elimination of ii instead of z' 2 gives for i z the same equation except that 12 is substituted for ii. Equation (5) is a homogeneous linear differential equation of the fourth order. The complete solution has four arbitrary con- stants, and any solution that has four arbitrary constants is complete. Instead of proceeding directly to a solution of (5) by introduc- ing an exponential with t in the exponent, it is somewhat more convenient to make our substitutions in (1) and (2) as was done in Chapter VII. We shall make no use of (5) further than to note that the complete integral has four arbitrary constants. 99. First Step in the Solution of (1) and (2). Let us begin the treatment of the pair of simultaneous equations (1) and (2) by letting ii = Ae ki , i 2 = Be kt (6) These values, substituted into (1) and (2), give, after division by c* A(L,fc + R, + -L) = MBk (7) and B(L 2 k + fa + -^) = MAk (8) Taking the product of (7) and (8), we obtain t + C&i* + ^2 + - = MW (9) 96 ELECTRIC OSCILLATIONS [CHAP. IX Dividing (9) by LiL 2 k 2 , we obtain, in terms of abbreviations next given, the equation where, as in previous chapters, M ;--w :. '-vis ^ n) ;-VVv <* = ' --S . ^ 12 > ft! 2 = =7-, 2 2 = ~ (13) i/lv/l JL/2vy2 Among these abbreviations note that the quantities fii and ft 2 are related to the corresponding co and a by the equations as may be seen by reference to (viii) at the beginning of Chapter II. Equation (10) is an equation of the fourth degree that k must satisfy, in order for (6) to be solutions of the original differential equations. In (10) the quantity r, defined by (11) is called the coefficient of coupling of the circuits. The quantities ai and a z are the logarithmic decrements per second, or damping constants, of the separate circuits when each is alone and uninfluenced by the other. 12i and 2 2 are the undamped angular velocities of the two circuits respectively when they are uninfluenced by each other. In equation (14) coi and co 2 are the free angular velocities of the separate circuits. It is seen that the undamped angular veloci- ties fii and fl 2 are equal to the free angular velocities in those cases in which ai 2 /2coi 2 and a 2 2 /2 2 2 are negligible in compari- son with unity. 100. Note on the Constants A and B. Returning now to equation (10), let us designate the four k's that are roots of (10) by ki, k z , & 3 , and & 4 . Then by (6) for each of the k's there will be a corresponding A and B, to which we shall give subscripts 1, 2, 3, and 4 identical with the respective subscripts of k, ob- taining *! = An6 knt , it = B^, where n = 1, 2, 3, 4. CHAP. IX] THE FREE OSCILLATION 97 Applying to these solutions the principle of additivity, we shall have as the complete integral of the differential equations (1) and (2) the following zi = A 1 e klt + A* k * + A,e k3t + A# k * (15) = 2A n e knt and likewise t, = 2B,/"', (16) where n = 1, 2, 3, 4 The constants A n and are arbitrary constants of integration. Although there are eight of these constants only the four A' a are independent of each other, for each B is related to the cor- responding A by an equation of the form of (7) or (8), in which we must give A and B either of the common subscripts 1, 2, 3, 4. Calling any one of these common subscripts by the generic designation n, we have from (7) and (8) fcn + Rl + ~ = MB n k n (17) B n l^k n + # 2 + - = MAnkn (18) Either of the relations (17) or (18) may be used to determine B n from A nt but if both (17) and (18) are used they give no more restriction than one alone, for the two equations are not inde- pendent, as their product has been used in determining k n . The eight arbitrary constants are thus reduced to four by having four relations among them. These four relations are obtained by giving n successively the values 1, 2, 3, and 4. The four arbitrary constants to which the eight are reduced are to be determined by the initial conditions in any specific problem. We shall postpone the determination of these con- stants A n and B n to the next Chapter, and shall proceed in this chapter to a discussion of the values of fci, & 2 , k 3 , and & 4 , which are the roots of the fourth degree equation (10). 101. Expression of the Roots k as Complex Quantities, and the Currents as Periodic Functions of the Time. Expanding (10) by multiplying the factors together, we obtain , ! 2 I ~ - , 2(q 1 Q 2 2 98 ELECTRIC OSCILLATIONS [CHAP. IX Let us now write the four values of k that are the roots of (19) in the complex forms fca - -<+ X, k, = -a" + jo" \ k 2 = -a'-X, & 4 = -a"-X'J They can be written in this form for if any root is a complex quantity, the conjugate complex is also a root. Real roots, if they exist, must therefore be two or four in number. To cover this contingency of real roots it is only necessary to make a/ or co", or both, imaginary. The a's always remain real. With the use of these complex roots, equations (15) and (16) can be transformed into sin ('< + "/( (23) Therefore, ti = J'ie" a ''sin (o't + 'i) + 7"i-"' sin (co r/ < + ^"0 (24) provided A i + A 2 = /'i sin \ . , A 3 + A 4 = / r/ i sin ^, j(A* - A,) = I", cos In order for (25) to be satisfied by real values of I\ and ^>'i, it is seen that A\ -j- A 2 must be real and A\ A 2 must be imaginary; that is to say, A\ and A 2 must be in general conjugate complexes. This looks like an additional restriction on the arbitrariness of A i and A 2 that we have imposed by the transformation. But, as a matter of fact, this limitation is imposed by the equations (15) CHAP. IX] THE FREE OSCILLATION 99 if we require that the current i\ be real and if we assume that co' is real, for this assumption gives at once (23) that requires the conjugate relation of A\ and A 2- If on the other hand co' is imaginary, let co' = ju h , where COA is real, then cos co' = cosh co A , and j sin co' = sinh co&, so that, in this case, (23) shows that both AI and A* are reals. With these two A's real, (25) shows that both I\ and S 1 2 + a 2 S 2 2 ) 2 + (ax + a 2 ) (aA 2 + a 2 S 2 2 ) (V + S 2 2 ) - (ai + a 2 ) 2 /Si 2 iS 2 2 + zS 1 S 2 (a 1 + a,) (a^! 2 + whence a'a"D(l - r 2 ) = Now let us subtract ia 2 D from the left-hand side of this equation, and from the right-hand side this same quantity with D replaced by its value from (50), and note that the difference obtained for the left-hand side is -z^D/S^ by (38). We thus obtain ~*** D = r 2 (aiSi 2 - a 2 S 2 2 ) 2 + zSiSz n - a 2 ) Replacing D by its value from (50) and collecting terms we obtain Z 2 , * ~~ CHAP. IX] THE FREE OSCILLATION 105 If we introduce the abbreviations equation (51) becomes z* + Az 2 + Bz + C = (53) where A = 2 (54) Equation (53), in tofocA A, 5, and C have the values given in (54), gives the value of z in terms of known constants of the circuits. In these equations, x, 5'i and 62 have the values given in (52). It is to be borne in mind in using these equations that if ai and a 2 are independent of Si and S 2 then 5i and 5 2 are dependent on Si and $2 and may be dependent on x* 109. If the Original Circuits Are Oscillatory when Each is Alone, All the Real Roots of (53) Are Negative. As a step toward fixing the limits of 2, we shall show that all the real roots of (53) are negative provided each of the two original circuits is oscilla- tory when it is alone and uninfluenced by the other circuit. If the original circuits are both oscillatory, 5i/27r < 1, and 6 2 /27r < 1 (55) To prove that the real roots of (53) are negative it is only necessary to show that the coefficients A, B, and C are all positive. Since x + 1/x, where x is positive cannot be less than 2, it is seen that condition (55) makes A positive. It is seen also that always C is positive, since it is a perfect square. The remaining coefficient B is more difficult to treat, but may also be shown to be positive under the limitations (55) as follows : Taking B from (54) add and subtract 26i6 2 /7r 2 , obtaining The last parenthetical expression may be written in the form 106 ELECTRIC OSCILLATIONS [CHAP, ix (x l) 2 /x. Grouping this last term with the first and grouping the fourth term with the third, we obtain The only term or factor in this equation that is doubtful as to sign is the expression within the brace; but by (55) 4 . (57) and, since x is positive, it is also apparent that ' '" : By .taking the product of (57) and (58), it is seen that the expression within the brace in the equation (56) for B is positive, and hence B is positive. We have thus proved that, if each of the original circuits is oscil- latory when standing alone, all of the coefficients of the cubic equa- tion (53) are positive, and that in consequence all of the real values of z are negative. We shall next be able to assign certain limits to the value of z, that will simplify the calculation of this quantity. 110. Determination of the Limits of the Value of z for Coupled Circuits Oscillatory when Alone. In the preceding section we have shown that z is negative provided the original circuits are oscillatory when alone. We can now establish outside limits of the value of z by very simple operations. To begin, let us take the original definition of z, equation (38) , multiply both sides of that equation by SiS 2 , and partly replace $i 2 $ 2 2 by its value from (37), obtaining (haaSi'S,* - a'a"S'*S"* ( . 8i8& = - ^ (59) Let us now make use of the algebraic generalization that for any two real quantities x and y then by this relation alone CHAP. IX] THE FREE OSCILLATION 107 Replacing the right-hand side of (60) by its value from (36) we have - (61) This quantity substituted into (59) gives - (62) Let us recall that z is negative, and let us divide both sides of (62) by SiS 2 , and make use of the abbreviations x = &/&, 5i = ai&, 6 2 = a z S 2 , (63) then we obtain 0?*? --/* (64) 47T 2 The inequality (64) gives the limits of the value of z, provided the original circuits are oscillatory when not coupled. 111. Reduction of the Cubic Equation for z to a Quadratic Equation over an Important Range of Constants. In equation (53) we have given a cubic equation for the determination of z, and we have shown that z is negative, and that it has the limiting values specified by the inequality (64), provided the original circuits are oscillatory, that is, provided g), B = 4(r 2 + u), C = r*u (72) The cubic equation (53) for z, in the isochronous case, becomes z 3 + 4z 2 (l - v) + 4z(r 2 + u) + 4r 2 w = (73) This factors into ^ (z - 4t>) + (z + u)(r 2 + 2) = (74) From this factored form we can make a discovery of a new fact in regard to the limit of z. We have already shown in the general case the relation (64), which in the isochronous case (since x = 1) becomes > z = ? - u (75) This inequality may be otherwise written in the form z < 0, and z + u ^ (76) This fact applied to (74) shows that the first term of that equation is negative or zero, and therefore the last term must be positive or zero to make the sum zero. We have just shown in (76) that one of the factors (z + u) of the last term is positive or zero, and hence the other factor of that term is positive or zero; that is, z + r 2 ^ (77) This result is important in determining the signs and the limiting values of expressions to follow. We shall now examine the condition under which the cubic 110 ELECTRIC OSCILLATIONS [CHAP. IX equation for z reduces to a quadratic. This condition as stated in (66) may be written z 2 B, which in the isochronous case, by (72), becomes z 2 4(r 2 + M) (78) Now by (75) and (77), z 2 < u 2 and also z 2 < r 4 so that (78) is met if either u 2 4(r 2 + u) or r 4 4(r 2 + u) (79) Either of the alternative conditions of (79) is sufficient to reduce the cubic to the quadratic. If u < r 2 the first of the alter- natives is met if u 2 Su-, tfiat is (gl ~ [^ < < 1 (80) 6ZTT If T 2 0. By (77) this condition is always fulfilled with oscillatory cir- cuits, and, therefore, by the note following (48) and (49) the signs in these two equations are correct for this case. These two equations (48) and (49), by the isochronous con- dition Si = $ 2 = $, reduce to a/ = f 1 - vrr ^ (84) a = 2(1 - where _ (i - ry - .) 4v + u Ln the system of two circuits that are separately tuned to the same undamped periods, the resultant undamped periods when the cir- cuits are coupled together and allowed to oscillate freely are given by the equations (83), and the resultant damping constants are given by (84) and (85) in terms of 6 defined by (86). The values of u and v are defined in (71). z is given by (73-) and is usually given with sufficient accuracy by (82) . 114. Application to Two Numerical Cases of Quasi Isochronism. As the first special case of quasi isochronism, let us take the following numerical values. Let 5i = 0.37T, 6 2 = O.lTT (87) In this case the values of u and v become u = o.Ol, v = 0.0075, 4v + u = 0.04 (88) With these numerical values (82) becomes r 2 + 0.01 z = 1.985 " V 1 " (r 2 + O.Ol) 2 CHAP. IX] THE FREE OSCILLATION 113 In this numerical case (84) and (85), on multiplying both sides by S/ir, become and a"S 0.2 1-T 2 _ 1 + Vl - 0} (89) (90) .6 aS/ir .6 .7 FIG. 2. Quantities proportional to resultant damping constants plotted against coefficient of coupling T in special case in which Si = O.STT, 62 = O.ITT, and Si = S 2 = S. where (1 - r 2 )(0.03 - z) 0.04 (91) Computations were made for various values of T. The method of making the computations consists in first determining z by the use of the equation following (88) and then computing &'S/ir and a"S/ir by the use of (89), (90) and (91). The values of S'/S and S"/S may be computed directly from (83). 114 ELECTRIC OSCILLATIONS [CHAP. IX Table I. Computed Values of Damping Constants and Undamped Periods of the Quasi Isochronous System of Two Circuits with Various Values of r. Given b^ = O.STT, 6 2 = 9 a"S a'S S' S" S' S" r T z tr IT s S *Vi +7 SVi- r .0000 .0000 -0.00000 0.300 0.100 1 .0000 1 .0000 1.000 .000 .0278 .00077 -0.00077 0.296 . 104 1 .0002 0.9994 0.986 .011 .0578 .00334 -0.00333 0.283 0.118 1.0011 0.9962 0.973 .025 .0802 .00646 -0.00636 0.264 0.139 1 .0037 0.9931 0.965 .034 0.092 .00842 -0.00810 0.249 0.154 1 .0071 0.9887 0.957 .036 0.101 0.0103 -0.00922 0.237 0.167 1.0143 0.9809 0.965 .033 0.109 0.0119 -0.00961 0.232 0.173 1.0216 0.9731 0.970 .029 0.121 0.0146 -0 .00980 0.232 0.175 1 .0323 0.9621 0.976 .025 0.187 .0348 -0.00996 0.246 0.168 1 .0735 0.9151 0.984 .014 0.200 .0400 -0.01000 0.250 0.167 .0810 0.9065 0.986 .012 0.300 0.0900 -0.01000 0.286 0.154 .130 0.8439 0.990 .007 0.400 . 1600 -0.01000 0.333 0.143 .176 0.7796 0.993 .005 0.500 0.2500 -0.01000 0.400 0.133 .219 0.7107 0.995 .003 0.600 0.3600 -0.01000 0.500 0.125 .260 0.6353 0.996 .002 0.700 0.4900 -0.01000 0.667 0.118 .299 0.5497 0.996 .002 0.800 0.6400 -0.01000 1 .000 0.113 .337 0.4487 0.997 .002 0.900 0.8100 -0.01000 2.000 0.105 .374 0.3172 0.998 .001 1.000 1.000 -0.01000 infin. 0.100 .414 .0000 1.000 1.000 Table II. Computed Values of Damping Constants and Undamped Periods of the Quasi Isochronous System of Circuits with Various Values of T. Given 5i = 0.037T, 5 2 = 0.01*- T 7-2 z a'S a"S S' S" S' S" IT V S S SVTT~r SVT^r 0.000 .000000 -0.000000000 0.01000 0.03000 .0000 1 .0000 1 .0000 1 .0000 0.001 0.000001 -0.000001000 0.01005 0.02995 .0000 1.0000 .9995 1 .0005 0.002 0.000004 -0.000003999 .01020 .02980 .0000 1 .0000 0.9990 1 .0010 0.004 0.000016 -0.000015996 0.01083 0.02917 .0000 0.9999 0.9980 1 .0019 0.006 0.000036 -0.00003598 0.01200 .02800 .0001 0.9999 0.9970 1 .0029 0.008 0.000064 -0.00006394 0.01390 .02601 .0001 .9999 0.9960 1 .0039 0.010 0.000100 -0.00009974 0.01940 .02057 .0002 0.9997 0.9953 1 .0047 0.012 0.000144 -0.00009997 0.01996 .02024 .0031 0.9964 .9970 1 .0026 0.015 0.000225 -0.00009999 0.01970 0.02030 .0053 0.9941 0.9978 1.0017 0.020 0.000400 -0.00010000 0.01960 0.02041 .0084 0.9911 0.9984 1.0011 0.030 .000900 -0.00010000 0.01941 0.02061 .014 0.9855 0.9989 1 .0006 0.040 .001600 -0.00010000 0.01923 .02083 .019 0.9803 0.9991 1 .0004 0.050 .002500 -0.00010000 1 .01905 0.02105 .024 0.9750 0.9993 1 .0002 0.1 0.01 -0.00010000 0.01818 0.02222 .048 0.9487 0.9995 1 .0000 0.2 0.04 -0.00010000 0.01666 0.02500 .095 0.8943 0.9997 0.9999 0.3 0.09 -0.00010000 0.01538 0.02856 .140 0.8365 0.9997 0.9998 0.4 0.16 -0.00010000 0.01437 .03333 .183 0.7744 0.9998 .9997 0.5 0.25 -0.00010000 0.01333 0.04000 .224 0.7088 0.9998 .9996 0.6 0.36 -0.00010000 0.01250 0.05000 .264 0.6318 0.9998 0.9994 0.7 0.49 -0.00010000 0.01176 0.06667 .303 0.5474 0.9998 0.9993 0.8 0.64 -0.00010000 0.01111 . 10000 .341 0.4468 0.9998 0.9989 0.9 0.81 -0.00010000 0.01052 0.20000 .378 0.3155 0.9999 0.9978 1.0 1.00 -0.00010000 .01000 infin. .414 0.0000 1 .0000 1 .0000 CHAP. IX] THE FREE OSCILLATION 115 Table I contains the results of the calculation with various values of T. These results are plotted in the curves of Figs. 2 and 3. As a second example of the quasi isochronous system, we have computed the case in which 1 = 0.03*-, 5 2 = O.Olr (92) .37T .ITT s z s .1 .2 .3 .4 .5 .6 .7 T FIG. 3 .^Quantities proportional to S f and S" plotted against T in special case in which 61 = O.STT, 5 2 = O.ITT, and Si = S = S. The results in this case are recorded in Table II and some of the significant values are plotted in Figs. 4 and 5. Although the scale in Figs. 4 and 5 is different from the scale in Figs. 2 and 3, it is seen that the case with the decrements given in (92) has general characteristics in common with the case with the larger decrements given in equation (87). 115. Discussion of the Results in the Numerical Cases of Isochronous Circuits, with Derivation of Limiting Values of z. Certain significant facts are apparent from Tables I and II, compiled for the two sets of specific values of the decrements. 116 ELECTRIC OSCILLATIONS [CHAP. IX One of these facts is that for small values of r 2 , z is approximately equal to r 2 . This may be derived theoretically from the cubic equation (74) for z, which by transposition of the first term to the right and division by z + u gives v - z) Z = -r 2 + 4(3 + u) ' If z is to become approximately r 2 the second term of the right-hand side must be small, and the equation must still be .030 .01 M .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 14 FIG. 4. Same as Fig. 2 except that di = 0.037T, 5 2 ="0.0l7r, and that scale is changed. approximately correct when z in the fraction is replaced by r 2 , giving * - -T{ 1 - 1 , approximately. (93) Now by (77) z + r 2 must be positive, so that (93) can be employed only when r 2 is less than u, and since the fraction of (93) was obtained by replacing z by r 2 , it is seen that for (93) to be applicable the fraction in (93) must be small in comparison with unity. If these conditions are fulfilled z becomes approximately CHAP. IX] THE FREE OSCILLATION 117 equal to r 2 . In symbols, these statements may be written as follows : then Z = T 2 (94) (95) 1.07 1.06 1.05 1 04 .^ ^ _^s ^ s^ ^s ^ 1.03 1.02 1.01 1.00 .99 60 |.98 & ^ .96 .95 .94 .93 .92 .91 s'/s ^ ^ ^ X [^ / ^\ X \ ^ ^ s"/s ^ ^ \, \ ^ $1= <5o= .03 7T .01 7T \ V. ^*v .01 ,02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 T FIG. 5. Same as Fig. 3, except that 5i = 0.037T, 5 2 = O.Olir, and scale is changed With the quasi isochronous system of circuits, and under the conditions expressed in (94) z may be equated to r 2 as given in (95). In subsequent sections we shall designate the case in which (94) and (95) are fulfilled as the r-case. Another fact apparent from Tables I and II is that with in- creasing values of T, z approaches in each case a definite limit, and this definite limit in each case is seen to be u. 118 ELECTRIC OSCILLATIONS [CHAP. IX This result may also be established analytically, as follows : Transposing the first term of (74) to the right, and dividing the resulting equation by r 2 + z, we obtain , z 2 (4v - z) which reduces to u provided the last term is negligible and positive. In detail, let us replace z by u in the last term, obtaining z = -t* 1 - U l * , approximately. (96) But, since z + u is positive by (76), this can only be true provided u is less than r 2 , and since we have replaced z by -^u, we must require also that the last term in (96) be negligible. In symbols, we have If ,-'-u! _ ^ (106) 47r 2 1 4?r 2 4 j 4?r 2 then z = - r 2 (107) 120 ELECTRIC OSCILLATIONS [CHAP. IX To obtain the damping constants in this case, let us substitute (107) into (86), obtaining (1 - T 2 )(V + T 2 ) which substituted into (84) and (85) gives (108) 2(1 - r 2 ) , approximately. In the r-case of isochronous circuits, as specified in (106), equa- tions (108) and (109) give the damping constants in the coupled system. In the values marked "approximately" we have neglected a quantity twice as large as that specified as negligible in (106). The values of u and v are given in (71). Taking up next the undamped periods in this r-case and re- placing z by r 2 in (83), we obtain S' 2 = S 2 , and S" 2 = S 2 (1 - r 2 ) (110) These results may be inaccurate, since in (83) the radical involves the sum of z and r 2 and also involves z 2 . We can obtain a closer approximation by employing for z equation (93), giving + T 2 ) 4(u - r 2 ) Now adding to this 2 2 /4 = r 4 /4, approximately, we have z + r 2 + z 2 /4 = ^ " This inserted into (83) gives r 2 ) u r (111) (112) Equations (111) cwd (112) gra't>e the values of the undamped periods (squared) for the two isochronous circuits in the r-case, CHAP. 1X1 THE FREE OSCILLATION 121 as specified in (94), or (106). The values of u and v are given in (71). 119. r-Case, Continued. Limits Approached as r 2 Approaches Zero. In the preceding section we have given equations for the damping constants and undamped periods in what has been called the r-case, as specified 'by (94), or (106). Let us DOW suppose that r 2 is small enough to be neglected in (108), (109), (111), and (112); then these equations reduce to a' = a 2 , a" = a 1} S' = S" = S (113) as may be seen by making r 2 = 0, and replacing u and v by their values. The condition under which r 2 is sufficiently near zero to make (113) substantially correct, may be derived by examining (108). Expansion of the radical in (108), second form, gives u) 2() ') + U) ^ (H4) provided T . 2-- 2 / tt '- a ff (115) 4y + w (ai H- a 2 ) 2 v ' In making these reductions we have used the definitions of u and v given in (71), and have used also the definitions (52) of 61 and 5 2 with Si = $2 = S for this special case of isochronous circuits. The remaining step of reducing a' to a 2 and a" to ai, as given in (113), consists in substituting (114) into (108), and making r 2 negligible in comparison with 1. Equations (113) give the damping constant and undamped periods in the isochronous system, provided r 2 is negligible as specified in (115). 120. Summary of Results with the Quasi Isochronous System of Two Magnetically Coupled Circuits. Considering first the damping constants, and having reference to Figs. 2 and 4, it is seen that for small values of r 2 , as specified in (115), a' = a 2 , a" = a\. 122 ELECTRIC OSCILLATIONS [CHAP. IX Under this same condition of small r 2 , with however, a some- what larger possible value of r 2 , reference to the Tables I and II, and to the curves of Figs. 3 and 5, and to the analysis of the preceding section, shows that substantially ct/ o// o O=O = O. For larger values of r 2 , such as are specified in (99) and desig- nated the w-case, a' and a" are given by (101) ; namely ';". >;. a/ = liTT) and a " = 2Ti^r Referring to Tables I and II, and to Figs. 2 and 4, it is seen that this latter condition is attained for values of r greater than about twice the values of r at which the a' curve and the a" curve come nearest together to form a neck in the figures. For this same range of values of r, in which r is greater than twice the value at which the neck is formed by the a' and a" curves, S f and S" are given by (102) and (103), and in the special case of small values of u (that is, small values of (61 5 2 ) 2 /4?jr 2 ) these quantities are approximately given by (104), which is r, and S" = For values of T intermediate between those values that give the simplified expressions for damping constants and periods, the exact expressions involving z must be employed. CASE II. THE GENERAL CASE WITH NUMERICAL COEFFICIENTS 121. Statement. If we take the general case of two mag- netically coupled circuits, such as are shown in Fig. 1, and sup- pose that the two separate circuits, when each is standing alone have the undamped periods Si and $ 2 and the damping constants ai and a 2 , the equations (41) and (42) specify the values of the undamped periods that coexist in both of the circuits when they are coupled together with a coefficient of coupling T. The equations (45) and (46) give the damping factors in the two oscillations of the coupled system. Both of these pairs of equations involve a quantity z. The exact value of z is given by the cubic equation (53) which has coefficients A, B, and C defined in (54). If we know the coeffi- cient of coupling r, the decrements 61 and 6 2 of the original cir- cuits, and x, which is the ratio of S 2 to Si, we can compute z CHAP. IX] THE FREE OSCILLATION 123 from (53), and can then proceed to solve completely the problem of finding the periods and damping factors of the coupled system. Instead of using the cubic equation (53) for z, it is usually sufficiently accurate to use the values of z given by (69). The test of this point is specified in (66) . We shall now proceed to compute S', S", a', and a" for four different values of r 2 , and shall allow the ratio of *S 2 to Si to be varied by varying S 2 , while Si is kept constant. With this con- ,OJt3 1.3 FIG. 6. Circuits not isochronous. Values of correction factor ( z) for various values of 82/81 and for various values of T. dition, if di and a 2 are supposed to remain constant, 61, which is aiSi, will stay constant, but 5 2 , which is a 2 2 will vary. We shall therefore assign a fixed numerical value 0.3ir to 61, and shall as- sign a fixed value O.!TT to 5 2 at S 2 = Si. That is 61 = O.STT, and a 2 Si = O.ITT. 122. Computation of z in the General Case of Two Magnetic- ally Coupled Circuits with Given Values of Si, a 2 Si, and with Various Values of r 2 and Various Values of the Ratio of S 2 to Si. We shall take in our numerical illustration Si = 0.37T, aA = O.ITT, then 6 2 = O.ITTO; (113) 124 ELECTRIC OSCILLATIONS [CHAP. IX where, as in (63) . x = S 2 /Sj. (114) The coefficients A, B, and C of (54), in this numerical case, become A = 1.97z + B = 4r 2 + - 2 + 0.98z 2 - 1.94 x C = r 2 (o.01z 3 +^~ - 0.06z) (115) (116) (117) The quantity r 2 is given four values; namely, 0.1, 0.01, 0.025, and 0.001. The first computation consisted in determining z. For this purpose the reduced equation (82), or (69), has been sufficient for all values of the computation, except for two values that are indicated in the table, where it was found necessary to use the cubic (53) instead of the reduced equation. The results for z are given in Table III, and are plotted in the curves of Fig. 6. Table III. Computed Values of the Correction Factor z in the Special Case in Which 5i = O.STT, a 2 Si = O.ITT for the General Case with S 2 /Si =x, and with Four Different Values of r 2 x = S 2 /Si Values of z for r"- = 0.001 1-2 = 0.01 T 2 = 0.025 r* = 0.1 0.76923 0.000231 0.002113 0.004587 0.011006 0.83333 0.0003287 0.002892 0.006250 0.012031 0.90909 0.0006382 0.005264 0.009028 0.012031 0.95238 0.0008865 0.007436 0.010424 0.011313 0.96154 0.0009358 0.008066 0.010619 1 0.011104 0.97087 0.0009728 0.008687 0.010641 0.010858 0.98039 0.0010013 0.010108 0.010568 0.010592 0.99010 0.0009760 010165 1 0.010314 0.010302 1.00 0.0009991 0.009203 0.009950 0.009991 1.01 0.0009659 0.008009 0.009478 0.009648 1.02 0.0009160 0.007245 0.008945 0.009325 1.03 0.0008499 0.006410 0.008175 0.008978 1.04 0.0007794 0.006053 0.007792 0.008622 1.05 0.0007147 0.005206 0.007214 0.008403 1.10 0.0003955 0.002891 0.004750 0.006529 1.20 0.0001199 0.000997 0.001967 0.003683 1.30 0.0000419 0.000382 0.000816 0.001882 1 In computing these two values all the terms of the cubic equation (53) were used. CHAP. IX] THE FREE OSCILLATION 125 123. Computation of S' and S" in the General Case with Numerical Constants. Having computed the values of z recorded in Table III, we shall next make numerical computations of $' and S". For this purpose, we shall divide both sides of (41) and (42) by Si, and replace S%/Si by x, obtaining S' (118) Using the values of x, 2, and r 2 given in Table III, the values recorded in Tables IV, V, VI and VII in the columns marked S'/Si and S"/Si were obtained. These values are plotted in Figs. 7 to 10. For comparison, to show the effect of the damping constants in modifying the periods, there is recorded in parentheses after each value of S f /Si and S"/Si the value obtained by regarding z as zero. In Fig. 8 the dotted curve is a graph of values obtained by neglecting z, while the continuous line curve is the graph of true values with z considered. 124. Computation of a' and a" in the General Case with Numerical Constants. Continuing with the same set of special values, we have next computed the values of ratios expressing a' and a" in terms of known quantities. For the formulation of this problem, let us first examine the equation (47), which is used to determine the algebraic sig- s of certain damping constant equations to be employed. Dividing both sides of the inequality (47) by Si, and replacing Sz/Si by x, we obtain (QI& + o*Si)(l + x 2 + xz) ^ Q _ ^ 2 . - > OjSi + a z SiX 2 (120) Replacing aiSi by its special value O.STT, and a 2 Si by its special value O.!TT, we obtain 0.2(1 + x* + xz) > (l-r 2 )(0.3 + O.lz 2 ) (121) as the criterion for determining the signs in (48) and (49), which we are going to employ. If (121) is fulfilled, the signs in (48) 126 ELECTRIC OSCILLATIONS [CHAP. IX 1.4 1.3 1.2 02 .8 ys, S'/S 5 t - .37T Oa^-.lTT T 2 - .001 7s l ,9 10 1.1 1.2 1.3 .8 FIG. 7. Curves of ratios of undamped periods for circuits not isochronous, with 5i = O.STT, 0,281 = O.lir, T 2 = 0.001. 1.3 1.2 .9 X neglect * /! neglec ngz T2-.01 .neglecting^ .9 .1.0 1.1 X S.IK. 1.2 1.3 CHAP. IX] . THE FREE OSCILLATION 127 .8 .9 1.0 1.1 1.2 1.3 .7 FIG. 9. Curves of ratios of undamped periods for circuits not isochronous plotted against S/Si, forr 2 = 0.025. 1.4 1.3 1.2 1.1 .9 .8 S^-STT t z Si*=-.l 72 =.1 .8 .9 1.0 1.1 1.2 1.3 _ o / 128 ELECTRIC OSCILLATIONS [CHAP. IX and (49) are correct. We shall next examine (48) and (49) with a view to using them in the present numerical case. Multiplying (48) through by Si/ir, and replacing a^Si by O.Sir and a z Si by O.ITT, we obtain a'S l = 0.2 _ 1 / 0.16 _ 4(0.03) TT 1 - T 2 " 2 \ (1 - T 2 ) 2 1 - T 2 a?(l ~ T 2 ) = i^{i-vr^\ ' ' . where In like manner, from (49) we obtain In using these equations it is to be borne in mind that, if (121) is not fulfilled, the signs before the radicals in (122) and (123) are to be interchanged. 125. Criterion Values. Applying the criterion inequality (121) to the present numerical cases it is found that the signs given in (122) and (123) are correct for all values of x greater than a certain limiting value for each value of r 2 . These limit- ing values are as follows: r2 Limiting value of x. Signs in (122) and (123) are correct for x greater than 001 999 010 987 025 961 .100 805 Keeping these criterion values in mind, equations (122) and (123) were used in computation of the values of a'S\/v and a"Si/ir recorded in Tables IV to VII, and plotted in the curves of Figs. 11 to 14. 126. Examination of Results in the General Case with Nu- merical Constants. The results contained in Tables III to VII will now be examined. The given constants used in the computa- tion of these tables are 61 = O.Sr, a 2 $i = O.lx, while the coeffi- cient of coupling had four different values whose squares are r 2 = 0.001, r 2 = 0.01, r 2 = 0.025, r 2 = 0.1. CHAP. IX] THE FREE OSCILLATION 129 127. Examination of z. Table III contains values of z for various values of x ( = S z /Si), and for the four different values of r 2 . These results are plotted in Fig. 6. It will be seen that in each case z has a maximum. For the two smaller values of r 2 (i.e., for r 2 = 0.001 and r 2 = 0.01) the maximum value of z is approximately equal to r 2 , and this maximum value occurs at a value of x a little less than unity. For the two larger values of r 2 , the maximum value of z is much smaller than r 2 and occurs at a value of x considerably different from unity. 128. Examination of the Undamped Periods. The values of the undamped periods, in the form of their ratios to Si, are given in Tables IV to VII, for different values of x ( = S 2 /Si) and for the different values of r 2 . Each of the tables corresponds to a particular value of r 2 . In these tables the quantities in parentheses are the values that are obtained if we consider z to be zero, while the values not in parentheses are the values obtained by giving z its proper value, and taking account of its effect on the resultant periods. A comparison of the values not in parentheses with those in parentheses shows the amount of the error that would be made in this numerical case of rather large damping if z were entirely neglected. The effect of the z differs with the coefficient of coupling r and with the ratio x of the undamped periods of the original circuits. From Table IV, in which r 2 = 0.001, it is seen that the effect of z is inappreciable for large and for small value of x (that is, for values in which the original circuits are widely out of syn- chronism), but at x = 1 (i.e., with the circuits synchronous) the effect of z in this case is to modify the computed periods by about 1 per cent. From Table V, in which r 2 = 0.01, it is seen that at x = 0.98 the effect of z is to modify the computed values by about 4 per cent. In this case also, the effect of z is hardly appreciable for large and for small values of x. Table VI, with r 2 = 0.025, shows that the effect of z is to modify the computed periods by about 2 per cent, for x in the neighbor- hood of 1, with this effect decreasing toward the small values of x and almost inappreciable at the large values of x. Similarly, Table VII, for r 2 = 0.1, shows that the effect of z 130 ELECTRIC OSCILLATIONS [CHAP. IX Table IV. Computed Values Involving Damping Constants and Un- damped Periods in the General Case with Various Values of x = 82/81. Given <5i = O.STT, a 2 Si = O.ITT and r 2 = 0.001 Values in parentheses are values obtained by regarding z as negligible. x = S 2 /Si a'Si/r a"Si/ir S'/Si 8"/8i 0.76923 0.3000 0.1004 1.0007 (1.0009) 0.7683 (0.7682) 0.8333^ 0.2999 0.1005 1.0006 (1.0010) 0.8324 (0.8320) 0.90909 0.2980 0.1024 1.0007 (1.0022) 0.9080 (0.9066) 0.95238 0.2969 0.1035 1.0005 (1-0046) 0.9514 (0.9476) 0.96154 0.2966 0.1038 1.0001 (1.0053) 0.9609 (0.9559) 0.97087 0.2966 0.1039 1.0000 (1.0067) 0.9705 (0.9639) 0.98039 0.2966 0.1039 0.9992 (1.0083) 0.9807 (0.9718) 0.99010 0.2966 0.1038 1.0003 (1.0114) 0.9893 (0.9784) 1.00 0.1039 0.2965 0.9987 (1.0038) 0.9972 (0.9938) 1.01 0.1037 0.2967 1.0072 (1.0104) 1.0023 (0.9977) 1.02 0.1034 0.2970 1.0136 (1.0160) 1.0059 (1.0037) .03 0.1030 0.2974 1.0199 (1.0218) 1.0096 (1.0079) .04 0.1029 0.2978 1.0270 (1.0285) 1.0135 (1.0128) .05 0.1025 9.2981 1.0332 (1.0343) 1.0163 (1.0154) .10 0.1007 0.2997 1.1014 (1.1025) 0.9982 (0.9973) .20 0.0994 0.3010 1.2013 (1.2014) 0.9986 (0.9983) .30 0.0990 0.3014 1.3009 (1-3009) 0.9988 (0.9987) Table V. Same as Table IV, Except That r 2 = 0.01 x - S/ Si a'Si/r a"Si/7r S'/Si S"/Si 0.76923 0.2900 . 0.1141 1.0052 (1.0077) 0.7614 (0.7610) 0.83333 0.2857 0.1183 1.0071 (1.0105) 0.8233 (0.8205) 0.90909 0.2705 0.1336 1.0089 (1.0194) 0.8966 (0.8874) 0.95238 0.2532 0.1508 1.0087 (1.0294) 0.9395 (0.9205) 0.96154 0.2472 0.1569 1.0073 (1.0322) 0.9499 (0.9270) 0.97087 0.2404 0.1637 1.0065 (1.0307) 0.9599 (0.9327) 0.98039 0.2116 0.1925 0.9960 (1.0395) 0.9795 (0.9386) 0.99010 0.1923 0.2118 0.9925 (1.0439) 0.9925 (0.9437) .00 0.1672 0.2368 1.0121 (1.0489) 0.9831 (0.9487) .01 0.1520 0.2520 1.0260 (1.0544) 0.9794 (0.9532) .02 0.1443 0.2598 1.0363 (1.0604) 0.9793 (0.9572) .03 0.1370 0.2670 1.0470 (1.0667) 0.9788 (0.9607) .04 0.1340 0.2701 1.0559 (1.0716) 0.9801 (0.9658) 1.05 0.1279 0.2762 1.0666 (1.0808) 0.9796 (0.9666) 1.10 0.1134 0.2907 1.1152 (1.1212) 0.9815 (0.9762) 1.20 0.1037 0.3003 1.2094 (1.2109) 0.9879 (0.9867) 1.30 0.1006 0.3031 1.3086 (1.3091) 0.9884 (0.9881) CHAP. IX] THE FREE OSCILLATION 131 Table VI. Same as Table IV, Except That r 2 = 0.025 x = S*/Si o'Si/T o"Si/ir S'/Si S"/Si 0.76923 0.2766 0.1336 .0079 (1.0168) 0.7538 (0.7470) 0.83333 0.2653 0.1450 .0175 (1.0242) 0.8087 (0.7983) 0.90909 0.2386 0.1716 .0263 (1.0404) 0.8747 (0.8630) 0.95238 0.2142 0.1960 .0358 (1.0542) 0.9080 (0.8920) 0.96154 0.2051 0.2051 .0399 (1.0579) 0.9154 (0.8975) 0.97087 0.1969 0.2123 1.0422 (1.0619) 0.9198 (0.9028) 0.98039 0.1892 0.2210 1.0464 (1.0651) 0.9252 (0.9068) 0.99010 0.1802 0.2300 1.0512 (1.0711) 0.9293 (0.9128) .00 0.1719 0.2383 1.0573 (1.0762) 0.9339 (0.9176) .01 0.1640 0.2463 1.0639 (1.0818) 0.9375 (0.9219) .02 0.1570 0.2532 1.0708 (1.0876) 0.9405 (0.9260) .03 0.1488 0.2615 1.0787 (1.0937) 0.9429 (0.9298) .04 0.1449 0.2653 1.0861 (1.1014) 0.9457 (0.9346) .05 0.1399 0.2704 1.0941 (1.1069) 0.9477 (0.9367) 1.10 0.1222 0.2881 1.1362 (1.1437) 0.9559 (0.9465) 1.20 0.1070 0.3033 1.2265 (1.2291) 0.9662 (0.9642) 1.30 0.1019 0.3084 1 . 3208 (1 . 3217) 0.0719 (0.9713) Table VII. Same as Table IV, Except That r 2 =0.1 x = St/Si o'Si/x a"Si/7r S'/Si S"/Si 0.76923 0.2346 0.2098 1.0481 (1.0552) 0.6963 (0.6916) 0.83333 0.2139 0.2304 1.0570 (1.0732) 0.7502 (0.7262) 0.90909 0.1855 0.2589 1.0917 (1.1013) 0.7901 (0.7833) 0.95238 0.1688 0.2756 .1116 (1.1218) 0.8128 (0.8052) 0.96154 0.1655 0.2790 .1162 (1.1263) 0.8172 (0.8099) 0.97087 . 1620 0.2824 .1213 (1.1312) 0.8214 (0.8142; 0.98039 0.1586 0.2858 . 1266 (1 . 1363) 0.8256 (0.8186) 0.99010 0.1552 0.2892 .1322 (1.1417) 0.8297 (0.8227) 1.00 0.1519 0.2926 .1381 (1.1473) 0.8336 (0.8269) .01 . 1485 0.2960 1.1441 (1.1531) 0.8375 (0.8310) .02 0.1455 0.2990 1.1504 (1.1591) 0.8412 (0.8349) .03 0.1424 0.3020 1.1569 (1.1653) 0.8447 (0.8386) .04 0.1395 0.3049 1.1633 (1.1715) 0.8481 (0.8423) .05 0.1376 0.3068 1.1701 (1.1779) 0.8514 (0.8457) .10 0.1250 0.3195 1.2045 (1.2106) 0.8643 (0.8599) .20 0.1127 0.3317 1.2847 (1.2875) 0.8863 (0.8836) .30 0.1020 0.3424 1.3702 (1.3718) 0.9001 (0.8991) 132 ELECTRIC OSCILLATIONS [CHAP. IX on the periods is small for large values of x. The effect of z is about 1 per cent, on computed periods for values of x between about 0.95 and 1 .02. For small values of x the effect of z is smaller than in the neighborhood of # = 1, but is still considerable for the smallest value of x used in the computations. It will be interesting to compare the effect of z, which is the effect of the damping constants, on the resultant undamped periods S' and S", with the effect of the damping constants OQ the original periods T\ and T 2 of the circuits if not coupled. Let us note that for any oscillatory single circuit the undamped period and the free period have respectively the values S = 27T/12, T = 2W/W = 27r/Va 2 -a 2 (124) whence a 2 /a 2 = Vl - a 2 S 2 /47r 2 = = 1 6 2 /87r 2 , approximately Using the values of 61 pertaining to this numerical example , we have fr = 1 - 0.011, ^ i while if 61 were zero Si would be equal to T i} so that the effect of the damping in this circuit alone is to modify its period by about 1 per cent. For the other circuit with the decrement dz, which is smaller, the effect would be less. It appears, therefore, that in the coupled system, the effect of the decrements in modifying the periods is as much as four times as great as with a single circuit standing alone (compare Table V). Let us refer now to the curves of Figs. 7 to 10. In these curves S'/Si and S"/Si are plotted as ordinates and x ( = S z /Si) is plotted as abscissae. We have adhered to the convention that of the two quantities S' and /S", the greater shall x be designated S'. In Figs. 7 and 8, for r 2 = 0.001 and r 2 = 0.01 respectively, the curves consist of two lines that cross; and the upper part of each of these lines has been designated S'/Si and the lower part S"/Si to conform to the convention that S'>S". In Figs. 9 and 10, which are for r 2 = 0.025 and r 2 = 0.01 respectively, the two curves do not cross or touch, and the curves for S'/Si and S"/Si are widely separated. The curves in these cases of the CHAP. IX] THE FREE OSCILLATION 133 larger coefficients of coupling are very similar in character to the corresponding period curves in which the resistances were con- sidered to be zero, as in the dotted curves of Fig. 8. The values in the present cases, as given in Tables VI and VII, in which the decrements are rather large, differ by as much as 2 per cent, from the values obtained by neglecting the resistances. A criterion can be obtained theoretically that will determine in any particular case whether the curves of S r and S" meet, as in Figs. 7 and 8, or do not meet, as in Figs. 9 and 10, but this in- vestigation is here omitted. 129. Examination of Damping Constants. Tables IV to VII contain values of a' SI/IT and a' f Si/ir for various values of x ( = Sz/Si) and for four values of r 2 , as indicated in the headings to the tables. Here, as always, a' is the damping constant in the coupled system belonging to the undamped period S', which is the larger of the resultant undamped periods, and a" is the damping constant in the coupled system belonging to S", which is the smaller of the resultant undamped periods. Curves corresponding to these damping constants are plotted in Figs. 11 to 14, with x as abscissae, and with a! SI/IT and a" SI/IT as ordinates. In Fig. 11, which is for the case of r 2 = 0.001, it is seen that for a range of x extending nearly up to # = 1, a' SI/IT is approxi- mately equal to 0.3 (which is the value of aiSi/ir = di/ir 0.3 in this numerical case). The same quantity is approximately equal to 0.1 (that is, approximately equal to a^Si/ir) for a range of x extending from x = 1 on up to the largest value of x given. The curve of a"$i/r does the same thing over a reversed pair of ranges. We may express this result as follows: In this special case of r 2 = 0.001, we see that I.Ifx< 0.99, a' = i and a" = a^ approximately, II. Ifx > 1, a' = a 2 and a" = a 1} approximately, III. Between x = 0.99 and x = 1, a' and a" undergo transition, IV. At x = 0.995 the damping constants a' and a" are equal. These simple relations are incident to the looseness of the coupling in this case. The curves of Fig. 12 show that with the larger coefficient of 134 ELECTRIC OSCILLATIONS [CHAP, ix .30 .28 .26 - -20 .14 .12 .1 !=37r = ITT = 001 .9 1.0 1.1 . 1.2 1.3 FIG. 11. Ratios involving resultant damping factors for nonisochronous circuits plotted against 82/81 for 5i = O.STT, aaSi = O.ITT, r 2 = 0.001. .8 .9 1.0 1.1 1.2 i.3 FIG. 12. Same as Fig. 11, except thatr 2 = 0.01. CHAP. I .32 .30 .28 .26 94 X] THE FREE OSCILLATION *~* ^ '3/ 7T ^ ^a 'SJ / / \ / .22 .20 .18 .16 .14 .12 .10 \ / ,=, 3T \ z d: Si= T Z- 1 7T 0^ / \ / / "T \ / ^r \ ^a 'Sjr \ ^^ ^ a' 8j/f 8 .9 1.0 1.1 1.2 1.3 X =-82/81 135 % 1 CO "8 FIG. 13. Same as Fig. 11, except that r 2 = 0.025. .34 .32 .30 .28 .26 .24 s .22 .18 .16 .14 .12 .10 ^^ a ^^ 'SJ* ***' x ^ / f / 3,= =.37T / =.1 \ / X \ \ \ \ S s^ ^v ^^ a '5; ITT ' ^ ^^^ .8 .9 1.0 1.1 1.2 X=S 2 /S l FIG. 14. Same Fig. 11, except that r 2 = 0.1. 1.3 136 ELECTRIC OSCILLATIONS [CHAP. IX coupling (with r 2 = 0.01) the region of transition is spread out so that it embraces practically the whole plotted range of x. The intersection point in this case is at about x = 0.987. With still larger coefficients of coupling (with r 2 = 0.025 and r 2 = 0.1), the curves of Figs. 13 and 14 show that the inter- secting points are still further shifted toward the smaller values of x. These points appear at x = 0.961 and x = 0.805 respectively, as has been previously determined. HI. THE LOOSE COUPLED SYSTEM 130. Determination of the Oscillation Constants When r = 0. We can best obtain the result in this case by letting r 2 = in the original fourth degree equation (10), which then factors into k 2 + 2ajc + ft! 2 = 0, or k 2 + 2a 2 k + Oa 2 = 0, whence replacing either ft 2 by a 2 + co 2 , we have k 2 + 2aik + ai 2 = -co! 2 , or k 2 + 2a 2 k + a 2 2 = -co 2 2 . Extracting the square roots we have k = ai jcoi, or k = a 2 jco 2 . From the definitions of k given in (20) we see that a' = i, or a 2 , a" = a 2 , or ai (126) co' = coi, or co 2 , co" =co 2 , or cox (127) By a combination of these equations we obtain also S' = Si, or S 2 , S" = S 2 , or Si (128) The ambiguity of these values can be removed by use of the convention that S' is greater than S", so that while if Si > S 2 , a! = ai, co' = coi, S' = Si Cff ^ ^2i W " CO 2* O ~*" O9 if Si < S 2) a' = a 2 , co' = co 2 , /S' = S 2 a" = 01, co" = coi, S" = >Si (130) CHAP. IX] THE FREE OSCILLATION 137 For convenience we may also here write the values of the periods and wavelengths, as follows: if St > S 2 , T' = Ti, T" = T z , X' = Xi, X" = X 2 (131) while if Si < S 2 , T' = T 2 , T" = Ti, V = X 2 , X" = Xi (132) Equations (129) to (132) give the values of the oscillation con- stants of the coupled system in terms of the values of the constants of the system not coupled, provided r 2 is effectually zero. The next Chapter will treat of the amplitudes in the system of two magnetically coupled circuits. CHAPTER X AMPLITUDE AND MEAN SQUARE CURRENT IN THE INDUCTIVELY COUPLED SYSTEM OF TWO CIRCUITS 131. Continuation of Preceding Chapter. In the preceding chapter the discussion was confined mainly to the periods and damping constants of the coupled system of two circuits related as shown in Fig. 1 of Chapter IX. However, in equations (15) and (16) we have given the general expressions for the currents i\ and i 2 in the two circuits respec- tively in the form knt where n = 1, 2, 3, 4 CD (2) We have found that the four &'s, ki, k 2 , k$ and & 4 , are the four roots of a fourth degree algebraic equation, (19) Chapter IX, and that these roots may be written as two pairs of conjugate imaginary quantities as follows : fci = -a' + j', k 3 = -a" + jf", k z = a' n We may now perform useful eliminations among equations of the type of (6) and (7) as follows: If we take the sum of the four equations comprised in (6) when n is given respectively the values 1, 2, 3, and 4, we obtain (8) n whence, by (5), + fliOi + ^ S^ 2 = (9) and by transposition 2^1 - - Ci,Qi (10) A similar treatment of equation (7) gives 2l - (11) 140 ELECTRIC OSCILLATIONS [CHAP. X If we next take equation (6), and divide it by k n , we shall have four equations, corresponding respectively to n = 1, 2, 3, and 4. If now we add these four equations, we obtain which in v*?w of (5) and (10) reduces to LiQi CiRi 2 Qi + 7^ 2V4 = 0. This, by transposition, gives A -/!. n *-i 2 . As an approximation to the value of J", let us first neglect all of the squares of the a's and the product of two a's in comparison with the squares of the co's or in comparison with the product of two co's, except where there appears differences of the squares of the o>'s. The term marked (1), within the brace of (30), is of the order o> 2 /2a. The coefficients, within the brace, of the trigonometric quantities that occur in the other terms have order as follows: Table of Order of Coefficients of the Trigonometric Quantities in Various Terms of (30) Term No. Order Order relative to the order of term (1) (2) co/4 a/2co (3) CO/4 a/2co (4) wV3a 1 (5) CO/2 a/co We shall next examine the trigonometric quantities by which these coefficients are to be multiplied. Let us call the trigonometric quantities in terms (2), (3), (4), and (5) respectively, F Z) F 3 , F 4 , and F 5 . In these trigonometric quantities we shall expand the an ti tangents of 'those quantities known to be large (that is, of the order of w/o) by the well known formula tan-'x = |-^+^-. . . . , where z*> 1 (31) and shall neglect terms of the order of a 3 /o> 3 in comparison with o/co. 1 In the extreme case in which (to' a*") 2 happens to be negligible in comparison with 2(o' -f- a") 2 - To cover all contingencies we estimate its order as large as it can ever be. CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 145 This gives . r o . co' - co" 3a' 2(a" - a')l F 2 = sin 2tan~ 1 , -- , -- 7 -- V~i rr ' a" a' co' co' + co" J . co' - co" 3a" . 2(a" - a' -i - 2tan r2a' 2a" , 2(a" - a') F 3 = sin I ria' 2a" , V(a" - a') co' - co" F 4 = cos 77-+ , , 77^ tan- 1 .. ^ Leo' co" co' + co" (a" + a')J . r . co' - co" , a' + a" 2a' 2a"-| F 5 = sin 2 tan" 1 -77 7 H . 77 7- 77- L a" a' co' + co" co' co" J Let us now continue our omission of squares or higher powers of a/co in comparison with unity, and expand the above expres- sions, with replacement of sin (a/co) by a/co and cos (a/co) by unity. This process gives F 2 = sinhtan-i'^l - [' + ^f^Plcos a" a'J L co' co' + co" J r2a' 2a" , 2(o" - a')1 L ' - " 1 , TT + i i ^ Sln tan" 1 . , TT L w' w" ' + a" J L (a " + o') J - sin[2tan- 1 " ~ " ,1 + L a OL J r2a' . 2a" a' + a"l T ^ ,co' - o"l 7- H 77- 7 ; T> cos 2 tan- 1 -77 > L co co co + J L a a \ The cosine terms in the expressions for F 2 , F 3 , and F 5 have as multipliers quantities of the order of a/co, and since the coeffi- cients by which these F's are to be multiplied in forming / (see Table of Coefficients) are of the relative order of a/co, these cosine terms will be neglected leaving to' co " F 6 = F 3 = F 2 = sin (2 tan- 1 t >) a a 2(a" - a') (a/ - co") " (a" - a') 2 + (co' - co") 2 10 146 ELECTRIC OSCILLATIONS [CHAP. X The remaining F, F 4 , written out by the formulas for the sine and cosine of an antitangent, gives 2a' 2a" 2(a" - fl ' -- ' " " _ \/(" + a') 2 + (' - w") 2 ((/-o/')[2(a a i tt "~ / / I //\ 1 (33) V(a" + a') 2 + (co' - co") 2 If now we introduce these several results into (30) and at the same time replace the O's by co's, we obtain Term No. '-*{+ (1) (g/-")Ua" - a') 2 + ("' - ") 2 J - ' 2 - + a 1 ) - 2(a (a" + a') 2 + (co' - co") : where (3) (34) J = (i&t, Jo ~ (co' + co") V(a ;/ - a') 2 + (co' - co") 2 Equation (34) /or J {/^es /&e integral of the square of the secondary current for a complete discharge under the condition that the square of each of the damping constants a is negligible in comparison with the square of the angular velocities co. No other approximation has been made. The result is in terms of the damping constants and angular velocities of the coupled system. . 139. Value of the Integral of the Square of the Secondary Current for Two Circuits of Small Damping, Nearly in Resonance and Very Loosely Coupled. Under the conditions given in this caption, the expression for the time integral of the square of the current in the secondary circuit reduces to a simple form. Assumptions are to be made as follows : Assumption I. The damping constants are supposed to be so small that their squares are negligible in comparison with the squares of the angular velocities. This assumption is fulfilled CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 147 by circuits even when the damping constants are large enough to cut the amplitude of current to one-half in one oscillation. The introduction of this assumption permits the use of equation (34) for J. Assumption II. The coefficient of coupling is supposed to be so small that we may with close approximation take a! = 0,1, a" 0,2, u' = o>i, w" = a> 2 o! = 0,2, a" = fli, a/ = a>2, to r/ = fc>i (36) as in equations (129) and (130), Chapter IX. Assumption III. The two circuits are assumed to be nearly in resonance so that co 2 is nearly equal to coi, and, except in difference terms we shall replace coi 2 , co 2 2 and coico 2 by a common quantity co 2 . Also we shall assume C0 2 - CO! < < 2CO (37) Referring now to .equation (34) it is seen that these assumptions make the term marked "Term No. (2) " negligible in comparison with the term No. (1), since the quantity in the square bracket in No. (2) cannot be greater than J^. Also it is seen that in Term No. (3) the term in the numerator subtracted from co'co" (a" + a') is negligible. In the remaining terms, making the substitutions called for in Assumption II, we obtain from (34) the following simplified approximate value of J; j = H^ { 1+1 4 (' + ") j (38) 4 ai a 2 (Q,\ -f- & 2 ) -\- (coi co 2 ) where H - - / (39) - ai) 2 + (i - co 2 ) 2 Equation (38) reduces to ff 2 co 2 (a! + a 8 ) (PI ~ Substituting for H 2 its value from (39), we obtain (ai + a 2 ) 2 -f (i - co 2 ) 2 Equation (41) grapes ^e z;a/we o/ J (which is defined as J = i i Jo 148 ELECTRIC OSCILLATIONS [CHAP. X in the case of a secondary circuit very loosely coupled to a primary circuit, when the condenser in the primary is charged with a quantity of electricity Qi and allowed to discharge. The two circuits are supposed to have damping constants whose squares are negligible in comparison with the squares of the angular velocities, and the circuits are supposed to be not more than 5 or 10 per cent, out of resonance. The next section shows a method of using (41) to obtain the decre- ment of an unknown circuit. 140. Determination of the Decrement of an Unknown Circuit by Measuring the Integral Square Current in a Secondary Circuit Loosely Coupled with the Unknown Circuit. One of the usual methods of measuring the logarithmic decrement d\ of an oscillatory circuit is repeatedly to charge and discharge the con- Wave Meter X II Fio. 1. For determining decrement of circuit I. Circuit II is a wavemeter with variable condenser 2, and a current-measuring instrument at A. denser C\ (Fig. 1), or inductance LI of the given circuit, and to make wavelength measurements and integral square current measurements in a loosely coupled standard secondary circuit (II) of small decrement d 2 . The standard circuit is usually a wavemeter, or, if calibrated to read directly in decrements, a decremeter. The approximate formulas for obtaining decrements by this method are derivable from (41). If we call the value of J when w 2 = wi the resonant value of J, indicated by J r , we have from (41) ,i Jr ~ + a,j CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 149 By dividing (41) by (42), we obtain J r (di ~h dz) 2 ~r~ (&>i co 2 ) 2 (43) 1 \">l I 2) 2 Let us now recall that 2irai and at the same frequency j rf 2 i also = Xr/X, where X r = the wavelength setting of the wavemeter at resonance, X = its wavelength setting for the reading /. In terms of these quantities equation (43) becomes 1 y = whence i+^-^J!,.^}- (44) in which that sign before the radical is to be taken that makes di + d 2 positive. A simple way of applying the formula is as follows: Plot a resonance curve of J against X, as is illustrated in Fig. 2. Then if we take the two values of X (X and X& say) that give the same value of J, and call X a - X 6 = AX (45) we shall have from (44) j i j i di + d z = + and 150 ELECTRIC OSCILLATIONS whence by addition and division by 2, [CHAP. X \ r + d 2 = and since \ a \b = X r 2 approximately, we may write TrAX / J \ r \ J r - J Ii + d 2 = (46) That is, to obtain di -f d z , we take the width AX of the resonance curve, Fig. 2, in meters wavelength at any height J, divided by the resonant wavelength X r in meter and multiply by TT and by the square root of J/(J r J). Jr \b \a FIG. 2. Illustrating equation (46). This formula is particularly easy to apply at the point where J = Jr/2, for the formula then becomes + d 2 = X, (47) where AX^ = the difference of the two waveleng-ths that give J one-half of its resonance value. EXCITATION BY DISCHARGING THE PRIMARY INDUCTANCE 141. Initial Conditions When the Current is Produced by the Discharge of an Inductance in the Primary Circuit. As has been pointed out in Chapter II, it is the practice in many CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 151 electrical measurements and in some small transmitting stations to excite the current oscillations by isolating a current in the primary inductance and allowing the current to subside. We have referred to this method of excitation as excitation by the discharge of an inductance. The discharge of the inductance is effected in practice by the use of an electromagnetically driven interrupter as shown at J in Fig. 3, where is illustrated a coupled system operated in this way A current from the battery B is sent through the inductance LI, and when this current has a certain value /i, which is prac- tically steady, the feed current is opened at J. We have then the initial conditions when t = 0, f i = I lt i t = q\ = C \R\Ii, q<2. = Primary (48) Ih Secondary u FIG. 3. These conditions, so far as they pertain to a single circuit are discussed in Chapter II. With these initial conditions, we are now to determine the values of A n and B n in the equations ii = SA n e*"<, i, = 2B n e knt (49) By integration of (49) we obtain = 2, (50) (51) Now introducing the initial conditions (48) we obtain A = /i, S n = 2A n /k n = - CiRJt, VB n /k n = 142. Manipulation of the Initial Conditions. To obtain further relations concerning A and B, we shall make use of the equations (6) and (7). If in (7) we make n successively 1, 2, 3, 4, we obtain four equations, which added together give f 5 + a-'z'B - MSA,,. rC n C 2 /Cn 152 ELECTRIC OSCILLATIONS [CHAP. X This equation, by (51), reduces to ! . (52) which is a new equation in terms of B^. Let us now take equation (6), multiply each term by k n , and sum up for the four &'s; and let us perform a similar operation on (7). These two operations yield L^A n k n + R^An + ~ 2 ^ Oi K n and By (51) these two equations reduce to LiSAnfc. = M2B n kn, L^Bnkn = M2A n k n (53) Solving the two equations of (53) as simultaneous, we obtain 2B n kn = 0, SAJfe n = (54) Collecting results, so far as concerns B, we have ZBnkn = 0, SB n = 0, 2Bn/k n = 0, 2B n /k n 2 = MCJi (55) It will not be necessarj^ to go through the detail of solving these four simultaneous equations, as we can obtain the result by a direct comparison of these equations (55) with the corres- ponding equations (14) obtained with the condenser-discharge method of excitation. If in (55) we let BI = Yi/ki, B 2 = F 2 //c 2 , etc., equations (55) in terms of Y n will be of the same form as (14), with only the Qi of (14) replaced by /i. It thus appears that if we substitute /i for Qi in the values of B n given in (17) to (20), and divide the result by k n , we shall get B n of the present problem. This gives j? MCJ^Skzkskj ~ (h - fa) (fa - fa) (fa - fa) The other quantities B 2 , B 3 , B* can be obtained from (56) by advancing the subscripts of the fc's. In order now to put our result into trigonometric form we may take the result (23) of the previous problem, multiply it by /i and divide it by Qifa, and, since CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 153 obtain for the present BI, Bl = _^^ >(--tan-^) (57) 2jco Qi A similar treatment of the other B's, and their combination to form izj gives ^ <-"< sin L"t + ^ - tan-' -^) } . (58) CO V a / 1 In this equation H, _/ i "A co'V ' 4co" co' l co" ,+tan-i-^-tan-i ( r + a") 2 + (co' - a/') 2 co 7 co" co' + co" \ H- v> 2 - tan- 1 , + tan" 1 ^ -tan- 1 , j + a") 2 + (co' + co") 2 " (59) 154 ELECTRIC OSCILLATIONS [CHAP. X This expression is exact. It gives the integral of the square of the secondary current of the coupled system excited by discharging the primary inductance originally traversed by a current I\. If, now, we neglect the squares of the damping constants in comparison with the squares of the angular velocities, this equation, by the employment of processes similar to those used in deriving (35), reduces to 1 1 J = l Qi 2 .I4o' ' 4o' (' - co") 2 in (2 sin 2 tan- fl " (60) H = _ _ (a/ + co") V(a" - a') 2 + (' - a/') 2 (a' + a ") 2 + (' - a/') 2 where Equation (60) gwes Z/ie integral of the square of the secondary current, in a coupled system, excited by discharging the primary inductance originally traversed by a current Ii, in case the squares of the damping constants are negligible in comparison with the squares of the angular velocities. No other approximation has been made. The Qi that occurs in (60) has no meaning, in this case, and is eliminated by the Qi occurring in H in (61). If next we assume the circuits very loosely coupled and assume that they do not depart from synchronism by more than a few per cent., and apply the assumptions and methods employed in deriving (41), we find ( , 16oia, (dj + 2 ) 2 + (i - o> 2 ) 2 where, as before J = Equation (62) gives the value of the time integral of the square of the secondary current in a coupled system excited by a discharge of the primary inductance originally traversed by a current /n. In obtaining this simplified result the squares of the damping con- slants have been neglected in comparison with the squares of the angular velocities, and the coefficient of coupling has been assumed CHAP. X] AMPLITUDE, MEAN SQUARE CURRENT 155 to be so small that the damping constants and angular velocities of the coupled systems are the same as these constants for the circuits uncoupled, as expressed in (36). Also the circuits as supposed to be near enough to synchronism to make (37) applicable. It is seen that the value of J divided by J r , which is the value of J at resonance, reduces approximately to the same value as with the condenser-discharge method of excitation (compare (41)), so that the method of decrement measurement illustrated in Fig. 1 and the text of Art. 140 applies also to the inductance method of excitation. CHAPTER XI THEORY OF TWO COUPLED CIRCUITS UNDER THE AC- TION OF AN IMPRESSED SINUSOIDAL ELECTRO- MOTIVE FORCE In the treatment of two coupled circuits the discussion up to the present has been confined to the free oscillation that takes place when the system is given a charge and is allowed to dis- charge. It is proposed now to treat the two circuits, when one of them has operating within it, or upon it, a sinusoidal electro- motive force. 1 FIG. 1. Two coupled circuits with impressed e.m.f. 144. Form of Circuit to Which the Analysis Applies. The form of circuit to which the analysis is to apply exactly is shown in Fig. 1, where the circuit I contains a condenser, an inductance and a resistance and a source of sinusoidal electromotive force, indicated at e. Coupled with the circuit I is a secondary circuit II, contain- ing also inductance, resistance, and capacity in series with one another. The constants of the circuits are Li, Ci, Ri for the primary, 1 This problem without condensers in the circuits was first treated by MAXWELL, Phil. Trans., 155, 1864. With condensers it was treated by BEDELL & CBEHOBE, Physical Review, 1, p. 117 and p. 177, 1893 and 2, p. 442, 1894. See also OBERBECK, Wied. Ann., 55, p. 623, 1895; and PIERCE, Proc. Am. Acad., 46, p. 291, 1911. 156 CHAP. XI] TWO CIRCUITS FORCED 157 and L 2 , C 2 , R% for the secondary circuit. M is the mutual in- ductance between the two circuits. 145. The Differential Equations. Let the e.m.f. impressed upon the primary be e = E cos ut = real part of E*? wt (1) Taking, now, the fall of potential around each of the circuits, and equating it to the impressed e.m.f., we obtain the following differential equations involving the currents in the two circuits: 1 * rfl+ L* + ^-f-**. (2) where in equation (2), for simplicity, we have replaced the actual impressed e.m.f., E cos ut, which is a real quantity, by a complex quantity Ed = #(S ut + j sin o>) (4) The result is that the solutions that we shall now obtain will give complex quantities for the values of ii and z' 2 . Of these complex values of i\ and i 2 , the real components will be the solu- tion of the given problem with E cos ut as the impressed e.m.f. 146. Nature of the Solution. The complete solution of the pair of equations (2) (3) is obtained by adding the particular integral to the complementary function. The Complementary Function in i\ and z" 2 is the general solu- tion of the system (2) (3) with the right-hand side of (2) replaced by zero. This we have obtained in Chapter IX in the form of (21) and (22), Art. 101. Such a solution for i\ and i z with the arbitrary constants undetermined is to be a part of the solu- tion of our present problem. The Particular Integral of the pair of equations is any pair of values of i\ and i z that will satisfy the simultaneous equations (2) and (3). 147. Determination of the Particular Integral. It appears that in order to meet the term involving the exponential in jut on the right-hand side of (2), we shall probably need such an ex- ponential in our value of ii and i 2 . Let us try setting (5) 158 ELECTRIC OSCILLATIONS [CHAP. XI where co is specifically the o> of the impressed e.m.f., and is not an unknown quantity to be obtained from the constants of the circuits as was the k in the exponentials in kt employed in Chapters VIII and IX. To see if the assumed solutions are correct, let us substitute (5) and (6) in (2) and (3), obtaining 4"' (7) + 3 L^ - - - JM*A = (8) In these equations let us designate the Reactances of the separate circuits by Xi and X 2 ; that is, let . and (10) It is seen that the exponential factors of (7) and (8) divide out ; and our assumed solutions prove to be correct provided (7) and (8) are satisfied. These reduce to . (#1 +JX3A jMwB = E. (11) and (fii + JXJB - jMuA = (12) and completely determine A and B } as we shall soon show. The complex quantities Ri -f- jXi and R z + JX* that occur in (11) and (12) and, for a given impressed frequency, are con- stants of the Circuits I and II, and are usually designated by a small z with proper subscript : __ p i AV (13") 22 = ^2+^2 (14) These quantities are called complex impedances. As further abbreviations it is customary to designate the magnitudes of z\ and 2 2 by capital Zi and Z 2 defined by The quantities Z and Z 2 are called impedances. CHAP. XI] TWO CIRCUITS FORCED 159 Returning now to the relations (11) and (12) between A and B, these equations in terms of Zi and z z become ziA jMuB = E, z z B - jMwA = 0, (17) E ' * (18) whence and JMA zz 1 - E 1 where, as an abbreviation, z'i ~'Vf ^^ (19) In terms of z'i, our equations (5) and (6) become (20) .(21) Since in (20) the quantity z'\ occurs as a divisor of the com- plex e.m.f. to give the complex current, we may call z'\ the apparent complex impedance of the primary circuit. We may anatyze z'\. into its real and imaginary parts by replacing z\ and z 2 by their values (13) and (14). Then (19) becomes z'i = Ri + jXi + - p , % . Rationalizing the second term, we obtain + ^ R 2 + j\X, - ^X z \ (22) (23) where 'i - i + ^R* (24) and X\ = X l - If we should replace /i, of equation (20) by its value as given 160 ELECTRIC OSCILLATIONS [CHAP. XI in (23), we should see that the current for the primary circuit would be the same as it would be if the secondary circuit were not present, provided the primary resistance were changed to R'i and the primary reactance to X\. These quantities R\ and X\ are called respectively the apparent resistance and apparent reactance of the primary circuit. It may be noted that the apparent resistance is greater than ihe true resistance; but, since X 2 may be positive, negative or zero depending on the relative values of L 2 co and Cw, the ap- parent reactance may be greater than, equal to, or less than, the true reactance of the primary circuit alone. If now we introduce a quantity called apparent impedance, indicated by Z'i, and defined by ' (26) and also introduce the abbreviation */! = tan- 1 ^ (27) /t i we may write (23) in the form -,' . _ y Jv'i /oo > \ z i Z ir (2&) We are going to use this equation in determining the real component of i\. In like manner, for the determination of i z , we may employ z 2 = Z 2 / tairl i 2 (29) and j = *' V2 (30) Substituting (28), (29) and (30) into (20) and (21), and taking the real part of the results, we obtain >'i) (31) cos (ut - , and X 2 , it be required to determine what adjustment of the Primary Reactance Xi is necessary in order to make the secondary current a maximum. That is, instead of adjusting the secondary reactance X 2 we are going to adjust the primary reactance Xi to give the maximum current amplitude in the secondary circuit. The result in this case can be obtained by inspection, for Z 2 does not involve X\. In the denominator of (33) only Z\ involves Xi, and we must choose Xi to make Z'i a minimum. By (34) it is seen that this is attained by making the expression in the last parenthesis in (34) zero; that is *V 71/f 2 2 - = -~^- (Partial Resonance Relation P). (37) A 2 Z 2 Equation (37) gives Partial Resonance Relation P, which determines the value that Xi musf have in order for the secondary current amplitude to be a maxium for the given fixed values of X 2 , M 2 co 2 and Z 2 . 164 ELECTRIC OSCILLATIONS [CHAP. XI 153. Note Regarding Effect of Resistances on Partial Reson- ance Relations P and S. In equation (36), Zi contains Ri as one of its terms, while in (37) Z 2 contains R% as one of its terms. The resistances do not enter otherwise in these two expressions. It is to be noted then that the resistance of the secondary circuit has no effect in determining the adjustment that must be given to the secondary reactance to make the secondary current a maximum; and the resistance of the primary circuit has no effect in determining the adjustment that must be given to the primary reactance to make the secondary current a maximum. 154. Secondary Current Under Partial Resonance Relation S. Let us obtain next the current amplitude in the secondary circuit when the secondary reactance is adjusted to the partial resonance relation S, as given in (36). To do this let us substitute the value of X 2 from (36) into (35) and extract the square root of (35) to get the denominator of (33). In making this substitution Z 2 2 of the right-hand side of (35) must be decomposed into R^ + X^, so that the X 2 2 may be replaced. When we have made this substitution we shall have imposed upon 7 2 the condition (resonance relation S) for a maxi- mum; therefore we shall write the resulting value of 7 2 as We obtain . ]s = which reduces to M<*E 1 2maxJs ~" " , ti<2.6\ H Equation (38) gives the current amplitude in the secondary circuit, when for fixed values of the other constants of the circuits, X* is set at the value to give a maximum secondary current amplitude. Expressed otherwise, (38) gives the amplitude of secondary current under partial resonance relation S. 155. Secondary Current Under Partial Resonance Relation P. In like manner, if we substitute (37) into (33) and designate the resulting value of 7 2 by [7 2max .] P , we obtain CHAP. XI] TWO CIRCUITS FORCED 165 Equation (39) gives the amplitude of secondary current under Partial Resonance Relation P; that is, under the condition that for fixed values of the other constants of the circuits, Xi zs set at the value to give maximum amplitude of secondary current. H. THE OPTIMUM RESONANCE RELATION 156. The Optimum Resonance Relation. For given values of certain constants of the coupled system we have found two different adjustments, one of the primary reactance, and the other of the secondary reactance, that would give a maximum amplitude of secondary current. In order to get the biggest possible current in the secondary circuit, it is apparent that we should, if possible, satisfy the Partial Resonance Relation S and the Partial Resonance Relation P both at the same time. It is somewhat more instructive to proceed by another method as follows : Equation (36) tells us what value we must give to the reactance Xz, of Circuit II, for a given X\ t Z\, E, M, and o>, in order to obtain a maximum amplitude of current in Circuit II. If now we take a different set of values of these constants Xi, Zi, we shall require a different value of X Z) and shall get a different maximum value of secondary current. We may now ask ourselves which of these several combinations of adjustments will give a maximum of the maxima of secondary current amplitude. To determine this let us suppose that X 2 is always automatic- ally given the value that satisfies resonance relation S, so that (38) is kept satisfied, even as we vary X\, and let us determine the value of Xi that under this condition will give a maximum Of [/2m.ls- This is attained mathematically by setting equal to zero the derivative of the denominator of (38) ; that is ~ JAo'fln'aZi , m - ~' Now by definition z, = so + AV = X l /Z 1 . 166 ELECTRIC OSCILLATIONS [CHAP. XI This put into the second form of (40), gives, after multiplica- tion by Zi, = Xifl 2 From this it follows that one or the other of the following equations 'is the condition of the required maximum of [/2max.]s; to wit: Either X l = O (I) Si = -z? We are now to decide which of these two conditions, (I) or (II), is correct for determining the required maximum of [/2 ma x.]s- Let us first replace Zi 2 by its value Xi 2 + Ri 2 , which, sub- stituted into (II), gives XS = f-W" 2 ~ flifli) (IF) (41) Hz Equation (II') is equivalent to (II). Let us examine two cases. Case I. Let M 2 C0 2 < flxfls. In this case the proper resonance relation is (I), for if M 2 co 2 is less than R\Rz, Condition (II') makes X\ imaginary and is there- fore unattainable. Case II. Let M 2 co 2 > RiR 2 . By substitution of Conditions (I) and (II) severally into (38) we find that Condition (I) reduces the denominator of (38) to = A (say); while Condition (II) reduces this denominator to 2M\/3]R; = B(say). Now B is seen to be less than A, because twice the product of any two real quantities is less than the sum of their squares. Hence in this case Condition (II) gives a larger amplitude of secondary current than does (I). If M 2 co 2 = R iR 2 , Conditions (I) and (II) reduce to the same condition as may be seen by comparing (II') with (I). It thus appears that under the limitations of Case I, Condition CHAP. XI] TWO CIRCUITS FORCED 167 (I) gives the largest attainable secondary current; and under the limitations of Case II, condition (II) gives the largest attainable secondary current; and, if Jf 2 co 2 = RiR 2 , Conditions (I) and (II) are both appropriate for giving the largest possible current, in the second a^ circuit. These results have been attained by supposing that, while seek- ing the optimum condition, we have kept (36) always satisfied ; so (36) must be fulfilled simultaneously with (I) when (I) is optimum and simultaneous with (II) when (II) is optimum. .Combining (36) with (I) and (II) in the two cases we have respectively the results following. If MW < RiR 2 (42) then X l = 0, and X 2 = (43) gives the largest attainable amplitude of secondary current. We shall call this system of equations the optimum resonance rela- tion at deficient coupling. On the other hand, if #ifl 2 (44) the combination of (II) with (36) gives V E> *7 2 Xi Hi /r as the condition for the largest attainable amplitude of secondary current. We shall call the system of equations (44) and (45) the optimum resonance relation at sufficient coupling. In the interest of completion of nomenclature, if MW = #ifl 2 (46) we shall call the coupling critical coupling. Either (43) or (45) is the optimum resonance relation at critical coupling, since both reduce to the form (43) as may be seen from (41). // (42) is fulfilled, (43) is the condition for maximum amplitude of secondary current. If, on the other hand (44) is fulfilled, (45) gives this condition. If (46) is fulfilled, (45) and (43) reduce to the same value. 157. Value of Max. Max. Secondary Current Amplitude at Deficient Coupling. The case of deficient coupling is the case in which < R^. (47) 168 ELECTRIC OSCILLATIONS [CHAP. XI Then the appropriate settings of the two circuits for the greatest possible amplitude of secondary current is the adjust- ment that makes Xi = = X 2 ', (48) that is, each circuit is separately adjusted so as to make its undamped period equal to the period of the impressed e.m.f. From (38) the current obtainable under these conditions is 2 max. max. (49) // the circuits are so loosely coupled that M 2 co 2 < R\R^ then for a max. max. secondary current, the circuits should be tuned to satisfy (48), and the current obtained at this adjustment is given by (49). The current is seen to decrease with decreasing M, for if we differentiate (49) with respect to M we obtain a negative quantity for all values of MV less than RiR 2 . 158. Value of Max. Max. Secondary Current at Sufficient Coupling. In this case #i# 2 . (50) The appropriate setting of the two circuits for the greatest possible secondary current in this case is given by equations (45) which are here rewritten Xz _ Rz _ MW V " E> *7 2 \ O1 / AI 111 "i As an alternative expression, it has been seen that the Condi- tion (II 7 ) of equation (41) was equivalent to the Condition (II), preceding (41), which combined with the first part of (40) gives and (52) Equations (52) are together equivalent to (51) provided both radicals in (52) are given the same sign. Now from the second part of (51) we obtain CHAP. XI] TWO CIRCUITS FORCED 169 If we substitute this quantity into (39) and call the resultant current amplitude 7 2 max . max., we have E 2 max. max: (53) // the circuits are so closely coupled as to satisfy the condition for sufficient coupling as defined by (50), then in order to obtain a max. max. secondary current, the circuits should be tuned to satisfy conditions (51), or the equivalent conditions (52), and the current obtained at this adjustment is given by (53). It is seen that in the case of sufficient coupling (that is, when MWJ$ RiR 2 ) the value of the secondary current obtained is independent oj the mutual inductance. 159. Optimum Resonance Relation Equivalent to Fulfillment of Partial Resonance Relations S and P Simultaneously. Before passing to a further consideration of max. max. current ampli- tudes it is interesting to note that the simultaneous fulfillment of Partial Resonance Relation S and Partial Resonance Relation P results in the Optimum Resonance Relation. The Partial Resonance Relation S given by (36) is X l > while the Partial Resonance P given by (37) is V *7 2 V ' A 2 Z- 2 * Taking the product and then the quotient of these two equa- tions, we obtain Z X Z 2 = M 2 co 2 (54) and ^2 2 _ ^2 2 ^2 2 + ^2 2 #2 2 /p.r\ V 2 V2 " V 2 I D2 "D2 V/ The last step in (55) is taken by the law of division in the theory of ratio and proportion. Taking the square root of the first and last members of (55) and combining with (S) we have the optimum resonance rela- tion (51), which is the case of sufficient coupling. Note, however, (S) and (P) are attainable simultaneously only provided (54) is attainable, but since by definitions of Zi and Z 2 , 170 ELECTRIC OSCILLATIONS [CHAP. XI hence, by (54) (S) and (P) are simultaneously attainable only provided This is not quite correct, because there is another way of satisfying (S) and (P) simultaneously without leading to (54), and that is by making X l = and X 2 = (56) so that alternative to the optimum resonance relation (55) we have (56) as a possible optimum resonance relation. By work done above, it was shown that (56) is the actual optimum reso- nance relations, provided . ^ We have thus shown that the Optimum Resonance Relation is Equivalent to the 'requirement that the Partial Resonance Relations P and S be fulfilled simultaneously. Instantaneous Value of Secondary Current and of Primary Current at Optimum Resonance. Sufficient Coupling. Under the conditions for optimum resonance for sufficient coupling the apparent resistance and the apparent reactance of the primary circuit, as given in (24) and (25), reduce to R\ = 2R l} X'i = (57) whence the angle v\ as defined in (27) reduces to This equation gives the value of the instantaneous current in the primary circuit at optimum resonance and sufficient coupling. In this equation E cos ut is the impressed e.m.f., and the result is Jor the steady state. Next, to determine the secondary instantaneous current, let us take (32), replace its amplitude by (53), and also make 0'i = 0, as in (58), obtaining max. max. (60) CHAP. XI] TWO CIRCUITS FORCED 171 This equation gives the value of the instantaneous current in the secondary circuit at optimum resonance with sufficient coupling, in a steady state, under the action of an e.m.f. E cos cat impressed upon the primary. POWER EXPENDITURE IN THE COUPLED CIRCUITS 160. Power Expended in the Primary and Secondary Cir- cuits in the Coupled System at Optimum Resonance for Suffi- cient Coupling. If we multiply the instantaneous e.m.f. E cos cot by the instantaneous primary current (59) at optimum reso- nance (sufficient coupling), we obtain for the instantaneous power Pi supplied to the primary circuit If we take the time average of this power, over an integral number of half-periods, or over a time that is long in comparison with a half-period, and indicate the average so obtained by PI, the average of the numerator becomes E 2 /2. This value is the mean square of e, which mean square we may indicate by E*, and obtain Pi = E*/2Ri (62) This is the average power-input into the system of circuits, at optimum resonance with sufficient coupling. Next, let us examine the power converted into heat or radiated as electric waves from the primary circuit. This is the square of the current times the resistance of the circuit. If we call this power [pi] R , we have - ** (63 ) of which the average, indicated by replacing p by capital P, is [Pi]* = E^/^R, (64) Equations (63) and (64) give respectively the instantaneous power and the average power converted into heat in the primary circuit or radiated from it as electric waves, at optimum resonance with sufficient coupling. The difference between the power-input and the power con- verted in the primary circuit is the power communicated to the 172 ELECTRIC OSCILLATIONS [CHAP. XI secondary circuit. By taking (63) from (61) and (64) from (62), this is seen to be E 2 cos 2 co* Pl2= -RT and (66) Equations (65) and (66) give respectively the instantaneous power and the average power communicated to the secondary cir- cuit at optimum resonance with sufficient coupling. These values are seen to be the same as the corresponding quantities converted in the primary into heat or radiated from it. Let us now as an independent operation calculate the power consumed in the resistance of the secondary circuit. This is obtained by multiplying the square of the instantaneous second- ary current (60) by the secondary resistance Rz f and gives of which the time average is P 2 =Y 2 /4:R 1 (68) These equations (67) and (68) give the instantaneous power and the average power converted into heat or radiation in the secondary circuit. It is seen that the average value is the same as the average value of power communicated to the secondary from the primary, and the same as the average power consumed in the primary. A comparison of the instantaneous values (67) and (65) shows that the conversion into heat is not in phase with the transfer from the primary to the secondary. This is not surprising for the power, for a part of the time, is stored in the condenser and inductance of the secondary circuit. As a general conclusion from this investigation into power the important result is obtained that, with M 2 co 2 greater than R\R^, if we adjust the two circuits to such values as to give a max. max. of secondary current, then one-half of all the power communicated to the system through the impressed e.m.f. is dissipated in the pri- mary circuit and one-half is dissipated in the secondary circuit. This adjustment is, therefore, not a very efficient one, in general, for communicating power to a coupled system and dissipating it in a secondary load. If on the other hand, our problem is the reception of electric waves from a distant station by means of a coupled system of CHAP. XI] TWO CIRCUITS FORCED 173 circuits and the. affecting of an instrument in the secondary circuit, which instrument ' responds more actively the larger the secondary current, this adjustment though not efficient may give the maximum of response in the receiving instrument. It is to be noted, however, that we have assumed a constant amplitude of impressed e.m.f., and if the radiation from the receiving antenna affects the resultant impressed e.m.f., a proper correction has to be applied. We shall. next discuss the conditions for maximum efficiency of transfer of power to the secondary circuit through the coupled system. 161. Condition for the Transfer of Power to the Secondary Circuit with Maximum Efficiency. We must now go back to our original current equations (31) and (32), unmodified by the introduction of any resonance relations, and form the expressions for the average power expended in the secondary resistance and the average power expended in the primary resistance. This is done by taking the square of the respective currents and multiplying by the respective resistances and averaging as to time. If we merely write the ratio of these average power values, we obtain Pi Z 2 2 flj It is seen that, for a fixed value of M, co, R^ and Ri, this ratio of the average power expended in the secondary to the average power expended in the primary is a maximum when X z , comprised in Zz, is zero. That is, X z = (70) Equation (70) is the condition for a maximum efficiency of the transfer of power to the secondary circuit. Putting (70) into (69), it is seen that at maximum efficiency P~ r> r> 1 JtliKz To obtain from this expression the efficiency at maximum efficiency it is only necessary to form from (71) the ratio Pz/ (Pi + Pz). This is done by taking the reciprocal of (71), adding unity to both sides, and again taking the reciprocal. This gives W (72) 174 ELECTRIC OSCILLATIONS [CHAP. XI Equation (72) gives the efficiency of the transfer of power from the impressed e.m.f. to the secondary circuit when the secondary circuit is adjusted for the maximum efficiency of such transfer. The efficiency of the transfer is independent of the primary adjustment. 162. Condition for the Transfer of Maximum Power to the Secondary Circuit, The Transfer Being Effected at Maximum Efficiency. -If we want to get the maximum transfer of power to the secondary circuit at maximum efficiency, we need merely put the condition for the maximum efficiency of transfer (namely, X 2 = 0) into the amplitude equation (33) for secondary current and then adjust X\ to make the square of this amplitude a maximum. Putting X 2 = into (33) we obtain '" (73) It is seen by inspection that to make this a maximum, we require Xi to be zero. We have then Xi = = X*, or L 1 C l = L 2 C 2 = l/o> 2 . (74) Equations (74) are the conditions for a maximum transfer of power at maximum efficiency from the e.m.f. to the secondary circuit. In this equation co is the angular velocity of impressed e.m.f. 163. Comparison of Secondary Current at Maximum Power and Maximum Efficiency with the Secondary Current at the Optimum Resonance Relation. The amplitude of the sec- ondary current at maximum secondary power and at maximum efficiency of transfer of power is obtained by inserting (74) into (73). This gives _ _ MuE -- This is the secondary current at maximum secondary power transferred at maximum efficiency from the source to the secondary. Let us compare with this the secondary current at optimum resonance, with coupling sufficient, which by (53) is E 1 2 max. max. ~ CHAP. XI] TWO CIRCUITS FORCED The combination of this equation with (75) gives max, eflf. 175 (76) I 2 max. max. 1 + R\l Table I contains calculated values of this ratio for different values of M 2 a> 2 /RiR 2 . Table I. Comparison of Secondary Current for Two Sets of Conditions JM>w* * ff 'max. /2 max. eff. RiRz /2 max. max 1 0.50 1.00 2 0.66 0.93 3 0.75 0.87 4 0.80 0.80 5 0.83 0.74 6 0.86 0.70 7 0.87 0.66 8 0.88 0.62 9 0.90 0.60 00 1.00 0.00 In the first column of Table 1 are arbitrary values of the ratio of M 2 co 2 to R\Rz. Consistent with these ratios, the second column gives the maximum attainable efficiency of the transfer of power to the secondary circuit from the source of e.m.f. This efficiency increases as the ratio in the first column increases. In the third column is the ratio of the amplitude of the secondary current obtainable at maximum efficiency to the amplitude attainable at the adjustment for maximum secondary current. It is seen that at 50 per cent, efficiency this ratio is unity, while with increasing efficiency this ratio decreases toward zero. CHAPTER XII RESONANCE RELATIONS IN RADIOTELEGRAPHIC RE- CEIVING STATIONS UNDER THE ACTION OF PERSISTENT INCIDENT WAVES 164. Use of Persistent Waves. Persistent, or sustained, waves have recently come into extensive use in radiotelegraphy and radiotelephony. With these persistent waves, which are emitted by the sending station while the sending key is depressed, tens of thousands of oscillations may arrive at the receiving station even during the production of a single dot of the tele- graphic code. This permits the establishment of practically FIG. 1. FIG. 2. FIG. 1. Inductively coupled radiotelegraphic receiving station with detector D in series in a secondary circuit. FIG. 2. Closed system approximately equivalent to Fig. 1. a steady state at the receiving station, so that the mathematical deductions of the preceding chapter may be applied directly to the radiotelegraphic circuits. 1 165. In Respect to Resonance the Antenna Circuit is Ap- proximately Equivalent to a Closed Circuit Consisting of a Localized Inductance, Capacity and Resistance. With a re- ceiving station of the type shown diagrammatically in Fig. 1, 1 This chapter is adapted from PIERCE, "Theory of Coupled Circuits, Under the Action of an Impressed Electromotive Force with Applications to Radiotelegraphy," Proc. Am. Acad., 46, p. 293, 1911. 176 CHAP. XII] RADIO RECEIVING STATIONS 177 certain theory and experiments, not here presented, show that in respect to resonance relations, the system is substantially equivalent to the system of Fig. 2, with antenna replaced by a suitable localized capacity, inductance and resistance. The e.m.f . impressed upon the antenna by the incoming waves may be simulated by a source e of e.m.f., Fig. 2, in series in the primary circuit. In the form of receiving circuit illustrated in Fig. 1, the detector D is in series in the secondary circuit, Circuit II, and this whole system goes over into the system of Fig. 2. In case the detector is of high resistance, it may be advanta- geous to take it out of Circuit II, and place it along with a con- denser C 3 on a branch in shunt to C 2 . This arrangement is shown in Fig. 1 of Chapter XV and is there treated. At pres- ent we shall suppose the receiving station to be of the type of Fig. 1, and to be equivalent to the simplified system given in Fig. 2. All that we have developed in the preceding chapter we shall now assume to apply approximately to Fig. 1, and shall de- scribe our results in terms of the radiotelegraphic circuits of this Fig. 1. It is to be borne in mind that what we shall say applies with greater accuracy to the simplified circuits of Fig. 2. I. PARTIAL RESONANCE RELATIONS S AND P 166. Transformation of Partial Resonance Relations S and P. If Circuit I and the mutual inductance of the system is kept constant and the reactance X 2 of the secondary circuit is used in tuning to obtain a maximum of amplitude of current in Circuit II, the setting required is said to satisfy Partial Resonance Relation S. This relation is given in the previous chapter by equation (36), which is here rewritten A/ 2 oj 2 X 2 = - frr Xi (Partial Resonance Relation S). (1) On the other hand, if Xi is used as the adjustable member while all of the other members of the system of circuits are kept constant, the condition for a maximum amplitude of secondary current (in Circuit II) has been called in the pre- vious chapter Partial Resonance Relations P. The equation for 12 178 ELECTRIC OSCILLATIONS [CHAP. XII this resonance relation is given as (37) of the preceding chapter, and is here rewritten M 2 o) 2 Xi = ~^-X 2 (Partial Resonance Relation P). (2) It is proposed now to transform these two resonance relations by replacing Xi, X z , Zi and Z 2 by their customary values. given respectively in (9), (10), (15) and (16) of Chapter XI, This operation gives 1 v^ // L 2 co TJ = - Wiw/ (Resonance Relation S) (3) and LIU 7J = - 5 - (Resonance Relation P). (4) We shall now change the form of these equations so that the result is expressed in terms of angular velocities, decrements, and the coefficient of coupling. For this purpose, let ft! 2 = 1/LiCi, 12 2 2 = 1/L 2 C 2 , r 2 = M 2 /L,L 2 (5) and let (6) (7) The quantities Oi and fi 2 as defined by (5) are quantities that have been extensively used in Chapters VI, IX, and X and have been designated Undamped Angular Velocities. The quantity T, called Coefficient of Coupling, has also been extensively used in the previous chapters. The quantities 171 and r/ 2 , defined by (6) and (7), are new, and are seen to be respectively I/TT times the loga'rithmic decre- ments of the two circuits per cycle of impressed e.m.f. Introducing these various abbreviations into (3) and (4) we may write these equations, after a transposition of terms, in the forms (Partial Resonance Relation S) CHAP. XII] RADIO RECEIVING STATIONS 179 and (Partial Resonance Relation P) For any given fixed values of the other quantities that occur in these equations, and for fixed amplitude of the impressed e.m.f., equation (8) gives the value that the ratio 12 2 /co must have in order to produce a maximum of amplitude of secondary current in a steady state. Likewise, for the other quantities fixed, equation (9) gives the value that the ratio fii/co must have in order to produce a maximum amplitude of secondary current in a steady state. 167. Transformation of Partial Resonance Relations S and P into Forms Involving Wavelengths. As most radiotelegraphic frequency measurements are made in terms of wavelengths, it is proposed to make certain obvious transformations to express equations (8) and (9) in terms of ratios of wavelengths. It will be remembered that the wavelength X corresponding to a period T, of angular velocity co, has been denned by the equa- tion X = cT = 27TC/CO (10) where c is the velocity of light in free space (in meters per second, if X is in meters and T in seconds). We have also used in previous chapters the idea of an Un- damped Wavelength of a circuit, which ordinarily differs but slightly from the free wavelength X of the circuit, in that the Undamped Wavelength, designated by a Greek Capital Lambda A, is defined as A = 27rc/a (11) The undamped wavelength A of a circuit is the wavelength that the circuit would have if its resistance were removed without changing the inductance and capacity of the circuit. Giving to equation (11) subscripts 1 and 2, and dividing it into (10) we have !/ = X/Ai, Q 2 / = X/A 2 (12) In terms of the ratios of wavelengths, equations (8) and (9) may be written A (Partial Resonance Relation S) 180 ELECTRIC OSCILLATIONS [CHAP. XII and (\2 \ / \ 2 \ f 1 A 2 2 / \ Ai 2 / 1 1 (Partial Resonance Relation P) In these equations \ is the wavelength of the impressed e.m.f., A! and A 2 are the undamped wavelengths of Circuits I and II respectively. In applying (13) AI alone is supposed to be varied in obtaining the maximum of amplitude of secondary current. In applying (14) A 2 alone is supposed to be varied in obtaining the maximum amplitude of secondary current. .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 .X 2 /Ai,orfiI/w 2 FIG. 3. Resonant values of X 2 /A| for various values of te/A^ 168. Examination of the Partial Resonance Relation S in a Numerical Case. We shall now take a numerical case in which r and rji 2 are given, and shall employ the Partial Resonance Relation S, in the form of equation (13), to determine the value of X 2 /A 2 that is required, for various values of X 2 /Ai, in order to produce a maximum of amplitude of secondary current. CHAP. XII] RADIO RECEIVING STATIONS 181 We shall take, in the example, r = 0.30, and shall give to rji 2 the four values 0, 0.001, 0.01, and 0.1. Computed numerical values are contained in Table I. Where the numbers are omitted near the middle of the table, the values of X 2 /A.2 2 are given as negative by the formula, and are therefore impossible of realization, because they would make A 2 imaginary. Table I. Resonant Values of (X/A 2 ) 2 for Various Values of (X/AO 2 . Given T = 0.30, and Given Four Different Values of T\ 1 2 Following Partial Resonance Relation S (X/Ai) = (X/A 2 ) 2 for m 2 = o 7712 = 0.001 7J1 2 = 0.01 7712 = 0.1 0.0 0.910 0.910 0.911 0.918 0.2 0.888 0.888 0.889 0.903 0.4 0.850 0.850 0.854 0.883 0.6 0.775 0.776 0.789 0.862 0.8 0.550 0.561 0.640 0.812 0.9 0.100 0.182 0.550 0.919 0.95 0.640 0.955 0.97 0.752 0.973 0.98 0.827 0.982 0.99 0.181 0.911 0.991 1.00 1.000 1.000 1.000 1.02 4.500 2.290 1.170 1.017 1.03 3.000 2.360 1.250 1.027 1.05 2.800 2.290 1.360 1.045 1.1 1.900 1.818 1.450 1.082 1.2 1.450 1.439 1.360 1.129 1.3 1.300 1.297 1.270 1.143 1.4 1.225 1.224 1.212 1.139 1.5 1 . 180 . 179 1.176 1.129 1.6 1.150 .150 1.146 1.117 1.8 1.112 .112 1.111 1.097 2.0 1.090 .090 1.089 1.082 2.5 1.060 .060 1.060 1.057 3.5 1.036 1.036 1.036 1.035 5.0 1.023 1.023 1.023 1.022 10.0 1.010 1.010 1.010 1.010 oo 1.000 1.000 1.000 1.000 The numerical results of Table I are plotted in the curves of Fig. 3, with (X/Ai) 2 as abscissae and (X/A 2 ) 2 as ordinates. For r/i 2 = 0, the curve is an equilateral hyperbola with axes 182 ELECTRIC OSCILLATIONS [CHAP. XII at 1,1, as may be deduced directly by making rji 2 = in (13). When ?? i 2 = 0.001, the curve practically coincides with the curve for r/i 2 = except in the interval of abscissae between 0.9 and 1.1, where it sweeps from the third quadrant up through the point 1,1 and joins with the part of the curve in the first quadrant. At the bottom of the figure between the abscissae 0.9 and 0.98 this curve for rji 2 = 0.001 has a gap in it. In this gap the com- .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 A 2 A 2 or w 2 /fi! 2 FIG. 4. Resonant values of Al/X 2 for various values of A X /X. puted values of the ordinates are negative, and the value of A 2 is hence imaginary in this region. The curves for ??i 2 = 0.01 and 0.1 fall into coincidence with the equilateral hyperbola for large and for small values of abscis- sae; but in the neighborhood of the abscissa at 1 they cross over from the first to the third quadrant. The greater the value of 77 1 2 the greater the departure of the curve from the equilateral hyperbola. The whole course of these curves resembles the corresponding CHAP. XII] RADIO RECEIVING STATIONS 183 curves in optics, obtained when the index of refraction is plotted against frequency, in the neighborhood of an absorp- tion band. 169. Plot of the Partial Resonance Relation S in the Numer- ical Case in the Reciprocal Form. It is deemed worth while to plot the values of the reciprocals of Table I. This will give the curves in the form of (A 2 /X) 2 versus (Ai/X) 2 . These reciprocals are recorded in Table II, and are plotted in the curves of Fig. 4. Table II was obtained by taking the reciprocals of all of the numbers within the columns of Table I. Table II. Reciprocals of Numbers in Table I (Ai/X) (A 2 /X)2 for rji 2 = 7ji2 = 0.001 iji* = 0.01 771 2 = 0.1 GO 1.099 1.099 1.098 1.089 5.0 1.126 1.126 1.125 1.107 2.5 1.176 1.176 1.171 1.133 1.67 1.290 1.289 1.267 1.159 1.25 1.818 1.782 1.563 1.232 1.11 10.000 5.495 1.818 1.088 1.05 1.562 1.047 1.03 1.330 1.028 1.02 1.209 1.018 1.01 5.525 1.098 1.009 1.00 1.000 1.000 1.000 0.98 0.222 0.437 0.855 0.983 0.97 0.333 0.424 0.800 0.973 0.95 0.357 0.437 0.735 0.957 0.909 0.526 0.550 0.690 0.924 0.833 0.690 0.695 0.735 0.886 0.769 0.769 0.771 0.787 0.875 0.714 0.816 0.817 0.825 0.878 0.666 0.847 0.848 0.850 0.886 0.625 0.870 0.870 0.873 0.895 0.555 0.899 0.899 0.900 0.912 0.500 0.917 0.917 0.918 0.924 0.400 0.943 0.943 0.943 0.946 0.286 0.965 0.965 0.965 0.966 0.200 0.978 0.978 0.978 0.978 0.100 0.990 0.990 0.990 0.990 0.000 1.000 1.000 1.000 1.000 184 ELECTRIC OSCILLATIONS [CHAP. XII By reference to Fig. 4, it is seen that in terms of the coordi- nates of Fig. 4, the curves have lost their symmetry, with the exception of the curve with 77 x 2 = 0, and this has shifted its as- symp totes. The equation for this case of 77 1 2 = may be ob- tained directly as follows: If the damping term of (13) is negligible, the equation becomes (1 -X 2 /Ai 2 )(l -X 2 /A 2 2 ) = r 2 (15) Performing the indicated multiplications, then multiplying both sides of (15) by Ai 2 A 2 2 A 4 , adding 1/(1 -r 2 ) 2 , and again factoring, we obtain l ' 1 A / Ag2 1 \ T * n *n\ -- f^j (-$* -- r 7,; = (i^p This is seen to be an equilateral hyperbola with asymptotes at (Ai/X) 2 = 1/(1 - r 2 ) and (A 2 /X) 2 = 1/(1 - r 2 ) (16) Equation (15) or the alternative equation (15a) ^s a statemen of the Partial Resonance Relation S in the special case in which ?7i 2 is negligible. Equation (16) is the equation to the asymptotes to the hyperbola (15). 170. Note on the Partial Resonance Relation P. We have given in Tables I and II numerical calculations of the partial resonance relation S, and have plotted the results in the curves of Figs. 3 and 4. We shall not here present the corresponding results for the partial resonance relation P, since by the sym- metry of equations (13) and (14) it will be evident that the tables and curves will remain as they are except that the sub- script 1 will be replaced by 2 and the subscript 2 will be re- placed by 1, in order to change the results into values required by the resonance relation P. 171. Effect of Coefficient of Coupling T on Partial Resonance Relation S in the Case of 17 1 2 = 0. If the resistance of the primary circuit be so small that rji 2 is essentially zero, the partial resonance relation S takes the form of equation (15), which is the equation of an equilateral hyperbola in terms of (X/A 2 ) 2 versus (X/Ai) 2 , with asymptotes at (X/A 2 ) 2 = 1 = (X/AO 2 (17) CHAP. XII] RADIO RECEIVING STATIONS 185 A series of such curves computed for different values of r 2 are plotted in Fig. 5. The computed values are contained in Table III. 1.0 1.2 1.4 1.6 1.8 2.0 FIG. 5. Showing relation of A2 2 to Ai 2 for resonance relation S with various values of r 2 , and with 771 = 0. As may be seen from the equation (15) and from the numerical results, as r 2 is made smaller and smaller, the equilateral hyperbola approaches the asymptotes, and in case r 2 = 0, the hyperbola becomes two straight lines coincident with the asymptotes. 186 ELECTRIC OSCILLATIONS [CHAP. XII Table III. Resonant Values of (X/A 2 ) 2 for Various Values of (X/Ai) 2 and Various Values of T 2 , According to Partial Resonance Relation S. Given 171 = fX/Ai}2 (X/A 2 ) 2 for kA/**J r 2 = 0.001 r 2 = 0.005 7-2 = 0.01 T* = 0.05 T* = 0.09 0.0 0.999 0.995 0.99 0.95 0.91 0.2 0.999 0.994 0.99 0.93 0.88 0.4 0.998 0.992 0.98 0.92 0.85 0.6 0.998 0.985 0.97 0.88 0.78 0.8 0.995 0.978 0.95 0.75 0.55 0.9 0.990 . 0.950 0.90 0.50 0.10 0.95 0.980 0.900 0.80 0.00 0.97 0.967 0.835 0.67 0.98 0.950 0.750 0.50 0.99 0.900 1.00 1.02 1.050 1.250 .50 3.50 4.50 1.03 1.033 1.165 .33 2.66 3.00 1.05 1.020 1.100 .20 2.00 2.80 1.1 1.010 1.050 .10 1.50 .90 1.2 1.005 1.025 .05 1.25 .45 1.3 1.003 .016 .03 1.17 .30 1.4 1.002 .012 .02 1.13 .23 1.5 1.002 .010 .02 1.10 .18 1.6 1.002 .008 .016 1.08 .15 1.8 1.001 .006 .012 1.06 .11 2.0 1.001 .005 1.010 1.05 .09 2.5 1.001 .003 1.006 1.03 .06 3.5 1.000 .002 1.004 1.02 1.04 5.0 1.000 .001 1.002 1.01 1.02 10.0 1.000 .000 1.001 1.005 1.01 00 1.000 .000 1.000 1.000 1.00 II. OPTIMUM RESONANCE RELATION AS SUFFICIENT COUPLING 172. Case of Sufficient Coupling. Equations for Optimum Resonance in Terms of Angular Velocities. Let us next examine what we have called in the preceding chapter the optimum re- sonance relation, which is the condition for a maximum maximum of secondary current in the steady state under the action of an impressed sinusoidal e.m.f. The coupling is called sufficient coupling whenever the mutual inductance between the two circuits is large enough to make CHAP. XII] RADIO RECEIVING STATIONS 187 The equations for the optimum resonance relation under this condition has been given in suitable form in equations (52) of the preceding chapter. If in these equations we replace Xi and X 2 by their customary values, and if further we introduce the subscript "opt" to designate the optimum relation, we have >) and (18) opt. where we must use the same sign in both equations to obtain a consistent simultaneous pair of values. This follows from the fact given in equation (51), Chapter XI, that the ratio of X 2 to Xi must be positive. If now we divide both sides of (18) by LIO>, or L 2 o>, as required, and use the abbreviations given in (5), (6) and (7), we obtain , opt. 1 (20) opt. These equations give the optimum values of the undamped angular velocities QI and fl 2 relative to the incident angular velocity co. These optimum values are values that produce a maximum maximum secondary current amplitude. The equations apply to the case of sufficient coupling, for which MW > #i# 2 , i.e., r 2 > 171172 (21) 173. The Optimum Resonance Relation in Terms of Wave- lengths, at Sufficient Coupling. If, in equations (19) and (20) we replace the ratios of angular velocities by the reciprocals of the corresponding ratios of wavelengths, in accordance with equations (12), and make certain evident transformations, we obtain o P t. (A,y = - -pr \ A /opt. 1 + 1?2A/ 188 ELECTRIC OSCILLATIONS [CHAP. XII where, for a consistent pair of values, both equations must have the same sign before the radicals. These resonance relations (21) and (22) are optimum provided (23) 174. Calculation of the Optimum Resonance Relation in Certain Numerical Cases. In order to facilitate the optimum values of Ai/X and A 2 /X, let us extract the square root of (21) and (22) and write the results in the form opt. -, where = n J - \9itih (24) FIG. 6. Auxiliary curve to assist in calculation of optimum resonance ad- justment. \ is replaced by ^ denned in (25) . opt. provided A-; VI where ^ 2 = 7?2 -- (25) Table IV gives computed values of (Ai/X) opt . for various values assumed for opt. Using + sign Using sign 0.0 1.000 1.000 0.1 0.953 1.054 0.2 0.953 1.118 0.3 0.877 1.196 0.4 0.845 1.292 0.5 0.817 1.414 0.6 0.791 1.581 0.7 0.767 1.825 0.8 0.746 2.236 0.9 0.725 . 3.162 .0 0.707 Infinite .1 0.690 Imaginary .2 0.675 Imaginary .3 0.660 Imaginary .4 0.646 Imaginary .5 0.632 Imaginary .6 0.620 Imaginary .7 0.608 Imaginary .8 0.597 Imaginary .9 0.587 Imaginary 2.0 0.577 Imaginary 2.1 0.568 Imaginary 22 0.559 Imaginary 2.3 0.550 Imaginary 2.4 0.542 Imaginary 2.5 0.535 Imaginary 2.6 0.527 Imaginary As an example of the manner of using the auxiliary curves of 190 ELECTRIC OSCILLATIONS [CHAP. XII Fig. 6, in calculation of optimum values of AI and A 2 , let us take a special case. Suppose T 2 = 0.30 and 77! = 0.1, let us give various values to 772 and compute the corresponding optimum wavelength adjust- ments, with the results recorded in Table V. In compiling this table the values of v\ and rjir}2 } the proper adjustment for a maximum secondary cur- rent is materially influenced by the coefficient of coupling r, and every change of r requires a readjustment of the wavelengths of both of the circuits of the coupled system. m. CURRENT AMPLITUDE AT OPTIMUM RESONANCE 176. General Value of Secondary Current Amplitude. In equation (33) of the preceding chapter we have the general ex- pression for the secondary current amplitude, and this expres- sion, in view of (35) of the same chapter, may be written (26) where X\ t X^ Z\ t and Z 2 are the ordinary abbreviations for the reactances and impedances, defined as follows: Xi = Li - 1/Citt, X 2 = L 2 co - l/C 2 co (27) Z2 7? 2 I V 2 7 2 7? 2 I V" 2 /OQ\ 1 ' "'1 ~T~ -A- 1 , "2. JLl/2 ~T~ ** 2 \^"/ In these equations w is the angular velocity of the impressed e.m.f. ; E is the amplitude of impressed e.m.f. ; and M is the mutual inductance between the two circuits. 7 2 is the amplitude of the secondary current for any values whatever of the constants of the circuits. 177. Current Amplitude in Secondary Circuit at Optimum Resonance, with Coupling Sufficient. We have also seen in the preceding chapter that if M 2 co 2 > R 1 R 2 (29) or, otherwise expressed, if r 2 > 171172 . (30) the secondary current amplitude obtained at optimum resonance is, by Chapter XI equation (53), E (31) max. max. O,. / D D CHAP. XII] RADIO RECEIVING STATIONS 193 In this case the amplitude of secondary current is independent of the coefficient of coupling provided only (29), or (30), is fulfilled. 178. Current Amplitude in Secondary Circuit at Optimum Resonance with Coupling Deficient. The coupling is called deficient whenever r 2 < 771*72 (32) Under this condition, by equation (49) Chapter XI, the value of the amplitude of secondary current is MuE . max. -f M 2 C0 2 (33) 1.0 .8 S 3.6 .2 1.0 FIG 8. Relative values of max. max. secondary current for different values ofr' ' 771172' In terms of the ratio constants r, T?I and 772, defined in (5), (6) and (7) this can be written /, max. max. (34) In this case the amplitude of current depends upon the ratio of Table VI following contains a series of values of relative ampli- tude of / 2max . max. fo r various values of the ratio r/\A?i??2- These results are plotted in the curve of Fig. 8. 13 194 ELECTIRC OSCILLATIONS [CHAP. XII In this table and curve the relative amplitude of secondary current is arbitrarily designated as unity for r 2 = 771772. Table VI. Relative Values of I2 max . for Different Values of the Ratio r/vW Relative values of 7 2max max >1 1.000 1.00 1.000 0.90 0.995 0.80 0.974 0.70 0.937 0.60 0.880 0.50 0.800 0.40 0.690 0.30 0.551 0.20 0.385 0.10 0.198 IV. ON THE SHARPNESS OF RESONANCE AND THE POSSIBILITY OF AVOIDING INTERFERENCE 179. Ratio of Interference. If we have an electromagnetically coupled receiving station of the form of Fig. 1, and if we set our receiving station in the optimum resonance condition for a given desired wave of angular velocity co , we shall receive from this wave an amplitude of current / 2ma x. max. given by equation (31), if the coupling is sufficient, and by (33), if the coupling is de- ficient; where E is the amplitude of e.m.f. impressed by the wave of co , and where the co of (33) is to be replaced by co . If now at the same time someone else is sending electric waves with a different angular velocity co, and is at such a distance from us as to impress an equal amplitude of e.m.f., we shall receive from him an amount of interfering current given by (26). Let us now take the ratio of the interfering current to the desired current, and call this ratio the ratio of interference, indicated by Y. Then, on forming the indicated ratio, we have: If the Coupling is Sufficient (i.e., if M 2 co 2 > R\Rz) Y = (35) CHAP. XII] RADIO RECEIVING STATIONS 195 If now we designate by 77 and 7720 the values that 771 and 772 have when co = co , we have r> if r> if /Oj\ 7710 "l/J-'lCOOj ^720 H/2/ JLjyfjOQ \oOy If also we let ->K we shall have, by the fact that the circuits are in optimum reso- nance for angular frequencies co , by (19) and (20), the additional equations 1 -- 2 = yiovo, and 1 -- = 1720^0 (38) COo COo In terms of these ratio constants, we may change the form of Xi, as follows : 1/CiCO = Z/iCo(l - /to co fii l ---- \COo CO COo which by (38) gives /CO C0 1 --- \COo CO coo co coy (39) Likewise where CO COo . fJlO^OCOQ CO COo Ui = X j U% = Hr COo CO CO COO CO If now we divide the numerator and denominator of the frac- tion under the radical in (35) by Li 2 L 2 2 co 4 , and make use of the abbreviations above given, we obtain 2T 2 2 COo (42) 196 ELECTRIC OSCILLATIONS [CHAP. XII Equation (42) can be otherwise factored so as to give Y = = 2 (43) / (1710^2 + 1720^1) 2 + (r 2 ; + 7710*720 ~~ In equation (43) F is the ratio of interference at sufficient coup- ling. It is the ratio of the secondary current produced by the inter- fering signal of angular velocity co to the secondary current produced by the desired signal of angular velocity co . The two signals are supposed to be of such intensity as to impress equal amplitudes of e.m.f. on the receiving antenna. We shall next write out a similar equation for the ratio of interference at deficient coupling. // the Coupling is Deficient (i.e., if M 2 w 2 < RiRz), we obtain F by dividing (26) by (33), with co in (33) replaced by co . This gives (Rig* + MW) 2 co 2 /co 2 Now we introduce the condition that the constants of the cir- cuits are such that the system is in optimum resonance (with deficient coupling) with the angular velocity coo,* that is, by (48) Chapter XI, X l = = X* at co = co. These last two equations give 1 -- IV/coo 2 = = 1- 2 2 /co 2 (45) Dividing numerator and denominator of (44) by Li 2 L 2 2 co 4 , subject to the condition (45), we obtain from (44) = (46) CO coo 2 where _ fo __ coo coo to Equation (46) may be otherwise factored so as to take the form (77101720 + T 2 )CO/C0 CHAP. XII] RADIO RECEIVING STATIONS 197 Equation (47) gives the ratio of interference Y at deficient coupling. The quantity Y is the ratio of the secondary current produced by the interfering signal of angular velocity co to the secondary current produced by the desired signal of angular velocity o>o, when the signals are such as to impress equal amplitudes of e.m.f. on the receiving antenna. 180. Tables and Curves Showing the Ratio of Interference in a Typical Case. Values calculated for the ratio of inter- ference F in a specific case are contained in Tables VII and VIII. These two tables of values were obtained with = 0.03, 1720 = 1.00, r 2 = various values (48) the The various values employed for r 2 are indicated in headings to the columns in the Tables. Graphs of the values given in these tables are exhibited in Figs. 9 to 12. The tables and curves employ as parameter the value of X/Xo ( = WQ/CO) where Xo is the wavelength of the de- sired signal, and X the wavelength of the interfering signal. In all three of the figures the black dots are values obtained with the case of critical coupling (r 2 = 7710*720 = 0.03). Table VII. Values of the Ratio of Interference Y at Sufficient Coupling for Different Values of Relative Incident Wavelengths, and Different Coefficients of Coupling T. Given 7710 = 0.03, 7720 = 1.00 Ffor X/Xo T* = 0.51 r2 = 0.30 r2 = 0.15 7-2 = 0.06 ( + ) (-) ( + ) (-) ( + ) (-) ( + ) (-) 0.87 0.319 0.171 0.293 0.240 0.256 0.247 0.232 0.278 0.909 0.411 0.296 0.387 0.312 0.353 0.334 0.327 0.356 0.952 0.638 0.475 0.619 0.497 0.577 0.524 0.548 0.553 0.98 0.886 0.779 0.876 0.795 0.871 0.814 0.849 0.833 1.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.02 0.875 0.833 0.865 0.767 0.847 0.776 0.841 0.805 1.05 0.580 0.400 0.557 0.416 0.567 0.433 0.530 0.468 1.10 0.320 0.198 0.309 0.225 0.297 0.215 0.298 0.230 1.15 0.209 0.125 0.203 0.129 0.198 0.134 0.204 0.144 The columns headed (+) were obtained by using the plus sign in the expressions for MI and ut (41), and belong to the long-wave optimum ad- justment of the receiving circuits," while the columns headed ( ) were obtained by using the minus sign in equation (41) and belong to the short- wave optimum adjustment of the two receiving circuits. 198 ELECTRIC OSCILLATIONS [CHAP. XII f* |jB !-* 5.3 .2 ,1 \ \ .51 .88 .90 .92 .94 .96 .98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 X/X FIG. 9. Ratio of interference. X = wavelength of desired signal. A = interfering wavelength. Black dots = values obtained at critical coupling ( T 2 SB 0.03). Sign (+) designates use of long- wave optimum adjustment; sign ( ) designates use of short-wave optimum adjustment. Given no = 0.03, ,20 = 1.00. 1.0 1.12 FIG. 10. Ratio of interference. Heavy lines for r 2 = 0.30. Dotted lines forr 2 = 0.15. Top curves using long-wave optimum adjustment; bottom curves using short-wave optimum adjustment. Black dots obtained at critical coupl- ing. Given ij 10 = 0.03; 1/20 = 1.00. CHAP. XII] RADIO RECEIVING STATIONS 199 1.0 *". V (-) \\ \ \ ( ) V 4 (-> s \ ^ (+) \ \ x^ *x x N 1.0 .6 .92 .96 1.00 1.04 1.08 V\o FIG. 11. Same as Fig. 9, .except that r 2 = 0.06. \ .88 .92 .96 1.00 1.04 1.08 1.12 XAo FIG. 12. Ratio of interference at deficient coupling for T 2 = 0.01 and 7^=0. Black dots obtained at critical coupling. Given 7710 = 0.03, 7720 = 1.00. 200 ELECTRIC OSCILLATIONS [CHAP. XII Table VIU. Similar to Table VII, but with Deficient Coupling X/Xo Ffor r* = 0.03 7-2 = o.oi r 2 = O.OOJ 4.00 0.00102 0.00068 0.00052 3.00 0.00263 0.00177 0.00130 2.00 0.0111 0.0074 0.00555 1.50 0.0370 0.0246 0.0183 1.25 0.0984 0.0653 0.0483 1.15 0.180 0.119 0.0888 1.10 0.274 0.184 0.138 1.05 0.510 0.380 0.279 1.03 0.707 0.543 0.435 1.02 0.833 0.700 0.588 1.01 0.950 0.890 0.825 1.00 1.000 1.000 1.000 0.99 0.952 0.899 0.841 0.98 0.848 0.721 0.612 0.97 0.728 0.571 0.461 0.952 0.552 0.397 0.306 0.909 0.333 0.222 0.167 0.870 0.239 0.157 0.117 0.800 0.156 0.103 0.0760 0.667 0.0860 0.0558 0.0413 0.500 0.0460 0.0300 0.0222 0.333 0.0242 0.0159 0.0118 0.250 0.0168 0.0108 0.00820 By reference to Fig. 12, one sees that with deficient coupling a decrease of the coefficient of coupling always diminishes the interference for any wavelength of the interfering signal. With the coupling sufficient, as displayed in Figs. 9, 10, and 1 1, the ratio of interference for a given coefficient of coupling may be either greater or less than the interference with the smaller coefficient of coupling designated as Critical Coupling. In this case, with r 2 = 0.03, the coupling is critical, for then M 2 coo 2 = R iR 2 , or, otherwise stated, r 2 = rj 101720. With the coupling sufficient, the curve for the long-wave tuning in the neighborhood of resonance shows generally a larger interference than the curve of short-wave tuning, but if the range of wavelengths is sufficiently extended the two curves cross and show the reverse condition. Such a crossing point is shown at X/Xo = 0.885 on one of the pairs of curves CHAP. XII] RADIO RECEIVING STATIONS 201 of Fig. 10. A mathematical investigation shows that the curve of interference for long-wave tuning always crosses the curve of interference for short-wave tuning at the point given by the equation Xi = V 1 - i7ior?20 + r 2 V. MAX. MAX. SECONDARY CURRENT AND DETECTOR RESISTANCE 181. At Optimum Resonance with Coupling Sufficient the Total Heat Developed in the Secondary Circuit is Independent of its Resistance. At Sufficient Coupling; that is, when MW > #i# 2 , (49) the current obtained at optimum resonance has been found to be Tjl 1 2 max. max. == ~ 7^ _ (50) which shows the striking property of being independent of the mutual inductance between the circuits, provided only that Mu is great enough to fulfill the condition for sufficient coupling. If the resistances of the two circuits are independent of the frequency, the higher the frequency the smaller M can be and yet have (50) fulfilled. For this reason, high-frequency trans- formers may be coupled much more loosely than corresponding transformers for low frequency, and iron which is used to in- crease M in low-frequency transformers is not advantageous in high-frequency transformers. Another very interesting and important fact is the fact that can be obtained from (50) that the heat developed in the sec- ondary circuit at optimum resonance with coupling sufficient is independent of the resistance #2 of the secondary circuit; for if we multiply the square the secondary current by R%, we obtain for the power dissipated in the secondary circuit a quantity in- dependent of R%. This means that at optimum resonance with sufficient coupling there is as much heat developed in the secondary circuit when a low-resistance detector is used as when a high-resistance de- tector is used. If, therefore, the detector is an instrument whose 202 ELECTRIC OSCILLATIONS [CHAP. XII indications are proportional to the heat developed, a low-re- sistance detector would be as sensitive as a high-resistance detector if it were not for the fact that a low-resistance detector is a smaller proportion of the total resistance of the secondary circuit. Similar considerations apply to a detector of the electrody- namometer type. If the deflections of the electrodynamometer are proportional to n 2 /2 2 , where n is the number of turns of wire in the coil, and if the size of the channel of windings is fixed so that the resistance R of the detector is pl/s, I and s being the length and cross section of the wire in the coil and p the specific resistance of the material of the wire, then we have / = 2irrn, in which r is the mean radius of the windings ; and approximately S = A/n, where A is the area of the channel. Therefore, or Roon 2 ; whence, if the deflection D is such that Z)oori 2 / 2 2 , we have DRI 2 2 This gives for the circuit containing the electrodynamometer detector the same relations as with the thermal detector above specified. From the results here obtained, we may draw the following conclusions : // the detector is to be used in series with the secondary circuit, and if the indications of the detector are proportional to the square of the secondary current times the resistance of the detector, and if the resistance of the remainder of the secondary circuit is inconsider- able in comparison with the resistance of the detector, and if the e.m.f. impressed on the antenna by the incoming waves has an amplitude uninfluenced by the tuning of the secondary circuit, and CHAP. XII] RADIO RECEIVING STATIONS 203 if the efficiency of the detector is independent of its resistance, then the indications of the low-resistance detector will be as great as the indications of a high-resistance detector. The low-resistance detector will then be preferred to the high-resistance detector, be- cause resonance with the low resistance is sharper. This analysis is given in the effort to determine the theoretical limitation upon the choice of a detector for use in series in the secondary circuit of a radiotelegraphic receiving station. In practice, up to the present time, only detectors of compara- tively high resistance are found to be applicable to the reception of weak signals. The reason, in the form of an alternative, is apparent from the analysis here given, to wit: Either, the detectors of low resistance have a smaller efficiency in the conversion of the oscillatory energy into perceptible indications; Or, the low-resistance detector by permitting and requiring a larger value of ! (67) The reactances are Xi = Lico - 1/Cico, X 2 = L 2 co - l/C 2 co (68) where Ci is total primary capacity consisting of Cio and Ci2 in 220 ELECTRIC OSCILLATIONS [CHAP. XIII series, and C% is the total secondary capacity consisting of Czo and Cw in series. Therefore, 7T = 79 H 7? > 7T = TY H 77~ U.I L/io <~/i2 02 Vv20 ^12 The discussion concerning the case of the transformer coupling, given in the present chapter, applies exactly to the capacity coupling if we give to m the value in (67) in place of the value in (56). With this understanding the table of equivalences Table I may be retained. 199. Current Amplitude in Circuit II. The current-amplitude equations (57) and (58) must be changed by replacing Afco in the numerator 1 by l/Ci2. We thus obtain . 7 * = 2 by l/<7i2 2 o> 2 , equations (61) and (63) become the partial resonance relations for the capacity coupling, as follows: V c ) gves * opt - and ' op, ; (73) In the case of capacity coupling by a condenser C iz common to Circuit I and Circuit II, the value of X. 1 79 with a minus sign is not employed because amplitude is essen- tially possible. CHAP. XIII] A GENERAL RECIPROCITY THEOREM 221 In like manner the value of Xi given in (73) produces the largest value of 1 2 , for given values of d 2 , X 2 , Z 2 , and co. 201. Optimum Resonance Relation and Current at Optimum Resonance. In order to obtain maximum current amplitude / 2 when both X\ and X% are varied and adjusted, it is necessary to give to them such adjustments that both (72) and (73) are satisfied. Let us note that the product of (72) and (73) gives ZiZ 2 = c ^\ z v whence (note also (72)) (73) Either X^ = = X 2 (74) or ZiZz = ^ (75) The latter can be fulfilled only provided i = p., c^w ? HlK * Returning to (72) and (73), let us divide one by the other obtaining fj| - f (76) whence by Division of Ratios, and combination of results with (72) we have v > ^ 2 (77) Also by (74) Xi = = X 8| provided l r i i < ^1^2 (78) C/12 CO Equations (77) anrf (78) are E _ (79) 222 ELECTRIC OSCILLATIONS [CHAP. XIII Equation (79) gives the amplitude of current in Circuit II, when Circuit II has its optimum adjustment, with any values whatever of the other constants of the system. Let us now make the additional requirement that Xi shall also have its optimum adjustment. There will be two cases according as the capacity coupling is Sufficient or Deficient. First, with Coupling Sufficient, equation (77) gives Zl which introduced into (79) gives ET -I 1 2 max. max. = . - , provided ^ > RiR 2 (80) Second, with Coupling Deficient, we may still employ equation (79) but must satisfy (78) by making X l = 0. Then in (79) Zi reduces to jRi, and we obtain E 1 /2max. max. = -- T , provided n , , < RiR 2 (81) C^BA + JL When Circuit I and Circuit II are coupled together by having a common condenser Ci2 we may designate the coupling as Sufficient when and may designate the Coupling as Deficient when :; : /:-V;-,:;;;v"; c^ #i 2 2 , hence ZSZS > R 12 *, and by (90), therefore X^ = 0. Comparison of this result with (87) and (88) shows that both Xi and Xz must be zero. In this case of resistance coupling between the circuits I and II, we have only one case of optimum resonance, given by (89), which corresponds to the case of Deficient Coupling in the other examples of Transformer Coupling and Capacity Coupling. . 205. Secondary Current Amplitude at Optimum Resonance with Resistance Coupling. The general expression for amplitude /2 of current in this case, since this amplitude is essentially posi- tive, is obtained by replacing Mw by Rn in equations (57), and is * i 7' 7 7 7 Before passing to the case of optimum resonance, let us in- troduce merely the resonance relation with X 2 optimum as given in (88), which is equivalent to X 2 = 0. This gives r> w [^2max.]x 2 opt. = '1^2 and, by Table I, Ri2*Ri\ (91) Equation (91) gives the amplitude of current in Circuit II when all the constants, except X%, have any values, and X 2 has its opti- mum adjustment as specified by (88). CHAP. XIII] A GENERAL RECIPROCITY THEOREM 225 Let us now introduce the condition that Xi as well as X z shall have its optimum adjustment. By (89) this can be attained only by making Xi = 0, then by (88) X z automatically becomes zero. Making ^fi = in (91) we obtain 2 max. max. (92) Equation (92) gives the maximum possible value of 7 2 , in the case of two circuits I and II coupled by having a common resistance Riz. The adjustment that gives' this max. max. current is given by (89), and is seen to be an adjustment of each circuit separately to have its undamped period equal to the period of the impressed e.m.f. Note that the case of resistance coupling is always one of essen- tially Deficient Coupling. lii CHAPTER XIV RESONANCE RELATIONS IN A CHAIN OF THREE CIRCUITS WITH CONSTANT PURE MUTUAL IMPEDANCES. STEADY STATE 206. Statement of Problem. We propose now to utilize the Reciprocity Theorem of the preceding chapter to determine the resonance relations in a system of three circuits arranged in a chain with the couplings between the circuits in the form of pure mutual impedances, as denned in Art. 192. The purpose of this treat- ment is, first, to give an illustration of the simplicity resulting from the use of the Reciprocity Theorem to determine resonance relations, and, second,* to lay the foundations for solving im- portant problems relating to radiotelegraphic practice. FIG. 1. Chain of three circuits with transformer coupling. 207. Illustrative Forms of Circuits. Two forms of circuits to which the present analysis applies are shown in Figs. 1 and 2. In Fig. 1 the couplings in the chain of three circuits are made by transformers. In Fig. 2, which is analogous to a much-used type of radio receiving system, the coupling between Circuit I and Circuit II is by a transformer, while the coupling between Circuits II and III is by a common condenser C 2 s. In both figures 7 3 represents a resistance that may be regarded as the resistance of the detector. The two figures are both special cases of a chain of three cir- 226 CHAP. XIV] CHAIN OF THREE CIRCUITS 227 cults with pure mutual impedances as defined in the previous chapter. 208. Anticipatory Sketch of the Method. The method em- ployed in this problem will consist in obtaining three Variant Expressions for the current in Circuit III, when the e.m.f. is applied to Circuit I. These three forms will be found to be \faE\ \0yE\ \faE\ .' *7 *7i */! *7 7 Or 7 7 7 f 7 Z/aZ/ >i/j i Z/iZ/2 ^3 Z/jZ/ 2 ^3 (1) (2) (3) where the vertical lines enclosing the numerator indicates abso- lute value. FIG. 2. Chain of three circuits with one transformer coupling and one capacity coupling. The various Z's will be found to have the definitions given in Table I, Art. 211. The values of the various Z's will then be shown to be such that we obtain certain fundamental forms of the reso- nance relations by inspection. The principles underlying the method will now be established, first, by directly showing the identity of the denominators of (1), (2), and (3), and, second, by the use of the Reciprocity Theorem. 209. Direct Proof of the Identity of the Denominators of Equations (1), (2), and (3). Referring to Table I in Art. 211 for definitions of the various Z's, let us note by direct multiplication and substitution that + (4) From equations (4) it appears that equations (1), (2), and (3) are established as soon as we prove the correctness of any one of 228 ELECTRIC OSCILLATIONS [CHAP. XIV them. This last step is easy to take, but will be here omitted, as the step occurs in the use of the Reciprocity Theorem following. On account of the importance of the Reciprocity Theorem in itself, we shall now make use of it to deduce again the identity of equations (1), (2), and (3), and shall incidentally supply such steps as have been omitted in the above sketch. I. APPLICATION OF THE RECIPROCITY THEOREM 210. Notation. The notation employed here will be the same as in the preceding chapter, namely, as we go around the nth circuit, R n = the sum of all the resistances in series in the nth circuit including resistances common to neighboring circuits, if there be such; L n = the sum of all self-inductances in series in the nth cir- cuit, including self -inductances common to the nth cir- cuit and its neighbors if there be such, and including the self-inductance of any primary or secondary coil of a transformer if any such coil be in the nth circuit; l/C n = the sum of the reciprocals of all capacities in series in the nth circuit, including the capacities of condeners; common to the nth circuit and to neighboring circusits X n = L n U - 1/CnU, Z n * = R n 2 + X n ^ Z n = R n + jX n ' y Wi2 = complex mutual impedances between Circuits I and II = 2 12 + jMizu, where 2 12 = complex impedance common to Circuits I and II, if there be such, and MIZ = mutual inductance between Circuits I and II, if there be such. ' w&23, W34, etc. = similar quantity to m\^ but for other pairs of circuits. 211. Values of Complex Current Amplitudes and Complex Currents. By means of the general methods of Chapter XIII, it is seen that with the cosine e.m.f. applied to Circuit I, the currents in the three circuits are the real parts of the complex quantities *! = A*?*, i 2 = A 2 e' w , * = A^ (5) where o> is the angular velocity of the impressed e.m.f., and A\, A z , A 3 , satisfy relations of the form of (34), (33) and (32) of CHAP. XIV] CHAIN OF THREE CIRCUITS 229 Chapter XIII, with, however, all of the terms of subscripts higher than 3 made equal to zero. These relations written out here are w 23 A 2 A 3 23 R* + 2 2 - A 1 = m 12 E R'2 + j E R,' + 2 2 - ni23 c 23 (6) (7) (8) where the third members of (7) and (8) are written down from the general knowledge that any algebraic combination of complex quantities is a complex quantity of the form a + jb. Equations (7) and (8) require that R\, X f lt R'^ and X' 2 shall be given definitions consistent with these equations (7) and (8). By working out the values of the denominators in (7) and (8), and equating the two denominators for the same quantity in each case, we obtain the values of the primed quantities in the first column of Table I following: Table I. Equivalences for Three Circuits with Pure Mutual Impedances Forward equivalences Backward equivalences Two-way equivalences v Z 3 2 Z' 2 2 ZV = flV + XV / and in subsequent equations, since the two w's are pure imaginaries, as may be seen by reference to their formation, we have let Wi 2 = and where ft and 7 are real quantities. jy (9) 230 ELECTRIC OSCILLATIONS [CHAP. XIV In setting up Table I we have replaced Wi 2 and m 23 by their values (9). 212. Currents in Terms of Forward Equivalences. We may now write down the values of the currents ii, i 2 , i z with the use of the Forward Equivalences contained in column one of Table I. This is done by taking (6) , (7) , and (8) , in terms of the primed quantities, eliminating among them and substituting the results in (5), and then rationalizing and taking the real part of the result, obtaining Tjl ii = ~r cos (ut 3 Now by the Reciprocity Theorem, this current I\ is the same as we should get in Circuit III (that is, 7 3 ) if the e.m.f. were ap- plied to Circuit I; whence v v 0/7 Q-> with e.m.f. in Circuit I. ^1^2 & 6 CHAP. XIV] CHAIN OF THREE CIRCUITS 231 This is equation (2) above. As to equations (1), let us note that it has been already ob- tained in (13). 214. Current Amplitude I 3 in Terms of Two-way Equivalences. We have, remaining, one more form of expression (3) to obtain for 1 1. This may be obtained by the Theorem of Reciprocity applied to Circuits I and II. By the general equations of the form of (29), Chapter XIII, when the e.m.f. is applied to Circuit II, and when there are only three circuits in the chain, we obtain the relations ziAi mizAz = + z 2 A 2 rai 2 A 3 = E (13) = Replacing m ]2 by jfa m 23 by jy, and solving (13) for AI, A 2 , and A 3 , we obtain ^A 2 AI A 3 A 2 = jyA E E T 2 #'2 fiFTo- ( Sa y) (14) The last denominator of the Adequation is an abbreviation for the complex denominator preceding it in the Adequation. Equating the real and the imaginary parts of these two denomina- tors respectively, we obtain, on solving, the values of #'2 and X' 2 contained in the last column of Table I. These values are the equivalent resistance and reactance of Circuit II as in- fluenced by the two Circuits I and II, and are hence called the Two-way Equivalences of Circuit II. Now solving the Ai-equation and the A2-equation of (14) as simultaneous, rationalizing and taking the amplitude of the real part, we obtain /! = p , with e.m.f. in II (15) Compare with this the amplitude of iz in (10), which gives (3E ^-, with e.m.f. in I (16) 232 , ELECTRIC OSCILLATIONS [CHAP. XIV By the Theorem of Reciprocity these two quantities (15) and (16) are equal, hence Z'fZ, = Z\Z\ (17) Multiplying both sides of this equation by Z 3 , we obtain Z.Z'fZs = ZsZ'jZ'i (18) which makes (3) true if (1) is true. But we have already proved (1). We have thus shown that equations (1), (2), and (3) are three different ways of expressing the current amplitude in Circuit III under the action of a cosine e.m.f. applied to Circuit I, pro- vided the current has reached a practically steady state. The use of the Reciprocity Theorem has enabled us to obtain certain Equivalent Resistances, indicated by R'i, R' Z} R 3 , R 2 , R f 2, and certain Equivalent Reactances, indicated by X'i, X' 2 , ^3, ^2, X f z t all of which are tabulated with their values in Table I. By taking the square root of the sum of the squares of these resistances and the corresponding reactances we have formed, and included in Table I, the Equivalent Impedances Z'i, Z' 2 , Z 3 , Z 2 , Z' 2 . We have then written down in terms of the Equivalent Impedances three different expressions for the Current Amplitude 1 3. In these three expressions (1), (2), (3), the occurrence of Xi, X z , and X 3 , as will presently be shown, is such that certain fundamental forms of the resonance relations may be had by inspection. H. PARTIAL RESONANCE RELATIONS AND RESTRICTED RESONANCE RELATIONS WITH PURE MUTUAL IM- PEDANCES UNCHANGED 215. Nomenclature. We shall designate as Partial Reso- nance Relation re Xi the adjustment of X\ that makes 7 3 (say) a maximum when all the other members of the circuits are kept constant. In general a Partial Resonance Relation re a Variable will mean the adjustment of the variable that makes the amplitude of the current in the detector circuit (or work circuit) a maximum while all the other members of the system are kept constant. In certain cases the range of adjustment of a designated variable may not be sufficient to attain an absolute maximum CHAP. XIV] CHAIN OF THREE CIRCUITS 233 of the work current. In those cases we shall designate as a Restricted Resonance Relation re a Variable the adjustment of the variable that will make the current amplitude in the work circuit the largest that can be obtained with any adjustment possible to the variable under the limitations of the restriction. In case, for example, Xi is the variable under observation, we shall refer to the value of X\ that gives the greatest work cur- rent, subject to the restrictions of X i, as the Restricted Resonance Relation re X\, or Resonance Relation re X\ Restricted. 216. Resonance Relations for a Chain of Three Circuits With Pure Mutual Impedances Unchanged. We have already pointed out in the anticipatory sketch (Art. 208) the nature of the steps to be employed. Three forms of expression for 7 3 were given in equations (1), (2), and (3), and these three forms have now been derived and shown to be identical in value. Since the numerators are supposed to be constant, we can make 7 3 a maximum, by making the denominators a minimum. By definition of the various equivalences in Table I it is seen that the denominator Z 3 Z' 2 Z'i, of equations (1) involves Xi only in the factor Z'\. To make 1$ a maximum by varying Xi, it is necessary, therefore, only to make Z'i a minimum re X\. Since the resistances of the system are all constants, in it is seen, by reference to Table II, that this is attained by making X'i 2 a minimum re Xi Hence, if X i is unrestricted, the resonance condition is X\ = (20) (Partial Resonance Relation re Xi) On the other hand, if Xi is restricted, the resonance condition is X\ 2 = minimum (21) (Resonance Relation re X i Restricted) In like manner, since in the denominator of (3), Xz occurs only in the factor Z' 2 , we find, by similar reasoning, X'z = (22) (Partial Resonance Relation re Xz) and X'z 2 = minimum (23) (Resonance Relation re X 2 Restricted) 234 ELECTRIC OSCILLATIONS [CHAP. XIV Again, since in the denominator of (2) Xs occurs only in the factor Z 3 , we have X 3 = (24) (Partial Resonance Relation re X s ) and X 3 2 minimum (25) (Resonance Relation re X 3 Restricted) Equations (20), (22), and (24) give respectively the partial resonance relations re Xi, X%, and X$, when the mutual impedances are pure and unvaried. In case restrictions on any or all of the reactances prohibits the attainment of any or all of the partial resonance relations, we must substitute for any of the relations that is unattainable the corresponding Resonance Relation Restricted, as given in (21), (23), or (25). III. APPLICATION TO A CASE IN WHICH THE REACTANCES ARE ALL UNRESTRICTED 217. Optimum Resonance Adjustments. Adjustments for a Grand Maximum of Current Amplitude I 3 , When the Reactances are All Unrestricted. Let us now determine the adjustments that must be given to all three of the circuits, in order to obtain a grand maximum of amplitude 7 3 , under the condition that all of the reactances are unrestricted. This is done by solving (20), (22), and (24) as simultaneous. For this purpose we shall make constant use of Table I, Art. 211. Let us first solve (20) and (22) as simultaneous. By (22) V' O rv A 2 = U- By a comparison of the third and first columns of Table I, Art. 211, it is seen that the satisfaction of this equation requires Equation (26) is an alternative form of (22) . Returning now to (20), we may write it (by the definition of \ in the form / 2 Equation (27) is an alternative form of (20). CHAP. XIV] CHAIN OF THREE CIRCUITS 235 If now (22) and (20) are simultaneously true, their equivalents (26) and (27) must be simultaneously true; so that by replacing X' 2 in the numerator and denominator of (27) by its value from (26), we obtain (28) Equation (28) is a first step in the treatment of (20) and (22) as simultaneous. From (28) it follows that either X l = (29) (3* pXi* p*RS R 2 != zT " ~z^ ~zs This last equation is obtained by dividing (28) by Xi, and clearing of fractions, obtaining the equality of the first term to the second. The third member follows from the second by employing the definition of Zi 2 . Extracting the square root of (30) and combining the alterna- tive combination (29) (30) with (26), we obtain either Xi = and X' 2 = (31) nr ^' 2 ^' 2 ^ ^Q0\ xT = ~Rl""^f Equations (31) and (32) constitute a pair of results, one or the other of which must be fulfilled in order to make Xi and X 2 both optimum, while X s may have any value whatever. The quantity X s is involved in X' 2 and R' 2 (see Table I). A similar treatment of (22) and (24) as simultaneous gives either X* = and X 2 = (33) * 2 -^-- y2 x7 " fl 3 ~zT- Equations (33) and (34) constitute a pair of results, one or the other of which must be fulfilled in order to make X s and X 2 both optimum, while Xi (involved in X 2 and R 2 ) may have any values whatever. We come next to treat of the case where all three of the circuits are at optimum adjustment simultaneously. This treatment consists in solving the equations (31) and (32) as 236 ELECTRIC OSCILLATIONS [CHAP. XIV simultaneous with (33) and (34), while keeping in mind that the two pairs of equations are themselves alternative possibilities. We shall first show that (32) and (34) are not simultaneously possible, -as follows : Replacing the primed quantities in (32) by their values from Table I, we obtain for this equation #2 , i ,. R 1 * . A similar treatment of (34) gives for it X, X s X S ZS R s T R 3 ZS ~ Z 3 2 If we make these two results true simultaneously their latter parts lead to 2# 2 /# 3 = 0, which cannot be true. We may, therefore, exclude the simultaneous fulfillment of (32) and (34) as a possible compliance with the resonance requirement. We shall next examine (31) and (33) as a possible simultaneous resonance adjustment. This combination gives Xi = 0, Xi = 0, X' 2 = 0, X z = By definitions of X' z and Xz (Table I), these equations reduce to Xi = X z = X, = (37) Equations (37) is the result of treating (31) and (33) as simultaneous. Let us now examine the combination of (31) and (34). By (31) two of the numerator terms of (34) reduce to simpler values, and the combination gives , , (38) Equations (38) is the result of treating (31) and (34) as simultaneous. In like manner the combination of (33) and (32) gives Y ,X Z R Z R3 + T 2 2 X* = 0, and = - = CHAP. XIV] CHAIN OF THREE CIRCUITS 237 Equations (39) is the result of treating (33) and (32) as simultaneous. 218. Adjustments for Grand Maxima of I 3 Summarized and Designated Optimum Combinations. Current Amplitude I 3 Obtained at the Optimum Combinations. Conditions Under Which the Combinations are Respectively Optimum. We have given in equations (37), (38), and (39) three combinations of relations any one of which satisfies (20), (22), and (24) simul- taneously, and is a possible optimum combination. We shall now show that it is sometimes one and sometimes another of these combinations that is optimum. Let us designate the three combinations as follows: Optimum Combination (a), equation (37); Optimum Combination (0), equation (38); Optimum Combination (7), equation (39). The condition under which Combination (0) is attainable may be had by inspection of (38), by noting that Z 3 2 cannot be less than R^, whence RiR* + ff 2 ^ j/i. R\Rz N Rs 2 ' that is, R^RiR* < #i7 2 - RzP 2 . (40) The inequality 40 gives the condition under which Combination (|8) is attainable. A similar process shows the condition under which (7) is attainable, and gives RtftRs ^ R*P 2 - fli7 2 (41) The inequality (41) gives the condition under which Combination (7) is attainable. There is no restriction on the attainability of Combination (a). To find the current amplitude 7 3 under the three Optimum Combinations respectively, let us take /3 in the form given in equation (3). This is T 13 = ' 2 On substituting the combination of equations (37) info this, we have for the value of 7 3 , under the optimum combination (a), the value _ ( 3) 2 238 ELECTRIC OSCILLATIONS [CHAP. XIV Likewise in (42) substituting the optimum combination (/3) as given by (38) , we obtain after reduction ~ Again in (42) substituting the optimum combination (7) as given by (39), we obtain after reduction. T7* :: (45) It will now be shown that [Islp, whenever (/3) is attainable is larger than [Iz] a . This is done by multiplying the numer- ator and denominator of (44) by 7, which makes the numerator the same as the numerator of (43). A comparison of the resultant denominators now shows that [/.U ? [JiL whenever RiRtRs + PR* + 7 2 #i - ZvRJtz RiR* + /3 2 ? (46) The left-hand side may be expressed as a square thus * + 2 - TX^ 2 > 0, which is seen to be always fulfilled, since the quantities under the radicals are all positive. We have then the result that Combination (0), if attainable, gives a larger value of 7 3 than does Combination (a). In a similar way it can be proved that Combination (7), if attainable also beats (a). It is not necessary to compare (0) with Combina- tion (7) since the two are never both attainable in the same case, as may be seen by comparing (40) with (41). The results may now be further summarized in the following Key. 219. Summary and Key Concerning Grand Maxima of I 3 When the Mutual Impedances are Invariable, and When the Reactances are Unrestricted. I. Resonance Combination (a). If where the vertical lines indicate "absolute value," use Resonance Relation X l = X 2 = X 3 = (47) CHAP. XIV] CHAIN OF THREE CIRCUITS 239 and calculate the grand maximum of /a by II. Resonance Combination If RiRzR* use Resonance Relations and calculate the grand maximum of /s by ///. Resonance Combination (7). If use Resonance Relations X 2 _ RtR z + y* _ 0* x * = Oj x; " ~i^r ^7 2 and calculate the grand maximum of 7 3 by \I 1 = ^s summary, or key, contains the optimum resonance combi- nations and the grand maxima of current in Circuit III, obtained when the reactances Xi, X%, and X$ are unrestricted. CHAPTER XV RESONANCE RELATIONS IN A RADIOTELEGRAPHIC RECEIVING STATION HAVING A COUPLED SYSTEM OF CIRCUITS WITH THE DE- TECTOR IN SHUNT TO A SEC- ONDARY CONDENSER ' =-- . I. GENERAL RESULTS 220. Form of Circuits. In Chapters XI and XII there is given a theory of coupled circuits approximately applicable to a radiotelegraphic receiving station in which the detector is in series in the secondary circuit. The treatment is approximate in that the receiving antenna of practice had its capacity, in- ductance, and resistance distributed along the length of the antenna, while the system treated was idealized by replacing Detector, Stoppage Condenser FIG. 1. Radiotelegraphic receiving circuits with detector in shunt. the distributed constants of the antenna by a lumped capacity, inductance, and resistance. It is proposed now to undertake a similar analysis of the corresponding problem with the detector and a " stoppage con- denser" Cso in shunt to the condenser C 2 s of the secondary cir- cuit, and to attempt to determine under what conditions, if any, this arrangement is superior to the arrangement of Chapter XII, Fig. 1. 240 1 D II :> >L 2 :> 1 J III s C( CHAP. XV] DETECTOR IN SHUNT 241 The form of circuit constituting the subject matter of the present chapter is given in Fig. 1. If we idealize this circuit by replacing the antenna and ground by a lumped capacity, as was done in the previous chapters, we have the arrangement given in Fig. 2, in which the condenser Ci replaces the antenna and ground, and the local e.m.f . e replaces the e.m.f . impressed by the incident waves. If the waves are persistent and undamped, the current will arrive at a steady state even for the shortest dot made at the sending station. We shall seek, therefore, only the steady-state solution. III FIG. 2. Similar to Fig. 1, but with antenna circuit replaced by a closed circuit. 221. Notation. We shall give the various parts of the cir- cuits the designations indicated in Fig. 2. If we compare this notation with that of Fig. 2 of Chapter XIV, we shall see that the notation is the same except that M i 2 has now been simplified toM. The reactances of the three circuits are seen to be 1 X*! = LiO) 1/ClCO, Xz = L 2 CO 1/C 23 OJ, Xz = 1/C 3 0) (1) where 1 1 1 , 9 x a rTT (%) Using the methods of the preceding chapters if we let m\z and w 23 be the complex mutual impedances between Circuits I and II and Circuits II and III respectively, and refer to the definition of these quantities given in (18) of Chapter XIII, we see that and w 23 = l/jC^w (3) In order now to make Chapters XIII and XIV directly ap- 16 242 ELECTRIC OSCILLATIONS [CHAP. XV plicable to the present problem, we shall note that and 7 as used in (9) of Chapter XIV have now the values y y y * 3 rr rjt rjt ' ^7/70/70 rr rji 0/7 x"jO, i ) With equations (4) as definitions of & and 7, Table I of Chapter XIV (Art. 211) contains the Equivalences for the present case. 222. Current Amplitude I 3 in Circuit III. By equations (1), (2), and (3) of Chapter XIV, we may riow write the current amplitude 7 3 in Circuit III in three variant forms as follows: \(3yE\ \0yE\ \0yE\ ^7/70/70 rr rji 1S "3 Z/iZ/ 2 (5) (6) (7) Equations (5), (6), and (7) grave three variant forms of expression for the current amplitude 7 3 in Circuit III. In these equations |8 and y have the values given in (4), and the various Z's have the values given in Table I Art. 211. 223. Investigation to Determine the Resonant Values of the Stoppage Condenser C 3 o- The condenser C 30 is in practice ordinarily called the Stoppage Condenser. We shall now seek the value of C 30 (called optimum value) that gives the greatest current amplitude 7 3 in the detector circuit (Circuit III). The detector has any resistance R s . If we examine equation (6), we see that ft y, E, Z\ and Z 2 are independent of C 30 , which is involved in Z 3 alone. The optimum value of C 3 o is thus the value that makes (since Z 3 is positive) Z 3 2 = a minimum, re C 30 (8) By Table I, Art. 211, Z 3 2 = fl 3 2 + X 3 2 (9) in which, by reference to Table I it is seen that R 3 is independent of C 30 . We may, therefore, attain our optimum value of C 30 by making X 3 2 = a minimum, re C 30 (10) If possible, we shall choose C 30 to make ^ 3 = (11) Equation (II), if attainable, will give the Partial Resonance Rela- tion re C 30 . CHAP. XV] DETECTOR IN SHUNT 243 If, on account of restrictions, it is not possible to fulfill (11), we shall choose C 30 to make the value of X s 2 a minimum, and obtain what we have called the Resonance Relationre C 30 Restricted. We shall use the restricted resonance relation only when the partial resonance relation (11) cannot be attained, for if (11) can be attained it will give a larger 7 3 than could be had with the restricted relation that does not make ^T 3 zero. 224. Resonance Relations re C 30 . Restrictions. Let us now write down the abbreviated value of X 3 from Table I, Art. 211. It is X 3 = X 3 - ^f (12) Z/2 Replacing X 3 by its value from (1) and (2), and indicating the square, we have yo 2 T 1 1 7 2 *2 L~ c^~ -c^~~z^ Now C 3 o can have any positive value, so that the first term in the bracket can have any negative value. We see then that we can make ^ 3 2 = (14) provided the remaining terms in the bracket of (13) are positive;. that is provided If (15) is satisfied, there is some value of C 3 o that satisfies (14), and hence (14) is attainable and is the resonance relation re Cso. In (15) X z and Z 2 are defined in Table I, Art. 211. If, now, on the other hand, (15) is not satisfied, then the last two terms in the bracket of (13) are negative. The first term in the bracket is also negative, and by inspection it is seen that we shall make the whole bracket squared a minimum, by making the first term zero. Therefore, for X^ 02 a minimum we must make C 30 = infinity (16) provided 1 - "& < (17) // (17) is satisfied, equation (16) gives the optimum value of C 3 o. This is the Resonance Relation re C 3 o Restricted. 244 ELECTRIC OSCILLATIONS [CHAP. XV 225. Expansion of Resonance Relations re C 3 o. We shall now elaborate (14) and (15). To do this, we shall introduce two new abbreviations as follows: Let (18) B = L 2 co - ^r 1 (19) ^i To justify the designation of (18) in a form that is essentially positive, let us note that, if we recall that we can factor (18) into = # 2 2 + B 2 (21) which shows it to be essentially positive. Now making use of Table I, Art. 211, and equations (1), (2), (4) and (19), we have = 5 + 7 (22) Also, = A 2 + 25 T + T 2 (23) In terms of these results, we can express (14), by using (13), as follows 1 J 2 (B -f- 7) Therefore, = - 1 + ^LM_ (24) Csow A 2 + 2^7 + 7 2 This gives = - + R 1~ o.. (25) Replacing 7 by its value 1/C 2 3&>, we obtain from (25) (26 > CHAP. XV] DETECTOR IN SHUNT 245 Since in (24) the denominator of the last fraction is positive, and since 7 is negative, and C 30 co is positive, equation (24) and consequently (26) can be realized, only provided A* + By ^ (27) Replacing 7 by its value this last inequality can be replaced by - (28) Equation (26) gives the value of C 3 o^ for a maximum amplitude of 7 3 . This is the Partial Resonance Relation re C 3 o. It can be attained only provided (28) is satisfied. If (28) is not satisfied, we must use the Restricted Resonance Relation, given in (16); namely, C 30 = ' (29) We shall consider next the Resonance Relations re <7 23 . 226. Resonance Relations re C 23 . We shall now make an independent investigation of the resonance relations re CM, and shall begin with the current amplitude equation (6), which squared gives In this equation the quantities 7, Z 2 and Z 3 all contain C 23 , while the other quantities of the equation do not, so that for a maximum 7 3 2 with respect to CM, we must make ^2 ^3 n = a minimum, re (7 2 3 T 2 Now, by Table I, Art. 211, o 2 3 = o 4 , 7 2 3 - *s 7 H -- - -5 -- r ~rT whence T 2 T 2 Z2 2 / i-k n i -rr n\ i *- / i-k r> T- -7- o -cr- \ i o 7 2 246 ELECTRIC OSCILLATIONS [CHAP. XV In this expression -X" 3 still involves 7, and must be replaced by its value from (1). This gives, after simplification, =3 i 7 2 7 2 i \ Cao / J y\ \ C 3 o co 2 (31) To make this a minimum with respect to 7, let us set the de- rivative of it with respect to 7 equal to zero, obtaining jowv 7 m (32) whence, either 7 = co (33) I..*. __ 1 (34) " 42 Since 7 is negative, (34) can be attained, only provided B - 1 To ascertain whether (33) or (34) gives the larger value of current amplitude 7 3 , let us substitute these two values succes- sively into (31), and designate the results respectively by DI and D 2 , as temporary abbreviations, obtaining 2 >, (36) when 7 has the value given by (33) ; and (using (32) for (34)) = > 2 (37) when 7 has the value given by (34). It is seen by inspection that 7) 2 is less than or equal to DI, so that (34) gives more cur- rent amplitude 7 3 than does (33), and is to be used whenever it can be attained; that is, whenever (35) is satisfied. CHAP. XV] DETECTOR IN SHUNT 247 We may now replace 7 in (33) and (34) by its value (4), obtaining either C 23 = (38) or (39) (35) is satisfied, equation (39) grwes the value of CM that produces a maximum value of 7s. When (35) is not satisfied, equation (38) is to be used to obtain a maximum value of 1$. 227. Optimum Simultaneous Adjustments of Both C 30 and C 2 3. Resonance Combination L. We have now obtained in- dependently the optimum adjustments re C 30 as given in (26) and (29) distinguished by the criterion (28), and the optimum ad- justment of C 2 3 as given in (38) and (39) distinguished by the criterion (35). We shall next determine what simultaneous adjustments of both C 30 and C 23 are optimum, leaving C\ still arbitrary. This is done by treating these various equations as simul- taneous, keeping in view the criteria under which any of the respective combinations is attainable. Let us begin with the combination Cao = a and (40) C 23 = By reference to the descriptive matter concerning (29) and (33) we see that the equations (40) can be a proper resonance combination only provided this combination is inconsistent with (28) and (35) . To be inconsistent with these inequalities (28) and (35) we require that A 2 > ^ , when C 23 = 0, and B - I n -jg ^ ; , when C 30 A rv 7-101 ! These two relations merely require that B < (41) We have then the result that under condition (41) the optimum combination of values of C 3 o and C 23 is that given by (40). This 248 ELECTRIC OSCILLATIONS [CHAP. XV means that in this case C 3 o is short circuited and C 23 is open circuited or removed. Since in this case the capacities no longer enter, we shall designate this combination (40), under condition (41) as Resonance Combination L. 228. Optimum Simultaneous Adjustment of C 3 o and C 23 . Resonance Combination 0. Let us examine next the combina- tion of (38) with (26). If these two equations are simultaneous we have C 23 = (42) The restrictions under which the equations (38) and (26) were resonance relations are that (35) be not satisfied and that (28) be satisfied. That is, 7? 1 and whenCjs = 0. The second of these inequalities gives B > (43) and the first, on replacing C 30 co by 1/B, and inverting the in- equality, gives A 2 _ Rf + B* B ^ B This by (43) and (21) gives #2 ^ #3 (44) Under conditions (43) and (44), the combination (42) is the optimum resonance combination with respect to both C 3 o and C 23 . We shall call this Resonance Combination 0. 229. Resonance Combination A. Let us next investigate (39) and (26) as a possible combination. This requires extensive elimination. By partial division of the fractional part of (26) this equation gives R2 _ 42 CSQOJ = C 23 co H r-7 + T^TA CHAP. XV] DETECTOR IN SHUNT 249 Equation (45) is the equivalent of (26) . Let us now replace the first two terms on the right and the corresponding expression in the last denominator by its equiva- lent from (39), obtaining _ _i_ P2, 3000 = ~ i~~ ~~ T ~n Transposing the first term of the right-hand side to the left, and collecting these two terms over a common denominator, we obtain, after clearing of fractions By (21) A 2 - B 2 = fl 2 2 , which, introduced into the preceding equation, gives a perfect square on both sides. Taking the square root, we obtain Clearing this of fractions and solving, we obtain E> O (47) The substitution of (46) into (39) gives The conditions under which these results can be attained are the conditions that make the radicals real, and make the numerator of (47) positive. These are A 2 ^ R2Rz, and B > 1 2 ^ R2 A * _ R ^ (48) By (21) the latter gives This equation combined with (48) gives for the complete condition B > 0, and R 2 < Rs < ^4 ( 49 ) 250 ELECTRIC OSCILLATIONS [CHAP. XV Under conditions (49) equations (46) and (47) give the optimum resonance combination with respect to both CZQ and Czz- We shall call this Resonance Combination A. 230. Resonance Combination B. There remains one other possible combination; namely, the combination of (28) and (39). This combination gives Cao = and (50) C 23 o> = B/A* We shall call (50) the Resonance Combination B Examination shows that the only restriction on this is B > 0, so that Resonance Combination B as given in (50) is applicable coextensive with Resonance Combinations and A. It can be shown, however, that where either or A is attainable the Resonance Combination B is inferior as a resonance relation, as follows : Taking the general equation (31), introducing in turn Combina- tion and Combination B as given in equations (50), and calling the results D and D B , we have Do = (# 2 + # 3 ) 2 (51) and DB _ - BW + A*' + (Bi . + Rs)2 (52) Let us note for future use that by (21) this can be written x-x (53) n D B =( Referring now to (51) and (52) it is seen that DO is the smaller whenever the fraction of (52) is positive; that is, whenever Since by (21), the right-hand side is greater than R 2 2 , we have a fortiori that D Q is smaller, when # 3 < R* (54) It thus appears that gives a larger 7 3 than does (B), whenever is applicable, as may be seen by comparing (54) with (44). We shall next show that Resonance Combination B as given in (50) is inferior to Resonance Combination A, whenever A is attainable. This is done by comparing the current Is at Com- CHAP. XV] DETECTOR IN SHUNT 251 bination A with that at the combination given in (50). Com- bination A is given in (46) and (47) . By (46) 1 - R * A * R* rw cj& ~W ' whence by transposition r> 9 i & f c. a \ 3 + C~V = ~E^~ ^ ) Equation (56) is an alternative statement of (46). Let us next note by transposition of (47) and multiplication by (46), that we obtain whence B ) = (57) Equation (57) is a partial expression of Combination B, and is true whenever (46) and (47) are true. Introducing (56) and (57) into (37), we obtain on expanding terms n R * , 2 # R zB 2 , o . 2 # 3 C 30 2 R, (K * + K *> . R2 2B 1 C 3 oco ^ C 30 2 co 2 ' which reduces to In this, let us replace the first factor on the right by its value from (55), obtaining Now making use of (21) this may be reduced to D 2 = 4# 2 #3. Identifying D 2 as the first member of (37), we have Z^/l = 4%<>fl 8 = D 2 (58) Equation (58) is the value assumed by the general equation (37) whenever the Resonance Relations A are fulfilled. 252 ELECTRIC OSCILLATIONS [CHAP. XV We shall now show that the right-hand side of (58), which is obtained with Resonance Combination A is smaller than the corresponding expression obtained with the Resonance Combina- tion B given in (50). We have already found that the result ob- tained with Combination B is ; < Z - = D B (59) where D B has the value given in (53). We see then that D, < D B , whenever This inequality reduces to (A 2 - R^Rs) 2 > 0, which is always fulfilled. We have then the result that the denominator (proportional to D 2 ) in the expression for 7 3 is less with the Combination A than that (proportional to D B ) with the Combination B, so that whenever Combination A can be realized it is to be preferred to Combination B. We have then the result that Resonance Combination B is to be used only when B is greater than zero, and when neither Combina- tion nor Combination A can be fulfilled. An examination of this fact leads to the conclusion that Reso- nance Combination B as given in equations (50) is valid only when >0,and^4 < #3 (60) Before summing up these results in a Key, let us obtain ex- pressions for the current amplitude 7 3 for these several Reso- nance Combinations, L, 0, A, and B. 231. Amplitude of Current I 3 for Resonance Combinations L, 0, A, and B. To obtain expressions for the amplitude of current for these several resonance combinations, we may employ equation (6), which squared may be written 7 3 2 = - f i CHAP. XV] DETECTOR IN SHUNT 253 where, as a temporary abbreviation, D = Z -^l (63) In case of the Resonance Combination L, we can find D, which we shall then call D L , by substituting (40) into (31), bearing in mind that 1/7 = - CW (64) This gives D L = (# so that by (61) the current in this case becomes (64a) Equation (64a) gives the Current Amplitude in Circuit III for the Resonance Combination L. To get the current for Resonance Combination 0, we have already obtained D in the form of D in equation (51), so that by (61) \T 1 M " E ffi*\ 3max.max.Jo- Z^Rf+Rj This is the current amplitude for Resonance Combination 0. Likewise for Resonance Combination A, we use the value of D given in equation (58) and obtain ^ This is the current amplitude for Resonance Combination A. For Resonance Combination B, we use the value of D given as D B in (53), and obtain This is the current amplitude for Resonance Combination B. 232. Summary and Key to Results for Optimum Values of C 23 and C 3 o and for Maximum Values of I 3 , with Arbitrary Val- ues of XL We are now prepared to give a summary of results obtained up to the present. In this we shall use the following abbreviations, which have already been defined: 254 ELECTRIC OSCILLATIONS [CHAP. XV (68) (see (18)), B = L 2 u - (see (19)), (69) A\ R* = R* + M *f 2 Rl (see Table I, Art. 211) (70) *i* Among these quantities there exists the relation A 2 = # 2 2 + 5 2 (see (21)) (70a) A key of optimum relations and amplitudes now follows. I. If B ^ 0, Resonance Combination L, L. The optimum C 2 s and CZQ are C 23 = 0,C 30 .= co (71) and the max. max. current is rr ' L II. If B ^ 0, there are three combinations, 0, A, and B. 0. When # 3 ^ Rz, the optimum C 23 and C 30 are C 23 = 0, C 30 o> = l/B (73) and the max. max. current is " 3 max. maxJ ~ % A. When R 2 < R* < the optimum C 2 s and C 30 are B - A RMA 2 - C 23 co = ~ \/! 3 (A 2 - B,^0 4 2 (75) and the max. max. current is MuE (76) CHAP. XV] DETECTOR IN SHUNT 255 B. When ^4 < #3, the optimum 23 and Ca are = B/A.\ C 30 = <*> (77) and the max. max. current is / D E> \ A + * equations and the several criteria under which the equa- tions are applicable are given in terms of quantities A, B, Rz, and Zi, all of which involve Xi. For any given Xi the criteria in the form of inequalities enable us to select the proper Resonance Com- bination and to compute the value of Is max max . 233. Abbreviations in the Form of Ratio Quantities. For purposes of calculation, it is desirable to introduce into the pre- vious equations certain ratios of the obvious electrical constants or variables of the circuits. As in previous chapters let M* R, R 2 ( 79 ) ;-r = (80) The last of these is a new ratio, taking account of the third condenser of the system, and combining it arbitrarily with the second inductance. In addition to these ratios let us employ also the following: <" 9 V (81) (82) JJ>2* |1| |i| < 83 > o A 2 + . n . T . 2r 2 (77i772 Jl) /OA\ -* H ^ ' (84) L 2 2 2 co P = ^ (85) Hz 256 ELECTRIC OSCILLATIONS [CHAP. XV In some of the computations we shall replace also the inverse ratios of angular velocities by ratios of wavelengths, by writing CO _ O a ' X where X = wavelength of impressed e.m.f., AI, A 2 , A 3 = undamped wavelengths corresponding to the un- damped angular velocities fii,fi 2 ,fi 3 , respectively. 234. Summary and Key in Terms of Ratio Quantities. In terms of this set of ratio quantities, the summary given two sec- tions back can now be put into the following forms suitable for computations : 1. lib ^ 0, L. Use the resonance relations C 23 = 0,-C,o = oo and calculate the current by rE x. -I L II. If 6 > 0, 0. When (87) rW (88) p I- 3 max. max.J . When use the resonance relations C 23 = 0, and ^ 2 = ^ = | and calculate the current by rE use the resonance relations (89) CHAP. XV] DETECTOR IN SHUNT 257 flo 2 X 2 AL L = / x r ^ 2/ _ 2 (92) Pja 2 - pr]i(r) 2 + ~^j J and calculate the current by rE *max. max.J A (93) 2\/RiRz\ r 2 + r 2 \fji B. When r2 use the resonance relations A 2 ^ o= - (94) and calculate the current by r P \ ^2 /r 1 , + T 2 M2) VRiRs V pt]ir) 2 a |- a (95) 7n terms of the abbreviations (81) to (87) ^e several equations of this summary give the relations for calculating the optimum ad- justments of CM and Cso, for any given value of Xi, or the related quantity Ji. There are contained also in the summary the values of Izmax. max. obtained when these respective adjustments are made. Before proceeding to a theoretical determination of the adjust- ment also of the Circuit I to give what may be called a grand maximum of current, we shall give an illustration of the results up to the present by the aid of numerical computations. II. COMPUTATIONS IN A SPECIAL CASE 236. Power Developed in the Detector for Various Adjust- ments of the Primary Circuit, with Optimum Adjustment of Secondary Condenser C 23 and Stoppage Condensers C 3 o, in a Special Case in Which r 2 = 0.1, 771 = 0.03, t/ 2 = 0.01. If we take the squares of the current equations (88), (90), (93) and (95) 17 258 ELECTRIC OSCILLATIONS [CHAP. XV and multiply them by RiRz, we may obtain values of I^RiRz/ r*E 2 , which are proportional to the power developed in the de- tector, whose resistance is R z . This we shall do in a series of special cases in all of which T 2 = 0.1 77! = 0.03 772 = 0.01 with ? = 10 4 , 10 3 , 10 2 and 10 (96) The values of T, 771, and 772 are approximately those attainable in practice in radiotelegraphic receiving. As to the resistance 7 3 of the detector, reliable experimental values of this quantity are not at present available, and in fact this resistance is a function of the current, and is complicated by an action of rectification. Nevertheless, it is possible that experiment may subsequently separate out from the complicated action of the detectors a term of the character of pure resistance, and also new types of detectors, more nearly approaching constancy of resistive action, may be discovered. These calculations may then be of great importance in pointing the way to proper design of receiving apparatus. Table I gives a series of calculation of relative power developed in the detectors of various resistance R z relative to Ri. The quantity called relative power is arbitrarily defined as follows : Relative Power = -jr ( 97 ) Table I was made as follows: Taking various arbitrary adjustments of Ji of the primary circuit, values of fii/w were computed by (81). The result was put into terms of relative wavelengths, by employing the relation where Ai is the undamped, or forced wavelength, defined in Art. 66, Chapter VI. The values of the generalized wavelength divided by the impressed wavelength X, corresponding to the assumed values of Ji are put into the first. column of the table (Table I). Next, corresponding to the various values of Ji, the several CHAP. XV] DETECTOR IN SHUNT 259 Table I. Power Developed in the Detector R 3 at Optimum Adjustment of C 2 s and C 30 for Various Settings of the Antenna Undamped Wavelength Ai Relative to the Incident Wavelength X and for Various Ratios of Detector Resistance to Secondary Resistance. Given r 2 = 0.1, rji = 0.03, 772 = 0.01 Relative power developed in R 3 Ai r X Jl -' Kz 10 3 102 10 00 1.0 0.548 B 0.576A 0.576A 0.576A 1.58 0.6 0.889 B 1.14 A 1.14 A 1.14 A 1.41 0.5 0.941 B 1.37 A 1.37 A 1.37 A 1.29 0.4 0.889 B 1.62 A 1.62 A 1.62 A 1.20 0.3 0.691 B 1.92 A 1.92 A 1.92 A 1.12 0.2 0.265 B 1.63 B 2.20 A 2.20 A 1.104 0.18 0.191 B 1.35 B 2.25 A 2.25 O 1.084 0.15 0.095 B 0.80 B 2.33 A 2.26 1.05 0.1 0.0292B 0.2755 1.74 B 1.86 O 1.045 0.09 0.033LL 0.313L 1.85 L 1.71 L 1.04 0.08 0.0408L 0.380L 2.05 L 1.473L 1.03 0.06 0.0658L 0.585L 2.30 L 0.935L 1.02 0.04 0.117 L 0.952L 2.31 L 0.578L 1.01 0.02 0.221 L 1.52 L 2.05 L 0.377L 1.005 0.01 0.283 L 1.78 L 1.87 L 0.316L 1.00 0.00 0.367 B 1.95 B 1.76 0.2840 0.976 -0.05 0.660 B 2.47 B 2.47 A 0.8940 0.967 -0.07 0.774 B 2.45 B 2.45 A 1.30 0.953 -0.10 1.00 B 2.40 A 2.40 A 1.86 0.933 -0.15 1.24 B 2.33 A 2.33 A 2.26 O 0.921 -0.18 1.40 B 2.25 A 2.25 A 2.25 0.913 -0.20 1.46 B 2.22 A 2.22 A 2.22 A 0.877 -0.3 1.59 B 1.92 A 1.92 A 1.92 A 0.850 -0.4 1.49 B 1.62 A 1.62 A 1.62 A 0.816 -0.5 1.30 B 1.37 A 1.37 A 1.37 A 0.792 -0.6 1.12 B 1.14 A 1.14 A 1.14 A 0.707 -1.0 0.573 B 0.576A 0.576A 0.576 A criteria of the "Key" in terms of ratio constants were investi- gated, and the proper formulas for the computation of 7 3 max max> were selected, thereby imposing upon the system the requirement of an optimum adjustment of C 2 s and Cso. By (97) the relative power was then computed, and placed in the last four columns of the table, with each numerical value designated by a letter indicating the formula employed in the calculation. 260 ELECTRIC OSCILLATIONS [CHAP. XV 236. Discussion of Results for Relative Power. The results are plotted in Fig. 3, with relative power as ordinates and relative primary wavelength as abscissae. The separate curves marked respectively 10 4 , 10 3 , 10 2 , and 10 are for the ratio of resistances Rs/Rz equal to these values respectively. It is to be noticed that each of these curves has two maxima, except the 10 2 -curve, which has three maxima. Various parts of the .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 FIG. 3. Plot of Table I. The quantities 10, 10 2 , 10 3 and 10 4 give the values of R s /Rz for the separate curves. various curves of this figure (Fig. 3) were computed by various formulas, in accordance with the criteria relations of the "Key." The heavy black line serving as a sort of upper boundary of the figure was computed by the formula corresponding to Case A. In Case A the computed value is the same for all ratios Rs/Rz of resistances, so that wherever the criterion of Case A is satisfied by any adjustment of the circuits, the curve obtained comes into coincidence with this heavy bounding line. Each of the curves marked 10, 10 2 and 10 3 has a maximum near its junction with the A -curve. The curve marked 10 4 does not have any A -values CHAP. XV] DETECTOR IN SHUNT 261 within the range considered. The curve marked 10 2 has its third maximum (the middle one ) on a part of the curve calculated by the L-formula. This is very interesting, for in this region the condensers of the secondary and tertiary circuit are inoperative, one being zero and the other infinite, or short circuited. For .9 1.0 1.1 1.2 1.3 1.4 1.5 U6 FIG. 4. Optimum values of A 2 /X for various values of Ai/X. The 10, 10 2 , 10 3 , 10 4 attached to the various curves gives the value of Ra/Rz for each curve. this particular set of constants we have an efficient tuning system without any secondary condensers ! Certain other facts regarding these curves will be presented in a theoretical discussion to follow a presentation of tables and graphs of the optimum resonance relations in our special case. 237. The Resonance Relations in the Special Numerical Case. It is proposed now to give numerical results concerning the 262 ELECTRIC OSCILLATIONS [CHAP. XV optimum adjustments of C 23 and C 30 in the special case under con- sideration. Instead of tabulating the capacities it is more con- venient to tabulate A 2 /X and A 3 /X, where A 2 /X = 27rc\/Z/ 2 C 23 /X (99) A 3 /X = 27rcVWVX (100) .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Ai/X FIG. 5. Optimum values of As/X for various values of Ai/X, for different values 2 as designated by numbers attached to the separate curves. With these definitions it is seen, as has been repeatedly pointed out, A 2 /X = co/12 2 (101) As/A = o>/n 3 (102) CHAP. XV] DETECTOR IN SHUNT 263 These values are computed by the aid of the formulas for the resonance relations in the various cases given in the "Key/' and are tabulated in Tables II, III, and IV. The results are plotted in Figs.- 4 and 5. The several curves are numbered with numbers giving the ratio of Rz/Rt taken as the bases for the calculations. No especial comment will be given, except that these curves permit a determination of the optimum value of the two con- Table II. Resonance Relations in Case R 3 /R2 = 10 4 . Optimum Values of A 2 /X and of A 3 /X, for Various Values of Ai/X. Given r 2 = 0.1, Tji = 0.03, 772 = 0.01 Given Calculated optimum values Ai/X Jl A 2 /X A 3 /X Formula 0.707 -1.00 0.953 0.808 -0.60 0.927 0.815 -0.50 0.912 0.845 -0.40 0.895 ( 0.877 0.912 -0.30 -0.20 0.866 0.818 i q B 0.953 -0.10 0.714 0.976 -0.05 0.598 1.000 0.00 0.285 1.005 0.01 0.000 1.01 0.02 ! | 1.03 1.036 1.042 0.06 0.07 0.08 i 2 1 I ! L 1.046 0.085 1.049 0.091 0.397 1.051 0.095 0.657 1.053 0.100 0.968 1.061 0.110 1.325 1.068 0.125 1.60 a i 1.085 1.116 1.194 0.150 0.200 0.300 1.56 1.38 1.22 1 B 1.290 0.400 1.153 1.414 0.500 1.118 1.570 0.600 1.095 00 1.000 1.053 264 ELECTRIC OSCILLATIONS [CHAP. XV densers CM and C 30 in the several numerical cases, and point the way to a further theoretical examination following. Table IH. Case R 3 /R 2 = 10 3 , 10 2 , and 10. Optimum Values of A 2 /X For Various Values of Ai/X. Given r 2 = 0.1, 771 = 0.03, 772 = 0.01 Given Calculated optimum values Ai/X Ji Values of A2/X for Rs/Rz equal 103 102 10 00 1.0 1.035 A 0.993 A 0.842A .58 0.6 .072A 1.020A 0.829A .41 0.5 .090A 1.030A 0.819A .29 0.4 .122A 1.053 A 0.783 A .20 0.3 .170A 1 . 086A 0.717A .12 0.2 .3805 1 . 166A 0.382 A .104 0.18 .4905 1 . 190A 0.0000 1.084 0.15 1 . 5565 1.217A 0.0000 1.05 0.10 0.9685 0.9685 0.0000 1.045 0.09 1.04 0.08 1.03 0.06 Zero L Zero L Zero L 1.02 0.04 1.01 0.02 1.005 0.01 1.00 0.00 0.2875 0.0000 0.0000 0.976 -0.05 0.5655 0.135A 0.0000 0.967 -0.07 0.6325 0.352A 0.0000 0.953 -0.10 0.650 A 0.482 A 0.0000 0.933 -0.15 0.732 A 0.616A 0.0000 0.921 -0.18 0.762A 0.662 A 0.0000 0.913 -0.20 0.781A 0.691A 0.242A 0.877 -0.30 0.838A 0.773A 0.510A 0.850 -0.40 0.870A 0.814A 0.609A 0.816 -0.50 0.892A 0.844A 0.666A 0.792 -0.60 0.907A 0.861 A 0.701A 0.707 -1.00 0.936A 0.897A 0.764A The formula used in each case is that given by the letter following the number given in the table. CHAP. XV] DETECTOR IN SHUNT 265 Table IV. Case R 3 /R2 = 10 3 , 10 2 , and 10. Optimum Values of A 3 /X for Various Values of Ai/X. Given T Z = 0.1, rji = 0.03, 772 = 0.01 Given Calculated Ai/X Ji Values of As/X for Rt/Rz equal 103 102 10 oo 1.0 0.200A 0.356A 0.630A 1.58 0.6 0.226A 0.403 A 0.716A 1.41 0.5 0.237A 0.430A 0.765A .29 0.4 0.267 A 0.474A 0.842 A .20 0.3 0.313 A 0.556A 0.992 A .12 0.2 00 B 0.748A 1.33 A .104 0.18 CO B 0.820 A 1.50 .084 0.15 CO B 0.948 A 1.61 .05 0.10 oo B oo B 3.48 O 1.045 0.09 1.04 0.08 1.03 0.06 oo L oo L oo L 1.02 0.04 1.01 0.02 1.005 0.01 1.00 0.00 00 B 1.00 1.00 0.976 -0.05 00 B 0.620A 0.636O 0.967 -0.07 00 B 0.578A 0.673O 0.953 -0.10 0.295 A 0.525A 0.7220 0.933 -0.15 0.267A 0.476A 0.7800 0.921 -0.18 0.254A 0.456A 0.8070 0.913 -0.20 0.247A 0.440A 0.784A 0.877 -0.30 0.224A 0.395A 0.702A 0.850 -0.40 0.207A 0.369A 0.655A 0.816 -0.50 0.196A 0.354A 0.625A 0.792 -0.60 0:191A 0.341A 0.606A 0.707 -1.00 0.181A 0.324A 0.572 A The formula used in each case is that given by the letter following the number in the table. III. THEORETICAL INVESTIGATION OF THE GRAND MAXIMA OF POWER 238. General Note on Grand Maxima of Power in the Detector. An examination of Table I gives some notion of the adjustment for a grand maximum of power in the detector. In the first 266 ELECTRIC OSCILLATIONS [CHAP. XV place the values in the table presuppose that the optimum adjustments of C 23 and C 30 have been made, and the numbers in the last four columns are max. max. values of relative power, so that the maxima of the several values give max. max. max. relative power. To avoid the use of the term max. max. max. we shall call these values the grand maxima. The table shows that when there are any 5-values, the grand maxima seem to fall on the B-sections of the curve or at a point near the junction of the 5-section with the A- section. When there are no B-values, the grand maximum seems to fall on the 0-section near its junction with the A -sec- tion. In one of the cases there is a third grand maximum on the L-section of the curve corresponding to R 3 /Rz equal to 10 2 . These inferences from the special-case curves are now to be corroborated by a theoretical investigation, in which the actual values of the grand maxima of power are to be discovered. 239. Investigation of Grand Maximum of Power with Respect to Resonance Combination L. Let us designate the relative power developed in the detector R s by the letter H, defined as in equation (97) ; that is H = Relative Power = -^^- 3 (103) T j Comparing this definition with (88), it will be seen that, for Resonance Combination L, H,. = (104) In this expression r and b involve the reactance constants of Circuit I. We propose now to find the value of Xi (or of the related quantity Ji) that will make H L a maximum, and we shall then determine the magnitude of this maximum value, which we shall call the grand maximum with respect to Resonance Com- bination L. Whatever adjustment makes 7 3 2 a maximum, with r, # 2 , Rs and E fixed, will make H L a maximum under the same conditions. Referring to equation (5) and noting that in that equation Z'i alone involves Xi, and that in consequence the square of CHAP. XV] DETECTOR IN SHUNT 267 the current is made a maximum with respect to Xi, by making X\ 2 a minimum, we have X\ 2 = a minimum (105) for the determination of XL Writing out X\ by values from Table I, Art. 211, we have X f . ~ **** This must be solved as simultaneous with (87), in order to have all three variables of the circuit made simultaneously optimum, in those cases in which (87) is an optimum condition. We shall first make 30 = c, and C 23 = ~~ 1/7, following definition (4), and shall then make 7 approach minus infinity. The first step of this operation gives \rt _ V" _ 1 7 2 ) + 7 2 2 2 co 2 (107) As a second step, it is to be noted that as 7 approaches nega- tive infinity this expression approaches as a limit ' The minimum value of X\ 2 is then seen to be zero, which may be always attained if the condenser of the Circuit I is capable of taking all possible values. Setting X'i equal to zero, replacing 2 by its value from (4), and dividing by LICO, we obtain Ji = 8(1 + T2 p)2 + 1 (109) Equation (109) gives the value of J\ that produces largest Relative Power in the Detector, when the Resonance Relation L is fulfilled by C 2 s and C 30 . The magnitude of the grand maximum of relative power, obtained by substituting (109) into (104) is p where 6 = 1 + i72 2 (p + I) 2 (111) 268 ELECTRIC OSCILLATIONS [CHAP. XV In these equations P = R 3 /R* (112) Equation (109) gives the value of Ji at which occurs maximum power with the Resonance Combination L, and the value oftherelative power at this maximum is given by (110). 240. Investigation of Grand Maximum of Power with Respect to Resonance Combination 0. For this combination, by (73), C 23 = 0, C 30 a> = 1/B (113) In this case by (1) and (4) we may write Z 2 = L 2 o> + 7 (114) Z 8 = - B + y (115) introduce these quantities into (106), and take the limit as 7 approaches minus infinity, obtaining Replacing B by its value from (69), we have Z'i = Xi - - 1 ^ x ^ 2 (117) The solution of this equation for X\ = is either X l = (118) or Zi 2 = Expressing these results in terms of ratio constants, we have either J l = (119) or ,, " l2 + l2 To decide which of these conditions is to be used in a given case, it is only necessary to note that since CM is zero, we have the case of two circuits with the secondary circuit made up of the inductance L 2 , the capacity C 30 , and the resistance R s + R%. An examination along these lines, making use of Chapters XI and XII, shows that (120) is to be used whenever it is attainable. CHAP. XV] DETECTOR IN SHUNT 269 When it is not attainable (119) is to be used. The case for (119) is the case of deficient coupling, while the case for (120) is the case of sufficient coupling. The smallest value that Ji 2 can have is 0, so (120) is attainable, provided 2 ^ Tiyi . , , IX / 2 /10^ TJl ^ ~, :rr' 'I.e., f)l1^Z\P ~T~ *-) ^ T X-'-^-'-/ 7j2\P i 1J whence p < - 1 (122) TF/ien (122) ^ 4 s satisfied, the optimum value of Ji for Resonance Combination fc's grwen 6^7 (120). When (122) *s rioi satisfied the optimum value of Ji is (119). We shall now obtain values of the grand maxima of relative power in this 0-case. Substituting (90) into (103) we have 1 In case Ji = 0, r by its definition (83) reduces to rji, and then (123) becomes [//ma " ]o = .fio. ^ T M 2 ' f r Jl = (124) T"! | -" + ^;} In case if we first square the brace of the denominator of (123), and then replace r 2 , we have (125) TFe mot/ sum wp these results as follows: With Resonance Combination 0, if p ( = Rs/Rz) satisfies the inequality (122), the optimum value of J\ is given by (120). The value of this power is given by (125). //, on the other hand, p does not satisfy the in- equality (122), the optimum value of Ji is given by (119), and the value of the relative power at this maximum is given by (124). 270 ELECTRIC OSCILLATIONS [CHAP. XV 241. Investigation of the Grand Maxima of Power with Respect to Resonance Combination A. If we substitute (93) into (103) we shall have the Relative Power HA = \ - ~ (126) ^ + r ) \ri / In this equation, r 2 , which is denned by (83) contains the reactance constant /i, and it is seen by inspection that the adjustment of Ji that makes (126) a maximum is Ji = 0, and that the value of H A at this adjustment is [Hm ^' ]A = 4(w, + r) (127) The condition under which this maximum is attainable is had by setting Ji = in the criterion inequality given immediately preceding equation (91) in the " Summary and Key in Terms of Ratio Quantities," Art. 234. This operation gives 1+ < P < 1 + + m , 2 , (128) + r ) The Resonance Combination A is attainable if the inequality (128) is satisfied by p (= R^/R^), and the optimum value of Ji is J\ = 0. The value of the relative power at this adjustment is given by (127). 242. Investigation of the Grand Maxima of Power with Respect to Resonance Combination B. The substitution of (95) into (103) gives HB = r f { p(iy a + (129) Indicating the denominator of this expression by Z), we shall have in expanded form D } (130) CHAP. XV] DETECTOR IN SHUNT 271 It is required to find the value of Ji that will make D a mini- mum. Setting equal to zero the derivative of D with respect to Ji we have , r (131) i , : I 1 (r 2 a 2 ) 2 I dJ, The second brace is a common factor that cannot vanish. It may be divided out. Doing this and replacing the deriva- tives obtainable from (82) and (83), we have ' - 1 - I d+ *V. - r*l + 2W.V, (132) Clearing this of fractions, we obtain = rV{ (1 + 772 V! - r 2 + 2 P 77 2 2 Ji) - Let us write out the value of r z a 2 by using (83) and (84), obtaining r 2 a 2 = (1 +T72 2 ) (r?i 2 + JV) + r 4 + 2r z (rj m - Jj (134) Note also, by (83) r 2 = ^2 + jj ( 135) The values given in (134) and (135) substituted into (133) gives = P^JS 4- P 2 J 1 2 + p l j l + p (136) where Ps = (1 + r7 2 2 + pi7 2 2 ) (1 + ^ 2 ) P 2 = (1 + *7 2 2 + P^/2 2 ) (~ 3r 2 ) Pi = {1 + r7 2 2 + 2p77 2 2 } {(1 + 77 2 2 )77i 2 + r 4 + 2r 2 77i772} + 4r 4 p77i772{l + 772 2 } {r 2 .+ 771772} PO = r 2 (r 2 + 771772) 2 + T 2 77i77 2 p (r 2 + 77i77 2 ) r 2 77i 2 (137) When the quantities on the right-hand side of equations (137) are numerically known, the cubic equation (136) may be solved by "trial and error" or by other known methods of solving a cubic equation with numerical coefficients. The cubic equation (136) gives the value of Ji at which occurs a grand maximum (or a minimum) of relative power with respect to the Resonance Combination B. 272 ELECTRIC OSCILLATIONS [CHAP. XV From the solutions obtained for the cubic in any numerical case, one must decide by a separate investigation which of the solutions give maxima and which minima of power, and one must deter- mine the value of the grand maximum of power by substituting the resulting value of J\ into (129), in which a and r are functions of J l as defined in (83) and (84). We shall follow the exact treatment here given by approxi- mations that are useful in important cases. 243. Approximate Treatment of the Grand Maximum of Power with Respect to Resonance Combination B. Instead of employing the cubic equation (136) to determine the value of Ji at which occurs a grand maximum of relative power with respect to Resonance Combination B, we may obtain an ap- proximate result as follows : In the value of r 2 a 2 given in (134) let (138) and J! 2 + T< then by (134) r 2 a 2 = (Ji r 2 ) 2 approximately (139) In advance of a determination of Ji 2 we know that it is positive, so that conditions (138) are satisfied, provided TJi 2 1, Tj! 2 T 2 , rU>?2 < < T 2 /2 (140) The inequalities (140) give conditions under which the approxi- mation (139) is applicable. It may be that (140) is more restrictive as to rji and rj 2 than is necessary. From (139) this is seen to be the case when Ji 2 is sufficiently different from zero to add appre- ciably to r 4 on the right-hand side of (138). If now we substitute (139) into (133), and neglect further i? 2 2 where it occurs in comparison with unity, we obtain (7" 2 ) ( 2 21 2 ) f "1 A ~t \ which, factored and with r 2 replaced by its value from (135), gives = [J 1 - r 2 ) {(Ji - r 2 )[(l + 2 P 7 ?2 2 )J 1 - r 2 ] - 0?i 2 -|-Ji 2 )p??2 2 T 4 ~ (142) CHAP. XV] DETECTOR IN SHUNT 273 Setting these two factors separately equal to zero, and solving for Ji, we obtain either J l = r 2 (143) Ji = r* Equations (143) and (144) give approximate values of J\ at which grand maxima (or minima) of power occur with respect to the Resonance Combination B. We shall next show that (143) is the condition for a minimum, and that (144) is the approximate value for a grand maximum. This result will be incident to a determination of the magnitude of the Power-maximum. 244. The Magnitude of the Relative Power with Respect to Resonance Combination B. Before we introduced any approxi- mations into the examination of the Resonance Combination B, we found that J\, in order to give a grand maximum of power, must satisfy (133), which was subsequently put into the form (136). We may, therefore, utilize (133) as far as possible to simplify the power equation (129). Concerning ourselves par- ticularly with the denominator of (129), which is given in (130) we may write (130) in the form D = rW + 2x + - (145) where, as an abbreviation x = P (rV + rV?) (146) We may factor (145) so as to give Equation (147) gives the denominator of the power equation (129), before any resonance conditions regarding Ji are introduced. We shall now transform (133), by introducing the abbreviation x. This gives (148) Equation (148) is the equivalent of (133), and is the relation that Ji must satisfy to give a maximum (or minimum) value of power H B . 18 274 ELECTRIC OSCILLATIONS [CHAP. XV Substituting (148) into (147) in such a way as to eliminate r 2 a 2 we obtain -, , 2P772 2 /! (1 + IfcVl - T 2 Simplifying this expression, replacing x by its value from (146), with r replaced by (135), and introducing the resulting value of D into (129), we obtain r/7 i ' mK 1 + ttVi - r2 + W-filKl + 2 - 2 maxjB equation (150) /ie values of J\ given by the cubic equation (136) must be introduced. The resulting values mil be either maxima or minima of H. Only the maxima are to be selected, and are to be used with the other adjustments incident to the Resonance Combination B. 245. Approximate Magnitude of Relative Power with Respect to Resonance Combination B. In equations (143) and (144) we have found approximate values of J\ that give either a maxi- mum or a minimum of power with respect to Resonance Combina- tion B. We may write (144) Ji = r 2 + a (approximately) (151) where a + JfrW + *V + r \ 1 -f- p7?2 (152) Introducing (151) into (150), with neglect of 7/ 2 2 in comparison with unity, we obtain , ! _ _ _ " 2 2 2 2222 ^ It is seen from this equation that J\ = r 2 gives a minimum of power, for this is equivalent to making a = in (157), and gives relative power zero, to the accuracy of the approximations employed in deducing (153). Using the approximate resonance value of Ji given in (151), equation (153) gives an approximate value of the grand maximum of power with Resonance Combination B. We shall now make a collection of the several optimum Res- onance Combinations, which are the Resonance Combinations CHAP. XV] DETECTOR IN SHUNT 275 L, 0, A, and B, with, however, their corresponding optimum values of Ji also taken into account. 246. Collection of Optimum Resonance Combinations. In the "Summary and Key in Terms of Ratio Constants, " Art. 234, we have given a list of Resonance Combinations designated L, 0, A, and B. In the pages following the Key we have determined the value of Ji that will give grand maximum expenditure of power in the resistance R$ for each of the Resonance Combinations. When Ji is thus made optimum with the several Resonance Combinations, we shall designate the combinations Optimum Resonance Combinations L, 0, A, and B.- Incidentally there, have appeared two Optimum Resonance Combinations 0, which we shall refer to as Oo and Oi. In stating the various combinations, we shall need to introduce the optimum value of Ji into the statement of the optimum ad- justment of the Circuits I and II whenever these adjustments are functions of Ji. We shall also collect along with the Optimum Resonance Com- binations the values of the Grand Maximum of Relative Power for those combinations. We shall later, where possible, lay down rules as to which of the optimum combinations is to be used for any given relation among the resistances of the three circuits. L. The optimum resonance combination L is C 23 = 0, C 3 o=oo, J t = (154) and the relative power at this adjustment is where 0=1 + 7722(1 + p) 2 (156) P = R 3 /R 2 (157) . The optimum resonance combination is C 23 =0, ^ = 1, J l = (158) 276 ELECTRIC OSCILLATIONS [CHAP. XV and the relative power at this adjustment is r r r 1 2 \ 2 P \ This combination is not to be used when P<^-I Oi. The optimum resonance combination Oi is C 23 = A 3 2 1 1 + ) f^V~ -Vr^+l) ~ *' and the relative power at this adjustment is , = 1 A T 2 ' is combination can be attained only provided A. The optimum resonance combination A is X 2 - (^2 + r 2 ) 9i 1 + /W2 + r\ 2 VI '} * X 2 1 + M2 f 1 hM* + T 2 \ 2 p?72, 2 s p r + l^r/ --(^ 2 + - 2 ) Ji = and the relative power at this adjustment is (150) (160) (161) (162) (163) (164) (165) CHAP. XV] DETECTOR IN SHUNT 277 This combination can be attained only provided (166) + T 2 ) B. The optimum resonance combination B is accurately ^ = ^, C 30 = (167) with Ji a root of the cubic equation Po + PiJi + P 2 /i 2 + P 3 Ji 3 =0 (168) with the values of P , PI, P 2 , and P 3 given in equation (137). To obtain the values of [H m&x ] B) the values obtained for J\ are to be substituted into (150), and then the minimum values, if any appear, are to be discarded. To obtain the adjustment appropriate to Circuit II, the values of Ji that give maximum values of H are to be introduced into b and a 2 of (167), in accordance with the definitions of b and a 2 given in (82) and (83). B. Approximate : When Th 2 l, 7 7l 2 << r 2 ,andr 7l r 72<< r 2 /2 (169) the optimum resonance combination B is approximately Aa 2 = T?! 2 + r 2 a + a 2 X 2 r^On 2 + 2r 2 a + a 2 ) + fa + T^r 2 ) 2 + a 2 (17Q) Jl = T 2 + OL and the relative power at this adjustment is [#ma*.]* = j( T 2 + a )2^ a + yfa + rVjj^l _|_ pl?2 2) + p7?2 2 r 2}2 (171) where a = fcV + tiV + T \ 1 + pr 2 2 pr; 2 247. Comparison of the Grand Maxima of Power for the Sev- eral Optimum Resonance Combinations. By comparing the values of the relative power (H max ) for the several combinations, we are able to decide which combination gives the greatest rela- tive power for any given value of p ( = Rs/Rz) - The results are given in Table V, 278 ELECTRIC OSCILLATIONS [CHAP. XV Table V. Proper Optimum Resonance Combinations for Different Values of R 3 /R 2 Value of p ( = Rz/Rz) Use optimum resonance combination designated < p < - 1 + -^ Oi 771772 ~ 771772 Oo 1 1 T * x p ^ 1 1 r2 1 7?1 771772 - ' ' 771772 772(7717/2 + T 2 ) T 2 , T?! 771772 772^771772 ~r T ) Table V was obtained (by steps not here given) by noting first that the optimum combination A could be attained only when p was within the limits assigned in (166), which are the limits given in the third line of the table. By subtractipn of the denominator in the expression for Power in the A -case from the corresponding denominator in the -case, it was found that the denominator in the A -case was always the smaller, so that combination A, when attainable, gives more power than combi- nation . It was next noted that combinations A and Oi are never attainable together, since the upper value of p for which Oi is attainable is ^- 1+ i ' , - by (163). It was then shown by subtracting power-denominators that combination Oi always gives more power than combination so that Oo is to be used only when Oi and A are both unattainable. This range is given in the second line of Table V. We are left in doubt up to here whether B or L should replace Oi, , or A in the ranges corresponding to the first three lines of Table V. A subtraction of the denominators in the case Oi, Oo, and A successively from the denominator in the case of com- bination L, shows that the combination L is not superior to any of the other combinations within the ranges given in the table. As to combination B, it is in such a form that there is difficulty CHAP. XV] DETECTOR IN SHUNT 279 in determining by direct subtraction whether or not the power for the B combination is greater than that for the other combi- nations. We find, however, that the B combination gives Ji = 0, when P = 1 + + -T- -- r (173) which is the adjustment of Ji for the A-combination. Also at this adjustment the value of the power for the 5-combination agrees with the value of the power for the A-combination. The inference from this is that the ^-combination has applica- tion to values of p greater than the limit given in (173), and this inference is entered in Table V. IV. COMPUTATION OF OPTIMUM ADJUSTMENTS AND GRAND MAXIMA OF POWER IN A SPECIAL CASE 248. The Optimum Adjustments of the Primary Circuit (Cir- cuit I) in a Special Case, with r 2 = 0.1, TH = 0.03, ij 2 = 0.01. In the "Collection of Optimum Resonance Combinations," Art. 246, there are given formula for computing the adjustments of the constants of the circuits to produce maxima of relative power in Circuit III. We shall here give the adjustments of Circuit I, in the form of values of Ji, where /i = 1 - -p; (174) Ai~ where X = the wavelength of the impressed e.m.f. AI = the undamped wavelength of Circuit I. The optimum value of Ji, which is the quantity computed will be sometimes designated Ji op t.- With the values of r, 771, and 772 given in the caption, I have computed the values of J\ opt . for various values of the ratio Rs/Rz, where R% is the resistance of Circuit III containing the detector, and R z is the resistance of the Circuit II. The values employed for Rz/Rz extend from 1 to 100,000. Fig. 6 gives the values of Ji op t. at which the grand maxima of power occur for values of Rs/Rz up to 700. The different parts of the curves are labelled to accord with the optimum resonance combinations L, 0, A , and B employed in their computation. The actual amount of relative power for these adjustments are given in the next section. 280 ELECTRIC OSCILLATIONS [CHAP. XV -.4 100 200 300 400 500 600 FIG. 6. Values of J"i pt. for various values of R 3 /Rz. The letters attached to the curves indicate the resonance relations employed. Table VI. Optimum Values of Ji for Large Values of R 3 /R2, with the Given Values of r, 771, and r) 2 opt. 7,000 0.457 -0.257 8,000 0.471 -0.271 9,000 0.483 -0.283 10,000 0.493 -0.293 15,000 0.531 -0.331 20,000 0.554 -0.354 50,000 0.609 -0.409 100,000 0.632 -0.432 00 0.657 -0.457 CHAP. XV] DETECTOR IN SHUNT 281 Continuing the examination merely of the optimum values of Ji, Fig. 7 contains the same curves as Fig. 6, with, however, a different scale for Rs/Rz, and an extension of the results to values of R 3 /R 2 up to 7000. +o +L -1 -.2 -.3 -4 7 -o --O +B r> 1000 2000 3000 4000 5000 FIG. 7. Extension of Fig. 6. 6000 7000 Beyond the ratio of resistances R^/Rz equal to 7000, curves are not given, but the computed results are contained in Table VI. In all calculations involving Combination B the approximate equations (170) and (171) were employed. 282 ELECTRIC OSCILLATIONS [CHAP. XV 249. Magnitudes of the Grand Maxima of Power for Various Values of R 3 /R 2 . Given r 2 = 0.1, r il = 0.03, 7? 2 = 0.01. Using the formulas collected in equations (154) to (172) and employing resistance ratios from 1 to 50,000, values of the relative power expended in the detector were computed in the special case of r 2 = 0.1, r?i = 0.03, 172 = 0.01, with the results given in Figs. 8 and 9. FIG. 8. Grand maximum of relative power vs. R z /Ri- Fig. 8 is for the range of R 3 /R 2 from to 700. Fig. 9 is for the range of Rz/R 2 from zero to 14,000. The extension of the range to 50,000 is given in Table VII. An examination of the curves of Figs. 8 and 9, and Table VII shows that with this particular set of constants, r, 771, and 772, the detector in which the greatest power is developed has a resistance between 150 and 600 times the resistance of the secondary induc- tance coil, and that the optimum adjustment of the circuits comes under the cases of Optimum Resonance Combinations A and B. As the resistance increases beyond 600 times the resistance CHAP. XV] DETECTOR IN SHUNT 283 of the secondary coil, the power expended in the detector de- creases. With a different coefficient of coupling and different values of 771 and r/ 2 this optimum range of resistances for the detector is different. The problem is too diversified to permit of exhaustive nu- merical examination. Magnitude of Grand Maximum of Relative Power=Hmax. k J- 1 i ' (-J t i tO tO tO K t~ 4^ COO tO tf*. C5 OO O tO * ^^ -B \ ^->> ^^, ^^ \ "^ ^^ ^-v ^ ^ "^" "^< \ " . ^^ 2000 4000 6000 8000 10000 12000 140(M FIG. 9. Extension of Fig. 8. Table VII. Rel. Power in Detector at Opt. Adjustments for Large Values of R 3 /R 2 Relative power at /13//12 Positive maximum Negative maximum 7,000 1.15 1.82 10,000 0.97 1.60 20,000 0.61 1.11 50,000 0.30 0.58 In these Fig. 8 and 9 and in Table VII a maximum, of relative power is called a positive maximum, or a negative maximum according as a positive or a negative value of J\ is used in its computation. 284 ELECTRIC OSCILLATIONS [CHAP. XV V. SOME GENERAL CONCLUSIONS 250. Form of Circuits. The discussion in this chapter per- tains to circuits of the form of Figs. 1 and 2, Art. 220, in which the detector and a stoppage condenser C 30 are shunted about the secondary condenser C 23 . The detector may have any resistance Rz whatever, and none of the resistances of the various circuits are neglected. 251. When Should C 23 be Zero? The question arises as to when is it advisable to have a condenser at C 2 3, and when do the resonance devices of the previous Chapters XI and XII without such a condenser give larger secondary current and larger power development in the detector? The answer is found in a consideration of equations (158) and (161) and of Table V. It is seen in (158) and (161) that C 23 = for combinations and Oi, and in Table V it is seen that these combinations give grand maxima of power whenever f- 3 < + 1+ (175) KI 171172 Equation (175) gives the condition under which the tuning of Chapters XI and XII with the secondary coil, the detector, and a variable condenser in series (without the C 23 of the present chapter) , will give more power in the detector than any adjustment with the use of C 23 (note, C 2 of Chapter X is the C 30 of present Chapter) . 252. What is the Best Value of the Stoppage Condenser C 30 for Detectors of High Resistance. For detectors of sufficiently high resistance to make Rs/Rt :> 1 + + , 1?1 , 2 . (176) 172(171172 + T 2 ) the optimum resonance combination is combination B, and in this case the value of the stoppage condenser C 30 is infinite. We have then the interesting result that, except for possible requirements of the telephone receiver used as an indicating in- strument, the stoppage condenser should be infinite whenever the detector resistance is sufficiently large to satisfy (176). In our numerical example, (176) becomes R 3 /R 2 ^ 364.3. CHAPTER XVI ELECTRICAL SYSTEMS OF RECURRENT SIMILAR SEC- TIONS. ARTIFICIAL LINES. ELECTRICAL FILTERS 253. Utility. The study of the electrical transmission char- acteristics of various systems of circuits that consist of recurrent sections in the form of a chain is highly interesting and important. Circuits of recurrent sections are employed as artificial tele- phone and telegraph lines. 1 By properly choosing the sections a line similating telephone and telegraph lines or cables may be constructed and employed in electrical experiments in the place of the actual lines. Circuits of recurrent sections may also be employed as elec- trical filters 2 for eliminating disturbances from telephone and telegraph circuits. It is believed that such filters may come to have a wide application to the elimination of disturbances from radiotelegraphic receiving stations. Such filters have also in- teresting applications to bridge measurements and other labora- tory operations, in which it is desired to eliminate harmonics and other disturbances. Further, by properly choosing the constants of the sections the electrical artificial line may be employed to introduce pre- determined time retardation of electric currents in a way that gives time retardation practically independent of the frequency 1 An artificial line with resistances in series and condensers in shunt was patented by Varley in 1862, British Patent No. 3453. A similar line but with uniformly distributed capacity and resistance was patented by Taylor and Muirhead, British Patent No. 684, of 1875. A line with uniformly distributed inductance, resistance, and capacity was made and described by Pupin, Trans. Am. Inst. of El. Engineers, 16, pp. 93-142, 1899. Another form of uniformly distributed artificial line was constructed and described by Cunningham, Trans. Am. Inst. of El. Engineers, 30, pp. 245-256, 1911. For further references and for an extended treatment of the subject see a recent book by Kennelly, "Artificial Electrical Lines," McGraw-Hill, 1917, from which the above references are taken. 2 G. M. B. Shepherd, "Note on High-frequency Wave Filters," The Electrician, 71, pp, 399-401, 1913. G. A. Campbell, U. S. Patent No. 1227113, 1917. 285 286 ELECTRIC OSCILLATIONS [CHAP. over wide ranges of frequency. This has been utilized by the author 1 in an electrical compensator employed in determining the direction o sources of sound, particularly under water, in sub- marine boat detection and in submarine signalling. Similar devices are applicable to direction-finding by electric waves and to the elimination of interference in radiotelegraphy by directive receiving. The principles to be developed in the study of these systems of recurrent sections will serve to show their general application, and will serve also as an introduction to the study of electric waves on wires, to be treated in the next later chapter. I. GENERAL SYSTEM OF EQUAL SECTIONS 254. General Type of Circuits. Notation. The discussion will be limited to a system of recurrent sections that are all equal, except at the terminals of the system. A system of this character, but with considerable generality as to the nature of the sections, is shown in Fig. 1. The complex impedances z , zi, z% } and Z T may each consist of any combination of capacities, inductances, and resistances. Each of the complex impedances 22 is common to two circuits and is of the nature of a mutual impedance. The impedances z\ are not common to two circuits, but for the sake of generality there is assumed a mutual inductance 2 between the elements z\ of each pair of adjacent loops, but no mutual inductance between loops not adjacent. The complex impedances z and Z T are the impedances of the terminal apparatus at the two ends of the system. , . . ' What may be called the line proper, exclusive of the- terminal impedances, ends in a half section si/2 at each end. 255. General Equations. We shall designate the complex current through the non-common elements of the successive loops as ? , ii, iz) is, . . . i n - i, i n > These currents are supposed to be positive when in the direction of the arrows marked i , ii, etc. The current io flows through the terminal impedance ZQ at 1 Description in U. S. Navy Archives and in pending U. S. patent application. 2 This mutual inductance may be made zero to suit cases in which no such mutual inductance exists. CHAP. XVI] LINES AND FILTERS 287 the input end, and the current i n through the terminal impedance Z T at the output end. We shall treat the problem for only the steady-state 1 condition, under the action of a sinusoidal impressed e.m.f. If we let the impressed e.m.f. be replaced by an exponential expression e and if we let (1) to = ii = A***' is it is seen, as in Chapter XIII, that the complex amplitudes of current A\, A?, . . . A n will be required to satisfy the fol- lowing algebraic equations obtained from Kirchhoff's e.m.f. law: E = z' = bA + zA = .bAi + zA ^ 6A 2 + zA + bA = where, as abbreviations, 2 . = 6 = -i +'z"A z z (2) (3) Method of Making All of the Equations (2) Symmetrical. It will be noted that all of the equations (2) may be made sym- metrical if we write and A n i (z f - z)A - E b (z" - z)A n n +i (4) (5) With these definitions of A_i and A n+ i, which have no other 1 A treatment of the transient state is given by J. R. Carson, Proc. Am. Inst. of El. Engineers, 38, p. 407, 1919. ELECTRIC OSCILLATIONS [CHAP. XVI meaning than that given them by the equations (4) and (5), we may replace E in the first equation of (2) and z" in the last equation of (2), obtaining for the whole set (2) = 6^_i + zA + bA! = bA + zAi + bA = bAi -f- zA 2 + bA (6) Each of the equations (6) is now seen to be of the generic form = (7) Equation (7) is a generic equation showing the relation of the complex current amplitudes in adjacent sections of the system of the form of Fig. I. This equation in which m is to be given values FIG. 1. General system of recurrent equal sections. Complex impedances zi and zz may be of any character. corresponding to the subscripts in (6), when taken in conjunction with (4) and (5) enables us to obtain a complete solution of the problem of determining the currents in the steady state. 256. Solution for Complex Current Amplitudes. 1 The equa- tion (7) may be shown to hold for all values of m. For our pur- pose it will be sufficient to show that it holds for values of m from m = ltora = n + l. We have already seen in (6) that equa- tion (7) holds for values of m from and including to and includ- ing n. To show that the generic equation (7) holds for m = 1, let us write down the equation that results from makin'g m = 1, obtaining = 6A_ 2 + :-A_! + bA . 1 In the theoretical treatment of this subject I have followed the method outlined in G. A. Campbell's U. S. Patent No. 1227113, 1917. CHAP. XVI] LINES AND FILTERS 289 This amounts merely to a definition of A_ 2 in terms of A-i and AQ, and since A_ 2 has no physical meaning in the problem, we can make this definition, and shall make no further use of it. In like manner one can satisfy himself that (7) holds also for m = n -+- 1. We shall now proceed to a solution of (7), which is of a form known as a difference equation. The known method of treat- ing this equation consists in assuming that A m = Ge^ (8) where G is independent of m. Substituting (8) into (7), and giving to m successively the values m 1, m, and m + 1, we obtain = G{be k(m ~ 1} + ze km + be k(m + 1} } (9) whence it appears that G is an arbitrary constant. Now dividing (9) by Ge km , we obtain on transposition, or otherwise written (11) We may write this result in still a third form by solving (10) as a quadratic in *, and this gives (12) Let us note also that if k satisfies (10) then k also satisfies it, since k and k enter into (10) symmetrically. Therefore we have another solution of (7) in the form A m = He~ km (13) where H is also arbitrary and independent of m. In order to distinguish between k and k, both of which are, in general, complex quantities, we shall specify that k has its real part positive. Now, since (7) is linear and homogeneous in A m , the sum of the two solutions is a solution; hence A m = Ge km + He~ km (14) 19 290 ELECTRIC OSCILLATIONS [CHAP. XVI where G and H are both arbitrary and independent of each other and of m. Equation (14), since it contains two arbitrary constants, is known by the theory of difference equations to be the most general solution of the given difference equation (7). In (14) k has the value given by (10), (11), or (12). 257. Introduction of Terminal Conditions, and the Determi- nation of the Arbitrary Constants G and H. To obtain the values of the arbitrary constants G and H, let us substitute (14), with proper value of m, into (4) and (5). We thus obtain (z'-z)(G + H) E Ge~ k + He k - b b and Lre 1 ^'* r *' -f As abbreviations fie ~ v " ' " i b let us write , whence by (3) x = 17 whp.nop. hv (%} 11 -\\jre"'" -|- ne f z - z 2 - Zi/2 Viu; (17) (18) 6 X and z"- z ': b ZT #2 Zi/2 i **j w/ y L By transposition of (15) and (16) and by the employment of (17) and (18), we obtain G(e~ k - x) + H(e k - x} = - E/b (19) and Ge kn (e k - y) + He- kn (e~ k - y) = (20) As further abbreviations let us write e~ k X e k x Y = - V* ~ y (22) * - y then (20) gives G = HYe~ 2kn (23) The substitution of (23) and (21) into (19) gives H _ _A i_ b(e k - x) 1 - e 2kn XY and by (23), E Ye~ 2kn G = r7~r -9k n w (25) CHAP. XVI] LINES AND FILTERS 291 These values of G and H substituted into (14) gives for the complex current amplitude A m the value E km + 7 -*(2n - m) A m = - - XYe~ 2} " (26) b(e k -x) Equation (26) gives the complex current amplitude A m of the current in the mth section. X and Y are given by (21) and (22); x is given by (17); k, by (10), (11), or (12). 258. Analysis of the Complex Current Amplitude into a Sum- mation, Exhibiting the Effects of Repeated Reflection. The ex- pression (26) for A m may be put into a more interesting form by expanding one of the factors as follows : Introducing this into (26) we obtain E (27) A m = - b(e k -x) -km ~ m ) -f X Ye~ k (2n + m } + XY 2 e~ k(4n ~ m) + ......................... } (28) This is a variant equation for A m , the complex current ampli- tude in the mth section of a line that terminates with the nth section. In equation (28) it is to be noticed that the multipliers of k in the exponents of the successive terms are as follows : Term Multiplier in exponent Interpretation First m = the number of steps to the mth section e.m.f. direct. from Second 2n m = the number of steps to outer end and back to the mth section. Third Etc. 2n + m etc. = the number of steps to outer end, back to be- ginning end, and then to mth section, etc. These several exponential terms are consistent with the view that the first term is due to direct transmission from the source, while the succeeding terms are due to successive reflections of current from the terminals of the line. Each step from section to section, on this theory, multiplies the complex current amplitude by the constant (complex) factor e~ k . 292 ELECTRIC OSCILLATIONS [CHAP. XVI To account for the multipliers X and Y applied successively to the terms after the first, it is only necessary to suppose that Y is the complex reflection coefficient of the terminal of the line remote from the e.m.f., and that X is the complex reflection coefficient of the terminal at the e.m.f. 259. Complex Current Amplitude in the mth Section of an Infinite Line or of a Line with Non -reflective Output Impedance. If the total number of sections n is infinite, or if Y is zero, all the exponentials in (28), except the first, disappear, and we have A m = I e~ km , f orn = (40) where a and

m) CHAP. XVI] LINES AND FILTERS 295 If now our original impressed e.m.f. e is 6 = E sin ut, instead of e = E^ ut j the expression for i m would have the exponential with imagi- nary exponent replaced by a sine function, giving W i m = = ~ am sin [wt 9' - 4>m} (4i a ) *'' Equation (4 la) gives the real current in the mth section of a line without reflection at the output end, and shows that a is the attenuation constant and the retardation angle (of current) per section of the line. Determination of a and a = sinh- 1 , m , _ m , _ , 4 sin' (1 _ f/2 _ p 2) : (46) (47) Let us now write as an abbreviation V = 1 " U * " P2 (48) a = sinh- 1 { + f/ 2 + F 2 - 7} (49) then ? = sin- 1 { + *7 2 + 7 2 + V} (50) Equations (46) emd (47), or Jfte alternative equations (49) and (50) a, which is the real attenuation constant for the current per section of the line, and L> L> LK FIG. 3. Line of Type II. ^ M ^ FIG. 4. Line of Type III. The line in Fig. 4, designated Type III, is similar to Type II except that there is mutual inductance M between the parts of coils common to two loops. The condensers C% are tapped to the mid points of these coils. It is to be noted that while the inductance per loop is L$, this is not the inductance per coil. 300 ELECTRIC OSCILLATIONS [CHAP. XVI The inductance per coil will be called L, in Type III. The line of Type III is called a line of T-sections. 268. Reference to Type I. We shall now determine P for Type I. In this case the reactances are X l = - 1/Ciw, X 2 = L 2 co, M = (64) These values substituted into (53) give Po = 1 ~ 2L^ (65) Then by (61), a = 0, provided that is provided (66) If as an abbreviation we write LIT = fi2 (67) the inequality (66) becomes \<^<> (68) With a system of Type I, Fig. 2, which is supposed to have zero resistance, the attenuation is zero for all currents of angular ve- locity greater than $2/2, where $2 has the definition given in (67). On the other hand, for all currents of angular velocity co less than $2/2, it is seen by reference to (63) that the attenuation constant is (69) This type of circuit lets through without attenuation frequencies higher than a specified value, and attenuates frequencies lower than the specified value, and attenuates them more the lower their frequencies. Computations and curves will be given later. CHAP. XVI] LINES AND FILTERS 301 269. Reference to Type II. In case of the resistanceless system of Type II, Fig. 3, the impedances are Xi = Lico, X 2 = -1/Cjw, M = (70) so that p _ i LiC 2 w 2 ~ 2 ' In this type of system, we shall have, by (61), a = (72) provided -1'1-M*^+1 (73) that is, provided 2 > %> (74) where now 12 2 = I/LtCz (75) On the other hand, if I > 2, then a = cosh-' {^^ - l) (76) a resistanceless system of Type II, Fig. 3, ^e attenuation is zero for currents of all angular velocities co Zess than 212, wfore fi /ias the value given in (75). On the other hand, for currents of angular velocities greater than 212 the attenuation is given by (76) and increases with increasing value of the angular velocity. This type of circuit lets through frequencies lower than the speci- fied value without attenuation, and attenuated currents of frequencies higher than the specified value. 270. Reference to Type III. If a line of Type III, Fig. 4, is made up of elements of zero resistance, the reactances are X = Li, X* = - l/C 2 co, M = M (77) If we think of the inductance elements as made up of coils, as ABj having inductance L and tapped at their mid points for the attachment of the condensers, it is to be noted that the in- ductance LI is made up of two coils in series, each of which has the inductance of a half -coil A B. The mutual inductance M is 302 ELECTRIC OSCILLATIONS [CHAP. XVI the mutual inductance between two half coils; whence, if there is no magnetic leakage, M = Li/2 (78) If there is magnetic leakage, M < Li/2 (79) Let us now introduce the coefficient of coupling T, between two adjacent loops. Then r 2 = M*/VDiLi, whence r = M/Li < 1/2 (80) Now introducing (77) into (53) we obtain T " C 2 co L^co 2 - 2 Introducing this value of P Q into (61), we obtain a = 0, provided that is, provided - 2 ^ ' (83) In terms of r, the inequality ( 83) can be written co ^ (84) Corollary. If the coils, as AB, have no magnetic leakage, then by (78), equation (84) becomes oo ^ co ^ (85) With a line of Type ///, Fig. 4, having mutual inductance be- tween adjacent loops, currents are transmitted unattenuated for all angular velocities given by (83) . or (84). // the coils have zero CHAP. XVI] LINES AND FILTERS 303 magnetic leakage, then by (85) all possible frequencies are trans- mitted without attenuation. Such a line is not a good filter, but will be found useful for its retardation properties when it is desired to transmit with suitable retardation all frequencies. HI. RESISTANCELESS LINES. TERMINAL IMPEDANCE 271. Surge Impedance of the Three Types of Resistanceless Lines. In order to adapt a line to its terminal conditions, or to adapt the terminal conditions to the line, it is important to choose the constants so that the line will transmit as large a current as possible with the frequencies that it is desired to transmit. This means that reflection at the output terminal apparatus should be avoided and that the equivalent impedance of the whole line, with its non-reflective output apparatus should be adapted to the impedance of the input terminal apparatus. This requires the determination of the quantity that we have called 2*, where Zi = the surge impedance = the impedance by which a line of an infinite number of sections, or of a finite number of sections with a non-reflective output impedance, can be replaced without changing the current in the zeroth section. We have found a general expression for Zi in equation (34), which is (86) We shall now determine Zi for the three types of resist anceless line given in Figs. 2, 3, and 4. In all of these types, since the resistances are zero, we may write 21 = jXi, 22 = jX t , and b = j(Mu - X 2 ) (87) Introducing these values into (86), we obtain Zi = ^V- XS - 4Zi* 2 + 4AT 2 co 2 - 8AfcoX 2 (87a) for Rl = = /t2 Now introducing the values of Xi, X 2 , and M for Types I, II, and III respectively, as given in (64), (70), and (66), we obtain 304 ELECTRIC OSCILLATIONS [CHAP. XVI For Type I, Zi == V c! ; 4C?aT 2 (88) For Type II, i;Hj? For Type III, /L t + 2M \~cr -4- For Type III, with no magnetic leakage, 2M = Li, and zt = V < 2L 1 /C 2 = VLfC2 (91) where L = inductance per coil of Type III, Fig. 4. Equations (88), (89), and (90) give the surge impedance, or equivalent impedance of a line with non-reflective output terminal apparatus, for Type I, Type II, and Type III, respectively. Equation (91) gives the corresponding quantity for Type III if the coils have no magnetic leakage. In this case it is seen that z t is of the character of a pure resistance and is independent of the frequency. In (88), (89), and (90) z is also a real quantity, and is of the character of a pure resistance, but in general this equivalent resist- ance involves the angular velocity and is different for currents of different frequencies. It will be shown later that by choosing the inductances and capacities small, while keeping their ratios large the terms involving angular velocity can be made negligible over considerable ranges of frequencies. 272. Condition for Non-reflective Output. In (37), we have seen that the complex reflection coefficient Y at the output termi- nal of the line is Y = Zi Z T Zi + Z T This is zero, if Z T = Zi (92) Equation (92) shows that for no reflection at the junction of the line with the output apparatus the complex impedance of the out- put apparatus Z T must be equal to the surge impedance 2,- of the line. This is true whether the line has resistance or not. CHAP. XVI] LINES AND FILTERS 305 To adapt this result to the three special types of line used in the illustration, it is only necessary to replace z,- in (92) by its known values for the three types. 273. Condition for Non -reflection at the Input Terminal Apparatus. likewise, by (39), whether the line is resistanceless or not, we can make the complex reflection coefficient X at the input end zero, if we can make *o = * (93) To make the line non-reflective at the input terminal apparatus it is necessary to make ZQ, which is the impedance of the input appa- ratus, equal to the surge impedance z< of the line. This is true whether the line is resistanceless or not. IV. RESISTANCELESS LINES. RETARDATION COMPUTATIONS 274. Retardation per Section of Resistanceless Line. In equations (61), (62) and (63) we have found that if -1 < Po < + 1, a = 0, and

+ 1, a = cosh-^o,

radians, and the angular velocity of the current is o> radians per second, the time T is such that the system would describe an angle

, provided

, but since o> is given only as a factor in the quantity at the heading of the first column, we have divided the numbers on this column into the correspond- ing numbers in the column headed for Line of Type I. For Type I, if we take account of the resistance in the inductance coils, we have , z 2 = # 2 + JLw (101) 310 ELECTRIC OSCILLATIONS [CHAP. XVI These values inserted into (100), give after rationalizing, and equating real and imaginary parts, P ~~ 1 ~ 2C 1 a>(# 2 2 V L 2 2 a> 2 ) U = (103) Then by the use of (48), we obtain _ 4L.dc.' - 1 8Ci 2 2 (/2, + L 2 2 co 2 ) These values of U and V can be put into a form in which the relative size of terms is mbre evident, by introducing the abbreviations 772 = Rz/L 2 w, and = L 2 Cico 2 (105) then Introducing these values into (49) and (50), we have a = sinh" 1 (107) (108) = sin" 1 1 (109) Equations (108) and (109) give the values of the attenuation constant a and the retardation angle

= sin" 1 1 r=w< + -) + (i-w]* (1-49) (111) 20 \/2(l + r/2 2 ) (1 - 40)' (112) In order to make a and

1 (114) , for 40 < 1 312 ELECTRIC OSCILLATIONS [CHAP. XVI In case the ratio of the resistance of the coils to their inductive reactance is small and in case 40 is not too near unity, so that the conditions (113) are fulfilled, equation (114) gives a and

2 (121) 314 ELECTRIC OSCILLATIONS [CHAP. XVI substitute these values into the preceding equation, and sepa- rate the result into real and imaginary parts, we obtain P = I - U = r/i*/2 (122) and by the definition of V given in (48) These values of U and V introduced into (49) and (50) give a = - { 1 -- I (1 +K*) \ \ (124) = sn Equations (124) and (125) give the values of attenuation con- stant a and retardation angle v per section of the line for a Line of Type II. The abbreviations employed are given in (12). The same sign must be employed before the inner radical in both equa- tions, and that sign must be chosen to make a and

(129) A > (130) Equations (129) and (130) reduce to provided 8L 2 co 2 K\ /Co j /T 77" = -7T-\/F ?J an d V = "VLiCz (131) (132) In the case of small decrement and small value of LiduP, as stipulated in (132), approximate values for a and

2 (147) Let co = angular velocity below which the filter is to give high attenuation. Then by (147) and (146), we must make L 2 and Ci such that (148) Equation (148) gives one relation for determining Lz and Ci to comply with the cut-off requirement. We shall next find another relation determined by the re- sistance of the terminal apparatus. To avoid reflection the complex impedance z of the input apparatus and the complex impedance Z T of the output apparatus shall each equal the surge impedance of the line, which is 2; that is Zo = Z T = Zi (149) Now the value of Zi for this type of line is given in (110), and is a complicated function of the frequency. We cannot in general make 2 equal to Z Q and Z T for all values of the frequency. CHAP. XVI] LINES AND FILTERS 319 Let it be supposed that while we wish to cut off all frequencies of angular velocity less than co , we are also interested in trans- mitting especially the high frequencies for which the conditions (118) are satisfied. For these frequencies and is in the nature of a pure resistance independent of the frequency. We should need to make our terminal apparatus as nearly as possible a pure resistance, of value where RQ = resistance of input apparatus and of output appa- ratus, which are to be nearly pure resistances. Equation (151) is another relation for determining L 2 and Ci, and is obtained on the assumption that the line is to be non-reflective at the terminal apparatus for high frequencies. Elimination between (151) and (48) gives as the required con- stants of the line L 2 = # /2co , and d = l/2# coo (152) Equations (152) give the value of the inductance and capacity elements of the line to cut off angular velocities above co and to operate between an input terminal apparatus of resistance R Q (inductanceless) and an output terminal resistance of the same resistance. Now as to the resistance of the inductance coils used in the line, it is desirable to have this resistance #1 as low as possible, con- sistent with space available and cost. Let us suppose that the coils are wound of wire of such size as to give R 2 /L 2 = 2A (say) (153) Assuming this value, and making preparation to employ (108) to determine the performance of the computed filter, let us note that by (105) 2A co coo co (154) As soon as we specify the ratio of A to co , we can compute a and

to co . Let us now compute a numerical example, given 2A = 250 and co = 5000 (155) 320 ELECTRIC OSCILLATIONS [CHAP. XVI This means that the coils L 2 have 250 ohms per henry, and that we wish to cut off angular velocities below 5000 radians per second. The results are given in Table IV. Table IV. Performance of a Filter Computed to Cut Off all Angular Velocities Less Than co = 5000. Given R 2 /L 2 = 260 a>/coo a

. CHAP. XVI] LINES AND FILTERS 321 The angular velocity of the cut-off frequency is the value of co at which the last brace under the outer radical of (124) changes sign. This is approximately the value of co at which * = 4 (157) or by (121) coo = 2/L2 (158) where coo = angular velocity of cut-off frequency, which is the angular velocity above which the currents are highly attenuated. It is to be noted that we can determine the product o/I/iC? either by specifying the desired time lag T per section and using (156), or by specifying the cut-off angular velocity co and using (158). // we proceed by specifying T, we must make T small enough to raise the angular velocity of the cut-off frequency to give the operating range of frequency required of the apparatus. The next step in settling upon the essential constants of the compensator is the choice of the impedance of the terminal apparatus. The impedance of the input apparatus z , the im- pedance of the output terminal apparatus Z T and the surge im- pedance z t of the line must be equal to avoid reflections in the line, and to obtain a maximum transfer of energy to the output apparatus. If we operate the line in the region of frequencies in which (156) holds, then by (133) = ZO = ZT = Ro (say) (159) -2 The several impedances in (159) being equal to the radical expression are real quantities independent of the frequency, and must be of the nature of pure resistances. Ro = resistance of the input apparatus and of the output apparatus, which must be both inductanceless to avoid reflection. It may not be possible to utilize terminal apparatus of the nature of pure resistances and attain the results desired. In that case, we can not avoid reflections at the junction of the line with the terminal apparatus, and we shall sometimes need to make a compromise in practice. We shall not here enter into 'the nature of a profitable compromise, but shall proceed on the assumption that (159) may be fulfilled. Now eliminating between (159) and (156), we obtain L! = R T, C 2 = T/Ro (160) 21 322 ELECTRIC OSCILLATIONS [CHAP. XVI Equations (160) give the inductance and capacity per section of a line of Type II, designed to give a time-retardation of current by the amount T seconds per section, and designed to operate between non-inductive input apparatus of resistance RQ and non-inductive output apparatus of the same resistance. By equation (148) this line, if its elements have sufficiently low resistance, will let through with small attenuation all frequencies of angular velocity less than To compute the performance of such a line we need to specify T and also to specify the resistance Ri of the inductance coils, but this need be done merely by specifying the ratio of R\ to LI. We give in Table V, the computation of the performance of such a compensator with the specific values. T = 6.5 X 10- 5 seconds, and ^ = 250. J-ji Table V. Performance of a Compensator Computed to Give a Time-lag of T = 6.6 X 10~ 5 Seconds per Section. Given Ri/Li = 250 H n a r seconds -10o 770 123 0.00802 0.0505 6.55X10- 6 0.923 1,540 245 0.00812 0.100 6.50 0.922 3,080 490 0.00812 0.201 6.52 0.922 . 6,160 980 0.00825 0.403 6.55 0.921 9,240 1,470 0.0085 0.607 6.60 0.920 12,320 1,960 0.0088 0.825 6.75 0.916 15,400 2,451 0.0098 1.06 6.88 0.906 18,500 2,944 0.0101 1.29 6.97 0.903 21,600 3,438 0.0114 1.54 7.14 0.892 24,640 3,922 0.0135 1.86 7.54 0.873 27,720 4,412 0.0185 2.25 8.04 0.831 29,300 4,660 0.0258 2.54 8.70 0.773 30,800 4,902 0.127 3.01 9.80 0.281 30,954 4,927 0.133 3.02 9.80 0.264 31,108 4,951 0.171 3.05 9.80 0.180 31,416 5,000 0.216 3.07 9.80 0.115 33,880 5,392 0.339 3.09 9.15 0.033 36,960 5,883 0.487 3.10 8.40 0.007 38,500 6,128 0.837 3.12 8.05 0.0002 In the first column of Table V is the angular velocity of the current, which is determined by the angular velocity of the CHAP. XVI] LINES AND FILTERS 323 impressed e.m.f. The second column contains the frequency n corresponding to the given values of cu. The third column con- tains the attenuation constant per section. The fourth column contains the retardation angle per section. The fifth column contains the time-retardation per section of the line. The last column contains the ratio of the current in the tenth section to the current in the zeroth section. Notice that the time-retardation per section changes only about one per cent, in the range of frequencies between n = 123 and n = 1470. Over this range of frequencies the line can be used to introduce known amounts of time-lag by introducing various numbers of sections of the line. The attenuation for ten sections of the line in this range is slight since over 90 per cent, of the current gets through. As we pass up to higher frequencies, the time-lag per section changes considerably. At n = 4902 the cutting off effect of the line begins to make its appearance, and at n = 5883, the current in the tenth section is less than one per cent, of the current in the zeroth section. It is to be noted that by making T smaller, the time-lag per section can be made nearly constant for higher frequencies than those given in this table. In fact by making T sufficiently small this compensator action, by which is meant the introduction of time-retardation substantially independent of the frequency, can be made applicable to the ordinary ranges of radio frequency. CHAPTER XVII ELECTRIC WAVES ON WIRES IN A STEADY STATE 288. Two Methods. There are two possible methods of treating the propagation of electric currents along wires; namely: I. By considering the wires as a limiting case of an electrical system with recurrent similar sections, 1 utilizing the facts obtained in Chapter XVI; II. By building up directly the differential equations for the currents on the wires and solving the equations anew. 2 We shall employ the former of these methods. We shall treat the problem only for the steady-state condition. 289. Diagram, Notation, and Impedances. Referring to Fig. 1, suppose that we have two parallel wires, with a source of e.m.f . at e, having a complex impedance ZQ, and with an output apparatus at T, having a complex impedance Z T , let it be re- quired to find the current i at any time t and at any distance x from the e.m.f. The wires have certain resistance, and inductance, per element of length, and they have a certain capacity per element of length. Let there be a certain current i flowing out through the top wire at a distance x from the e.m.f., and, on account of sym- metry, let there be an equal current in the opposite direction in the lower wire at the same distance x from the e.m.f. As in Fig. 2, let us divide the wire into lengths Ax, and for each length Ax let us suppose a capacity Cz between the wires. 1 For an infinite line this method was employed by E. P. Adams, Proc. Am. Philosophical Soc., 49, 1910. 2 This method was employed in a special case by Sir William Thomson (Lord Kelvin) in an examination of the feasibility of the Atlantic Cable in 1855, published in Proc. Roy. Soc., May, 1855. The general problem of waves on wires was first treated by Kirchhoff, Pogg. Ann., 100, 1857. Further extensive work on the subject was done by Heaviside. Phil. Mag., 1876, and Electrical Papers, Vol. 1, p. 53. 324 CHAP. XVII] ELECTRIC WAVES ON WIRES 325 The wire is thus divided into elemental sections of length The shunt capacity per section is then, C 2 = CAz where C = capacity per unit of length of the wires. < X > j< *i > i 1 | > LU \ L r i ! i J I.. .] i_ h j . i \^ FIG. 1. Two parallel wires. FIG. 2. Resolution of two parallel wires into a system of elemental sections. Assuming that there is no current leakage between the wires, and designating the complex shunt impedance per section of the system as 22, we have 22 = - j/Gukx (2) ) ) : %m ^ Hi ill f : r 1 im \ \ ^ | o ) ) : FIG. 3. The rath section of two parallel wires. Treating the line as made up of sections of length Ax, equation (2) gives the complex shunt impedance per section, provided there is no current leakage between the wires. Let us consider next the series impedance in each of the sec- tions. A typical section is shown in Fig. 3. If we call this 326 ELECTRIC OSCILLATIONS [CHAP. XVII section the rath section, a current i m flows in the parts of this section not common to the next sections; that is to say, this current flows in each of the wires, as shown in Fig. 3. The complex series impedance of this section is zi = flAx + ?Xo>Ax (3) where R = resistance per loop unit of length = the resistance per unit length of outgoing conductor + resistance per unit of length of return conductor; L = inductance per loop unit of length = inductance of the two wires per unit length of the duplex system, when one of the wires is a return conductor for the other. Equation (3) gives the complex series impedance per section of length Ax. Here we may note one other simple relation. If x is the distance from the e.m.f. to the rath section, then x = raAx (4) 290. Attenuation Constant and Retardation Angle per Loop Unit of Length of the Wires. The system of Fig. 2 is an example of a line of Type II of Chapter XVI, and has the attenuation constant and retardation angle per section (that is-, per length Ax) given in (124) and (125), Chapter XVI, in which by (121), Chapter XVI, and (3) and (2) of the present chapter, 77! = R/Lu, t = LCo> 2 (Ax) 2 (5) Introducing these values into (124) and (125) of Chapter XVI, and calling the resultant quantities Aa and A _ f ax f jftx /g\ where a = , & = (9) Ax Ax in which for continuous values of x, we must take the limit of (9) and (8) as Ax approaches zero, giving Equations (10) give the attenuation constant a and the angle of retardation

4 II. In the range of frequencies in which # 2 /8L 2 o> 2 1 (16) equations (14) and (15) become R 1C a = (17) III. In the range of frequencies in which R/Lu > 1 (18) we may expand the radicals in (11) and (12) and obtain iRC^l a = 'WWI .j J-JW . -U / JLJ VJ ~fy~ I " ~W T 002 0^4" ~r f (1") z i it Zrc ort Leo -- 1 + + " + ' ' ' (20) IV. In the range of frequencies in which ^R 1 (21) (22) Equations (14) and (15) gwe respectively the attenuation constant a per unit length of the line and the retardation angle per unit length of the line^ provided (13) is satisfied. If (16) is satisfied, the corresponding values of a and are given by (17). Under conditions (18), a and /3 are given by (19) and (20) respectively. Under conditions (21), these quantities are given by (22). In these equations R, L, and C are respectively the Resistance, Inductance, and Capacity per loop unit of length. 292. Surge Impedance of the Line. If in (125a) of Chapter XVI, we replace Ri and LI by Rkx and LAz respectively and neglect terms involving (Az) 2 in comparison with unity, we shall CHAP. XVII] ELECTRIC WAVES ON WIRES 329 have for the surge impedance 2 of the continuous line, the value Equation (23) is the exact expression for the surge impedance of the continuous line in which R, L, and C are respectively the Re- sistance, Inductance, and Capacity per loop unit of length of the line. This becomes Zi = -X/7V provided R 2 /2LW 1 (24) It becomes Zi = V- j \l^ provided L 2 co 2 /2# 2 1 (25) \Cco 293. Reflection Coefficients. Condition for No Reflection. The complex reflection Coefficient X at the input apparatus by (39), Chapter XVI, is ' x = J-zpl; (26) where ZQ = impedance of input apparatus. Likewise the complex reflection coefficient Y at the output ap- paratus is F = ?4^?T (27) Zi ~T~ ZT where Z T = impedance of output terminal apparatus. 294. General Expression for the Complex Current Amplitude at a Distance x from the Impressed e.m.f., When the Length of the Parallel Wires from Input Apparatus to Output Apparatus is 1. To obtain this value, we shall use the general equation (28), Chapter XVI, with proper transformation to suit the smooth line problem. We have already found in (8) of the present chapter that km __ axjftx In this we have made the rath section a distance x from the e.m.f. The total length of the present line is to be I, and the total number of sections of the discrete line of equation (28), Chapter XVI, was n, so that if we replace ra by n and x by I, we have by the equation next above kn al j(il I oo\ c = c e (28) 330 ELECTRIC OSCILLATIONS [CHAP. XVII Also in Chapter XVI, equations (30) and (32), we have - b(e k - x) = z + Zi (29) Substituting these several values into (28), Chapter XVI, and designating the resulting value of A m by the ' A x , we have E (30) 20 2* _|_ -f In deriving this equation we have assumed that the impressed e.m.f. is e = E ' ^ ' CHAP. XVII] ELECTRIC WAVES ON WIRES 331 The values of R and X are resistance and reactance, respec- tively of z + z t ; that is, zo + z; = R + jX (36) Equation (34) gives the current at distance x from the source of e.m.f. for two parallel wires infinite in extent, or with a non-re- flective output impedance. The expression (34) may be looked upon as made up of the product of three factors as follows : E = = amplitude of current at x = 0. Zt e~ xa = attenuation factor, by which the current-amplitude at x = is to be multiplied to get the current-amplitude at x = x. sm{u(t fix/u)6') = the periodic factor, which is periodic in t and periodic in x. It may be noted that in the periodic factor &' = lag of current behind e.m.f. at x = 0. fix = lag of current at x = x behind current at x = 0. We may now obtain the velocity of propagation by noting that the periodic term at t = t z and x = x z will have the same value that it has at t = ti and x = Xi, provided (37) whence >//? = velocity of propagation (38) The quantity on the left of (38) is seen to be the velocity of propagation, because Z 2 ~ t\ is the time that must elapse for a given phase of the disturbance to travel from x\ to X*, and what- ever the values of x^ and x 2 the ratio of the distance to time is independent of the distance. Equation (38) shows that the velocity of propagation of the dis- turbance along the wires is v = a>/ft (39) where ft is given exactly by equation (12), and is further given in approximate form for special cases in (15), (17), (20) and (22). 332 ELECTRIC OSCILLATIONS [CHAP. XVII Although we derived v on the assumption of a non-reflective line the result is correct for any line, for the terms after the first in (30) give the same velocity v for each term. We must, however, when X and Y are complex quantities attribute to the reflected waves a change of phase at reflection, which is coefficient of j in the imaginary part of Y . OT = tan { -- ' - i - 7 \ - I real part of X and f coefficient of j in the imaginary part of X. , , real part of X Equations (40) and (41) give the angle by which the reflected current lags behind the incident current at the output impedance and the input impedance respectively. 296. Velocity and Attenuation of High -frequency Waves on Parallel Wires or on Two Concentric Tubes. The velocity of a sinusoidal current in the steady state on two parallel wires is By reference to the value of given in (12) it is seen that v is in general a function of the frequency. But by (16) and (17) it is seen that if u is sufficiently large to make R 2 /8LW < < 1, then v =l/\/LC (42) Equation (42) gives the velocity v of propagation along two parallel wires. The same equation evidently holds for propagation along two tubes, one inside of the other and coaxial with it. In (42) L and C are inductance and capacity per loop unit of length, and the unit of length must be the same as the unit of length occurring in the velocity. The inductance capacity and velocity must be measured in some consistent set of units. Formulas for the inductance and capacity of parallel wires and of concentric tubes are well known as follows: For two parallel wires in which the current is flowing only on the outside surface, one being a return wire, 4ju loge c.g.s. electrostatic units per centimeter of , L = - - length of wires d c centimeter of length of wires = 4/i log c.g.s. electromagnetic units per ,**-, CHAP. XVII] ELECTRIC WAVES ON WIRES 333 d 4 M log e - = r^ henries per cm. length (45) in which L = inductance per centimeter (loop) of length, r = radius of one wire, d = axial distance between wires, At = magnetic permeability of the medium between the wires, c = ratio of electromagnetic unit of quantity to electro- static unit of quantity = 3 X 10 10 cm. /sec. In the same case the capacity is C = - -7 c.g.s. electrostatic units per loop 4 loe: centimeter of length of wires ' ' c r k c.g.s. electromagnetic units per loop -, vj_j.^. U*W WV* VT A** M^AAW* V v*J.J.xv^J ^fV-/A J.W|_7 / A *7\ 4c 2 log e centimeter of length of wires /blO 9 = - -7 farads per loop centimeter ,.. 4c 2 log - of length of wires where C = capacity per loop centimeter of length of wires, k = dielectric constant of the medium between wires, c = ratio of units = 3 X 10 10 cm. /sec. In like manner for two coaxial tubes with the current only on the adjacent surfaces e.s.u. e.m.u. henries r> i Rt o 1 R* P 6r 1 OO P /Af\\ 2Atlog TT r, ?JLI iOg -5- ,. . (49) T e /LI n ,__ /t2 /ti centimeter j : Ji 10 9 of length e.s.u. e.m.u. farads k k &10 9 per loop c = z #2 centimeter in which R 2 = inner radius of outer cylinder, Ri = outer radius of inner cylinder. 334 ELECTRIC OSCILLATIONS [CHAP. XVII By taking the square root of the product of L and C in any one of the sets of units, for the case of the parallel wires or for the case of the coaxial tubes, we obtain by (42) , provided R 2 /8L 2 u 2 < < 1 (51) Equation (51) gives the velocity of propagation of high-frequency waves on two parallel wires or on two coaxial tubes. In this equation c, which is the ratio of the electromagnetic unit of quantity to the electrostatic unit of quantity, has been shown by experiment to be equal to the velocity of light. If the medium between the wires is a vacuum k = ju = 1, and v = c (52) that is, the velocity of the high-frequency waves on parallel wires or coaxial tubes is equal to the velocity of light, when the medium around the wires or between the tubes has dielectric constant and permeability unity. 1 As to the attenuation constant in this case of high-frequency waves, a substitution of C in farads and L in henries per unit length into (17) for a gives R Ik -. for parallel wires (53) 1 Direct experimental determinations of the velocity of high-frequency waves on wires have been made as follows: Observer Velocity in centimeters per second Published in Blondlot 2.930X10 10 2.980 2.980 3.003 2.954 2.994 2.998 2.998 2.995 2.999 1 Comp. Rend., 117, p. 543, 1893. Am. Journ. of Sci., 49, p. 297, 1895. Phys. Rev., 4, p. 81, 1896. Trowbridge and Duane Saunders For best determinations of the velocity of light see Book II, Chapter IV, Art. 42. CHAP. XVII] ELECTRIC WAVES ON WIRES 335 and a = -z\h -- 5- for coaxial tubes (54) 2 ^ 2 where k, n, d, R, Ri and R z have values given above. Equations (53) and (54) give the attenuation constants per loop unit of length for two parallel wires and for two coaxial tubes respectively. In these equations R is the resistance in ohms per loop unit of length, using the same unit of length that is applied to the attenuation constant. These equations apply only to cases of sufficiently high frequency to make R 2 /8L*u 2 negligible in comparison with unity. 297. Stationary High-frequency Waves on Two Parallel Wires Open-ended at Outer End and Non-reflective at Input End. Reference is made to Fig. 4. Let the length of one wire -2-lx FIG. 4. Showing direct and reflected distances. from the e.m.f . to the open end be I. The open end is equivalent to an infinite terminal resistance. Therefore by (27) Y = -1 (55) We shall now make the input impedance non-reflective, which by (26) and (24) gives X = 0; 2 = Zi = VL/C, provided (56) Also referring to (35) and (36) we have Z = 2VL/C, and 6' = (57) If now e = E sin at (58) and we take the sine part of (30), with attention to (55), (56) and (57), we obtain 7r ==r{r-~sm [a>(t-x/v)]-e-"W-*) sin [u(t-(2l-x)/v)]} LJ I \j (59) 336 ELECTRIC OSCILLATIONS [CHAP. XVII where v = c/Vki* (60) k and ju = dielectric constant and permeability of medium around the wires, and where Equation (59) 0wes /ie steady-state current at the distance x from the e.m.f. for the case of high-frequency waves on two parallel wires of length I open-ended at the outer end and non-reflective at the input end. The current is seen to be the resultant of two wave- systems one passing direct from the source of e.m.f., and the other reflected with a reversal of sign from the open end of the system. The out-going wave has traveled a distance x and the reflected wave has traveled a distance I -f- I x. It is to be noted that at the outer end of the wires, where x = 1, equation (59) gives i = 0. On the other hand, at x = 0, the current is to = (sin o>t - e~ 2al sin ( - 2l/v) } (62) 2\/ L/C Equation (62) gives the current at the input end of a line with non-reflective input impedance and with outer end of the line open. From equation (62) it may be noted that if e~ 2al is nearly equal to unity, we shall get the largest value of i Q , if we make the length of the line such that the second term is brought into phase with the first term; that is, if 2(d/V = 7T, 37T, 57T, . . . (63) If we multiply numerator and denominator of (63) by T, the period of the e.m.f., and note that a>T = 2>jr, and vT = Xi where Xi = the wavelength of the waves on the wires, we have, as the condition for a maximum value of i Q , I = \!/4, SXj/4, 5V4, . . . (64) When the attenuation factor e~ 2< * 1 is nearly equal to unity, we obtain a maximum amplitude of current at the input end when the length of each of the wires is an odd number of half wavelengths of the waves on wires; provided the outer end of the system is open- CHAP. XVII] ELECTRIC WAVES ON WIRES 337 ended, and provided the input impedance is non-reflective and is excited by a high-frequency sinusoidal e.m.f. 298. Stationary High-frequency Waves on Two Parallel Wires Non -reflective at the Input End and Terminated by a Con- denser C' at Outer End. Reference is made to Fig. 5. The output terminal impedance in this case is ZT = -3/C'<* (65) The input impedance and the surge impedance of the line are Zo = Zi = VL7C = Ro (say) (66) where L and C are [inductance and capacity per loop unit of length of the wires. Equation (66) in the condition for non- reflection at input end. Me' FiG. 5. Parallel wires connected at outer end through a condenser C". Introducing (65) and (66) into (27) we have for the reflection coefficient at the outer end of the line V "" u _ 2jtan-i(l/.BoC"a>) 1 ~ r> 1 r, we may write 27TC CO = (76) C' FIG. 6. Linear relation of X 2 to C". whence (75) becomes 1 X 2 1 By transposition, and replacing Cv 2 by 1/L, we obtain X 2 = 47r 2 c 2 aLC') + 47rW^^ (77) provided J_ 45 (78) Equations (77) and the inequality (78) may also be written X 2 - X 2 = B 2 C', provided i(^) '< < 1 where B* = WcHL, and X 2 = BHC/3 (80) 340 ELECTRIC OSCILLATIONS [CHAP. X Equation (79) gives the capacity C' that must be placed at outer end of two parallel wires each of length I, to bring the sysi to resonance with an impressed e.m.f. whose wavelength in j space is X. This equation applies accurately provided the condit stipulated in (79) is met. , 'If various values of X 2 and the coi sponding values of C' with fixed value of I are plotted the resul a straight line of the form of Fig. 6. 300. Approximate Application to a Coil of Distributed Capaci The result obtained in the form of (79) for the condition um which a system of two parallel wires with a condenser at the ou end is resonant to an impressed e.m.f., is found by experime to hold approximately for a coil attached to a condenser as Fig. 7. E.M.F. of wave length. X impressed here U C FIG. 7. Coil and condenser. If we apply to the coil an e.m.f. near its middle section, may be done by induction from another oscillating circuit, a if we give to the impressed e.m.f. various wavelengths X, a resonate by giving the condenser C r various values of capaci it is found that an approximate relation in the form of (79) hoi in that X 2 minus a constant X 2 is proportional to C", and the p of the result is similar to Fig. 6. This result can be accounted for by attributing to the c a capacity per unit length and an inductance per unit leng (of wire or of axial length) provided the product of these quj tities is constant for different sections of length. It is not 1 lieved that this is exactly the case, but is true to the degree approximation to which the linear relation of X 2 to C' is true. 1 J. C. Hubbard, "On the Effect of Distributed Capacity in Single La; Solenoids," Phys. Rev., 9, p. 529-541, 1917. CHAP. XVII] ELECTRIC WAVES ON WIRES 341 301. Difference of Potential Between Two Parallel Wires in Relation to Current Distribution Along the Wires. Returning now to the general problem of the transmission of electric dis- turbances along two parallel wires, we may note the following general relations that are true whatever the terminal conditions of the wires and whether the currents are in a steady state or not. We omit only from consideration the cases in which there is leakage of current across from one wire to the other in the region of length under consideration. Reference is made to Fig. 8. Let # be a distance along the wires measured from some arbitrary origin. Let Ax be an ele- ment of length at x. Let i be the current flowing into the ele- FIG. 8. Used to obtain relation of e to i. ment Ax at any time t, where i is some function of x and t; that is i = i(x, t) (81) where the i on the right-hand side indicates a functional relation. Equation (81) is a formal expression for the current flowing into the section Ax at x and t. To get an expression for the current flowing out of Ax, we need merely note that this current is at a distance x + Ax from the origin, and write i' = i(x + Ax, t), which expanded by Taylor's Theorem gives i' = i + ^ Ax + . . . (82) ox where the dots represent terms of higher order in Ax. Equation (82) is a formal expression for the current flowing out of Ax. 342 ELECTRIC OSCILLATIONS [CHAP. 5 If now we let e be the average excess of the potential of top wire over the potential of the bottom wire and note that capacity of a length Arc of the top wire is CAx, we have for charge on the top wire in the element of length Ax the value Ag = eC&x Now by KirchhofFs current law the excess of current flo? into Ax over the current flowing out is the time rate of incn of charge of Ax; that is dt The substitution of (81), (82) and (83) into (84) gives di . n de - Ax + . . . = CAx dx dt Dividing this equation by Ax and taking the limit as approaches zero, and noting that the terms of higher orde Ax disappear, and that the average value of e in the region proaches the actual value e at x, we obtain di de Equation (85) is an important differential equation connec the current i at any distance x at any time t with the differenc potential e between the wires at the same x and t. By continuing this process or reasoning, and applying Kirchhoff s e.m.f. law to the element of length Ax of both w: we can build up completely the proper differential equations the waves on wire and obtain all of the results obtained abov< the other method. We shall not do this, but shall merely rr application of equation (85) to a single case. 302. Distribution of Current and Potential Along Two Par* Wires, with the Outer End Open, and with a Non-reflec Input Impedance, Assuming Negligible Attenuation. Circ for this case are given in Fig. 4. If the attenuation constant negligible, the current may be obtained from (59) by repla< the exponentials by unity. This gives i = |L={sm u(t - x/v) - sin co[ - (21 - x)/v]} ( 2 v L/C CH*P. XV11] ELECTRIC WAVES ON WIRES 343 Substituting this value of i into (85), we obtain de = 1_ di_ dt C dx = ~ - = {cos w[t - x/v] + cos u[t - (21 - x)/v]}. Integrating this equation with respect to t and replacing v by its value l/\/LC, we obtain e = | {sin [ - x/v] + sin co[Z - (21 - x)/v]} (87) Equations (86) and (87) are /ie values of current and potential at distance x from the origin at time t, with the electrical system shown in Fig. 4. Let us next take the special case in which the amplitude of current on the wires is a maximum. By (63) and (64) this is the case in which the length of wires I satisfies the equation or (88) r*'-Xi/4, 3X,/4, 5Xi/4, . . ., ul/V = ir/2, 37T/2, 57T/2, . . . Iii this case (86) and (87) become 771 i = =={8111 w(t - X / v) + S1H 0>(t + X/v)} (89) 2\/L/C e= {sin a)(t x/v) sin u(t + x/v) } (90) By expanding the sines of the sum and difference terms and collecting, these equations become sin ut cos - (91) VL/C and e = E cos ut sin - (92) Equations (91) and (92) give the current and potential along two parallel wires of length an odd number of times the quarter wavelength of the waves on the wires, provided the outer end of the wires is open, and provided the e.m.f. is impressed through a non-reflective im- pedance at the input end. The current and potential are out of phase with each other in time and space. 344 ELECTRIC OSCILLATIONS [CHAP. XVII 07=0 It (a) The Wires (6) Current if I =\/4 4 8 (c) Potential If * = X t /4 FIG. 9. Stationary waves on wires. CHAP. XVII] ELECTRIC WAVES ON WIRES 345 303. Plot of Stationary Current and Potential Waves on Wires of 302. A plot of equations (91) and (92) for two different cases is given in Fig. 9. In this figure (a) represents the wires; (b) represents the current distribution along the wires if the wires are Y wavelength long; (c) represents the potential distribution in that case. The curves (d) and (e) show respectively the cur- rent distribution and the potential distribution if the length of each of the wires is % of a wavelength. In each of the diagrams the different curves correspond to dif- ferent times. For example, in (b) and (c) these curves are num- bered to 11. The curves numbered in the two diagrams are respectively the current and potential at t = 0. The curves numbered 1, 2,3 . . . show the values of current and potential at times equal to ^2? 21 2 > %2 of a whole period after t = 0. BOOK II ELECTRIC WAVES CHAPTER I ELECTROSTATICS AND MAGNETOSTATICS 1. Electric Intensity. In a field of electric force the force is said to have at every point a certain intensity, which is defined as the force with which a unit positive charge of electricity would be impelled if introduced at the point without changing the ex- isting distribution of force. In order not to change the existing distribution the exploring charged body must be a very small body with a very feeble charge, and the force per unit charge is obtained by dividing the force by the charge. The electric intensity is a vector, which we shall designate by E in Clarendon Type. Throughout this volume all vectors shall be designated by heavy-faced, or Clarendon, type; all scalars by light-faced type. The vector components of E in the direc- tions x, y, z shall be designated by E x , E y , and E z . The scalar magnitude of E shall be designated by E with components E x , E v , and E z ; unit vectors along the axes of x, y, z, shall be desig- nated by i, j, k, respectively. A plus or minus sign between vectors means a vector sum or difference. For example, E = E* + E v + E 2 = E x i + Eyj + #*k means that E is the vector sum of its components; that is, E is in magnitude and direction the diagonal of the rectangular parallelepiped with E z , E y , and E 2 as adjacent edges. The magnitude of E is seen to be given by the scalar equation in which the plus sign indicates ordinary addition. 2. No Simple Method of Computing E. In the most general case in which there are various conductors and insulators aggregated into a system there is no simple method of computing 347 348 ELECTRIC WAVES [CHAP. I the electric intensity E. We shall be able to arrive at the laws governing such a system only by successive generalizations from simpler systems. The generalizations made will involve the introduction from time to time of new assumptions which may not have been submitted to immediate experimental tests. Instead of resting on direct tests of the assumptions themselves, the validity of the assumptions may require to be established by tests made on the consequences of the assumptions. 3. Electrical Intensity Due to a Single Point Charge in an Infinite Vacuum. In this simple case where there is a single point charge in an infinite vacuum the electric intensity at any point distant r from the charge has the magnitude E = g/r (1) The direction of this intensity is the direction of r, so that the magnitude and direction of E is expressible in the vector equation E = ^U r (2) In these equations E = electric intensity at P in dynes per electrostatic unit charge, q = electric charge at in electrostatic units, r distance from to P in centimeters, U r = a unit-vector in direction of r from to P. The inverse-square law 1 for electric intensity, as expressed in equations (1) and (2), has been put into an integrated form and submitted to rigid experimental tests by Cavendish. 2 4. Effect of Dielectric on Electric Intensity. If into the field surrounding the point charge various dielectrics are introduced, the intensity is in general changed in a very complicated way. These various dielectrics are said to have different values of inductivity, or dielectric constant. 3 The inductivity, or dielectric constant, of the medium at any point will be designated by c, which is in general a function of the coordinates x, y, z, and in some media (those of a crystalline character) the inductivity is also different in different directions. // the medium is infinite in extent and is everywhere of the same 1 Due to Coulomb. 2 Left in manuscript published by Maxwell in 1879. 3 Attributed by Faraday to a "certain polarized state of the particles;" Experimental Researches, 1295, 1298, and 1304 (1837). CHAP. I] ELECTROSTATICS AND MAGNETOSTATICS 349 nductivity , the electric intensity is inversely proportional to the inductivity of the medium, and the law of force is given correctly by the equation er 2 with 'magnitude E = i w where 3 = intrinsic charge (defined in next section). This proposition is proved by the fact that it gives the proper value for the capacity of a condenser with homogeneous dielectric. 5. Definition of Intrinsic Charge. In the statement of the law of force immediately preceding, the charge q is designated as intrinsic charge. An Intrinsic Charge is a charge whose time derivative within a region gives the ordinary electric current flowing into the region. A body which contains an intrinsic charge will suffer a translation if placed unsupported in a uniform electric field. Intrinsic charges are to be distinguished from the induced charges, that are sometimes supposed to exist in dielec- trics, in the form of a union of positive and negative charges capa- ble of being oriented under the action of a uniform field, but undergoing no translation in such a field. In modern electron theory, intrinsic charges are supposed to be due to free electrons; and induced charges due to bound electrons. The motions of the free electrons throughout conduct- ors constitute the ordinary conduction currents of electricity. This subject will be considered later, but for the present the only charges referred to shall be the intrinsic charges. 6. Electric Induction. Related to electric intensity it is convenient to employ a second vector called Electric Induction, which we shall designate by D, with components D x , D y , and D 2 . Whether the medium is homogeneous or not the Electric In- duction at any point is defined as the product of the electric intensity at the point by the inductivity e of the medium at the point. In a non-crystalline, or isotropic, medium the dielectric constant is the same in all directions, and D = eE D* tf = e (5) D! = eE! 350 ELECTRIC WAVES [CHAP. I On the other hand, if the medium is crystalline (anisotropic) the dielectric constant at a given point has different values in different directions, and, in general, D x = XX E X + xy E y ' Dy = Cy X E x + CyyEy D z = ZX E X (6) 7. Definition of Flux of Induction. At any point P in a given field of force the electric induction has magnitude and direction that are functions of the coordinates of P. Suppose an element of surface dS to be drawn at P, and let the normal to dS have the direction N, Fig. 1. If the induction at P is D, the flux of induction through dS is defined as the product of dS by the normal component of D ; that is, d D = DdS cos (D, N) (7) where d D = the flux of induction through dS, D = magnitude of D, cos (D, N) = cosine of the angle between D and N. Fio. i. The flux of induction through any ex- tended surface S is obtained by integrating d$D over the entire surface: = fDdS cos (D, N) (8) 8. Proof of Gauss's Theorem for a Homogeneous Dielectric. We come now to an important proposition due to Gauss, concern- ing the flux of induction through a closed surface. Let us suppose that we have throughout a certain region a homogeneous di- electric of dielectric constant and that there is an intrinsic charge q of electricity concentrated at a point within the region, and let us draw within the homogeneous region any closed surface S completely enclosing the charge q, Fig. 2. At any point P on the surface the electric induction is in the direction of r and has, by equations (4) and (5), the magnitude D = q/r'< (9) CHAP. I] ELECTROSTATICS AND MAGNETOSTATICS 351 The total flux of induction outward through the closed surface is &> = fDdS cos 9 (10) where 6 is the angle between D and N. Now if dil is the solid angle subtended at q by dS, it is seen by the geometry of the figure that dS cos 6 = r 2 d!2 (11) whence, by substitution of (9) and (11) in (10), 4>D = q fdtt = 4irq (12) where 4> D = flux of induction outward through the closed surface. It thus appears that in a homo- geneous medium the flux of induc- tion outward through any closed surface is independent of the posi- tion of q within the enclosure. The limitation that q is to be concen- trated at a point may hence be removed, and the charge q may be distributed in any manner whatever within the enclosure. If on the other hand we have a charge g within the homogeneous medium but outside of the enclosure, Fig. 3, and if we draw a solid angle dti at 50, intercepting, from the closed surface, ele- ments dSi, dSzj etc., it will be seen that at every element dSi where the direction of r is into the enclosure, cos (r, N) is negative; therefore, FIG. 2. and at every element dS 2 at which r points out from the enclo- sure, cos (r, N) is positive; therefore, dS,cos(D,N)_ and that there are as many positive elements as negative ele- ments; hence the flux of induction outward through all the ele- 352 ELECTRIC WAVES [CHAP. I ments intercepted by dQ is zero. Therefore, the total flux of induction through a closed surface due to a charge outside of the enclosure is zero. For charges both inside and out, the result may be summed up as follows: Gauss's Theorem. The total flux of electric induction outward through any closed surface due to charges partly within the enclo- sure and partly outside of it is 4?r times the quantity of intrinsic electricity within the enclosures. FIG. 3. 9. Limitation Under Which Gauss's Theorem has been De- duced. In the preceding section we have started with a very limited experimental result that the electric intensity due to a point charge in a uniform medium is that given by equation (4). To this we have added the definition of induction given in equa- tion (5). From this limited material we have deduced Gauss's equation D = which is rigorously established for a uniform medium The derived result is less definitive of D than the original equation (4). This is evident from the consideration that with a given distribution of intrinsic charges the elementary equation (4) would determine one and only one value of the induction DI (say) at a given point; whereas Gauss's equation would be satisfied by DI plus any other vector Do such that the surface integral of Do over the closed surface is zero. 10. Assumption that Gauss's Theorem is Perfectly General. Equation (12), Gauss's Theorem, is in accord with the equation (4) and the definition (5) when the dielectric is uniform, and is CHAP. I] ELECTROSTATICS AND MAGNETOSTATICS 353 therefore in accord with experiments performed on uniform di- electrics; for example, experiments on the capacity of condensers. As the next step in our search for general laws of the electric field, we are going to assume that Gauss's Theorem without any modification whatever is perfectly general for every possible distribution of charges, conductors, and dielectrics at rest. The justification of this assumption is to be sought in a comparison of experimental results with deductions from the assumption. 11. Gauss's Theorem Expressed in Terms of a Point Relation. We shall next express Gauss's Theorem in terms of a point- relation. Let us take a point whose coordinates are x, y, and z, and for our closed surface, let us take the surface of the elemental volume AT = AzAf/A? (13) Let p be the intrinsic density of electricity at the point x, y, z, and let p be the average density in the elemental volume; then the total intrinsic quantity of electricity in the volume is = pAT (14) whence by Gauss's Theorem, equation (12), the flux of induction is A$Z) 4?rpAT (15) or taking the limit as AT approaches zero = 47T P (16) dT The left-hand side of this equation is seen to be the limit as the volume approaches zero of the flux outward of D from a small volume divided by the volume. This quantity is called the divergence of D. There follows a digression in which the diver- gence of a vector is obtained in a different form. 12. Digression on the Divergence of a Vector. Let $ A be the surface integral of the outward normal component of any vector A over a closed surface, and let it be required to find an analytical expression for the limit of the ratio of the surface integral to the volume as this volume approaches zero. In Fig. 4 is represented the element of volume AxAyAz with one of its corners at the point x, y, z. Let A be a vector whose components are analytic functions of the coordinates x, y, z. Let Ax be the average value of the ^-component of A over the 23 354 ELECTRIC WAVES [CHAP. I surface (1). This quantity is in the direction of the normal inward to the surface. The average value over the opposite dk x surface (2) is, by Taylor's Theorem, Ax+-~-"A:e + . . ., and is seen to be outward. Likewise the average normal component of A at the surface (3) is Ay inward, and that at the surface (4) is A y + ~jT L ^y -f- . . . outward. Similarly for the other two faces of the element, which are perpendicular to the z-axis, the average normal components of the vector are respectively i inward and A* +~^~ Az + . . outward. A ' === Aa; FIG. 4. Giving a minus sign to the normal vectors that are inward, and multiplying the magnitude of each of the normal terms by the corresponding area of the face of the element through which it acts, we have, as the total outward normal surface integral, the equation 4- IT + dA + \A V + + _ Az + dz . |A#A?/ I (17) Dividing by AzA?/A2 = AT and taking the limit as AT approaches zero we have Lim. \fAndSr} = AT= Ql Ar _ dr dA x dA y dA z dx " dy " dz (18) CHAP. I] ELECTROSTATICS AND MAGNETOSTATICS 355 4 where the derivative with respect to r $,s a partial derivative because A may be regarded as a function of x, y, z and r; so that the partial derivative with respect to r means the deriva- tive at a fixed point x, y, z. Equation (18) may be briefly written Ar = L~ST where ,. A dA x . dA y dA z f . div. A = h H (20) The divergence of a vector A is the flux of the vector outward from a small volume divided by the volume. It is a scalar quantity, has in general different values at different points, and may be obtained directly by performing the operation indicated in equation (20). i 13. Poisson's Equation. In view of equation (20) we may now express equation (16) as follows: div. D = 47rp (21) wherever p is finite. The divergence of electrical induction at any point where p is finite is 4r times the intrinsic charge density p at the point. Equation (21) is known as Poisson's equation. At all points in space where there is zero intrinsic charge density div. D = (22) 14. Gauss's Theorem Applied to a Surface Distribution. Sur- face Divergence. Suppose that there is an intrinsic charge distributed over a surface, with a surface density cr. At a point in such a surface p is no longer finite, so that Gauss's Theorem cannot be reduced to the divergence equation (21), but is pref- erably reduced to a new point relation as follows : 1 Assumptions have been made in sections 10 and 11 as follows: 1. In passing to the limit in deriving (16) it was assumed that the in- trinsic charge density p at the point x, y, z, is spatially continuous in such a way that for a sufficiently small region about x, y, z the average density differs from the density at the point by an amount less than any predetermined quantity. 2. It was assumed that D is a function of x, y, z of a form capable of being developed by Taylor's theorem. 356 ELECTRIC WAVES [CHAP. I At any required point on the surface (Fig. 5) let us mark out an element of surface AS, and through the periphery of AS, draw lines in the direction of the electric induction. These lines bound a short tube of induction, which we shall suppose to be terminated by the surface elements ASi and A$ 2 parallel to AS. Let h be the distance between ASi and A$ 2 . Over the convex surface of the tube the normal component of induction is every- where zero, since the induction is in the direction of the convex surface. Over the ends of the tube, let the average component of induction away from AS be Dini and D 2n 2. Then by Gauss's Theorem DmiAi + D 2 n 2 A 2 = 4-n-aAS (23) If now we allow h, the height of the tube, to approach zero, ASi and A$ 2 both approach AS as a limit; whence Dim + D 2 2 = 4*0 (24) FIG. 5. If now we allow the surface AS to shrink toward a point P on the charged surface, the average values in (24) may be replaced by their true values at the point, giving Dim + D 2n2 = 47T(7 (25) in which Di n i and D 2n2 are both drawn away from the charged surface. The sum of the two normal components thus drawn is called by Abraham and Foppl 1 the surface divergence of the vector D. The result (25) may be stated as follows: The surface divergence of induction at any point is 4ir times the intrinsic surface charge density at the point. As a corollary, the surface divergence of induction is zero at all points where there is no intrinsic surface charge. If instead of drawing the two normals both away from the surface under consideration, one of them be reversed so that they Abraham und Foppl: Theorie der Elektricitat, Vol. 1, p. 77, 1907. CHAP. I] ELECTROSTATICS AND MAGNETOSTATICS 357 point in the same sense through the surface, equation (25) becomes Di n i D 2n l = 47T0- or (26) that is, there is a discontinuity in the magnitude of the normal component of D amounting to 4, where a is the intrinsic surface density. 15. Analogous Treatment of Magnetic Field. In a field of magnetic force, the force at any point per unit magnetic pole is called the Magnetic Intensity and is designated by H. The unit magnetic pole is a pole that will repel an equal pole at a distance of one centimeter with a force of one dyne in vacuo. The product of the magnetic intensity by the permeability of the medium at the point is called Magnetic Induction, and is designated by B. B = fj,n Ey = fJ,Hy where (27) ju = magnetic permeability. The question whether there is or is not any intrinsic volume density of magnetism is open to disputation. It is proposed to limit the discussion in the present work to cases where this volume density is zero; so that reasoning similar to that used in the discussion of electrical quantities in the preceding para- graphs gives from the inverse square law for a uniform magnetic medium the result div. B = (28) and this is assumed to be universally true. Also in all cases that will come under our observation surf. div. B = (29) CHAPTER II MAXWELL'S EQUATIONS 16. Summary of Chapter I. The important results obtained in the preceding chapter are contained in the following equations, which are taken with their original numerical designations: div. D = 4?rp, wherever p is finite, (21), Ch. I. surf. div. D = 4, (25), Ch. I. div. B = 0, (28), Ch. I. surf. div. B = 0, (29), Ch. I. where D and B are respectively electric and magnetic induction at any point, p is intrinsic volume density of electric charge, and o- is intrinsic surface density of electric charge at the point. The electric intensity E can be obtained by dividing D by the dielectric constant c; the magnetic intensity H can be obtained by dividing B by the permeability /x. The above equations are not sufficient to determine D and B. 17. Note as to Additional Requirements. In addition to the divergence of a vector we need also its curl, which is a related vector to be later defined. These two quantities, divergence and curl, together with certain boundary conditions, are sufficient to determine a required vector. In electrostatics, where there are assumed to be no electric currents or motions of electric charges and no variations of D and B with the time, it can be shown that the curl of D and the curl of B are both zero. It can then be shown that a scalar potential function exists, and familiar methods are at hand for completely determining D, B, E, and H in cases where proper boundary conditions are given. When, however, we leave the field of electrostatics and enter upon the general problem, the curls of D and B are no longer zero, the scalar potential functions for these vectors have no existence, and the older theoretical investigations of Laplace and of Poisson are insufficient to describe the characteristics of the electro- magnetic field. 358 CHAP. II] MAXWELL'S EQUATIONS 359 The way to proceed under these more difficult conditions was pointed out by Maxwell in 1865-6, in a mathematical research which contained a prediction of the existence of electric waves, determined the velocity of propagation of the waves, and ex- plained the nature of light. 18. Further Experimental Relations for the Electromagnetic Field. In developing the theory of electric waves, we may make use of the following experimental laws : I. THE M.M.F. EQUATION. The work done by the magnetic field in carrying a unit magnetic pole once around a closed path, Fig. 1, linking positively with a closed circuit carry- ing a steady current I is W = 4irl (1) in which W is work in ergs per unit pole, and I is current in absolute c.g.s. electromagnetic units of cur- rent (absamperes). Throughout this volume, in order to obtain symmetrical results, we shall measure all electrical quantities in absolute c.g.s. electrostatic units, and all magnetic quantities in abso- FlG - 1. Arrows marked W -. , 7 , * ' -A a u an d I point in the direction of lute c.g.s. electromagnetic units, buch pos iti ve linkage. a composite system of units is called the Gaussian system. In Gaussian units, equation (1) becomes in which (2) the number of electrostatic units of quantity of electricity in one electromagnetic unit. 1 II. THE EM .F. EQUATION. The electromotive force pro- duced in a closed circuit, Fig. 2, by varying the flux of magnetic induction linking with it positively is V = - dt (3) in which V is in electromagnetic units (ab volts). If we put V 1 It is a characteristic of the Gaussian units that c always enters along with , whether t is expressed as in (4) or implied as in (2) implied in that the current / is quantity per unit time. 360 ELECTRIC WAVES [CHAP. II into electrostatic units so as to conform with the Gaussian system as above specified, we have i n _L (4) dt The direction of positive linkage is shown in the diagrams of Figs. 1 and 2. Equations (2) and (4) are called respectively the magneto- motive force equation and the electromotive force equation, ab- breviated M.M.F. and E.M.F. They are now to be transformed into point relations. For this purpose a system of rectangular axes is chosen as follows. 19. Choice of Axes. Follow- FIG. 2. Arrows marked V and B . point in direction of positive linkage, ing What now Seems to be the prevalent usage in electromagnetic theory, we shall adopt as our system of rectangular axes the system shown in Fig. 3, in which z points out from the plane of the paper toward the reader, when x is to the right and y is up- ward in the plane of the paper. This rule merely gives the relative orientation of the axes, and it is evident that the scheme of Fig. 4 is the same system of axes. A?/ \y (in) (out ) FIG. 3. Positive set FIG. 4 of axes. Also positive set of axes. 20. Transformation of Magnetomotive Force Equation into a Point Relation. Let us take any extended region (for example, the room of a building) and suppose that there are electric cur- rents flowing in conducting masses within the room, and let the current density at any point x, y, z be u with components u x , u v , and Ui, along the three axes respectively. As a special case u may be zero at some or all points. CHAP. II] MAXWELL'S EQUATIONS 361 Let us now consider the magnetomotive force around a rec- tangle AyAz, Fig. 5, drawn with one corner at the point x } y, z. The component of current density at the point x, y, z perpen- dicular to the area A^/Az is u x . The other components of current density, those in the directions y and z } contribute nothing to the M.M.F. around the area. Now the average value of u x over the area is different from u x at x, y, z, and we shall designate this average value by u x . The current through the area is then (5) FIG. 5. whence by equation (2) the M.M.F. around the area is W = (6) Let us now get a second expression for this M.M.F., W, by estimating directly from the geometry of the problem the work done by the magnetic forces in driving a unit magnetic pole around the area A?/Az. The magnetic force on a unit pole at the point x, y, z is H, with components H*, H y , and Hz along the three axes. The magnetic force and its components are different for different points of the region. Since the work by a force in displacing its point of application is the magnitude of the force times the displacement in the direction of the force, we shall have for the work of carrying a 362 ELECTRIC WAVES [CHAP. II unit magnetic pole in the positive direction around the rectangle the equation W = H y ky + P*A z- H'yby - H^z (7) in which H y is the average value of H y along the side (1), H f z is the average value of H z along the side (2), etc. Now "H v is a function of x, y, z, and At/, and may be written H y = H y (x,y, Also we may write H'y = H y (x, y, z which is the same function with z+ As substituted for z; whence by Taylor's Theorem, assuming proper continuity and writing only first order terms, H', = H,+ d te (8) In like manner, nT7 E'.-H. + ^AI, (9) Taking the right-hand side of (6) and equating it to the right- hand side of (7) after replacing H' y and H'z by their values from (8) and (9), we have Dividing through by At/Az and taking the limit as At/ and Az approach zero, we have A JJ AH Olz OJLlv c dy dz and by similar reasoning, 4:TTUy _ dH x dHi c dz dx dHy dH 3 c dx or briefly the vector equation 47TU curl H (12) CHAP. II] MAXWELL'S EQUATIONS 363 where curl H is a vector, the magnitude of whose x, y, and z components 1 are respectively , TT dH z dH y mrl - H = w -* -. ,_.._. dri x dH z curlsH = _____ curl z H = - - (13) dx dy and where curl H = i curl x H + j curlj, H + k curl, H (14) The vector equation (12) or the equivalent Cartesian equations (11) give a relation 2 between the electric current density at a point (in electrostatic units) and the magnetic intensity at the same point (in electromagnetic units) derived under the limitations: I. That the vectors u and H are continuous functions of the coordinates of the point; and II. That the current is of such a character that the original M.M.F. equation (2) is true. 21. Transformation of the Electromotive Force Equation into a Point Relation. We shall now transform the other funda- mental equation (4) into a form analogous to (12), and obtain a second set of Maxwell's equations. We can do this by the similarity of the equations (2) and (4), without going again through the details of a demonstration like the preceding. It is to be noted that W of (2) is a line integral of the magnetic intensity H around a closed curve. Likewise, in (4) the electro- motive force V, defined as the work by the field in driving a unit charge around a closed circuit, is a line integral of the electric intensity E around the circuit. Also the magnetic induction B is related to the flux of induction B in the same way that current 1 From the above analysis it is seen that the method of obtaining the component of curl H in any particular direction N at any particular point P is as follows : Surround P by a closed curve S in a plane perpendicular to N. Let AS be the area within the curve, then i^n P fHds cos (H, * f fHds cos (H, S)1 . -or that is, curl N TL is the line integral of H around the periphery of a small area perpendicular to N divided by the small area. 2 Maxwell: "Electricity and Magnetism," Vol. II. 364 ELECTRIC WAVES [CHAP, n density u is related to current /. This shows that by going through a process Similar to that employed in transforming (2), we should obtain from (4) the equations 1 dB x dE z dE v c dt dy dz IdBy = dE X dE Z c dt ' dz dx 1 dB z dE y dE x c dt dx or, briefly, in vectorial notation, 1 r)B (15) The vector equation (16), or the equivalent Cartesian equations (15), gives a relation 1 between the time derivative of the magnetic induction at a point and the electric intensity at the same point derived under the limitations: I. That the vectors B and E are continuous functions of x, y, z, and t; and II. That the original electromotive force equation (4) is a correct experimental law. Equations (15) or (16) may be called Maxwell's Magnetic Induction Equations. 22. Further Examination of the Two Curl -equations. In the preceding sections we have derived the equations = curl H (12) 20 1 r)Tl _ i ^ = curl E (16) 21 which may be called respectively the " current-density equation" and the " magnetic induction equation." These equations were derived subject to the assumption that the M.M.F. law (2) and the E.M.F. law (4) are correct and general. We shall now show that the current-density equation (12) of 20 cannot be true in general ; for the reason that the divergence of any curl (e.g., div. curl A) is zero, while the divergence of u is not zero except in a special case in which the quantity of elec- tricity flowing out of a given region in a given time is equal to the 1 Maxwell: "Electricity and Magnetism," Vol. II. CHAP. II] MAXWELL'S EQUATIONS 365 quantity flowing in. The separate steps of this demonstration will now be given. 23. Theorem. Div. Curl A = 0. Proof: r) r) r) div. curl A = -r-(curl x A) + (curly A) + (curl* A) ox dy dz dx \ dy ' dz J dy \ dz ' ~ dx !Tz{~dx ' ' ~dy~] = (17) The last step is conditioned on the equality of such quantities as d /dA t \ d /dA z . These quantities are equal provided it is permissible to change \ ^ the order of differentiation, and this is permissible provided the second order derivatives so obtained all exist.'" The conclu- sion then is that the divergence of the curl of any vector A is zero, provided A is of such a character that the several second order derivatives of each of its components all exist. 24. Application of This Theorem to (12) and (16). Taking the divergence of both sides of the current-density equation (12) and the magnetic-induction equation (16), we have, respectively div. u = (18) |( div - B )= Now (19) is true, for by 16 div. B = 0. There is, hence, no inconsistency in the magnetic-induction equation (16). On the other hand, we shall now show that (18) div. u = is sometimes true and sometimes not true; to wit, div. u = is true when and where there is no changing intrinsic charge density; but div. u does not equal zero when and where there is a changing intrinsic charge density. Let us proceed to a critical examination of div. u. 25. Examination of Div. u. If we take any small volume AT surrounding a given point P, the quantity of electricity per second flowing out of Ar is the surface integral of the outward normal component of u over the closed surface bounding Ar. This 366 ELECTRIC WAVES [CHAP. II quantity flowing out is also the time rate of decrease of the quantity of electricity within AT. Equating these two expressions we have Dividing by AT and taking the limit as AT approaches zero, we have Limit fy* w n ) 368 ELECTRIC WAVES [CHAP. II It is very apparent that the quantity here added to u is just sufficient to make the sum u' solenoidal; for by (20) and (21), dp d surf. div. u = -- > ot ot and by Art. 13, equations (21) and (25) (their time derivatives) 1 dDl dp 1 &D do- "- Summing these quantities, we have div. u' = 0, surf. div. u' = (23) Maxwell's Assumption is the assumption that in respect to "I J^T\ the Magnetic Field the quantity -. -- acts as a density of current, 4?r ot which he called displacement current, and which must be added to conduction current density u to give complete current density u'. 28. The Generalized Current Density Equation. With this assumption the current density equation (12) may be generalized into 4ru , 1 dD . _ + ci>r = curlH (24) which may be called Maxwell's Generalized Current-density Equation. The addition of the first two terms is a vector addition. It is apparent that there is no mathematical inconsistency in Maxwell's method of generalizing the conception of an electric current, in respect to its effect in producing or responding to a magnetic field. Whether or not this generalized current is related to the magnetic field intensity by an equation of the form of (24) is a question for experimental determination. The experimental test has never been adequately made on the assumption directly. The validity of Maxwell's Assumption rests on his prediction from it of the existence of electric waves, and on his prediction of the electromagnetic character of light. These predictions have been amply verified. 29. At any Surface of Electric or Magnetic Discontinuity the Tangential Components of E and H are Continuous. We need the proposition here stated, for the solution of problems pertaining to surfaces of discontinuity. It may be proved as fdllows: Referring to Fig. 7, at any surface of discontinuity in conductivity, dielectric constant or permeability, let us draw a CHAP. II] MAXWELL'S EQUATIONS 369 small elongated rectangle with its length a parallel to the surface of discontinuity, and let 6 be the width of the rectangle. Let EIT, EZT, Ez, and E be the average values of the electric intensity along the four sides of the rectangle; and let B be the average value of magnetic induction perpendicular to the rectangle; then we have by the E.M.F. equation (3) the result - - ~ (B ab) = - E lT a + E 3 b + E 2T a ~ E 4 b C ut If now we assume that B and E are everywhere finite, and let b approach zero, the left-hand side of the equation approaches zero; also E s b and E*b approach zero; hence [Eira = E*Ta], for b = and if we let a also approach zero, the average values of E along the sides a ap- proach the actual values at a point on the surface; whence E IT E, (25) Hence the tangential component of E is everywhere continuous. In like manner, if the current density FIG. 7. In proof of con- u is everywhere finite, it can be shown tinuit y of th e tangential f 1 1 - - IT T-I /. component of E. from the M.M.F. equation of the form of (2), with I replaced by a surface integral of u and with u' re- placing u to give the equation generality, that the tangential component of H is everywhere continuous. CHAPTER III ENERGY OF THE ELECTROMAGNETIC FIELD. POYNT- ING'S VECTOR 30. Summary of Chapters I and II. The important results obtained in the preceding chapters may be summarized in the following equations (in which partial derivative with respect to time is indicated by a dot over a symbol) : 47TU . D , TT , , x 1 = curl H (A) c c - - = curl E (B) c div. D = 47T P (C) div. B = (D) D = eE (E) B = M H (F) To these may be added an expression for' the current density u (derived from Ohm's Law) u = 7 E ] where (G) 7 = specific conductivity. J We have also derived the following surface relations that hold at surfaces of discontinuity surf. div. D = 4^ (//) surf. div. B = (7) E\T EIT (J) H, T = H, T (K) These equations will hereafter be designated by the letters ascribed after them respectively, instead of by the accidental numerical designations with which they first appeared. 370 CHAP. Ill] ENERGY -371 In the present chapter we shall treat certain general proposi- tions regarding the energy of the field. For this purpose we need at the outset a few theorems in vector analysis. 31. Scalar and Vector Product. Let A and B be any two vec- tors drawn away from a common point, Fig. 1. These vectors may be written A = A x i + A y j + B = B x i + B y j + (1) (2) where i, j, and k are unit vectors along the three axes respectively. If now we introduce the convention that p = J2 = k 2 = 1? ij = -ji = k, jk = -kj = i, ki = - ik = j, and take the product of A and B term by term, we obtain -fiJD -- ^\. %j x ~~\ *** y**3 y ~~\ -^*- z*-5 z l~" FIG. 1. Illustrating vec- tor product of A and B. A z B y )i+(A z B x -A x B z )j + (A x B v -A y B x )]s. (3) We may call AB the complete product of A by B. It is seen to consist of two parts, one of which, consisting of the sum of the first three terms, is scalar; and the other, consisting of the sum of three vector components, is vector. These two parts are called respectively the scalar product, to be designated by A*B (read "A dot B "), and the vector product, to be designated by AxB (read "A cross B"). Then A*B = A X B X -\- AyBy AxB= (AyB z - A z B y )i It is seen that A Z B Z (A z B x ~A x B z )j (4) (A x B y -AyB x )k (5) A-B = AB {cos (A, x) cos (B, x) + cos (A, cos (A, z) cos (B, z) } cos (B, y) + (6) = AB cos (A, B) The scalar product of two vectors is the product of their magni- tudes by the cosine of the angle between them. 372 ELECTRIC WAVES [CHAP. Ill To find the meaning of the vector product AxB, let us designate by I, m, n the direction cosines of A, and by I', m', ri the direc- tion cosines of B. Then the square of the magnitude of AxB is the sum of the squares of the i, j, and k components; that is [AxB] 2 = {(rnri - nm'Y + (nl' -In'Y + (lm f - ml') 2 }A 2 B 2 = {(w 2 + n 2 )/' 2 + (n 2 + Z 2 )m' 2 + (I 2 +m 2 )n' 2 - 2(mm'nn' + ll'nri + mm'll r )}A 2 B 2 . Now 1 = I 2 + m 2 + n 2 = I' 2 + m' 2 + n' 2 , whence the preceding equation becomes [AxB] 2 = A 2 B 2 {1 - (IV + mm' + nn') 2 } = A 2 B 2 {1 - cos 2 (A, B)} = A 2 5 2 sin'(A, B) therefore, [AxB] = AB sin (A, B) (7) This gives the magnitude of the vector product. Let us next determine its direction. This can be done by taking the scalar product of A and AxB, which by (4) may be written A-(AxB) = A x (A y B 2 - A Z B V ) + A V (A Z B X - A X B 2 ) + A,(A x B y - A V B X ) = 0. In like manner it can be shown that B-(AxB) = Hence by (6) the vector product AxB is perpendicular to A and to B. By the convention ij = k, etc., this perpendicular is to be drawn with respect to A and B so that a positive rotation about the product vector will turn A into the direction of B. Hence, the vector product AxB is a vector whose magnitude is the product of the magnitudes of A and B by the sine of the angle between them, and whose direction is the positive perpendicular to the plane of A and B. The product BxA has the opposite direction, so that BxA = - AxB (8) CHAP. Ill] ENERGY 373 If we make B and A identical, equations (7) and (6) show that AxA = (9) and A A = A 2 (10) 32. The Divergence of a Vector Product. The divergence of AxB may be found directly as follows: div. AxB = ~ (A y B z - A Z B V ) + OX d_ dz dB dz dx dx ^dA z _ A dB z Mf _ A Bx B dx dAy dz dz = B x cuA x A-f By curly A x curl^B A tf curlyB = B- curl A A curl B A z cur! 2 B (ID 33. Energy and Radiation. We shall now treat a very im- portant general proposition with respect to the energy and radiation of energy in the electromagnetic field. Let us take any point x, y, z, Fig. 2, and de- scribe at x, y, z an element of volume &y Suppose that there are current density u and electric and magnetic intensities E and H at x, y, z. Let us study the energy transformations taking place in the volume Ar. The electromotive force between the two opposite Ai/Ae-faces of the volume element is the average electric intensity E x times the distance Az. The current flowing between these faces is the average normal current-density u x times the area of one of these faces A?/Az. Whence the electrical power (energy per second) A; / / \ / / /^~ / x y* 7 Ax FIG. 2. 374 ELECTRIC WAVES [CHAP. Ill converted into heat or other form of power by the current in the ^-direction is EJI* Az AT/ Az. Likewise, the power expended by currents in the y and z-directions is EyUyAzA^/Az and E 2 u*AzA2/A2 respectively. Adding these three quantities we have for the electrical power converted into other forms of power in the element the value AP = (EJI* + EyU y + E,u,)Ar (12) Dividing by AT and taking the limit as AT approaches zero, we have, for the power converted per unit volume at x, y, z, the quantity dP En E E = E x u x + E y u v + E 2 u 2 = E-u. Let us now replace u by its value from Maxwell's equation (A), obtaining ^P ^ 1 r > (13) or 4?r 47T Now by the theorem expressed in equation (11) div. ExH = H curl E - E curl H. Substituting the value of E curl H from this equation into (13), we obtain ~ = /- H-curlE -- div. ExH - ~ E-f). dr 4ir 4ir 4?r Replacing curl E by its value from Maxwell's equation (5), we have ^ = -^H-B - -j^-E D - -^-div.ExH. or 4:r 4ir 4?r Sin,ce e and AI are independent of the time, B = MH, D = cE and the first two terms may be written as derivatives of squares ; and the last term, when multiplied by dr becomes by (19), Chap- ter I, a surface integral over the surface of the volume dY; so that dp = - - H * + E * dr ~ In this equation (ExH) n is the outward normal component of the vector ExH, and the integration contemplated in the last term of the equation is an integration extended over the sur- face of the volume. CHAP. Ill] ENERGY 375 In order to give an interpretation to the equation let us write, as abbreviations, O ' O \ / and s = ^ExH. (16) Then equation (14) becomes dP = - Udr - fs n dS (17) in which dP is the power, or energy per second, converted into heat or other form of energy within the element dr. This power is the sum of two terms, both with negative signs. We, there- fore, naturally look to these terms as the source of supply of the power that is converted. One of the terms is a volume term and, taken with its negative sign, it may be regarded as the time rate of decrease of the magnetic and electrical energy in the element of volume, so that U = energy per unit volume. The other term is a surface term, and taken with its negative sign, it is the time rate at which energy flows into the element through its surface. Then s n dS is the quantity of energy per second flowing through dS in the direction of the outward nor- mal, that is, S = energy par second flowing in the direction of s per unit area perpendicular to s. This vector s, defined in equation (16) is called Poynting's Radiation Vector, and was discovered by Professor J. H. Poynting. 1 The equation (15) for the energy density in an electromagnetic field, and the equation (16) for the flow of electromagnetic energy per second per unit cross section of the energy beam are very important quantities in the theory of electric waves. Although we have employed in the above derivation the general case in which there is an electric current of density u at the point x, y, z, it is seen that the whole demonstration holds when u = 0. 1 J. H. Poynting, Phil Trans., 2, p. 343, 1884. 376 ELECTRIC WAVES TCHAP. Ill We should then have dP = 0, and Udr = - fs n dS (18) This means that in this special case that the rate of gain of electrical and magnetic energy within the region is equal to the rate at which electromagnetic energy flows in through the surface. CHAPTER IV WAVE EQUATIONS. PLANE WAVE SOLUTION 34. Digression to Find Curl Curl A. In proper combinations of Maxwell's equations the work may be simplified by the use of a proposition in vector analysis concerning the curl of the curl of a vector. Let us designate the vector by A. Then let us perform elementary operations as follows: curl x curl A = -^ curl z A curl y A dy dz d tdAy dA x \ d /dA x dA z \ dy \ dx dy 1 dz\ dz dx / _ d 2 A x _ d 2 A^ d_ tdA v dA,\ dy 2 dz 2 T dx\~dy~ ' ' ~dz/ d 2 A Subtracting and adding 2 *> we have curl, curl A = - V 2 A X + A (div. A) (1) ox where as an abbreviation we have written the Laplacian operator . (2) In like manner, ourl y curl A = - V 2 A y + ~ (div. A) (3) curl, curl A = - V 2 A Z + ^- (div. A) (4) oz These three curl curl components may be collected into a single vector equation by multiplying respectively by i, j, and k and adding, with the result curl curl A = V 2 A + grad. div. A (5) where if ^ is any scalar quantity, and if i, j, k are unit vectors 377 378 ELECTRIC WAVES [CHAP. IV along the three axes, then gradient ^, which is abbreviated "grad ^," is defined by the equation and triangle square of a vector A is defined by V 2 A = V 2 A x i + V 2 ^ y j + V 2 A*k = V 2 A* + V 2 A y + V 2 A, (7) 35. Elimination Among the Electromagnetic Field Equations for a Homogeneous Isotropic Medium. In a homogeneous me- dium e, M, and 7 are constants. If the medium is isotropic, these quantities are also independent of direction. Under these con- ditions, Maxwell's Equations (A) and (B), Art. 30 may be written (8) - = curl E (9) C If now we take the curl of both sides of (8), we have curl E + - 4- ( curl E ) = curl curl H - c c at Replacing in this equation curl E by its value from (9), and re- placing curl curl H by its value from (5), we have, since div. H = 0, Now starting with the other Maxwellian Equation (9) and taking the curl of both sides of it, we obtain - - (curl H) = curl curl E c ot from which by (8) and (5) we obtain 47T7M <3E e/i d 2 E 7r - T gradp The equations (10) and (11) are vector equations and are true for each of the components of H or E, in a homogeneous CHAP. IV] WAVE EQUATIONS 379 isotropic medium. For example, taking the ^-components we have, after dividing by i, the scalar relations Similar expressions for the other components may be had by advancing the letters. Each component is thus obtainable in a differential equation not involving the other components, so that the problem may be completely solved in any cases in which the differential equation of the type of (12) or (13) can be solved. We shall not at present enter into the discussion of the general equations but shall consider certain important special cases. 36. Special Case in Which the Homogeneous Medium is an Insulating Medium Uncharged. In this case the conductivity 7 is zero and the intrinsic charge density p is also zero, so that each component of electric and magnetic intensity E x , E y , E z , H x , H y , H z satisfies an equation of the form where M is a generic expression for either of the components of electric or magnetic intensity. This equation. is of a type known in elasticity theory as the wave equations. 37. Special Case of a Plane Wave in an Insulating Homogene- ous Uncharged Medium. The equation (14) applies to this case, but this equation is to be still further specialized by making M a function of s and t alone, M = f(s, (15) where s = Ix -f my + nz (16) where /, m, n are the direction cosines of s; so that I = cos. of angle between s and x m cos. of angle between s and y _ n = cos. of angle between s and z ( 1 = J2 + m 2 + U 2 Equation (16) is the equation of all points x, y, z on a plane PQ (Fig. 1) perpendicular to s at its end; so that s is the perpendicular distance from the origin to the plane. 380 ELECTRIC WAVES [CHAP. IV For a fixed value of s, and at a fixed time, the value of M (15) is the same at all points of the plane. M is a generic symbol for each component of electric or magnetic intensity, so that each of these intensities is the same all over the plane s at a given time. As the time changes, these values of intensity in the plane change but remain of uniform value over the plane. If on the other hand, the time is considered fixed, and different values are given to s, each of the different values of s will repre- sent a different one of a series of parallel planes perpendicular to s, and over each of these different planes the intensity will be uniform but different from plane to plane. FIG. 1. Every point x, y, z, of the plane QP satisfies equation (16), in which s is the length of the perpendicular from to the plane. The field of electric and magnetic intensity may thus be called a plane field. In the case of the plane field the wave equation (14) reduces to a simpler form if we express V 2 M in terms of s thus: dM ds dM ds dx ds ' likewise, dx dx 2 d 2 M dy 2 d 2 M dz 2 = Z 2 ds 2 ds 2 ~ 9 ds 2 (18) CHAP. IV] WAVE EQUATIONS 381 Whence (14) becomes fM d*M _ 6 2 M c* ~W " ~W Each component of electric and magnetic intensity in the plane field satisfies an equation of the form of (19). This equation, for reasons that we shall soon see, is called the Plane-wave Equation. 38. Classification of Solutions of the Plane-wave Equation. Let us now undertake a solution of the plane-wave equation (19), in which M is a generic symbol for any of the electric or magnetic intensities. Two classes of solutions will appear. These we shall call Class I and Class II, described as follows: Class I. All solutions that reduce both sides of (19) to zero; Class II. All solutions that reduce the two sides of (19) to equality but not to zero. We shall find that only solutions of Class II are important for electric wave theory, but both classes will now be considered. 39. Solutions of Class L Let P be any solution of (19) of Class I. Then by definition of this class d 2 P An integration of these equations as simultaneous gives P = A + Bt + Cs + Dst (22) in which A, B, C, D are constants of integration. In all cases in which the intensities are restricted to finite values B = C = D = 0, and P = A (23) This constant A is also zero in all cases in which only fluctuating quantities enter into consideration. 40. Solutions of Class II. Returning now to the plane-wave equation (19), let us seek for solutions of Class II; that is, for solutions that do not reduce the two sides of the equation to zero. Any function of s + at (if a has the proper value) is a solution of (19), as may be seen by direct substitution as follows: 382 ELECTRIC WAVES [CHAP. IV Let M = G( 8+at ) (24) where G is a symbol for " function," and let us take the second derivatives of G with respect to s and t. For this purpose, let us designate the first and second derivatives of G with respect to its argument (s -f at) by G' and G" respectively. Then dM whence, equation (19) becomes ^ a*G" = G" (25) This equation is satisfied by G" = 0, which has already been treated in Class I. It is also satisfied by any function G what- ever, provided /X c 2 That is, a = + ~ (26) or It thus appears that in our attempt to find one functional solution of (19) we have found two; namely, and M= G + -= o where F and G are any functions of their respective arguments. Now since equation (19) is linear and homogeneous, the sum of the several solutions is a solution; that is (27) This solution is the complete integral of equation (19); for the term P includes all solutions that satisfy (19) by reducing CHAP. IV] WAVE EQUATIONS 383 both sides to zero, and the terms F and G being two arbitrary functions include all other solutions of the second-order partial differential equation with two independent variables. If we omit the P solution, which as we have shown in Art. 39, is of no importance where only fluctuating intensities enter into consideration, we shall have only the F and G solutions of (27). 41. Examination of the Plane -wave Solution. Velocity. In equation (27) is given the complete solution of the plane- wave equation (19). In this solution M is any one of the com- ponents of electric or magnetic intensity. The functions F and G may be different for the different components, but the argu- ments of these two functions remain always the same two arguments. The several functions are interrelated by Maxwell's equations and are further delimited by the boundary conditions at the source of the disturbance and at any surfaces of discontinuity that exist between different media. Without at present entering into these interrelations and limita- tions, we can discover certain interesting properties of the field by examining the general solution (27). We can, for example, obtain the result that the F and G parts of the field are disturb- ances that move with a finite velocity, and we can determine the velocity as follows : Let us confine our attention at first to the function F, and write We see that, whatever value M may happen to have all over the plane at the distance Si from the origin at the time t\, it will have the same value all over the plane s% at the time t z , provided c c si = ti = s 2 /= t 2 (28) for then That is, the time at which a given condition exists at two dif- ferent distances are related to the distances by the equation (28) or 2 Si C 384 ELECTRIC WAVES [CHAP. IV The distance traveled s 2 Si divided by the time to travel it (2 ~~ ti) gives the velocity; whence v = -= (29) is the velocity with which the condition at Si moves in the direction of increasing s. In a similar way it may be shown that the equation represents a condition moving in the opposite direction (the direction of decreasing s) with the same velocity. From the above discussion it appears that if we have an electro- magnetic field in which all of the components of electric and magnetic intensity are functions of s and t alone, where s is the perpendicular distance from an arbitrary origin, and if the intensities are supposed to remain everywhere and at all times finite, and if there are no constant components of intensity, the quantity P becomes 0, and each component of intensity consists in general of two superposed disturbances, or waves, moving in opposite directions along the axis of s with the velocity given in (29). The form of the functions F and G will depend upon the manner of the origination of the disturbance and upon the conditions at certain surfaces of discontinuity bounding the homogeneous region under consideration. In particular cases one of the func- tions G, say, may be everywhere zero, and the whole field will move forward in one direction with the velocity v. In other particular cases, as when we have a reflection of waves, both the forward-moving wave and the backward-moving wave will coexist and give an interference system. The importance of having the two functions in the solution is precisely this that it enables us to give a description of the phenomena of reflection when they occur. 42. Velocity in Free Space Equals the Ratio of Units, Equals the Velocity of Light. In space devoid of all matter, e = /* = 1 ; therefore, the velocity (29) becomes in empty space VQ = C, where c is the number of absolute c.g.s. electrostatic units of quantity of electricity in one electromagnetic unit of quantity. CHAP. IV] WAVE EQUATIONS 385 The prediction that electric waves in free space should have the value here given was made by Maxwell in his original writings on the electromagnetic theory of light. Before that time it was known from experiments that c, the ratio of the units, was ap- proximately the velocity of light. Maxwell himself made some of the measurements of the ratio of the units. Later experi- mental determinations of these quantities are given in the follow- ing table. Table I. Comparison of Velocity of Light with Ratio of Units 1 Velocity of light Observer 2.99853 X 10i 10 cm. /sec . Michelson 2.99860 Newcomb 2 . 99860 Perrotin 2.99852 Weinberg Average . . . 2 . 99856 Ratio of units Observer 3.0057 X 10 10 Himstedt 3.0000 Rosa 2. 9960 Thomson and Searle 2.9913 H. Abraham 3 . 0010 Hurmuzescu 2 . 9978 Perot and Fabry 2.9971 Rosa and Dorsey Average . . . 2 . 9984 44. Refractive Index for Electric Waves. To get the index of refraction for electric waves of any insulating medium of dielec- tric constant e and permeability AJ, it is only necessary to note that the velocity in this medium is while the velocity in vacuo is The ratio of these two velocities is the index of refraction n of the medium for the particular frequency at which e and /* are measured; that is, n = Vo/V = \/fJL (30) 1 For references to literature see Rosa and Dorsey : Bulletin Bureau of Standards, Vol. 3, Nos. 3 and 4, 1907. 25 386 ELECTRIC WAVES [CHAP. IV It is to be noted that the derivation of this equation assumes that the medium is non-conductive and that there are no motions of charged particles within the medium; for such a motion con- stitutes a current, and all such currents have been excluded from the special problem of the insulating medium. 45. The Plane Electric Wave in a Non-crystalline Homo- geneous Dielectric is a Transverse Wave, with its Electric and Magnetic Intensities Perpendicular to the Direction of Propaga- tion and Perpendicular to Each Other. Proof: Each compo- nent of electric intensity of the wave moving in direction of positive s is a function of s vt, and therefore of t s/v, where v = s = Ix + my + Let (31) E v = 9(t - s/v) E z = h(t - s/v) where /, g, and h are any functions of their argument t s/v. Let the derivatives of/, g, h with respect to t s/v be indicated by/', g' } h', and let us now determine the values of the components of H by Art. 30 Equations (B), the ^-component of which gives c dt dy dE, dz m n Integrating and multiplying by -- , we have (omitting the constant of integration as of no significance for the wave-field) H x = J- (mh - ng) 1, - nE v ) Likewise = A /- (nE x - IE,) I- (IE V - mE x ) (32) CHAP. IV] WAVE EQUATIONS 387 Let us now recall that Z, m, and n are the direction-cosines of s] that is, I, m, and n are the components along the axes of x, y, and z of a unit vector U s along s; whence by (5), Art. 31, equations (32) may be combined into the vector equation = Ji \M Us x E (33) where Us = a unit vector in direction of propagation s. This equation (33) gives the magnetic intensity H in magnitude and direction in terms of the electric intensity E for the case of a plane wave traveling in the direction s (or U) in a homogeneous insulating medium. In magnitude, it is seen by (33) H = JlE (34) Vji In direction H is 1 to E and 1 to s. To prove completely the proposition enunciated in the heading of this section, it remains to prove E also perpendicular to s. This can be done by starting with H x , H v , and H z as functions of (t s/v). The equations will be similar to (31) but with differ- ent functions. Then applying Maxwell's equations (A), Art. 30, of the type c dt dy dz we obtain E x = and similar equations for E v and E z ; whence vectorially E = - JU 8 x H = JH x U 8 (35) This equation agrees with (33) and shows in addition that E is _J_ to U. The conclusion from (33) and (35) is then that E, H, and Us are mutually perpendicular and are oriented with respect to one another in the same way as the axes x, y } z, in Fig. 3, Art. 19. The direction or propagation, which is the direction of U a , is also the direction of Poynting's vector s. 388 ELECTRIC WAVES [CHAP. IV Various nemonic rules have been suggested for remembering the orientation of E, H, and s. A simple one is as follows : E = east, s = south, H = high (upward). For the backward moving wave the rule for the orientation of the intensities with respect to the direction of propagation is the same; namely, equation (35). It is seen, however, that if we reverse the direction of one of the quantities E, H, s, we must reverse one other of them but not both, since any one of the vector quantities has as a factor the vector product of the other two. 46. The Instantaneous Electric Energy per Unit Volume is Equal to the Instantaneous Magnetic Energy per Unit Volume of a Single Plane Wave. This proposition follows at once, by squaring (34) and dividing by , which gives This equation, as well as (34) from which it is derived, holds true when there is a single plane wave moving in* one direction. It does not hold when there exists an interference system, as will be shown below. 47. Harmonic Solution for a Plane Wave, Plane Polarized, in a Homogeneous Insulator. Up to the present we have treated the problem of the plane wave by means of general functions, and we have shown that the electric and magnetic intensities and the direction of propagation are mutually perpendicular. Let us assume that the wave is plane polarized. This means that the direction of the electric and magnetic intensities do not change. We may choose the axes so that E is along the z-axis, and H is along the 2/-axis, then the direction of propagation will be the direction of the 2-axis ; and we may write E x =f(t-z/v) (37) and by (34) H v = J-/(* - *A) (38) \M where v = -= (39) CHAP. IV] WAVE EQUATIONS 389 It is now proposed to limit the problem by assuming that the electric intensity E x is a harmonic function of the time. By (37) it will then be a harmonic function of (t z/v), and may be writtten E x = Esm{u(t -z/v) + j (40) and by (38), H v = ~p-E sm{w(* --z/v)+ 0J (41) Ex A A A V V FIG. 2. Orientation of electric and magnetic intensities in a plane wave travel- ing in the z-direction in a homogeneous isotropic medium. where E = amplitude of E x ; o> = angular velocity in radians per second of the harmonic oscillation = 2ir/T, where T = period; = phase angle depending on choice of origin of time. Equations (40) and (41) give the electric and magnetic in- tensities of a harmonic wave moving in the ^-direction. It is seen that in such a wave the electric and magnetic intensities are in phase in time and space. At a given time the distribution of intensities for different values of z are given in Fig. 2; where, 390 ELECTRIC WAVES [CHAP. IV for simplicity of drawing, separate diagrams are made for the two intensities. To obtain the wave length X, we have only to note that the addition of X to z does not change E x or H y in (40) and (41). This means that or vT (42) As we have shown in the examination of the general functions of (t s/v), the whole diagrams of Fig. 2, except the axial line oz, are supposed to move forward in the ^-direction with the velocity v. If the observation is made at a fixed point on the axis, z = constant, the vectors of electric and magnetic intensity will fluctuate sinusoidally with the time. The plane of the wave is a plane perpendicular to oz and any such plane has all over it a uniform value of electric intensity, and of magnetic intensity, at a given time. CHAPTER V REFLECTION OF A PLANE WAVE FROM A PERFECT CONDUCTOR In the present chapter we shall treat the reflection of a plane electric wave from the surface of a perfect conductor. In Arts. 48 and 49 the wave will be considered to be harmonic and to be incident normally. In Arts. 50 and 51 the more general case will be considered, in which the incidence is oblique and the wave not limited to the harmonic form. In a later chapter cases in which the conductor is not a perfect conductor will be considered. FIG. 1. Electric wave E x , H y , traveling in the z-direction, incident normally on a perfectly conductive surface M . 48. Reflection of a Harmonic Plane -polarized Plane Wave from a Perfectly Conductive Plane at Normal Incidence. Let M, Fig. 1, be a perfect conductor with a plane surface in the #2/-plane through the origin of coordinates. Let a plane-polar- ized wave coming from the left of the surface in a dielectric medium of dielectric constant e and permeability ju be incident normally upon the surface, and let us choose the axes so that the ic-axis is in the direction of the electric intensity, and the 2/-axis in the direction of the magnetic intensity. 391 392 ELECTRIC WAVES [CHAP. V The characteristic of a perfect conductor is that the electric intensity within the conductor is zero. In the medium in contact with the conductor the tangential component of electric intensity is continuous with its value within the conductor, and therefore zero at all times. We have assumed the incident wave harmonic, but a single harmonic value for E x , such as is given in (40), Art. 47, does not possess the property of being zero at z = 0, and is therefore in- sufficient to represent the system of waves in the present prob- lem. By our general solution (27), Art. 40, we may add to the wave traveling in the ^-direction another wave traveling in the opposite direction, and with proper choice of intensities, phases, etc., it is possible to make the direct and the reflected waves annul each other as to electric intensity at the surface of the con- ductor. Since the incident wave is harmonic, the reflected wave to annul it must be also harmonic and of the same frequency and same phase angle. By proper choice of the origin of time we may make this phase angle < = 0, and write the solution E x = E l sin u(t - z/v) + E 2 sin (t - z/v) + J-E sin u>(( + z/v) (3) M V/l This equation for H v may, if desired, be independently derived by substituting the value (2) for E x , with E y and E z equal zero, into Maxwell's Equation (B), Art. 30. Equations (2) and (3) show that the magnetic intensity H v is made up of two harmonic wave-trains traveling in opposite directions, having equal amplitudes, and having the reflected magnetic intensity in phase with the incident magnetic intensity; while the electric intensity E x consists also of a direct and a CHAP. V] REFLECTION OF A PLANE WAVE 393 reflected wave of equal amplitude, but the reflected electric intensity is opposite in phase to the incident electric intensity. Let us now put equations (2) and (3) into a better form for their interpretation. Expanding the sine terms by the trigonometric formulas for the sine of a sum or a difference, we obtain E x = 2E cos cot sin (uz/v) (4) H y = 2 A /- # sin ut COS (az/v) Vn (5) 49. Plot of Stationary Wave System. A plot of the two inten- sities is given in Fig. 2, where, to obviate difficulty in plotting, - z-Axis FIG. 2. Stationary waves of electric and magnetic intensity at normal incidence on a perfectly conductive plane surface M . In the figure the period is represented as r. no effort is made to show that the magnetic and electric inten- sities are at right angles to each other. The equations (4) and (5) are thus seen to be the equations to two stationary wave systems. There are certain points in space where the electric intensity is always zero and certain other points where the magnetic intensity is always zero. These positions of constant zero-intensity are called nodes. Between the electric nodes and between the magnetic nodes there are points of maximum fluctuation of intensity, which are called loops. 394 ELECTRIC WAVES [CHAP. V Whereas, in the single free train of waves the electric and mag- netic intensities are exactly in phase in time and space; in the interference system, or stationary system, the electric and magnetic intensities are 90 out of phase in time and space. The wavelength in the incident wave by (42), Art. 47, is X = 2irv/w. The positions of the nodes in the stationary system are seen to be at the following values of z: The nodes of E x are at etc.; that is -z = 0, X/2, X, 3X/2, etc. (6) The nodes of H y are at -z = X/4, 3X/4, 5X/4, 7X/4, etc. (7) Loops exist halfway between these respective nodes. It is seen that the distance between consecutive electric nodes or consecutive magnetic nodes is half the wavelength of the incident wave. The distance between consecutive electric loops or consecutive magnetic loops is the same distance. Since the reflected intensities are equal to the incident in- tensities in amplitude, the perfectly conductive surface is a perfect reflector for electromagnetic waves. 50. Reflection of a Plane Wave from a Perfectly Conductive Plane at Arbitrary Incidence. Let the conductive plane, which we shall call the mirror, pass through the origin of coordinates and be perpendicular to the #-axis. Suppose a plane direct wave to be traveling in a medium of dielectric constant e and permeability /*, and in the direction of a line s with direction cosines I, m, n. Then any point x, y, z on the incident wave front W, Fig. 2, will satisfy the equation Ix + my + nz = s (8) where s is the distance from to W, In the Direct Wave, let the components of electric intensity by any functions /, g, h of (t s/v) ; that is E x = f(t -.s/v) } E v = g(t - s/v) (9) E z = h(t s/v) J where V = A / VAtC CHAP. V] REFLECTION OF A PLANE WAVE Then as in equation (32), Art. 45, H x = J-(mE 2 - nE v ) H y = ^-(nE x - IE Z ) HZ = -\l(lE y mEx) 395 (11) It is apparent that this direct wave alone is not sufficient, for the reason that the tangential components of electric force must be at all times zero at the mirror, and the values of (9) do not satisfy this condition. It is, therefore, necessary to sup- pose a reflected wave also to exist and to be superposed upon the direct wave. We shall assume the reflected wave to be also a plane wave and to be traveling in some unknown direction along a line i, with direction cosines l\ t mi, n\ t and shall show that with proper choice of Si and with proper intensities in the reflected wave, the proper boundary conditions are satisfied. The reflected wave may be expressed in terms of arbitrary functions fi, g\ and hi as follows: = M + = gi(t + = hi(t -i /k>i>J (12) A/- = A/- (m\E\z A/- (n\Ei x liEiz (13) with l\x + m\y + n\z = s\ (14) where Si is the distance OW\. Now by the conditions at the mirror, when we put x = 0, the total tangential electric force must be zero; that is, the sum of the direct and the reflected E y and E z values must be zero; hence (15) my + nz<> 396 ELECTRIC WAVES [CHAP. V where in these -equations y and z are coordinates of any point in the surface of the mirror. To make (15) true for all such points and for all values of -t, we must have for the operators g and gi, h and hi, the relations ffi = ~ and for the direction cosines, mi = -\-m \ ni = +n (17) Let us determine the other direction cosine Zi. By the fact that the sum of the squares of the direction cosines of a given line is unity, l\ is equal to plus or minus 1} but if it were plus I, then Si would be identical with s for any given point x, y, z and the total y and ^-components of E would by (16) be zero every- where at all times, and our incident wave would have only an Fio. 2. Illustrating a plane wave W traveling in direction S incident at angle o* incidence 6 upon a plane perfectly conductive surface M . z-component and would be traveling parallel to the mirror. This case is of no interest, as the problem is then, so far as concerns the dielectric medium, the same as that with the mirror absent. Excluding this case, equivalent to no mirror present, we have in all other cases h = -I (18) The equations (17) and (18) show that the electric radiation obeys the ordinary law of reflection of light; namely, the reflected ray is in the same plane with the incident ray and the normal to the mirror at the point of incidence, and the angle of reflection is equal to the angle of incidence. (Proof follows.) CHAP. V] REFLECTION OF A PLANE WAVE 397 This is seen by reference to Fig. 2. The angle of incidence 9 = cos" 1 Z. The angle of reflection 9' = the supplement of cos" 1 li 0. The equality of mi to ra and of n\ to n, makes the incident and reflected beam in the same plane perpendicular to the mirror. Returning now to the question of electric and magnetic in- tensities, we have found the form of g and hi in terms of g and h. It remains to find the form of /i. This can be done by employing the fact that the electric intensity is in the wave front in both the direct and reflected waves; that is, the components in the directions s and i are respectively zero. This means that If + mg + nh =0 (19) lifi + migfi + njii = (20) In view of (16), (17) and (18) the equation (20) becomes If i mg nh = 0, which added to (19) gives /i = / (21) 51. Intensities in Direct Wave and Reflected Wave, and Total Intensities at the Mirror. Summarizing the results, we have for the intensities of the direct and reflected waves and for the total intensities at the mirror the following equations: Direct Wave (22) E x = f(t - 8/v)} H x = ^p (mE z - nEy)] Ey = g(t - s/v) H y = ^ ( nE * - Z&) E n = h(t - s/v) H z = A /- (IE V - mE x ) 1 M Reflected Wave ^T > ^\<^ Ei, = f(t + 8i/v) Hi, = Ely = - g(t i si/v) Hiy = J * (njgfi, + /^ ia ) Ei z = - h(t -f i/0 ^i z = J-(-lEi v -mEi x ) (23) 398 ELECTRIC WAVES [CHAP. V Total Field at the Mirror by (9), (12), (16), (22) and (23) E x -f~ EI X 2E X H x + HI X = J? _1_ 7? fl H _1_ W O7 ii/j / -t~-C'ly vl fly \ fl\y ^il j TJT |_ TJT f\ TJ t IT ()TT Eiz -J- Hiiz U 1 z ~\~ -fl Iz tl z at x = (24) It is seen that the effect on the plane wave of the plane per- fectly conductive mirror is to double the normal electric intensity at the mirror and annihilate the tangential electric intensities; also to annihilate the normal magnetic intensities and double the tangential magnetic intensities at the mirror. In the space at any distance from the surface of the mirror the equations (22) and (23) permit the complete computation of the reflected wave in terms of the direct electric intensities where these are known. CHAPTER VI VITREOUS REFLECTION AND REFRACTION 62. Reflection and Refraction of a Plane Electric Wave by a Homogeneous Insulator. Suppose a plane electric wave in an insulating medium of inductivity ei, and permeability MI to be incident upon the plane surface of a second insulating medium of inductivity 2 and permeability /Z2. Let the surface between the two media be through the origin of co- ordinates and perpendicular to the x-axis, as in Fig. 1. Let us assume that the direct wave is traveling in the direction of Si with direction cosines Zi, mi, and n\\ and that there is a refracted wave traveling in the second medium in some direction s 2 (direction cosines Z 2 , ^2, n 2 ), and also a reflected wave in the first medium traveling in some direction s 3 (cosines 1$, Ws, nz) in the first medium is c O Boundary of Media X (Outward) FIG. 1. Concerning reflection and refraction at a boundary. The velocity of the waves V. the velocity in the second medium is (1) (2) It is required to find the directions of propagation of the re- flected and refracted waves, and their intensities relative to the incident intensities. The geometrical equations of the tfhree wave-fronts are respectively hx + l s x m 2 y m 3 y n 2 z = (3) 399 400 ELECTRIC WAVES [CHAP. VI We shall first write down the values of the electric intensities in the three waves respectively: In the Direct Wave E lx = fi(t - E ly = gi (t - EI Z = hi(t In the Refracted Wave In the Reflected Wave ft(t ~ = h 3 (t - (4) (5) (6) The magnetic intensities in these three waves are given re- spectively by the vector equations (cf. (33), Chapter IV): Ui x Ex HI 6 2 2 = * / (7) where Ui, U2, and Us are unit vectors in the directions of i, s z , and 83 respectively. In addition to the above equations we have by equation (26), Chapter I, the condition that at the boundary between the two media the normal component of electric induction is continuous, since there is no intrinsic surface charge, and this gives HlZ (8) This equation is true for all values of y and z in the surface between the two media; whence it follows that CHAP. VI] VITREOUS REFLECTION, REFRACTION 401 m, = m, _ m, Vi i s 21 = 2! = - 2 (10) #1 Vi Vz Now it is to be noted that l\ is the cosine of the angle of in- cidence of the ray = cos 61 ; lz is the cosine of the angle of re- fraction = cos 02,* and Is is the cosine of the supplement of the angle of reflection = cos 3 ; whence = sin0! (11) + n z 2 = sin 2 (12) VW + n 3 2 = sin 03 (13) And by taking the square root of the sum of the squares of (9) and (10) we obtain sin 0! = sin 3 (14) (15) sin e 2 Vz Equation (14) shows that the angle of reflection is equal to the angle of incidence. Equation (15) shows that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is the ratio of the velocity in the incident medium to the velocity in the refracting medium. These are the ordinary laws of reflection and refraction. To make these laws complete we need also to show that the incident ray, the refracted ray, the reflected ray and the normal to the surface are in the same plane. This can be seen to be true by noticing that the y and z axes have not yet been chosen. If we make the 2-axis perpendicular to the incident ray, n\ will be zero; and by (10) HZ and n s are also zero, so that all three of the rays are perpendicular to the z-axis, and are, therefore, in the same plane, which plane also contains x, since it is a concurrent perpendicular to z. In order next to determine the coefficient of reflection of the surface between the media, let us keep the orientation of axis above suggested. Then the three rays are in the rri/-plane, as shown in Fig. 2. Let us compare the energy incident per second upon any area dS with the energy transmitted through dS per 26 402 ELECTRIC WAVES [CHAP. VI second. The cross sections of the three beams with their bases on dS respectively are dA l = hdS ] dAz = l z dS \ (16) dA 9 = l s dS J By Poynting's Theorem (eq. (16), Chapter III), the energy flowing per second per unit cross section of either of these beams is (17) FIG. 2. Relation of areas of cross-section in the several beams. The energy flowing per second through any area dA per- pendicular to the ray is the area times the value of s; i.e., cdA _ __ ds = - : E x H. 4?r Substituting the values of the dA's from (16) and the values of of the various H's in terms of their E's from (7), we have for the energy per second at dS on the surface between the media, the values Incident Energy per Sec. = -^ -J l\E\ Reflected Energy per Sec. = (18) (19) CHAP. VI] VITREOUS REFLECTION, REFRACTION 403 Refracted Energy per Sec. = -- Z 2 #2 2 (20) where the subscript (o) indicates that values at the mirror are to be taken; that is, values with x = 0. Calling the ratio of the reflected energy per second to the in- cident energy per second the coefficient of reflection, indicated by r, we have by (18) and (19) and by the law of the conservation of energy, from (18), (19) and (20), by equating incident energy to reflected plus refracted energy and dividing out a common factor whence by transposition and division, (22) The equations (21) and (22) hold for any orientation of the electric vector in the plane of the incident wave. It is proposed now to determine the coefficient of reflection in terms of the index of refraction and angle of incidence alone, for two principal directions of polarization of the electric wave. This is done in Art. 53 for E perpendicular to the plane of incidence, and Art. 54 for E parallel to the plane of incidence. 53. Determination of Coefficient of Reflection when E is Perpendicular to the Plane of Incidence. In this case, since the plane of incidence is the xy-pl&ne, we have the E entirely in the ^-direction; that is, E = E z . As before, let us indicate by a subscript ( ) the value of E at the reflecting surface. From the continuity of the tangential component of electric force at the surface, since the whole force is tangential, we have EIQ + E 3Q = EIQ (23) Dividing by EI Q and substituting from (21) and (22), we obtain Vr = Jr J~- d - \ k \62Mi 1 + r - (1 - r) (24) 404 ELECTRIC WAVES This squared gives, after factoring, [CHAP. VI Dividing out a common factor, we obtain (25) Now /eiMi __ V*. \2M2 Vi so that (25) may be written k/*i*>i(l + whence _ - \/r) HIM 2^2 (26) O Boundary of Media FIG. 3. E perpendicular to the plane of incidence. In this equation, if 0i and 62 are respectively the angle of inci- dence and angle of refraction, Zi = cos 61 Z 2 = cos 02 = -Jl - 2 sin 2 0i, by (14). These values substituted in (26) give / 2 cos 0i tii A \ 9 sin 2 0i r = (27) jij cos 0i + MI A/ 1 ^ - sin 2 0i Now = > where ni and n 2 are the indices of refraction of incident and refractive media respectively. CHAP. VI] VITREOUS REFLECTION, REFRACTION 405 In all insulating media ^ = m = 1, so that (27) may be written cos 61 - J(- -V- sin 2 61 '-A/ 1 L COS 0] (28) sin 2 61 Equation (28) gives the coefficient of reflection r in case the elec- tric force in the incident wave is perpendicular to the plane of in- cidence. In this equation ni and n z are indices of refraction of incident and refractive media respectively and are not to be confused with direction cosines. 54. Determination of the Coefficient of Reflection when E is in the Plane of Incidence. In this case E z = 0, Fig. 4, and we have for the total electric intensity in each ray E = V#x 2 + E 2 o Boundary of.Media FIG. 4. E in the plane of incidence. and for the total magnetic intensity H = H z . The condition of continuity of the tangential component of magnetic intensity at the reflecting surface gives, since the whole magnetic intensity is tangential, the boundary condition #10 + #30 = #20 (29) Expressing now the coefficient of reflection in terms of #i , #20, and #3 , by replacing the E's in (21) and (22) by equivalent values in terms of the #'s taken from equations (6), we have r = 406 and ELECTRIC WAVES [CHAP. VI These values substituted in (29) give eiju 2 (30) which is the same as (25) except that the subscripts of c and /* are advanced, and therefore gives on simplification (cf. 26) (31) Replacing l\ and h by their values in terms of 0i, we obtain r = (32) ~ /*i cos 0! + M8 A/ 2 - sin 2 6 or in terms of indices, of refraction, when jm = /i 2 = 1, r = ) cos 0i sn (33) Equation (33) gives the coefficient of reflection r in case the electric force in the incident wave is parallel to the plane of incidence. In this equation n\ and n% are indices of refraction of incident and refractive media respectively. 65. Transformation of Equations (28) and (33). By the law of refraction (15), in view of definitions preceding (28), we have sin 61 sin 0o HI (34) where n\ and n^ = indices of refraction of incident medium and refractive medium respectively. From (34) I^i^0 2 = Jl.~ (^ 'sin 2 0! (35) cos 2 = CHAP. VI] VITREOUS REFLECTION, REFRACTION 407 Substitution of (35) into (28) gives cos 61 cos 6 2 cos 0i H - cos 9 2 HI Replacing by its value from (34), we obtain sin (0i - 2 ) sn 2 ) f For incident E perpendicular to . "ane of inci- dence. (36) Treating (33) in a similar manner, we obtain r = -tan (61 -6 2 ) tan (0i -f- 2 ) f For incident E 1 parallel to plane (37) I of incidence. v ' Equations (36) and (37) are known as Fresnel's equations. In these equations r is the ratio obtained by dividing energy per second leaving reflecting surface in reflected beam by energy per second in- cident on same surface. 01 = angle of incidence. 2 = angle of refraction. Equation (36) is for a plane incident wave with the electric force perpendicular to the plane of incidence. In optics such a wave is said to be polarized in the plane of incidence. Equation (37) is for a plane incident wave with the electric force parallel to the plane of incidence. Such a wave is said to be polarized perpendicular to the plane of incidence. The plane of incidence is the plane of the incident ray, the reflected ray and the normal to the reflecting surface. CHAPTER VII ELECTRIC WAVES IN AN IMPERFECTLY CONDUCTIVE MEDIUM 1 56. Wave Equations in a Homogeneous Imperfect Conduc- tor. It has been shown in Art 35, Chapter IV, that in a homo- geneous medium of conductivity 7, permeability M> and dielectric constant e, the magnetic and electric intensities satisfy the equations and /!_*/,. At? *-.. ^217 A** (2) 57. Relaxation Time. A question now arises as to the value of the intrinsic volume density p in such a medium. We can determine this matter by taking the divergence of equation (8) , Art. 35, remembering that the divergence of a curl is zero; we have iv.E + - -(div. E) =0 (3) c c dt or replacing div. E by its value in terms of p, 4Try_ dp _ Q v v Integrating this we obtain P = P ^_ ^ (4) where e is base of natural logarithms and Whence it appears that if p has the value p at some time reckoned as origin of time, p will decrease exponentially with 1 This chapter is based on ABRAHAM and FOPPL, " Theorie der Elelctrizi- tat," Vol. 1, p. 321, 1907. 408 CHAP. VII] IMPERFECTLY CONDUCTIVE MEDIUM 409 the time. The process is called relaxation, and the time for p to fall to one eth of its value is r, given by (5), and called the re- laxation time of the material. The relaxation time for any good conductor is so short that it has never been experimentally determined for any metal. Its determination for so poor a conductor as pure water is a matter of extreme difficulty. 58. Steady-state Plane Wave Equation. Equation (4) shows that after the lapse of a sufficient time, usually very brief, the value of p in any conductor is substantially zero, and we may omit the p term from (2) . Having thus simplified the equation (2), let us next restrict the wave field to a plane- wave field. Then E and H will be functions of t and s alone, where s is the perpendicular distance from the origin of coordinates to a plane over which the field is constant at a given time. Then if Z, m, and n are the direction cosines of s, s = Ix + ifay + nz (6) is the equation of any such plane, and the quantities V 2 H and /)-TT r)2T7 V 2 E reduce to -r-y and j, so that the wave equations (1) and (2) become s/felSjJSV...*! m c 2 \ e dt dt 2 1 ' ' ds 2 and /47TT M / ? \ ~dt" dt 2 ) " = ds 2 59. Limitation to Solution Harmonic in Time. Each com- ponent of electric intensity and each component of magnetic in- tensity must satisfy an equation of the form of (8). Let M be the generic designation for E x , E y , E z , H x , H v , H z , then ejj. /47TT dM d 2 M \ _ d 2 M ^ \ e ~dt ~ "W ) ~~ " ds 2 This equation is a form of equation that plays a fundamental role in telegraphy and telephony and is known as the telegraph equation, which has been the subject of much theoretical and practical investigation. We shall content ourselves with a treatment of the equation for the special case in which the solution involves the time har- 410 ELECTRIC WAVES [CHAP. VII monically. M will then be the real part of the quantity that can be written in the form M = e iut F(s) (10) where F(s) is some function of s but not of t. Designating the second derivative of F with respect to s by F", and substituting (10) in (9), we have jo, - *() = F"() (11) Since F" is a complete derivative, (11) is an ordinary differential equation of the second order with constant coefficients, and its solution may be written in the form F(s) = ae ks (1.2) which substituted in (11) gives w = *( j - *> 2 ) (13) C while a is completely arbitrary. It is seen that k is a complex quantity. Let us break up k into real and imaginary parts by setting k = -~(x+jn) (14) c where x and n are both real quantities, and x is positive to avoid infinite values of M. From (14) and (13) we are to determine x and n. 60. Determination of x and n. Substituting (14) in (13) we obtain x + jn = Vwj - 1 = VvVlyTj - e (15) where T = 27r/co = the period. (16) Squaring and equating real and imaginary parts, we have X 2 - n 2 = - ev. (17) and 2 x nj = 2ypTj (18) Subtracting (18) from (17) and extracting the square root, we obtain __ X ~jn = V/TV - 2jTj - (19) CHAP. VII] IMPERFECTLY CONDUCTIVE MEDIUM 411 The product of (19) and (15) gives This compared with (17) gives, by addition and subtraction and by omitting signs inconsistent with the condition that x and n shall be real and x shall be positive, the result } (21) - <) (22) ft M may now be expressed in terms of x an d n by combining (14), (12) and (10), and is (ox? M = ae~ c.0*- s / c > (23) where a is an arbitrary constant and is in general a complex quantity. The real part of (23) is also a solution of the given differential equation, and may be written in the form M = Ae ~ ^ cos{o>( - ns/c) + 4>] (24) where A and < are both arbitrary constants. A solution of the form of (24) is the most general harmonic solution of angular velocity w of the given differential equation (9) ; for the assump- tion that the solution is a harmonic function of the time with angular velocity o> reduces the equation to the form of (11), which is an ordinary differential equation of the second order, so that any solution that contains two arbitrary constants, is the general solution. 61. Extinction Coefficient, Velocity, and Index of Refraction. Each component of electric and magnetic intensity in a har- monic wave in a homogeneous conductive medium satisfies an equation of the form of (24) with, however, in general a different value of A and for each component. It is seen that the intensities are attenuated as the wave penetrates deeper and deeper into the conductor, and that the attenuation is determined by the factor e c which may be called the Attenuation Factor. The quantity x 412 ELECTRIC WAVES [CHAP. VII is called the Extinction Coefficient of the medium for the given frequency of oscillation. The exponential term is expressed in the rather complicated form here given, so that x shall be a quantity symmetrical in form with n. A verbal description of the extinction coefficient may be had by substituting CO = 27T/7 7 where T is the period of oscillation, and X = cT, where X equals the wavelength in vacuo; then the attenuation factor given above becomes or e ~ x , if s = so the extinction coefficient x is the logarithmic decrement of ampli- tude for a traversed distance equal to ^- of a vacuum wavelength. Returning now to (24), let us see next the significance of n. Apart from the attenuation factor, M is seen to be a function of t s/(c/ri)\ therefore, the velocity of propagation of a given phase of the wave is v = c/n (25) where c is the velocity of the wave in vacuo. Hence n is the index of refraction of the conductive medium for the particular frequency. By substituting the value of n from (22) in (25), we have for v i-- s = (26) = c ^ e* + 47 2 7 72 - (27) (28) The values of x t n an d v may be simplified for certain special cases by expansion of the radical expressions with neglect of small terms. Examples follow. CHAP. Vll] IMPERFECTLY CONDUCTIVE MEDIUM 413 62. Special Case of Small Conductivity. If y 2 T 2 is negligible in comparison with 2c 2 v = - (29) n = \/M (30) (31) -*? = <-*? (32) In this special case of low conductivity, the velocity v, the index of refraction n, and the attenuation factor e~~T are all independent of the frequency of oscillation. 63. Special Case of Large Conductivity. If, on the other hand, the conductivity is so large in comparison with the dielec- tric constant that e is negligible in comparison with 47 T, (33) (34) (35) (36) In this special case the velocity, index of refraction, and attenuation factor all involve the square root of the period of oscillation. 64. Relation of H to E. Each component of E can be ex- pressed in the form of (23), where only the real part is to be taken. The y and 2-components are in which s = Ix -{- my + nz. (The direction cosine n is not to be confused with the index of refraction n.) Now by Maxwell's Equation (B), Chapter III, taking the z-component _ /j dH X = dE Z dEy c dt ~ dy ' dz 414 ELECTRIC WAVES [CHAP. VII Integrating with respect to t, we obtain -^(x + ttf) Zff m = - - : - (mE. - nE v ) c ja) H x = -(-xj + n)(mE. - nE v ) z - nE v ] (37) The factor ar*** 1 indicates that the real part of (37) may be obtained by taking the real parts of E g and E y and retard- ing their phase angles by tan" 1 ^) . If we indicate such a retarda- \/X\ tan" 1 w, we have the real equation H x = - Vn 2 -\- x 2 { (mE s - nE y ) \tan~ 1 } (38) The expression in braces is seen to be the re-component of the vector product U,xE \tan~ 1 where U, is a unit vector in the s-direction. There are similar components for H v and H z ; so that the total vector H may be written H = Vn 2 + x 2 U. x E\tan-' g) (39) This equation means that H is the positive perpendicular to s and to E, that the magnitude of H is - Vw 2 + x 2 times the magnitude of E, and that H lags behind E in phase by the angle whose tangent is x/n. Written trigonometrically, with the aid of (24), if the magni- tude of the resultant electric intensity is -i^l ( i E- Ae c co s a>( - n*/c)+ - tan~ ] ( ~ ) [ (41) CHAP. VII] IMPERFECTLY CONDUCTIVE MEDIUM 415 65. Poynting's Vector. Transmission and Absorption of Energy. We shall next determine the amount of energy flowing per unit cross section per second in the direction of s. The general form of Poynting's vector is s = ~E xH 4r which gives for the problem under consideration s = A 2 e - -7^ cos a cos { a tan" 1 4?r n where a = u(t ns/c) + U s = a unit vector in the direction s. Expanding the second cosine factor, and taking the time average, indicated by s, we obtain i. /l (42) (43) <7T fj, & where A 8 = amplitude of E at . Equation (42) or (43) gives the average rate of flow of energy per second per unit area within the conductor. It is easy to obtain from this expression (42) the average rate at which energy is absorbed in the conductor. The absorbed energy per unit volume per second indicated by P is the decrease of s per unit distance, "^ (44) A 8 = amplitude of E at 5. where, again, The same result may be obtained by taking the time average I/ of electromotive force per unit length times current-density. 416 ELECTRIC WAVES [CHAP. VII Equation (43) gives the average power transmitted per unit area and equation (44) gives the average power absorbed per unit volume. 66. The Reflection of a Harmonic Plane Polarized Wave from a Plane Imperfectly Conductive Surface at Normal Incidence. In Chapter V the reflection from a perfect conductor has been considered. It is proposed to investigate now the reflection at normal incidence of a plane harmonic wave from a surface of a conductor of any conductivity 7, dielectric constant c, and perme- ability M- Let the surface of the conductor be through the origin of coordinates and in the ^-plane, Fig. 1, and let the x-axis be in the direction of the electric intensity. Let a plane elec- tric wave traveling in a vacuum in the 2-direction fall upon the conduc- FIG. 1 Illustrating a plane t j ye sur f ac e of which the COnduc- wave incident normally on the . surface of a medium of any con- tlVlty, dielectric Constant and per- ductivity T, dielectric constant e mea bility are respectively 7, c, and /*. and permeability ju. T J . , . , , \ ,, ,. Indicating by subscript (i) the di- rect wave; by (2) the transmitted wave, and by (3) the reflected wave, we have In the Direct Wave (incident) Vacuum AI cos co(t z/c) #1 = A: and Poynting's vector S j - - ~~L ^L i of which the time average is 0)(t Z/C) (45) (46) In the Reflected Wave Ez x = -A 3 cos {&(t + z/c) + < 3 ( H 8l , = - A 3 cos {o>( + z/c) + 0s} C 40 TT (47) (48) (49) (50) CHAP. VII] IMPERFECTLY CONDUCTIVE MEDIUM 417 In the Transmitted Wave _x* * cos [ 2 \ fJL/ M whence ' tan ** = "- ' ' (60) Now taking the sum of (58) and (59) and making wt = 0, we have 2A l = A 2 (^-i^ cos 2 + ~ sin I /A /* and by (60) this reduces to 2A. - Therefore, and by (57) v r = .0* ~ *)|, where, by (21) and (22), (62) (63) Equation (61) gives the coefficient of reflection r at normal inci- dence of a harmonic electric wave of period T from the plane surface of a homogeneous body of conductivity 7, dielectric constant e, and permeability p. in contact with a vacuum. 67. Special Case for Conductivity Zero. The equation (61) is true in general for normal incidence whatever the value of the conductivity. If 7 = 0, x 0, and with n = 1, this reduces to n 2 = 5 iVe 2 + 4T 2 ? 72 + * a + ) 2 which is the equation to which (28) and (33), Chapter VI, derived for vitreous reflection, also reduce when the incidence is normal, CHAP. VII] IMPERFECTLY CONDUCTIVE MEDIUM 419 i.e., 0i = and the first medium is a vacuum. (N. B. The quantity n of (63)_reduces in this case to the familiar index of refraction n = \X;ue.) 68. Special Case of a Good Conductor. In this case if we assume negligible in comparison with 4y T, we have by (34) and (35) n = x = whence the coefficient of reflection r of (61) becomes + ZnwT + 2yT This law has been tested for the reflection of long heat waves from metals in some experiments by Hagen and Rubens 1 and has been found to agree with the facts within the limits of the errors of measurement for the metals tested, except bismuth. 69. Phase Changes at Reflection at Normal Incidence. In equation (60) we have obtained the value 02 = tan' 1 - (65) E fj. -f- n This angle < 2 is the angle of advance of the phase of the trans- mitted electric intensity over the phase o e the incident electric intensity. The corresponding angle for the transmitted magnetic intensity is To obtain the phase angle of the reflected wave, we may use equation (58), which for wt = ir/2 becomes A 3 sin 03 = A 2 sin < 2 - In view of (56) and (57) this may be written n r HAGEN and H. RUBENS, Ann. der Physik. (4), Vol. II., p. 873, 1903. 420 ELECTRIC WAVES [CHAP. VII which by (65) gives, after proper transformations, *,= *,- tan-' x . + 2 ff_ M , (66) This angle fa is the angle of advance of phase of the electric or magnetic intensity of the reflected beam over the incident beam, by reflection at normal incidence. CHAPTER VIII ELECTRIC WAVES DUE TO AN OSCILLATING DOUBLET 70. Doublet Consisting of an Electron Oscillating in a Positive Atom. One conception of an oscillating doublet based on the Thomson Atom 1 is illustrated in Fig. 1. This system is supposed to consist of a large positively charged and practically immovable positive sphere of uniform charge density, within which a small negatively charged body (an electron) is oscillating about its position of equilibrium at the center of the sphere. Let the dis- tance of the electron from the center of the atom be p. Let the charge of the electron be e, and the charge of the positive sphere be +e. If every element of the sphere at- tracts the electron with a force inversely pro- portional to the distance from the element to FIG. l. A doublet the electron, the total force on the electron eiTtlo^ 8 ^ c^pabte 1 of will be proportional to the distance p and oscillating within a uni- proportional to e 2 , and will be in the line charged solid joining the electron with the center of the sphere; that is, Restoring force = A = Ke 2 p The static energy of the system will then be W. = JAdp = \ K*p> =\K {/() } (1) where f(t) = ep = moment of the doublet (2) K = restoring force per unit distance per unit charge. The kinetic energy of the system is where M = - 2 (4) 1 SIR J. J. THOMSON, "The Corpuscular Theory of Matter," London, 1907. 421 422 ELECTRIC WAVES [CHAP. VIII In modern electron theory the mass m and therefore the quan- tity M in this expression for the kinetic energy is a constant only provided the velocity of the electron is small in comparison with the velocity of light. We shall need this assumption later for other reasons. The total energy of the system is U = (5) 71. Alternative Conception of Doublet Leading to Equivalent Results. An alternative type of oscillator lead- ing to the same form of energy equation is illus- trated in Fig. 2. Two bodies A and B of large mutual capacity are connected by a short wire of zero resistance, and electric currents are supposed to flow between A. and B giving them at any time equal and opposite charges q. The capacities of the bodies A and B are supposed to be so large that the capacity of the connect- ing wire may be neglected. Then the same cur- rent i will flow throughout the length of the connecting wire, and i = q. If C is the mutual capacity of A and B, the static energy of the system will be w> = \ ? The energy in the inductance L, which is the inductance of the connecting wire, is W L = \ Li* ..Q, -o FIG. 2. Dumb- bell doublet. Whence the total energy of the system is (6) If p is the distance apart of A and B, and we write the moment of this system we have which is of the same form as (5). (7) CHAP. VIII] DUE TO AN OSCILLATING DOUBLET 423 In this alternative type of doublet, the distance between A and B must be small in comparison with the wavelength of the free oscillation of the system, so that the distributed capacity in the lead wire L may be neglected. 72. Oscillations with Constant Energy. If, with the first type of doublet, we assume the energy U constant we shall have V = = Kff + Mff, which divided by / and integrated gives / = A l cos (co * + 0) (8) where AI and are arbitrary constants and A similar treatment of the second type of doublet gives the same value of /, but with The oscillation in either case would go on undiminished with constant amplitude and frequency, if the system did not radiate or receive any energy. We shall next show how to calculate the energy radiated as electromagnetic waves from an oscillator of these types. But we shall arrive at the result only by an indirect and somewhat tedious process. 73. Treatment of a Polarized Spherical Wave. In this we shall follow the method of Hertz. 1 Without at present enter- ing into a consideration of the source of the waves, let us consider an electromagnetic field in which the component of magnetic intensity in the ^-direction is zero; that is H, = 0. We shall assume that the medium is homogeneous'everywhere except near the origin of coordinates, where there will be located an oscillator of, as yet, an undefined character. In any part of the medium, whether homogeneous or not, the ^-component of Maxwell's Equation (B) gives = ^ _ *j* (11) dx dy 1 HERTZ, "Electric Waves," translated by D. E. Jones, Macmillan and Co., 1893. See also Planck, " Warmest rahlung," Earth, p. 100, 1906. 424 ELECTRIC WAVES [CHAP. VIII It follows that for the two components E x and E y a scalar function V exists such that dV Ty (12) as may be proved by a cross differentiation that leads to (11). Let us next assume that outside of the source the medium has no intrinsic charge, so dE x , dE v dE x rv ~dx "~dyr"~dz from which by substitution from (12), dE z dz dx 7_ &V 2 ' A,2 (13) An examination of (13) suggests making V the ^-derivative of some function F so that the equation (13) can be integrated. Let dF V = - dz Then from (12), (13) and (14) we have (14) dz* (15) Let us now write down two of Maxwell's Equations (A), Chapter III, which are, for H z = 0, and for u x = u y = C dt dE V dz dH x (16) C dt dz Substituting from (15) into (16) and integrating we obtain Hy H f = c d d?/ a a^ 7 C dt dx (17) CHAP. V1I1] DUE TO AN OSCILLATING DOUBLET 425 Equations (15) and (17) show that, without any assumption other than that p = H z = u x = u y = 0, we have been able to express all of the components of electric and magnetic intensity in terms of the derivatives of F, which is a scalar function of x, y, z, and t\ so far as we have seen up to the present F may be any such function. F is, however, not completely arbitrary, for the ^-component of Maxwell's Equation (5), .Chapter III, is dE z , dE v + c dt dy dx~ 2 dy^d^ 2 ^dy ~dz 2 ~dy by (15) Replacing the left-hand side of this equation by its value from (17), we have dy \dyr which integrated with respect to y gives In performing this integration we have neglected the arbitrary functions independent of y, which the integration gives as addi- tive terms to v (18). ! These may be added ad lib., and when added give an equation for F less restrictive than (18). If we restrict F to (18) we shall have it at least sufficiently restricted. We may say then that given any scalar function F satisfying equation (18), and performing on it the operations indicated in (15) and (17), we shall obtain for points outside of the region of intrinsic charge a set of possible values of electric and magnetic intensities that will make H z = u x u y = 0. We shall now put a further restriction of F; namely, we shall assume F a function of only t and the distance r from the origin of coordinates: F = F(r, t) (19) where r = x 2 2 z 2 426 ELECTRIC WAVES [CHAP. VIII Preparatory to substituting (19) in (18), we have dF = xdF dx ~ r dr d 2 F = ldF x 2 dF x 2 d 2 F dx 2 ~ r dr r 3 dr + r 2 ~dr~ 2 with similar terms for the y and ^-derivatives, giving V9zr 2dF . d 2 F 1 a 2 / _\ A 2 F = --- -- = -- 1 rF \ r dr ^ [dr 2 rdr 2 \ I This result substituted in (18) gives The integration of this equation as in 40 gives where (22) v = Let us confine our attention to the value of F given by the first of these terms, the /-term, which is a spherical wave of F traveling in the positive direction of r with the velocity v. In differentiating (22) for substitution in the equations of E x , . . . Hx, . . . , let us call and df(t - r/v) d(t - r/v) ' J It is to be noted that So that we can express all of the derivatives of / in terms of /; for example CHAP. VIII] DUE TO AN OSCILLATING DOUBLET 427 df = _f dr v dF^ z_ dz '' r a ^ dxdz ./ v r 2 v *r_ e r , 4,- 1 dz r 3 r 2 ~*~ rv^ v 2 ! ?1 2 _ r*v r 2 v (23) Substituting these values in (15) and (17), we obtain 3f r r rt; (24) / , / - + - H z (25) Equations (24) and (25) give the values of the electric and magnetic intensities at the point x, y, z in terms of the coordinates of the point and in terms of f and its time derivatives. It is to be noted that xE x xH x E X H +yE y + yH y + zH, = + E Z H* = (26) (27) (28) Whence H is perpendicular to r in Fig. 3, and (since H z = 0) toz. Hence H is tangent to the sphere and also tangent to the sectional circle normal to the z-axis. H is perpendicular to E, but E is not perpendicular to r, and hence is not tangent to the spherical surface. 428 ELECTRIC WAVES [CHAP. VIII Let us transform our equations to spherical coordinates, Fig. 3, and let $ = the longitude of the point x, y, z, 6 = its colatitude, r = its distance from the center p = the radius of the small circle in plane perpendicular to z. . Then r = \/x* + y 2 + z* (29) P = Vx 2 + y 2 (30) Let us now determine the components of H and E along , 6 and r in the direction of the increasing value of these coordinates. These components will be designated by the use of , 9 and r as subscripts. FIG. 3. Spherical coordinates. = H y cos < H x sin TT X TT y = Jtly fix c r r esjn_e c r r v The and r-components of H are zero. r ~ 'r + v r + r 3 I r v 2cos0 ' (3D (32) CHAP, vill] DUE TO AN OSCILLATING DOUBLET 429 E e = E p cos - E g sin 9 = (E x - + E y 2} cos - E z sin 9 \ p -'p/ w ~ i y ~ i \ w i y jr i )^j_ _j_ w __ i_ j_ * " l " 4 n 4 2 ^ " r 2 3/ r rv v r >in9[/ /_ /I /v. *.2 ' ', .2 I (33) The -component of E is zero. Equations (31), (32), and (33) give the values of the components of H and E along the spherical coordinates. It is seen that H is in the direction of the parallels of latitude, and that E has a component in the direction of the radius r, and another component in the direc- tion of the meridianal line. Let us now investigate the electric and magnetic field in the neighborhood of the origin, in order to determine the character of the oscillator that could give rise to the field under con- sideration. 74. Proof that the Field Here Given is the Field Due to a Doublet at r = 0. In the equations for the components of H and E, let us investigate the field at distances r from the origin, and suppose that r is so small that [/][/! (34) v r where the symbol < < means "is negligible in comparison with." . The meaning of this assumption becomes clear when we con- sider / to be a periodic function of the time with angular velocity co; then the amplitude of / is co times the amplitude of /. Thus (34) becomes fc-^i vT r r X/27T (35) Under these conditions, the fourth and fifth equations of (23) 430 ELECTRIC WAVES [CHAP. VIII \2 \2E 1 show that A 2 F is negligible in comparison 1 with -, and that d 2 F the E z of (24) reduces to , so that by (24) E x , E y , and E z are dF respectively the x, y, and ^-derivatives of the same quantity - ; whence we see that the electric force at this position near the origin of coordinates has an ordinary static potential function dF * - --T- ( 36 ) OZ and by the second equation of (23), neglecting small terms, we obtain - . . :..'; *-/ - __.. (37) We shall now show that this is the potential due to a doublet at the origin with the moment e/, provided the square of the length of the doublet is negligible in comparison with 4r 2 . FIG. 4. In Fig. 4 suppose two charges e and e separated by a dis- tance p, lying along the direction of the z-axis, and suppose that the point P is distant r from the origin of coordinates midway between the charges, then the electrostatic potential at P is e e 7*1 f 2 e e ( _ P CQS9 \ / pe cos e c(r2 _ P 2 cos ' G ) 9 2 F 1 Unless ^-7 becomes small, as it does for certain relations of z to r. In that case the whole force component E z becomes negligible in comparison with E x or E v . CHAP. Vlll] DUE TO AN OSCILLATING DOUBLET 431 Let us now impose the condition that p 2 4r 2 (38) then, since cos 6 = z/r (compare Fig. 3), *.-? (39) Comparing (37) with (39) it is seen that if = 1, the potential^ of the electromagnetic field at points near the origin of coordi- nates is the potential of a doublet ^ of moment (cf. (2)) pe = / (40) If, on the other hand, the dielectric constant of the medium is different from unity, the moment of the doublet must be pe = f (41) in order to have a field continuous with the dynamic electro- magnetic field at points near the oscillator. The conclusion is that the electromagnetic field given by the dynamic equations (24) and (25), or the alternative polar ex- pressions (31), (32) and (33), satisfies the boundary condition imposed by a doublet of moment e/at the origin; but this doublet must be so short that the square of its length p 2 4ri 2 where, by (35), ri X/27T To cause an error of less than one per cent, in the computations, p ^ .002 X/27T ^ X/3000, approx. This means in the case of a doublet of the type described in Art. 70 that the velocity of the moving electron must be not greater than 1/3000 of the velocity of light. In the alternative type of doub- let described in Art. 71 the length between the capacities A and B, Fig. 25, must be not greater than 1/3000 of the radiated wave- length. We may now continue with the problem under these limita- tions. 75. Electric and Magnetic Intensities at Great Distance from the Oscillator. Let us now consider the electric and magnetic 432 ELECTRIC WAVES [CHAP. Vlll intensities at a point distant r from the oscillator, where r is so great in comparison with the wavelength that and a fortiori This means for / a harmonic or nearly harmonic function of the time that r X/27T. Under these conditions, equations (31), (32) and (33) become e sine f(t - r/v) ft = r cv p - sme /"( ~~ r / v ) r ~v*~ E r = in comparison with E e . where f(f) equals the moment of the doublet divided by the dielectric constant. In vacuum, and sufficiently approximate in air t (42) _sine/(*-r/c) r E r = (43) where /() = the moment of the doublet. The electric and magnetic intensities, when the dielectric surround- ing the oscillator is air, are equal to each other, and inversely propor- tional to the distance from the oscillator when this distance is large. The two intensities are directly proportional to the sine of the angle between the direction of the oscillator and the direction of the radius to the point under consideration. The electric in- tensity is in the direction of the meridional lines from the pole to the equator. The magnetic intensity is in the direction of the par- allels of latitude. 76. Power Radiated through a Large Sphere. If we consider a large sphere with the oscillator as center, we can apply Poynt- ing's Theorem and obtain the power radiated through any surface element of the sphere or through the whole sphere. The energy radiated per second (that is, the power radiated) CHAP. V11I] DUE TO AN OSCILLATING DOUBLET 433 through an element of surface dS of the sphere is by (16), Chapter III, u4S = ^E x H dS (44) = ~ E e H* dS numerically (45) Substituting for E e and H^ their values from (43), we obtain ds (46) with direction of r. The element of surface dS = r 2 sin GdQd(f>. This value substituted in (46) gives for the total power radiated through the sphere at great distance from the origin the value Total power radiated = -^L I d I sin 3 QdQ 47r*; 3 J Jo momentofdoublet and the /in (47) is'f(t - r/v). Equation (47) gives the total power (energy per second) passing through any distant sphere with the oscillator as center, and with an infinite medium of dielectric constant e. When the dielectric is air, (47) and (48) become Total power radiated = ^ (/) 2 (49) where f=f(t~r/c), and f(t) = moment of the doublet. 77. Power Radiated by a Sinusoidal Oscillator in Air or Vacuum. Let us next take the special case, in which the medium has unity dielectric constant and where the / of the dynamic electromagnetic field is assumed sinusoidal in the form / = A sin co( r/c). In this case through a distant sphere by (49) ^ , . 2A 2 o) 4 sin 2 co(f-r/c) Total power radiated = _ 3 , 28 434 ELECTRIC WAVES [CHAP. VIII of which the time average is where 3c 3 X = wavelength = . CO 78. Radiation Resistance of Sinusoidal Oscillator. For the oscillator described in the preceding section the moment of the oscillator is / = A sin ut = Iq where / is the length of the oscillator regarded as of the alter- native type of Art. 71. The current in such an oscillator is Au cos co * = =- 2ircA COS at ~~XT~ The mean square current is 27TVA 2 // we define the radiation resistance R of the oscillator as the mean power radiated divided by the mean square current, we have R = E. S. units (25) o CA One electrostatic unit of resistance equals 9 X 10 11 ohms, so that the radiation resistance in ohms becomes R = L ohms (53) Equation (53) gives the radiation resistance of an oscillating doub- let whose length I (or, as we have previously called it, p) is negligible in comparison with the wavelength X of the radiated wave. The application of this formula to a radiotelegraphic antenna, as has been made by Riidenberg, 1 is without theoretical justifica- tion, except in a very special case. We shall, in the next chapter, discuss at length the radiation from a radiotelegraphic antenna. 1 Riidenberg: "Annalen der Physik," 25, p. 453. CHAPTER IX THEORETICAL INVESTIGATION OF THE RADIATION CHARACTERISTICS OF AN ANTENNA 1 79. Introduction. For the proper design of a radiotelegraphic transmitting station it is important to know the radiation charac- teristics of different types of antenna. For example, if a flat-top antenna is to be employed, the ques- tion arises as to what is the best relation of the length of the horizontal part to the length of the vertical part, when the excitation is to be produced by a given type of generator. It may be known in a general way that the greater the vertical length, the^reater the radiation resistance; it may also be known that the greater the horizontal length of the flat-top the greater the capacity of the antenna will be, and the greater will be the amount of current that can be made to flow from certain types of generator. Now these two quantities, radiation resistance and applied current, are both factors in determining the output from the antenna. For a given generator, with known characteristics, the problem of getting the greatest output of high-frequency energy is a problem in the determination of the maximum value of the product of current square and radiation resistance of the antenna. But this is not the whole problem, for there comes also into con- sideration the question as to how much of the radiated energy is radiated by the horizontal flat-top in what may be a useless direction. Again, of the energy radiated from the vertical part of the antenna, how much of it contributes to the electric and magnetic forces on the horizon, where the receiving station is situated ? For the solution of these various problems it is important to know the radiation characteristics of the antenna in the form of certain functional relations. These relations should be known 1 This chapter was originally published by the author in the Proceedings of the American Academy of Arts and Sciences, Vol. 52, pp. 192-252, 1916. Certain errors in the original publication are here corrected. 435 436 ELECTRIC WAVES [CHAP. IX even when inductance is added at the base of the antenna for pro- viding coupling or for increasing the wavelength to adapt it to the generator. These quantities 'should be known theoretically, since the ordinary measurements of these quantities do not permit us to distinguish radiation that is useful from the useless radiation as heat losses and from the radiation in useless directions. It is the purpose of this chapter to give a treatment of this problem. Such a treatment is, so far as I know, up to the present entirely lacking, but the method here employed is that developed by Abraham 1 in a very remarkable paper entitled Funkentele- graphie und Elektrodynamik. In that paper, Abraham obtained theoretically the characteristics of a straight oscillator vibrating with its natural fundamental and harmonic frequencies. The present work is an extension of Abraham's method to the much more difficult problem of an antenna with a flat-top and with added inductance at the base. 80. Inadequacy of the Conception of an Antenna as a Doublet. Apart from the brilliant investigation by Abraham, all other attempts at the treatment of the radiation from an antenna assume that the antenna is a Hertzian Doublet. 2 This is only a very crude approximation to the facts, for the derivation of the electromagnetic field about a doublet assumes that the length of the doublet is negligible in comparison with a quantity that is itself neg- ligible in comparison with the wavelength. Hence, the doublet theory will apply in all of its essentials to an antenna, only provided the length of the antenna is not greater than one three thousandth of the wavelength emitted (see Art. 74). Of course, it may be that at great distances from the oscillator, the theory that it is a doublet may not introduce any large errors into certain problems such as the propagation over the surface of the earth; but the present treatment shows that the doublet theory does introduce large errors into computations of such quantities as the electric and magnetic field intensities and the radiation resistance of an antenna. It seems probable that other problems also should be revised in such a way as to replace the conception of the antenna as a doublet by the view of it as an oscillator that has a length comparable with one quarter of the wavelength. 81. Method of the Present Investigation. In the present in- vestigation, a doublet of infinitesimal length is assumed at each 1 M. Abraham: Physikalische Zeitschrift, 2, 329-334 (1901). 2 See Chapter VIII of present volume. CHAP. IX] CHARACTERISTICS OF AN ANTENNA 437 point of the antenna. This is the device used by Abraham. These elementary doublets are free from the objection regarding their lengths, as they are of infinitesimal lengths, while the wavelength is that due to the whole antenna and therefore is enormously large in comparison with the lengths of the elemental doublets. The electric and magnetic forces due to each of the doublets is determined at a distant point and is summed up for all of the doublets of the antenna, with strict regard to the difference of phase due to the different locations of the different doublets. Such a process performed for all points of a distant sphere surrounding the antenna gives the total electric and magnetic forces at all points on the sphere. Then by integrating Poynting' s Vector over the entire sphere, we obtain the total power radiated, and from this we compute the radiation resistance and other characteristics of the antenna. The effect due to the vertical portion of the antenna and to the horizontal flat-top portion are computed separately, so as to give information as to how much energy is radiated with its electric force perpendicular to the horizon and how much parallel to the horizon. In deciding as to the proper distribution of the elemental doublets along the antenna, the form of the current curve from point to point of the antenna is assumed independently. This process is not entirely above reproach, because Maxwell's equa- tions, if they could be properly applied to the problem, would themselves give the distribution that is consistent with the applied electromotive force at the base of the antenna and with the shape and form of the antenna. This step of accurately deriving the distribution is, however, at the present time not possible of mathematical execution. The distribution here assumed for the current in the antenna, 4s a function of the time and of the position along the antenna, and is given in the next section. 82. Assumed Current Distribution. The form of antenna to which the whole discussion is devoted is illustrated in Fig. 1, and consists of a vertical portion of length a and a horizontal flat-top portion of length b. These quantities a and 6 may have any relative values whatever. At the base of the antenna is an arbitrary inductance L for varying the wavelength. 438 ELECTRIC WAVES [CHAP. IX The current at any point P' of the antenna is assumed to be given by the equation ;,i' , . 2-7TC . . 27T/Xo 7 \ ... i = I sin Z-sm \- - I) (1) where c = velocity of light, X = natural wavelength of the antenna without inductance, X = the wavelength with the inductance, i = the current at the point P', I = length measured along the antenna from the inductance to the point P'. ^ -b => _^_ FIG. 1. Type of antenna. An inductance L not shown in this figure is supposed to be inserted between the antenna and the ground G for varying wavelength. The character of the assumed distribution is as follows: The factor sin -^ t means that the current is sinusoidal in time at every A point of the antenna, with the angular velocity 2ir _ 2wc _ 2irc T =: cT = X The meaning of the other factor (2) T 4TT /AO 7\ T / \ / sin y {- - I) = J (say) is illustrated in the diagrams (a), (b) and (c) of Fig. 2. If there is no inductance, X = X , and the factor becomes J = I cos (3) (4) This is illustrated in (a). CHAP. IX] CHARACTERISTICS OF AN ANTENNA 439 In the case with added inductance, \ ^ X , and we must keep the general form of J given in equation (3) . This equation for positive values of I gives the upper half of the diagram (b). When I is supposed negative the curves obtained continue along the dotted lines of (b) and do not give a figure symmetrical with the upper half. To produce proper symmetry the absolute value of I must be employed in equation (-1) when it is applied to the distribution of the image to take account of reflection. FIG. 2. Assumed distribution of current in the antenna. It is also to be carefully noted that when I = 0, equation (1) becomes T . 7rXo . 27rC, IQ = I sin TJ sin -r-t ,* A A so the amplitude at the base of the antenna is IQ = I si (5) (6) Now, finally, when the antenna has a flat-top it is assumed that the top part of the antenna is bent over without any significant 440 ELECTRIC WAVES [CHAP. IX change in the magnitude of the current at the various points, as illustrated in (c). When the equation (1) is to be applied to the vertical portion of the antenna, we shall call I = z' (7) where z' = vertical distance from the ground of the point P' on the antenna. . When the equation is to be applied to the horizontal part of the antenna, we shall call. * I = a + x' (8) where x' = distance along the horizontal part of the antenna to any point P" on the flat-top. The discussion will now be divided into several Parts: Part I. Electromagnetic Field Due to Vertical Portion of the Antenna; Part II. Field Due to Horizontal Portion of the Antenna; Part III., The Mutual Term in Power Determination. Part IV. Computations of Radiation Resistance. Part V. Field Inten- sities and Summary. PART I FIELD DUE TO VERTICAL PORTION OF ANTENNA 83. Coordinates. Let the origin of coordinates be at the point of connection of the antenna to the ground. Let the 2-axis be vertical. About this vertical, axis as polar diameter, let us construct a system of spherical coordinates in which the position of any point P is given by its distance r from the origin, and the , angles 6 and <. 6 = the angle along meridional -lines from the pole, = the angle along parallels of latitude from a vertical plane of reference whose position is at present immaterial. This system of coordinates with the positive directions of the angles indicated is given in Fig. 3. , If z' is the vertical ordinate of any point P' on the vertical portion of the antenna, and r the distance from P' to P, and if the distance OP is large in comparison with z', we mav write (see Fig. 4) r = r<> z' cos 6 (9) CHAP. IX] CHARACTERISTICS OF AN ANTENNA 441 '84. Field Due to a Doublet at P'. At a distant point P the electric and magnetic intensities due to a- doublet of length dz'and charges e and e at P' is, by Hertz's theory, given in Art. 75, dEe = dH = sin 9 (10) FIG. 3. A set of spherical coordinates. The coordinates of P are TQ, 0, . where f(t) = the moment of the doublet = e dz r , where e is in electrostatic units-, (11) dE e = the electric intensity in electrostatic units, which is en- tirely in the direction of 0; that is, of the meridional lines ; dH = the magnetic intensity in electromagnetic units, which is entirely in the di- rection of the parallels of latitude; r = distance P'P in centimeters, v c = velocity of light in centi- meters per second. The -two dots over the /in (10) indicate the second time derivative. -TIG. 4. In writing equation (10), the slight difference in the direction of the perpendicular to r from the direction of the perpendicular to r is neglected in view of the 442 ELECTRIC WAVES [CHAP. IX largeness of r in comparison with the length z' measured on the antenna. Also the r which should occur in the denominator of (10) has been replaced by r , which can be done without appreciable error for large values of r. The same substitution cannot be made in the argument of / in (10), for there r determines the phase of the oscillation, and this phase changes through an angle of TT for a half wavelength, independent of the distance from the origin. 85. Expression of the Field in Terms of Current. . . e We shall next express the moment of the doublet and the intensities of the field in terms of the current i at e the point z'. To do this we shall think of the current as delivering a charge -\- e to one end of the element of length dz' and a charge e to the other end of dz' in a certain time. A neighboring doublet has a different current and delivers different charges + e\ and e\ partly counteracting the charges of the given doublet, _ and leaving just the charge e e\ that actually occurs dz' FIG 5 on the wire. This is represented in Fig. 5. With this view of the case, when i is in e.s.u., i = e, and Whence, by substituting the value of i from equation (1) into equation (12) we shall have, in view of (7) and (9) 2?r / sin 2ir , dEe = dHj, = - - cos (ct r + z cos 0)- ACTo A 27T/X A By integrating this expression from z 1 = to z f = a, we obtain the electric and magnetic intensities at the point P due to direct transmission from the vertical portion of the antenna. Indicating this integration, we have . / a Ee = HA = - I cos -^ (ct r + z' cos 0)- Xcr J X 2_ /\ \ " / **0 i /\7/ / ~t A\ sin - ( T + z f )dz r (14) CHAP. IX] CHARACTERISTICS OF AN ANTENNA 443 By reflection from the earth, which we shall regard as a perfect reflector, we have intensities that must be added to the above. These intensities may be obtained by considering the radiation to come from an image point at a distance z' below the surface. The effect of this is obtained by changing the sign of the z' in the cosine term of equation (14), but as was pointed out in Art. 82 the sign of z r in the sine term must remain. We obtain thus for the intensities due to the reflected wave emitted by the vertical portion of the antenna the value 2ir I sin 6 a 2w , , E e = H* = - - I cos (ct r z' cos $) Jo A (15) Adding the equation (15) for the reflected intensities to the direct intensities of (14), remembering that if A and B are any two angles cos (A - B) + cos (A + B) = 2 cos A cos A (16) we obtain for the total intensities at P the equation 4?r / sin 6 2ir , N f /2-jrz' A E0 = H = - cos (ct r ) I cos ( -r cos 6 ACr A Jo \ A / sin Y" (x ~ Z J dz' (17) which resolves into 4?r / sin 6 2ir , , N E Q = H = - - cos (ct r ) Acr A 2irz' cos 6 2-n-z' . COS COS -r dz A A 2irz' cos . 2irz' 2irz cos 6 . 2 the squares of sines and cosines in the integrands of (24) may be avoided, and equation (24^ written , cos IB f T/2 cos (2A cos 9) do sin 2 B P /2 . +-- 2-J. ~^ ~ sin ' then r /2 j0_ r -d* i r/_i_ , i Jo sin Ji 1 - u 2 2J \1 + M + 1 - _ir_^_ If du _lf +1 _^_ " 2j 1 + u + 2j_x 14- 1* T 2J- 1 1 4- ti With this operation as a model, two of the other integrals of (25) may be written, respectively r /2 cos (2 A cos 0)^0 = 1 r +1 cos(2A^)^ sin " 2j_ i 1 + u /2 cos (A cos 0) d0 _ 1 + X cos (Au) du sin0 ~2_! l+u Another of the integrals, examined in more detail, gives ' /2 cos 0sin (2A cos 0) d6 sin T Jo 1 u sin (2Au)du 1 - W 2 = s I ( r^ -- T-T ) sin (2Aw) d 2J \1 - it 1 + u/ 1 f 1 sin (2Au) du .if" ^in (2 AM) 2 Jo " 1 + U "*" 2 J 1 + M _1 r 2J du l + u CHAP. IX] CHARACTERISTICS OF AN ANTENNA 447 Similarly, the remaining integral becomes r /2 cos 0sin (A cos 0) dO _ 1 C + 1 sin (Au) du sin0 ~2j_i ~~^T+~u~ Returning now to equation (25), we shall integrate the third and fourth terms, setting them first, and shall substitute (26) to (30) for the other terms, obtaining 2/ 2 . f 2ir , J f sin 2 . sin 2 B sin 2A axis instead of the 2-axis. This arrangement is shown in Fig. 8. The plane of the zero value of S is now to be fixed as the plane of the x and 2-axes. Now using the current distribution of equation (1), we must replace I by a + %', which gives, when treated as (12) was treated, x" 2 - - 2x'x - 2za + (48) 454 ELECTRIC WAVES [CHAP. IX The fictitive current at P" is just equal and opposite to that at P', with, however, a different distance from the point P, so we may write > 2c?r7 [ 2ir I . x' 2 - 2xx' + 2za + ft = ^eosj..^ --- __ Whence by addition, employing the trigonometric relation cos (a + /?) cos (a /3) = 2 sin a sin /?, equation (47) becomes , 47r7 sin ^ . f 2T / z' 2 - 2zz' + a 2 \ dEj, = dH? = -- r-^ 1 sm { ^ ( ct r -- - -- --- ) r cX I X \ 2r / In this equation we may as usual replace -r by A. Also we A may make an approximation as follows: For large values of r x' 2 - 2xx' + a 2 xx' ^ - = -- = x' cos ^. &TQ TQ x' 2 I a 2 In making this approximation the neglected term is = --- , Zr Q and this is to be neglected even in the phase angle, because its value is absolutely small. We have then JTJ 47r/ sin ^ . Az . [2ir , . } all's, = -- - - sm sm j (ct r + x cos ^) [ TQ [ A J ; (50) This equation may be shortened up by writing r = (ct - r.) (51) and To obtain the total electric and magnetic intensities due to the flat-top, the equation (50) must be integrated for all the doublets and their images between the limits x' = and x' = b CHAP. IX] CHARACTERISTICS OF AN ANTENNA 455 where b is the length of the flat-top. This integration is expressed in the following equation. 47r/siniA . Az r . / . 2irx' ,\ . E+ = HS = r - sin sin IT H r cos \j/ J sin (B - ^-] dx f (53) \ \ I To perform the integration let us introduce a change of variable by putting n 2 ^' X , e = B r then dx = ^ ds \ 2ir and the limits of integration become for x' = 0, s = B, for x' = 6, s = 0. Equation (53) then becomes 2/siniA . AzC. &j, = H^ = sin I sm(r + B cos y s cos \l/)sin s ds r^c TQ JB 21 sin \l/ . AzT . N C = -.sin - sin (T + B cos ^) I cos (s cos i/O sm s ds r^c TQ L JB cos (r + B cos i/O I sin (s cos ^) sin s ds (54) JB J The expressions of this equation may be integrated by the use of formulas 360 and 359 of B. O. Pierce's Tables and give 21 Az\ . ..r Ej, = Hz = = :sm sm(r + B cos i/O cos s cos (s cos i/O on w r<> I L -|0 cos \f/ sin s sin (s cos ^) JB 27 sin cos (T + B cos i/O cos \f/ sin s cos ( s cos \f/) cos s sin (s cos ^) Ayr ( in - sin (r + B cos ^) | 1 + cos B cos TQ L rocsini/' r f5 (cos + cos ^ sin B sin (5 cos i/O f r + cos (T -f B cos \l/) I cos i/' sin B cos (5 cos ^) cos B sin (B cos \f/) \ Az r f i sin sin r I cos B cos (B cos i/O 1 rc sm + cos T J cos ^ sin B sin (B cos \f/) \ (55) 456 ELECTRIC WAVES [CHAP. IX Equation (55) gives the electric and magnetic intensities due to the flat-top at any distant point whose coordinates are r = distance of the point from the origin, z = vertical height of the point above the earth's surface, ^ = angle between r and the z-axis; this z-axis being parallel to the flat-top. The quantities, A, B, and r are defined by equations (20) and (51). We shall next discuss the total power radiated from the antenna. 92. Concerning Power Radiated from the Total Antenna. It is to be noticed that the electric and magnetic intensities due to the flat-top of the antenna and those intensities due to the vertical portions of the antenna are directed along the meridional and latitudinal lines of two systems of polar coordi- nates with their poles one quadrant apart. This does not make the re- spective intensities perpendicular to each other, and it becomes necessary to resolve one set of these intensities along and perpendicular to the other set of intensities. At a given point on the sphere about the origin of coordinates, the quantities , 0, 2 and ^ are oriented in a manner represented in Fig. 9. If we let a = angle between \f/ and then also a = angle between and S. It is also apparent that Angle between S and 6 = a | Q Angle between ^ and = ~ a Let us now resolve E# and H^ into components along B and perpendicular thereto (that is, along ) obtaining for the 0- components E = E cos a (a - |j = H cos a - = X sn a, CHAP. IX] CHARACTERISTICS OF AN ANTENNA 457 and for the ^-components /37T \ _ . E#, v = Ef cos I-gr a j = E+ sin a HZ,*? = HZ cos a. Adding these quantities to the corresponding components of the intensities due to the vertical part of the antenna, we obtain for the total intensities, which are designated by primes, the values E' Q = E e + E# cos a, E' v = .E^sin a, H Q = HZ f&D- ft H' v = H 9 + HZ cos a. All of these intensities are perpendicular to r . To get the power radiated through an element of surface dS perpendicular to 7*0, we may make use of Poynting's vector, in the form where the cross between the vectors means the vector-product. This vector-product, expanded, gives 9 H v + E+ Hz + 2 cos a E 9 HZ dS (56) We have already found the first term of this power and have obtained its integral all over the aerial hemisphere. This integral we have called the power radiated from the vertical part of the an- tenna. We shall call the second term above (56), when properly integrated, the power radiated from the flat-top. The third term, since it contains both sets of coordinates, may be called power radiated mutually. These designations are merely for conven- ience in paragraphing the mathematics involved. 93. Power Radiated from the Flat-top. Let us now enter upon a determination of the power contributed by the second term of the right-hand side of equation (56), and integrate this 458 ELECTRIC WAVES [CHAP. IX term over the aerial hemisphere; that is, the hemisphere above the surface of the earth regarded as a plane. The element of area of this hemisphere is dS = r 2 sin ^ d$ d2 (57) Thisi s to be substituted in the required term involving E^ and H?', but these quantities involve the coordinate z, which must be replaced by its value in polar coordinates z = r sin ^ cos 2 (58) Besides (57) and (58) we are also to substitute the values of E* and H z from (55) into the terin ^H^)dS (59) Ef and H? are identical, by (55); the product will give certain terms involving sinV, other terms involving cosV, and still other terms involving sin r cos r\ where r has the value given in (51). If we take the time average for a complete cycle, or, if we prefer, for a time that is large in comparison with a complete period, we have av. sinV = av. cosV = J; while the average of the product av. sin r cos r = 0. The integral form of (59) then becomes, if p = the time average of radiated power, p = ^_ I __JL cos 2 + cos 2 i/'sin 2 5 + 1 - 2 cos B cos (B cos i 2ircJ sm\ls l 2 cos ^ sin B sin (B cos 7 -V2 We shall first perform the integration with respect to /, 1 cos (2A sin ^ cos S) sin i// cos '"' - ' Jv /7r/2 = I d2 J-ff/2 r/2 !!. _ - I cos (2A sin \f/ cos 2 " -r I /" 1 /V2 = I cos (2A sin ^ cos 2) rfS -- I cos (2A sin ^ cos 2 2j_/2 2j ( , (61) = ^ _i I cos (2A sin ^ cos 2)dS (62) ^ 2J CHAP. IX] CHARACTERISTICS OF AN ANTENNA 459 This last step consists in changing the variable of the first integral of the right-hand side of (61) by putting 2' = 7T + 2, which makes the limits ~ and TT without any other change, except the change of 2 to 2'. But since this is the variable of integration, the prime may be omitted, and the terms of (61) added, giving (62). Equation (62) may now be integrated for Formula (11), Art. 121 of Byerly's Fourier's Series and Spherical Harmonics giving for the integral of (62) r/2 y- v (A sin ^ cos 2) } =9-9 J (2A sin $) (63) C*/2 Irf2| J-7T/2 -7T/2 where J is the BesseFs Function of the zeroth order, with a development of the form 1 I 2 2 4 2 2 2 4 2 6 2 H- (64) Before substituting in (60) let us simplify the general trigono- metric factor in the brace of (60) by placing cos 2 \f/ by 1 sin 2 \f/, and letting k = 2A } as in (42), we then obtain Jp (k sin \f/) sin \l/ ~ 4cJ 2 sin 2 \f/ sin 2 2 cos B cos (B cos \f/) 2 cos ^ sin B sin ( B cos \f/) \ d\f/ k G sin 5 \L k 2 sin sn 2 2 2 2 4 2 2 2 4 2 6 2 2 - sin 2 \j/ sin 2 5 - 2 cos B cos 2 cos ^ sin B sin (B cos cos (65) or + 2 cos B 5) ( - 1) fe 2>4262 ^ ^cos ( cos 460 ELECTRIC WAVES [CHAP. IX , ~ k n r* -f 2 sin 2} B( 1) 2 024252 ~2 I sin"" 1 ^ cos ^ sin (5 COS I/' ) C?^ I n = 2,4,6,... (66) Treating these several integrals separately, we have rrl r sm"- 1 \j/d\ls = I sin"- 1 \f/d\l/ + I sin"- 1 \l/d\j/ Jo J* 2 rrl sin"- 1 \l/d\J/ + I cos"" 1 \f/d\js Jo 2-4.6... .-2, (68) by B. O. Pierce's Tables, Formula No. 483. Likewise I sin" +1 Now by Byerly's Fourier's Series and Spherical Harmonics equation (9), Art. 121, sin"- 1 1 cos (B cos ^) ^ = - tt-1 .7 n^i (B) ( 69 ) where Jn-i (B) is a Bessel's Function of the order (n 1) /2, and 2 F l^) is the Gamma Function of ^. For the last integral of (66), we have by Problem 2 and equa- tion (9) of the same article of Byerly's Fourier's Series rsin"- 1 \j/ cos \p sin (B cos \f/) d\j, . B C* - I sin n+1 \j/ cos (B cos ^) rf^ w Jo B 2 CHAP. IX] CHARACTERISTICS OF AN ANTENNA 461 Substituting these various integrations (67), (68), (69), and (70) in (66), we have 1 2 r_ n c n i n P = fc[ X - 4 (- D + 2 sin'BS (- I)* 2 cos n = 2, 4, 6,- oo B = is between and ~- This result may be expressed in a power series by expanding the BessePs Functions by equation (6), Art. 120 of Byerly's Fourier's Series, giving n-l B~2~ (72) + ; (73) (74) and ^tt'.'.'n+l ^ Note that Kl) ,.4.C.. -2 462 ELECTRIC WAVES [CHAP. IX Putting these values in equation (71) we obtain sin 2 (n+1) cosB-\l -. rt/ B * . N + Si "2(n+ 1) ~ t ~2!2 2 (n + l)(w + 3) *. 1 \ 3!2 3 (n + l)(ra + 3)(n + 5)~ r " j r > r> A I - 1 B 2 + B smB- n + 1 2 (n + B 4 1 1 2 2 2(n-f - l)(n + 3)(n + 5) 1 2 3 3 (n + l)(n + 3)(n + 5)(n + 7) H where n = 2,4,6, ... (76) Equation (76) may be further improved for purposes of calcu- lation by expanding the trigonometric functions in power series and collecting the terms. For this purpose sin 2 _ 1 - cos 2B _ J5 2 2 2 4 2 4 6 _ 2 6 8 2 4 ~2! ~ 4! 6! 8! B sin B = B 2 - ^ + - - ... (79) Equations (77), (78) and (79) substituted in (76) will give - l)!r , F n (B) (80) where F n (B) is a polynomial in B, B 2 , B 4 , etc., where the co- efficients of the several powers of B are contained in the table of page 464. In this table the bottom row of terms gives the coefficients of the powers of B, when the summation indicated in (80) is per- formed with n = 2, 4, 6. . . o> . The various terms in the columns were employed in obtaining the last row by addition. CHAP. IX] CHARACTERISTICS OF AN ANTENNA 463 The coefficient of B 10 is not contained in the table, because of its numerous terms, but its value when summed up is 255ft 4 + 6084n 3 + 51396n 2 + 177264n + 193536 10! (n + l)(n + 3)(n + 5)(n + 7)(n + 9) Substituting the values of the coefficients multiplied by the corresponding powers of B and summing up as indicated in equa- tion (80), we obtain for the power the expression 11 6 13 8 B 1Q 3780 + 56700 ~ 93555 "* 56 B * BI I" ooi/2n TTon er rrnr ~ 1 1120 6480 T 83160 77395500 r 7?4 R6 *7 D8 ^ I 7.6 I ** K I __i_ 1 (Q-\\ 145360 24960960 "" 6 !34720 '} ^ This equation gives the average power radiated in the aerial hemi- sphere from the flat-top of the antenna regarded as a separate radiator with the distribution that it has under the fundamental assumptions of the problem. The current is to be measured in absolute electrostatic units, and the power is in ergs per second. In this equation ^ = 2?rfr X 4?ra k = 2A = -^. It remains to find how this power is modified by the mutual effect consisting of the interference between the waves emitted from the vertical portion of the antenna and the waves emitted from the horizontal part. This is the subject matter of Part III. 464 ELECTRIC WAVES [CHAP. IX 4 +. p $ s G 1 t + t 4 t + 4 xE CO CO xE CO 1 g r4 co" 4 4 rH co 4 co" rH ^^ 55 4 s~^ |: 4 4 91 S 4 P -f- 7 - 1 ^_ |H^ rH 7 X I x ^y 4 4 ' rH 4 ; 4 rH * + g I I 1" f I ? a f^ g CO s ^ CO s 23 rH Jg 3 i j! 1 00 * N C and Fio. 10. r . By Fig. 3, z = r cos (83) dS = r 2 sin dB d (84) In the spherical triangle of Fig. 10, a is represented, as de- fined, as the angle between and \f/, while opposite to a the side is 7T/2, The important trigonometric relation in a spherical tri- angle is as follows: I. The cosine of any side is equal to the product of the cosines of the two other sides plus the continued product of the sines of these sides and the cosine of the included angle. 30 466 ELECTRIC WAVES [CHAP. IX By this proposition, referring to Fig. 10, we see that cos ^ = cos n cos 6 + sin = sin 6 cos < 2i A = sin cos (85) By the same proposition 7T cos = = cos cos ^ -f- sin sin ^ cos a; A cos cos \l/ /OCN cos a = . , . , , (86) sin sm 2 ^ or cos a _ cos cos $ x,,-, sin \l/ sin sin \f/ and by (85) this becomes cos a _ cos cos ) 1 sin 2 cos 2 J This is a very complicated expression involving the integral of an integral. We shall first proceed to perform the integration with respect to 0. L t V = r 2 *cos sin ( sin cos 0) d' which by dropping the primes and substituting in (92) and (91) gives sin 8 cos ) cos 2 * r-/ 2 cos sin (B sii y/ ^ 4 J o 1 -sin 2 0cos 2 Now expanding in series as follows: /D a D a 3 sin 3 B cos 3 . sm (B sm cos 0) = B sm cos \- 3! B 5 sin 5 e cos 5 and ^ ^TT n = 1 + sin 2 cos 2 + sin 4 cos 4 + . (93o) 1 sm 2 cos 2 and by taking the product of these two series we obtain Jir/2 r d B sin cos 2 L BS \ i B T;T sm 3 e cos 4 3! ' 5! + ... ; C94) Integrating (94) by formula 483 of B. O. Pierce's Tables, we obtain 468 ELECTRIC WAVES [CHAP. IX V = 27T B sin ft-- , l-3-5-7 + 24^8-3! 5!--7l Sm (95) We shall next proceed to perform the second integration with respect to 4> indicated in (89) . For abbreviation let us write cos 2 < d(j) f 2 - cos 2 < d C" /2 cos J. 1 - sin 2 cos 2 J o 1 - si sin 2 B cos 2 by reasoning similar to the above. Expanding the denominator by (93a), we have /V2 I W = 4 I d + . . . Jo } (If we need it, this integral can be obtained by direct integration in the form 1 1 W = 27TJ cos (1 + cos0) but the expanded form is more useful for our purpose.) Now substituting (95) and (96) in (89) we obtain 2/ 2 f * p = - I dB cos B sin (A cos 6) j cos B cos (A cos B) sin B cos B sin (A cos B) cos G ^ (B - sin B) sin + . . . (97) CHAP. IX] CHARACTERISTICS OF AN ANTENNA 469 To evaluate this expression we must obtain the following integrals : /* - sin (2A cos 9) /ncn dB sm n cos - L - ;r - - (98) =f -f "-r /2 sin cos 2 sin 2 (A cos 6) (99) /2 d0 sin n 8 cos sin (A cos 6) (100) where n = 1, 3, 5, 7, . . . / 3 is the simplest of these integrals and will be considered first. By expanding sin (A cos 6) in series we have r*/ 2 f t A 3 cos 4 6 A 5 cos 6 1 I 3 = I d0 sin" e | A cos 2 + ^ - ... [ which by Byerly Int. Calc., Art. 99, Ex. 2, may be integrated in Gamma Functions as follows: + . . . (101) If we note that /n + 2 \ _n^n in\ 1 V 2 l ) 2 2 \2/ r /rc + 4 \ _ n + 4 n + 2 n /n\ 1 \~2~ + V " 2 2 2 \2/ 22 470 we obtain ELECTRIC WAVES [CHAP. IX 3-1 3! n(-f 2)(n-f4) 5-3-1 5! + 2) A 2 4) 4 2 (n + 4) (n + 6) 6-4-2(n + 4) (n In like manner (102) 2A 2n(n + 2) (2A) 4 (2A) 2 2(n + 4) (2A) -ss-f. - (103) 4 -2(n + 4)(n + 6) 6 -4 -2(n + 4) (n + 6)(n + 8) Now taking up integral 7 2 from equation (99), let us write it r /2 j* fl I 1 - cos (2A cos 0) 1 / 2 = dB sin n cos 2 { ^ -h Jo I Z J and expanding cos (2A cos 0) in series, obtain 1 f ,J - (2A) 2 cos 2 (2a) 4 cos 4 . 1 / 2 = a N0 sm"0cos 2 a ^ ^^ -+...[ ^J L I 2! .4! j This equation, integrated in Gamma Functions between the limits and 7T/2 gives r (^P) r it i 21 (2 A) 4! 2 r(-- 6 + i) CHAP. IX] CHARACTERISTICS OF AN ANTENNA 471 2 A 2 n(n + 2)(n 2 4-2 (n + 6) 6-4-2(n+6)(n + 8) (104) Employing the values of I\, 7 2 , Is found in equations (103), (104) and (102) we may write the expression for the mutual power in the integrated form 2/2 (2A)' r ? f/D i>\ FI 2 [B {(B - smB)^ [I - S G-4-2-5-7-9 6-4-2-7-9-11 (2A) 6 I 6-4-2-9-11-13 -I 8-6-4-7-9-11 l-3/ p & . p \ 2 A 2 [3 5(2A) 2 , (2A) 4 " 2-4\ "3! -/B--5-7L2 4-2-9 + 6-4-2-9-11 " (2A) 6 I 8-6-4-2-9- 11-13 "* 9(2A) ( 6-4-2-11-13 8-6-4-2-11-13-15 A 4 2-5-7 G-4-2-5-7-9 472 ELECTRIC WAVES [CHAP. IX 1-3 /_ B 3 V \2A r, A 2 , A 4 3! - SmB )F5 L 1 ~ 2^7+ 4^7^9 - ^ _ + 1 6-4-2-7-9-11 ^ 'J 1-3-5 / B 3 , JB 6 ' 3! +5T- Sm + 4-2-9-11 8-4-2-9- 11 -13 (104) 11 If now we recall that G = A + B, it will be seen that the equation (104) is entirely in terms of A and B and /. For purpose of computation it is found advisable to expand all of the trigonometrical expressions in powe'r series and then perform with them the indicated operations. This was done with considerable labor and gave the following expression for mutual power: p = MM.0261J!? 4 - .00586J5 6 + .000515B 8 + A 3 j.00555 3 - .00317 5 + .000442B 7 - .00002975 9 + . . 1 } + A 4 i -.003435 4 + .000808B 6 - .00007465* + . . . } + A 5 j -.00106 3 + .000603J5 5 - .0000828B 7 + .0000055B 9 -. . . | + A 6 J.00126B 4 - .00028B 6 + .0000238B 8 - . . . )] (105) This equation gives the time average of the power radiated in the aerial hemisphere by the mutual effect of the fields from both parts of the antenna and is the correction to be added to the power radiated by the two parts, estimated as independent of each other. The current I is in absolute c.g.s. electrostatic units, and the power is in ergs per second. 96. Summation of Flat-top Power and Mutual Power. We have obtained in equation (81) the time average of flat-ton radiated power, and in equation (105) the time average of mutual radiated power. If we replace the k of (81) by its value in CHAP. IX] CHARACTERISTICS OF AN ANTENNA 473 terms of A, the two expressions may be added together. At the time of the addition we shall reduce the units to the practical system of multiplying the right-hand sides of both power equations by 30 times the velocity of light in centimeters per second (i.e., by 30 c), and obtain p = 60/ 2 [A 2 {.05955 4 - .01167 6 + .000974B 8 - .0000458B 10 + . + A 3 1 .0055J5 3 - .00317B 5 + .000442B 7 - .00002975 9 + . . - A 4 j.010585 4 - .00204B 6 + .000171B 8 + .0000082B 10 + .-.,.] - A 5 j.001065 3 - .000603B 5 + .0000828B 7 - .0000055 9 +. . .1 .001965 4 - .000325 6 + .0000335 8 - . . j + . ] (106) This is the total power contribution of the flat top by virtue of its individual and mutual action. The power is in watts, and the current I is in amperes. Certain Tables computed in the next Part of this chapter make calculations with this series comparatively simple. IV. COMPUTATIONS OF RADIATION RESISTANCE 97. Equation for Radiation Resistance. If a = length of vertical part in meters, , 6 = length of horizontal part in meters, AO =' the natural wavelength of the antenna in meters, A = the wavelength in meters of the antenna as loaded with inductance at its base, A 7rX 474 ELECTRIC WAVES [CHAP. IX we may obtain the radiation resistance of the antenna by dividing the power radiated by the mean square of the current at the base of the antenna. This mean square current at the base of the antenna is by (5) -,_/ 2 sin 2 fa/2) 2 Performing this division as to the flat-top power employing equation (106) and adding the result to the radiation resistance for the vertical portion as given in equation (44) we obtain for the total radiation resistance of the antenna the equation ~ R * C S q ~ Rz Sin q sin 2 (g/2) r*A z + nA* - r*A* - r,A 5 + r 6 A + . . .} (107) This is Radiation Resistance in Ohms, where , ~ 15 \ 2 + 2 (2AY - i-? 6 + 2 j M 3!2 (2A) 5!4 (2A) Tf6~ ( } ~' ' 'I 2 4- 2 2 4 4 2 4- 2 4 6 6 2 + 2' - 8 72 + 2? - 9 817 r 2 = 120 !.0595 4 - .01167B 6 +.000974B 8 -.00004585 10 + . . .} r 3 = 120 {.0055B 3 -.00317 5 +.000442B 7 - .00002975 9 + . . . } r 4 = 120 { .0106B 4 - . 00204B 6 + .0001715 8 - .0000082B 10 + . . .} TS = 120 J.00106B' -.000602B 5 +.000083B 7 - .0000055B 9 + . . .} r 6 = 120 I.00196B 4 - .00032B 6 + .0000335 8 - . . . } (108) 98. Tables of Coefficients of Radiation Resistance. There follow in Tables I and II the values of the coefficients Ri, R 2 , R^ CHAP. IX] CHARACTERISTICS OF AN ANTENNA 475 7*2, 7*3, r 4 r 5 , r 6 , for various values of A and B respectively. These tables have been computed by the equations (108). Table I. Coefficients RI, R 2 , and R 3 2A X/4o Ri Ri R 3 0.1 31.416 0.04998 0.049919 0.002498 0.2 15.70 0.19971 0.19870 0.01994 0.3 10.47 0.44848 0.44344 0.06700 0.4 7.85 0.79521 0.78107 0.1579 0.5 6.28 1.2383 1.20634 0.3060 0.6 5.236 1.7759 1.6969 0.5241 0.7 4.488 2.4055 2.2602 0.8232 0.8 3.927 3.1240 2.8786 1.2137 0.9 3.491 3.9290 3.5403 1.696 1.0 3.141 4.8165 4.2315 2.300 1.1 2.854 5.7837 4.9383 3.009 1.2 2.616 6.8232 5.6442 3.823 1.4 2.241 9.150 7.000 5.90 1.5 2.092 10.3392 7.611 6.999 1.6 1.962 11.64 8.15 8.35 1.732 1.812 13.415 8.798 10.113 1.8 1.743 14.40 9.10 11.20 2.00 1.570 17.241 9.550 14.354 2.20 1.427 20.15 9.55 17.80 2.236 .403 20.778 9.508 18.470 2.40 .307 23.22 9.00 21.42 2.60 .207 26.37 7.90 25.20 2.642 .189 27.053 7.60 25.927 2.80 .121 29.40 6.22 29.05 3.141 .000 34.45 2.12 35.64 Table II. Coefficients r 2 , r 3 , etc. B X/46 r 2 ra rt rs n 1.4 1.112 18.36 0.282 2.34 0.047 0.806 1.2 1.31 11.09 0.370 2.00 0.079 0.409 1.0 1.57 5.85 0.330 1.05 0.054 0.211 0.8 1.96 2.48 0.209 0.459 0.038 0.090 0.6 2.61 0.858 0.065 0.152 0.022 0.0362 0.4 3.93 0.177 0.042 0.032 0.0074 0.0062 0.2 7 85 0.0092 0.005 0.002 0.001 0.0004 0.37 4.23 0.130 0.019 0.0232 0.0060 0.0043 0.57 2.75 0.703 0.101 0.125 0.0194 0.0127 0.77 2.04 2.234 0.218 0.400 0.040 0.0752 0.97 1.62 5.260 0.317 0.937 0.061 0.180 1.17 1.34 10.18 0.367 1.822 0.073 0.356 1.37 1.15 17.20 0.280 2.990 0.059 0.504 476 ELECTRIC WAVES [CHAP. IX 99. Curves of Resistance Due to Radiation from the Flat- top. We shall now proceed to discuss the curves of radiation re- sistance of variously proportioned antennae when employed at various wavelengths relative to the natural wavelength. As pre- 1.4 FIG. 11. Radiation Resistance of horizontal top portion of antenna plotted against values of B. The separate curves numbered .1, .2, .3, etc. to 1.0 are for values of A = .1, .2, .3, etc. to 1.0. liminary, the resistance due to radiation from the flat-topped portion of the antennae is first computed. The equation for this is the summation of terms in (107) containing the small r's as factors; that is, Ro = due to (109) CHAP. IX] CHARACTERISTICS OF AN ANTENNA 477 flat-top in which A = q - T" FIG. 12. Total Radiation Resistance plotted against values of B. The separate curves through the origin are for designated values of 7. Separate curves not passing through origin are for different values of A + B. Since the coefficients (small r's) are functions of B only, as given in Table II, it follows that when A and B are given, the value of the flat-top R may be computed. The results of the computations for various values of A and B are plotted in Fig. 11. 478 ELECTRIC WAVES [CHAP. IX In this figure values of B are the abscissae, while the flat-top resistances in ohms are ordinates. The separate curves num- bered .1, .2, .3, etc., to 1.0 are for values of A = 0.1, 0.2, 03 etc. to 1.0. The outside end-points of these several curves, through which a limiting curve is drawn, are determined by the equality of the A+3 o .1 FIG. 13. Enlarged view of some of the curves of Fig. 12. impressed wavelength X and the natural wavelength of the an- tenna X ; that is, by the value of A + B = v/2, which is the largest value A + B can have for the fundamental oscillation of the antenna. 100. Curves of Total Radiation Resistacne. The next step consists in computing the radiation resistance of the vertical portion of the antenna, using the first three terms of equation (107), and employing a large number of values of A and B. To these values of resistance due to the vertical portion of the an- tenna the corresponding resistance of the flat-top are added so CHAP. IX] CHARACTERISTICS OF AN ANTENNA 479 as to give the total resistance of the antenna for various values of A and B. Curves of resistance for various values of A + B are then plotted in Fig. 12, with values of B as abscissaB and values of resistance as ordinates. Figure 13 is an enlarged view of some of the curves that are on too small a scale to read in Fig. 12. Then to make the family of curves more useful for ready reference a series of curves are drawn through all the points which have a common ratio of length of flat-top to length of total antenna. This ratio is designated by 7, where B b 6 = length of flat-top a = length of vertical part of antenna. These 7-curves all pass through the origin. Next as a final step the curves of Fig. 14 are taken from the curves of Figs. 12 and 13 with the new set 1.4 3.0 3.4 3.8 2.2 2.6 X/Xo FIG. 14. Total Radiation Resistance plotted against X/X . The separate curves marked 0, .2, .3, etc. are for values of 7 =0, 0.2, 0.3, etc. of parameters. These curves of Fig. 14 are the final curves of total radiation resistance, and are in terms of the ratio of the wavelength employed to the natural wavelength (that is X/X ) and the ratio of the length of flat-top to total length of antenna (that is 7). Fig. 15 is merely a magnified view of certain of the curves that are too small to read on Fig. 14. 480 ELECTRIC WAVES [CHAP. IX 101. Total Radiation Resistance of a Straight Vertical Antenna at Various Wavelengths Obtained by Inductance at the Base. As an example, let it be required to find the total radiation resistance of a straight vertical antenna for various wave- lengths obtained by adding various inductances at the base. For this case 7 = 0, and from the 7 = curve of Figs. 14 FIG. 15. Magnified view of some of the curves of Fig. 14 with the larger values of X/X . and 15 R may be directly read. The values which were used in plotting this curve are given in Table III, where they are com- pared with the corresponding values computed on the assumption that the oscillator is a Hertzian doublet. This latter assumption 1 1 This result is obtained by taking equation (53) of Art. 78, and noting that the power is radiated only in the upper hemisphere, whence 407T 2 Z 2 R X 2 ohms; CHAP. IX] CHARACTERISTICS OF AN ANTENNA 481 Table III. Resistance of a Straight Vertical Antenna for Different Values of Wavelength Obtained by Inductance at the Base X/Xo ratio of wavelength to natural wavelength R, radiation resistance in ohms computed by present theory Radiation resistance in ohms computed on doublet theory .00 36.57 98.7 .12 26.40 78.7 .21 21.70 67.3 1.31 17.65 57.5 .43 14.28 48.2 .57 11.62 40.0 1.74 9.10 32.6 1.97 6.92 25.4 2.24 5.19 19.7 2.62 3.78 14.4 3.14 2.58 10.0 3.93 1.65 6.40 5.26 0.90 3.60 7.85 0.30 1.16 15.70 0.082 0.40 31.42 0.01 0.10 gives R = 160 x 2 It is seen that the departure of the present theory from the doublet theory is very large for the straight vertical antenna, as should be expected. It should be noted that the first value in the column of resist- ances computed by the present theory agrees with the value for this case computed by Abraham in the work cited in Art. 89. This one value, for the fundamental oscillation, is the only value arrived at by Abraham and is the case of a straight vertical antenna oscillating with its natural frequency. Abraham's other computed values are for the harmonic vibrations with more than one loop of potential always without loading the antenna by inductance, and without any flat-top extension of the antenna. For convenience Table II at the end of the book contains computed values of Total Radiation Resistance for Flat-top but I is length of whole doublet, and therefore is 2a, whence R = 160 X 2 31 482 ELECTRIC WAVES [CHAP. IX Antennae of various ratios of horizontal length to vertical length and for various ratios of wavelength X to natural wave- length \ . 102. Comparison of Computations on the Present Theory with Dr. Austin's Values for the Battleship " Maine." Figure 16 gives the Radiation Resistance of the Antenna of the Battle- ship "Maine" as computed by the present Theory in comparison with Dr. Austin's measured values of the total resistance of this antenna, and in comparison with values computed on the doublet 400 800 L200 1600 2000 2400 X FIG. 16. Total Radiation Resistance vs. Wave length for the Antenna of the Battleship "Maine." Black dots are Dr. Austin's observed values; heavy line, computations by present theory; light line, computations by doublet theory. theory of Hertz. The black dots of Fig. 16 are Dr. Austin's observed values. The heavy line was obtained by computation by the present theory, and the weaker line, by computation re- garding the antenna as a doublet of half-length equal to the ver- tical height of the antenna. It is seen that the departure between the present theory and the doublet theory is not so great as in the case of the straight vertical antenna, for the reason that the doublet theory becomes more nearly correct as the half-length of the oscillator becomes small in comparison with the wavelength. CHAP. IX] CHARACTERISTICS OF AN ANTENNA 483 Neither of the theories gives a rising value of the resistance with increase of wavelength, and, as Dr. Austin has pointed out, his rising values for long waves are probably not due to radiation from the antenna but possibly to dielectric hysteresis in the ground beneath the flat-top. I do not give more extended comparisons with experimental values at the present time, because I am now making some ex- periments to see how much reliance may be placed in antenna resistance measurements made by buzzer methods of excitation in comparison with measurements made by excitation with gaseous oscillators and other methods of continuous excitation. 103. Example of Different Methods of Constructing an An- tenna that Will Have a Specified Resistance for a Given Wavelength. Let it be required to construct an antenna that will have a given resistance (4 ohms, say) for a given wavelength (2000 meters, say). To solve this problem, it is only necessary to look up the four ohm point on the different 7-curves of Figs. 14 or 15, and find the corresponding value of X/X . We can then find the X of the antenna, since X is given. Dividing the X by 4 we obtain the total length of antenna. The value of 7 gives the fractional part of this length which is to be horizontal. The complete result is tabulated in Tab]e IV. Table IV. Constants of the Different Antennae that have 4 Ohms Re- sistance at 2000 Meters Y X/Xo Xo Total length, meters Vertical length, meters Horizontal length, meters Intensity factor in horizontal plane 0.8 1.075 1860 465 93.0 372.0 0.275 0.7 1.39 1435 359 107.7 251.3 0.300 0.6 1.67 1198 299 119.6 179.4 0.310 0.5 1.94 1030 258 129.0 129.0 0.312 0.4 2.18 916 229 137.4 91.6 0.313 0.3 2.32 861 215 150.5 64.5 0.314 0.2 2.44 820 205 164.0 41.0 0.315 0.0 2.52 793 198 198.0 00.0 0.320 The question as to which of these antenna 1^o choose for the given purpose is now chiefly a problem in economics. The economic question is, which, for example, is cheaper: Two poles or towers 93 meters high and 372 meters apart, or one tower 484 ELECTRIC WAVES [CHAP. IX 198 meters high? This of course pre-supposes that it is designed to use a flat-top antenna instead of some other type, such as an umbrella. The problem is, however, not wholly economic because the lower antenna would be preferable as a receiving anterina on account of its weaker response to atmospheric disturbances. There is also the further question as to which of the tabulated antennae will give the greatest vertical intensity of electric and magnetic force on the horizon at a distant receiving station. This is the subject matter of the next Part (Part V). PART V FIELD INTENSITIES AND SUMMARY 104. The Electric and Magnetic Intensities at a Distant Point in the Horizontal Plane. Equation (19) gives the values of the electric and magnetic intensities at a distant point due to the vertical portion of the antenna. If we replace / of that equation by its value in terms of 7 from equation (6), and make cos = 0, we have the intensities in the horizontal plane in terms of 7o, which is the amplitude of the current at the base of the antenna. This gives cos B - cos G 2/n 2?r E = H+ = - - cos (at r ) . cr X | sin 2X (111) The quantities outside the square brackets are constant for a given distance r and a given amplitude of transmitting current Jo- The relative intensities are therefore determined by the factor in the square brackets, which we may designate by ^ _ cos B cos G . TrXo (112) Sm 2X Using the values of B, G } given in equation (20) and the value of 7 in (110), this equation (112) becomes /7T\Q\ 7rX COST \9\) COS 2X X = L. - (113) . TfXo Sin 2X CHAP. IX] CHARACTERISTICS OF AN ANTENNA 485 This quantity X we shall call "The Intensity Factor in the Horizontal Plane." It is to be noted that the electric and magnetic intensities in the horizon plane are not effected by radiation from the flat-top; for, by equation (55), the field intensities from the flat-top are zero for 2 = 0; that is, all over the horizontal plane through the origin. In Fig. 17 the Intensity Factor in the Horizontal Plane is plotted for various values of 7 and various values of X/X . Taking from these curves the values of the intensity factors corresponding to the values of 7 and X/Xo of Table IV we obtain the results in the last column of Table IV. It is seen that the intensity factor is slightly smaller for the larger values of the relative length of flat-top. This diminished value of the intensity factor should be compensated by the use of a slightly larger trans- mitting current. The required in- crease of current may be easily computed by equation (111). .8 -> FIG. 17. Relative intensity of the vertical component of Electric Force in a horizontal plane at a given distance from various antennae and for a given amplitude of transmitting current. 105. Summary. This chapter contains a mathematical theory of the flat-top antenna. The process employed consists in the integration of the effects of an aggregate of doublets assumed to be distributed along the antenna so as to give a current distri- bution described by equation (1) and illustrated in Fig. 2. The electric and magnetic field intensity due to each of the doublets is determined by the Maxwell and Hertz Theories for 486 ELECTRIC WAVES [CHAP. IX all distant points in space. These field intensities are summed up for all the doublets with strict allowance for the differences of phase due to different doublets; the summation gives the resultant field intensities. Then by Poynting's theorem the power radiated from the antenna through a distant hemisphere (bounded by the earth's surface assumed plane) is computed by the integration of a number of intricate expressions. From the radiated power the radiation resistance is obtained by dividing by the mean square of the current at the base of the antenna. Tables of coefficients for computing radiation resistance are given, and curves are plotted of the calculated values of radiation resistance for different ratios of the length of the flat-top to the total length of the antenna and for different relative wavelengths obtained by loading the antenna with inductance. Table II at end of volume gives for ready reference computed values of Radiation Resistance for Various Antennae used at various wave- lengths. Curves are also given for determining the relative electric and magnetic field intensities in the horizontal plane for differently proportioned antennae variously loaded. Various equations developed in the treatment may find application to problems in the design of radiotelegraphic stations. Although this investigation was undertaken in ignorance of a simple case investigated by Professor Max Abraham, by a similar fundamental method, his work was discovered early in the course of the treat- ment and served as a check on one of the resistance values here given. APPENDIX AND TABLES APPENDIX I MATHEMATICAL NOTES Note 1. Proof that the Sum of Two or More Solutions of a Homogeneous Linear Differential Equation is a Solution. Let us take for example the equation Suppose that i = i\ is a solution (2) and i = it is another solution (3) to prove that i\ + i% is a solution. By condition (2), ii substituted for i in equation (1) reduces the right hand to zero : that is Likewise, condition (3) gives Adding equations (4) and (5) and distributing the differen- tiations (which can be done only when the derivatives are of the first degree) we obtain T d z (ii -f iz) , D d(i\ -f- iz} . (i\ + iz} / x di* ^ dT~ C ^ whence it appears that the sum of ^ and iz substituted in the original equation satisfies it; that is, the sum of the solutions is a solution, as was to be proved. If we have a third solution it can be combined with the sum of the first two solutions, just as the first solution was com- bined with the second so that the sum of any number of solutions is a solution. Note 2. The Sum of Multiples of Several Solutions of a Homo- geneous Linear Differential Equation is a Solution. If i = i\ is a 489 490 ELECTRIC WAVES solution, equation (4) is true. Multiplying equation (4) through by any quantity A\ t we obtain and, if A\ is independent of t (i.e., a constant) we may introduce it within the sign of differentiation (only provided all the deriva- tives enter only to the first degree) and obtain T d z (A 1 i 1 ) , , , = L ~~di^~ } ~ R ~~dT ~c~ which is our original equation (1) with A\l\ substituted for i. Therefore, i A\i\ is a solution of (1). Likewise, if i = iz is a solution, it can be proved that Aziz is a solution, and by the proposition above their sum is a solution. The conclusion is this. // we have a linear, homogeneous dif- ferential equation with constant coefficients, and we find several solutions of the equation, we may take any number of the solutions, multiply each by any arbitrary constant and add together the mul- tiples and obtain thereby a result which is a solution of the original differential equation. Note 3. Proof that the Number of Independent Arbitrary Con- stants in the Solution of a Differential Equation Cannot be Greater than the Order of the Differential Equation. As a first step toward the proof of this proposition, let us consider the for- mation of some differential equations by the elimination of constant from a relation between the dependent variable, the independent variable, and the arbitrary constants. Example 1. Given y = Ax (9) in which A is an arbitrary constant; to form an equivalent dif- ferential relation between y and x, not containing A. This can be done by the elimination of A between (9) and its derivative equation. Only one derivative equation is necessary; namely, the equation obtained by taking the first derivative of (9) . This derivative equation is |- A (10) Eliminating A between (9) and (10) we obtain APPENDIX 491 The differential equation (11) is an equation of the first order. It is of the second degree. The degree of the equation cannot be determined by the number of arbitrary constants in the solution. On the other hand, the number of arbitrary constants determines the minimum order of the resulting differential equation. The differential equation cannot be of an order lower than the first, when the solution contains one arbitrary constant, for in order to eliminate the constant two equations are required the given equation (9) and some derivative, which results in a differential equation of order at least as high as the first. Example 2. Given y = A,e klt + A*f* + A,e kst , (12) in which t is the independent variable, and A\ 9 At, and A 3 are arbitrary constants, to form a differential equation of which (12) is a solution. To eliminate the three arbitrary constants, four equations are necessary: for example, the equation (12) and three equations obtained by taking successive derivatives of (12) . The successive derived equations are + AJc*** + AJc* k * (13) * (14) f = Aifci'e* 1 ' + AJcJefi* + A 8 fcV* (15) Cut Now an elimination of the arbitrary constants from (12), (13), (14) and (15) gives - (fci + k 2 + fa) + (A?!** + kfa + k 2 k s ) - k^kty = (16) which is a differential equation of the third order. It is apparent that the three constants of (12) cannot be elimi- nated without using at least three of the derived equations, and arriving at a differential equation of at least the third order. In like manner, if we have a functional relation containing n arbitrary independent constants, and we eliminate the constants by using the derived equations, we shall finally arrive at a dif- ferential equation of at least the nth order. We have said at least the nth order, for it is apparent that, if 492 ELECTRIC WAVES we had wished, we might have used higher derivatives than the nth in order to eliminate the n constants. The conclusion is: The solution of a differential equation cannot contain more arbitrary, independent constants than the order of the differential equation. Note 4. A Solution Containing n Independent Arbitrary Con- stants is the Most General Solution of a Linear, Differential Equation of the nth Order with Constant Coefficients, and Em- braces Every Other Solution as a Special Case, Obtainable by Giving Specific Values to the Constants. We shall prove this proposition first for the case in which the differential equation is homogeneous. Taking t for the independent variable and y for the dependent variable let y= AJM) + AJ*(t) + . . . A J n (0 (17) be a solution of a linear, homogeneous differential equation of th e nth order, and let this solution contain n arbitrary, independent constants A\, A 2, . . . A n . To prove that any other function y = Mt) (18) cannot be a solution unless derivable from (17) by giving proper values to some of the constants. For if there is such a solution (18), then y = A,/i + A 2 / 2 + . . .A n / n + A r / f (19) is a solution by Note 2, where A r is a new arbitrary constant- But by Note 3 this cannot be for it is impossible to have in the solution more independent arbitrary constants than the order of the equation. Therefore, (18) cannot be a solution unless it be a special case of (17). It may be such a special case, for in that case it would not bring with it a new arbitrary constant A r . The proof thus far holds only provided the linear, differential equation is also homogeneous, for only in case of the homogeneous linear equation does the proposition of the additivity of multiples of solutions (Note 2) apply. Next let us treat the case in which the original linear, differential equation is not homogeneous. The general form of this equation may be written (20) APPENDIX 493 in which P, Pi, P 2 , P n are constant coefficients. For reference let us write down the equation Suppose that we have a solution of (20) containing n independent arbitrary constants, A\, A 2 , . . . A, in the general form . . . + A n f n (22) in which /i, / 2 , . . . / n are functions of t. If there is any other solution of (20) not comprehended in (22), let it be 2/2 = fr(t) (23) If (22) and (23) are both solutions of (20), then y = 2/i - 2/2 (24) is a solution of (21), for yi reduces the left-hand member of (20) to f(t), and 3/2 reduces the left-hand member of (20) to the same f(t) ; and by subtraction y = yi y 2 reduces this member to 0, and therefore satisfies (21). Also by Note 2, y = A r (yi - 2/2) (25) where A r is any arbitrary constant, is a solution of (21). That is y = ArAJi + A r A 2 fz + . . . + ArAJn + AJ r (26) must be a solution of (21 )j which is impossible, because it con- tains n + 1 arbitrary constants, unless f r (l) is a special case of 2/i. We have the result that if we have of equation (20) any solution containing n arbitrary independent constants it is the general solution, and contains any other solutions as a special case obtainable by giving specific values to some of the arbitrary constants. Whether the original linear, differential equation is homogene- ous or not, we have proved the proposition stated at the head of this note. When the equations are not linear it is proved in books on differential equations that the general solution of the nth order equation has n arbitrary constants but that there are certain singular solutions which are not derivable from the general solution by giving specific values to the arbitrary constants. 494 ELECTRIC WAVES In employing the criterion of this note as a test of the generality of the solution, care must be taken to ascertain that the n arbi- trary constants are independent. If they are not independent the solution is not the general solution. Note 5. General solution of the equation ^ + Pi = f (t) (27) where p is a constant. For reference write down the equation g + pi = (28) Let i = T 2 be any solution of (28), where !T 2 is a function of t. If we indicate the time derivatives of T? by T'^ we shall have by (28) T\ + P T 2 = (29) Now let the complete solution of (27) be written in the form i = T^ (30) where TI is also a function of t. Then by (27) T'lT* + TiT\ + p^T* = f(t) (.31) whence by (29) T\T 2 = f(t) (32) Integrating we obtain Therefore, by (30) (33) Now T 2 is any solution of (28). The simplest solution may be used, and (33) will still be true. The simplest T 2 that is a solu- tion of (28) is r 2 = e~ pt . This substituted in (33) gives i = Ae-** + e~ pt fe pt f(t) dt (34) In performing the integration indicated in equation (34) no constant of integration is to be added, since the only arbitrary allowable for the solution of a first order equation is already comprised in A. APPENDIX 495 Equation (34) is the complete integral, or general solution, of (27). Note 6. General solution of the equation (35) where L, R, and C are constants. Differentiating, we obtain in which v is the time derivative of v. Replacing i in (35) by -57 we obtain also For reference write down the auxiliary equation We have seen in early chapters of the text that i = e Kt is a particular solution of (37), where 7? k = ~ + - = ~ a + ja (38) where a = R/2L II R 2 = \LC 4L 2 ' Let us now write the solution of (36) in the form of i = Te kt , (39) where T is some function of t, and substitute this solution directly in (36). We obtain, after division by e kt , ^r- = LT + (2kL + R)T (40) where T and T are the first and second time derivatives of T. The simplicity of this equation arises from the fact that = T {l g (e* 1 ) + Rj t (e"') + (e*>) } (41) 496 ELECTRIC WAVES because i e kt is a solution of (37) when k has the value given in (38). Equation (40) which we have derived from (35), when inte- grated, gives, after division by L, T + (2k + ^) T = B 1 + j- lve- ( 42 ) where BI is a constant of integration. By (38) the coefficient of T is 2ja. This equation is of the form of (27) and by (34) gives T = Aie-* + e- 2iu * I JBie 2 ^ + ^ \ ve-kt dt \ dt (43) Integrating the BI term and replacing Bi/2ju by A 2 we obtain T = Atf-V" 1 + A 2 + I e 2 ^ ve+-& dt dt and since Cr C. -i I \e 2 ^ I ve+-& dt J we have I^Z!! / ^2^-^ ^ rf ; ( 44 ) The integration indicated in the last term can be carried one step farther by integration by parts fudv = uv J*vdu ,,2j u t dv = e 2 ^ dt, v = r- 2?co u = fw*-*<* dt, du = ve at whence f,2jut r. i r. fudv = ~ v0*-&dt - ^- 2juJ 2jw J Therefore (44) becomes 6 +jw I ve at-jt dt _ e -ji I f,e+^ d^ | (45) J J where R ~ _ "' /T~ R 2 w ~ VLC ~ 4f?' APPENDIX 497 In equation (45) A and B are arbitrary constants, and no further arbitrary constants are to be introduced after the indi- cated integrations are performed. Equation (45) is the formal solution of the differential equation (35), and gives i directly when v is given as a function oft, provided the indicated integrations can be performed. It is evident from a comparison of (36a) with (36) that the solution for q differs from that for i (45) only in having different arbitrary constants and in having v replaced by v, giving J ve at ~^ dt - e-S* \ ve at +^ dt\ (46) Equation (46) is the formal solution of the differential equation (36a) , and gives q when v is known as a function of t } provided the indicated integrations can be performed. BI and B% are arbitrary constants and no further arbitrary constants are to be introduced after performing the indicated integrations. Expression of i in terms of v instead of v. The integrations indicated in equation (45) may be performed by parts in such a way as to replace v by v. This is done as follows: fwp-te dt = uv - fvdu, where dv = vdt, u = e**-'* Likewise fve at +^ dt = ve at+jat -(a + ju)Sve ai+jai dt whence (45) becomes e -at [ f C } p-j^- (a +jw)e-J t I ve^+i^dt - (a - jco)e^ I ve at ~^dt (47) 2jLu r j j Further Transformation of Equations (46) and (47). We may now change the expressions for q and i into definite integrals with the constants explicitly determined by the following proc- ess, taking (47) as a sample. We may write the identity, em- ploying a change of variable, l l t f =t 32 498 ELECTRIC WAVES where v t > means v (which is a function of t) with its t everywhere replaced by t'. If now on the right-hand side we add and subtract the same quantity, we obtain tve + **dim f ***t**+**'# J Jt' = This last term is a constant, which when introduced into (47) will merely change the constant A\ to A'\ say. Making a similar transformation of the last integral of (47), it is to be noted that in (47) the multipliers of the resulting integrals may be introduced under the integral signs, since the integrations are now with respect to t' instead of t. So that (47) becomes i = e~ ( 27X0, 2/Lco j t >=o This now becomes by changing to trigonometric function i = Ie~ at sm M + *- cos co (t-t')dt' This compared with (49) shows that 7 sin (art + i) = Q {a> cos (art -f ^2) a sin (a>^ -f ^2) 1 , and this equation is true for all values of t. Letting art = Z \~ d CO "T 6T /o co coJ . 1 (57) V' CO 500 ELECTRIC WAVES Therefore, (58) / \ CO Equations (50) and (54) give the required values of i and q where the constants, I and 0i have the values given in (57) and (58). In these equations I and Q Q are the values of currejit and charge, respectively, at the time t = 0. Note. In case 7 = QQ = (57) and (58)) become indeterminate, but (56) shows that in that easel = 0. Note 7. Solution of the equation in the Critical Case in which R 2 = 4L/C (60) In view of equation (60), equation (59) may be written This equation may be reduced to one of a lower order by separat- . Rdi . , Rdi Rdi , . ,. ^. .. , .. ing 7-7- into Try-r + TTf-r., and indicating operations as follows : ._d_(M,Ri\ ,R_td dt \dt T 2L/ " r 2L U 2L Whence , /dt , fi*\ \d< ^ 2L/ _ Rdt di Ri 2L * d< + 2L Integrating, we obtain /di , /?i\ /^^ . log U + 2L) = -2L + - 8 (62) in which B is a constant of integration. Let B = log A z ; A 2 being an arbitrary constant, then (62) gives di . jRz _ dt + 2L = Az 2L ' which is of the first order, and may be integrated by the use of the formal equation (34) of Note 5, giving APPENDIX Rt Rt C , Rt . Rt and, therefore, Rt 501 (63) in which AI and A% are arbitrary constants of integration. Equation (63) is the complete integral, or general solution of (59) in the Critical Case. Note 8. Solution of the equation V = T " i "p 4 (64) in which V is a constant. The solution of this equation may be obtained directly by substituting q for i and V for v in equation (45) of Note 6. A more elementary method of solving (56) is by inserting a new variable z = q CV, when (64) becomes d*z dz (65) which has already been solved (see Chapter II) with the follow- ing results: In general In critical case where and - 4- * I 2L T \4L 2 Lc R \R Z 1 fc2= ~2L~ \4L^" L whence by the value of z q = B^ Lc CV This is the solution in case R 2 4L/C whence by the value of z q= (Bi+ B z t)e-2l + CV 502 ELECTRIC WAVES Table H Relation of Capacity-inductance Product to Undamped Wavelength and Frequency of a Circuit, Together with Squares of Wavelengths Units. X is in meters, n is in cycles per second, L is in micro-henries, C is in microfarads. Formulas Employed in Calculation. X = 3 X 10 8 X 2ir\/LC X 10~ 6 (1) This last factor comes from the fact that a micro-henry is 10~ 6 henries, and a microfarad is 10~ 6 farads. The product involves 10~ 12 , of which the square root is 10~ 6 . By squaring and transposing equation (1), we obtain L x C = 28145X 2 X 10~ n (2) In computing n, the formula employed is n = (3 x 10 8 ) ^- X (3) Accuracy. The values in the table were computed and checked on a calculating machine and are accurate to the last figure given. Rule for Extending Range of the Table. If we annex one zero at the end of wavelength values, (a) we must annex two zeros to values of X 2 , (b) omit the last digit from values of n, (c) displace decimal point two places to right in the L x C values. 1 A table of this character prepared by Mr. Greenleaf W. Pickard has been issued by the Wireless Specialty Apparatus Company of Boston. Mr. Pickard's table has only three significant figures in values of L X C, and four significant figures in values of n. The utility of Mr. Pickard's table has led me to compute and publish the present table, which is augmented by the inclusion of the X 2 values, and which is accurate presumably to all of the figures given. TABLE I 503 X X 2 L X C n X X 2 LXC n 100 101 102 10000 10201 10404 0.0028145 0.0028711 0.0029282 3000000 2970297 2941177 136 137 138 18496 18769 19044 0.0052075 0.0052825 0.0053599 2205882 2189781 2173913 103 104 105 10609 10816 11025 0.0029859 0.0030442 0.0031030 2912621 2884616 2857143 139 140 141 19321 19600 19881 0.0054379 0.0055164 0.0055955 2158274 2142857 2127660 106 107 108 11236 11449 11664 0.0031624 0.0032223 0.0032828 2830189 2803738 2777778 142 143 144 20164 20449 20736 0.0056752 0.0057554 0.0058361 2112676 2097902 2083333 109 110 111 11881 12100 12321 0.0033439 0.0034055 0.0034677 2752294 2727272 2702703 145 146 147 21025 21316 21609 0.0059175 0.0059994 0.0060819 2068966 2054795 2040816 112 113 114 12544 12769 12996 0.0035305 0.0035938 0.0036577 2678571 2654867 2631579 148 149 150 21904 22201 22500 0.0061649 0.0062485 0.0063326 2027027 2013423 2000000 115 116 117 13225 13456 13689 0.0037222 0.0037872 0.0038528 2608696 2586207 2564103 151 152 153 22801 23104 23409 0.0064173 . 0065026 0.0065885 1986755 1973684 1960784 118 119 120 13924 14161 14400 0.0039189 0.0039856 0.0040529 2542373 2521008 2500000 154 155 156 23716 24025 24336 0.0066749 0.0067618 0.0068494 1948052 1935484 1923077 121 122 123 14641 14884 15129 0.0041207 0.0041891 0.0042581 2479339 2459016 2439024 157 158 159 24649 24964 25281 0.0069375 0.0070271 0.0071153 1910828 1898734 1886792 124 125 126 15376 15625 15876 0.0043276 0.0043977 0.0044683 2419355 2400000 2380952 160 161 162 25600 25921 26244 0.0072051 0.0072955 0.0073864 1875000 1863354 1851852 127 128 129 16129 16384 16641 0.0045395 0.0046113 0.0046836 2362205 2343750 2325581 163 164 165 26569 26896 27225 0.0074778 0.0075699 0.0076625 1840491 1829268 1818182 130 131 132 16900 17161 17424 0.0047565 0.0048300 0.0049040 2307692 2290076 2272727 166 167 168 27556 27889 28224 0.0077556 0.0078494 0.0079436 1807229 1796407 1785714 133 134 135 17689 17956 18225 0.0049786 0.0050537 0.0051294 2255639 2238806 2222222 169 170 171 28561 28900 29241 0.0080385 0.0081339 . 0082299 1775148 1764706 1754386 504 TABLE I Continued X X 2 L XC n X X 2 L XC n 172 173 174 29584 29929 30276 0.0083264 0.0084235 0.0085212 1744186 1734104 1724138 216 218 220 46656 47524 48400 0.0131313 0.0133756 0.0136222 1388889 1376147 1363636 175 176 177 30625 30976 31329 0.0086194 0.0087182 0.0088175 1714286 1704545 1694915 222 224 226 49284 50176 51076 0.0138710 0.0141220 0.0143753 1351352 1339286 1327434 178 179 180 31684 32041 32400 0.0089175 0.0090179 0.0091190 1685393 1675978 1666667 228 230 232 51984 52900 53824 0.0146309 0.0148887 0.0151488 1315790 1304348 1293104 181 182 183 32761 33124 33489 0.0092206 0.0093227 0.0094255 1657459 1648352 1639344 234 236 238 54756 55696 56644 0.0154111 0.0156756 0.0159425 1282051 1271186 1260504 184 185 186 33856 34225 34596 0.0095288 0.0096326 0.0097370 1630435 1621622 1612903 240 242 244 57600 58564 59536 0.016212 0.016483 0.016756 1250000 1239669 1229508 187 188 189 34969 35344 35721 0.0098420 0.0099476 0.0100537 1604278 1595745 1587302 246 248 250 60516 61504 62500 0.017032 0.017310 0.017591 1219512 1209677 1200000 190 191 192 36100 36481 36864 0.0101603 0.0102676 0.0103754 1578947 1570681 1562500 252 254 256 63504 64516 65536 0.017873 0.018158 0.018445 1190476 1181102 1171875 193 194 195 37249 37636 38025 0.0104837 0.0105927 0.0107021 1554404 1546392 1538462 258 260 262 66564 67600 68644 0.018734 0.019026 0.019320 1162791 1153846 1145038 196 197 198 38416 38809 39204 0.0108122 0.0109228 0.0110340 1530612 1522843 1515152 264 266 268 69696 70756 71824 0.019616 0.019914 0.020215 1136364 1127819 1119403 199 200 202 39601 40000 40804 0.0111457 0.0112580 0.0114843 1507538 1500000 1485il49 270 272 274 72900 73984 75076 0.020518 0.020823 0.021130 1111111 1102941 1094891 204 206 208 41616 42436 43264 0.0117128 0.0119436 0.0121767 1470588 1456311 1442308 276 278 280 76176 77284 78400 0.021440 0.021752 0.022066 1086956 1079137 1071429 210 212 214 44100 44944 45796 0.0124119 0.0126495 0.0128893 1428572 1415094 1401869 282 284 286 79524 80656 81796 0.022382 0.022701 0.023021 1063830 1056338 1048951 TABLE IContinued 505 X X 2 LX C n X X 2 LXC n 288 290 292 82944 84100 85264 0.023345 0.023670 0.023998 1041667 1034483 1027397 360 362 364 129600 131044 132496 0.036476 0.036881 0.037292 833333 828729 824176 294 296 298 86436 87616 88804 0.024327 0.024660 0.024994 1020408 1013513 1006712 366 368 370 133956 135424 136900 0.037703 0.038114 0.038531 819672 815217 810811 300 302 304 90000 91204 92416 0.025331 0.025669 0.026010 1000000 993377 986842 372 374 376 138384 139876 141376 0.038947 0.039369 0.039791 806452 802139 797872 306 308 310 93636 94864 96100 0.026354 0.026699 0.027047 980392 974026 967742 378 380 382 142884 144400 145924 0.040214 0.040641 0.041069 793651 789474 785340 312 314 316 97344 98596 99856 0.027397 0.027750 0.028104 961538 955414 949367 384 386 388 147456 148996 150544 0.041503 0.041936 0.042369 781250 777202 773196 318 320 322 101124 102400 103684 0.028460 0.028820 0.029181 943396 937500 931677 390 392 394 152100 153664 155236 0.042809 0.043248 0.043692 769231 765306 761421 324 326 328 104976 106276 107584 0.029547 0.029913 0.030278 925926 920246 914634 396 398 400 156816 158404 160000 0.044137 0.044582 0.045032 757576 753769 750000 330 332 334 108900 110224 111566 0.030650 0.031021 0.031401 909091 903614 898204 402 404 406 161604 163216 164836 0.045482 0.045938 0.046394 746269 742574 738916 336 338 340 112896 114244 115600 0.031776 0.032153 0.032536 892857 887574 882353 4Q8 410 412 166464 168100 169744 0.046850 0.047312 0.047773 735294 731706 728155 342 344 346 116964 118336 119716 0.032918 0.033307 0.033695 877193 872093 867052 414 416 418 171396 173056 174724 0.048241 0.048708 0.049175 724638 721154 717703 348 350 352 121104 122500 123904 0.034084 0.034478 0.034872 862069 857143 852273 420 422 424 176400 178084 179776 0.049648 0.050121 0.050599 714286 710900 707547 354 356 358 125316 126736 128164 0.035271 0.035671 0.036071 847458 842697 837989 426 428 430 181476 183184 184900 0.051078 0.051556 0.052040 704225 700935 697674 506 TABLE I Continued X X 2 LX C n X X 2 LXC n 432 434 436 186624 188356 190096 0.052524 . 053014 0.053504 694445 691244 688073 510 515 520 260100 265225 270400 0.073205 0.074649 0.076104 588235 582524 576923 438 440 442 191844 193600 195364 0.053993 0.054489 0.054984 684932 681818 678733 525 530 535 275625 280900 286225 0.077576 0.079059 0.080559 571429 566038 560748 444 446 448 197136 198916 200704 0.055485 0.055986 0.056487 675676 672646 669643 540 545 550 291600 297025 . 302500 0.082071 0.083599 0.085139 555556 550459 545455 450 452 454 202500 204304 206116 0.056994 0.057500 0.058012 666667 663717 660793 555 560 565 308025 313600 319225 0.086695 0.088263 0.089847 540541 525714 530974 456 458 460 207936 209764 211600 0.058525 0.059037 0.059555 657895 655022 652174 570 575 580 324900 330625 336400 0.091443 0.093056 0.094680 526316 521739 517241 462 464 466 213444 215296 217156 0.060073 0.060596 0.061120 649351 646552 643777 585 590 595 342225 348100 354025 0.096321 0.097973 0.099642 512821 508475 504202 468 470 472 219024 220900 222784 0.061643 0.062172 0.062701 641026 638298 635593 600 605 610 360000 366025 372100 0.10132 0.10302 0.10473 500000 495868 491803 474 476 478 224676 226576 228484 0.063236 0.063771 0.064306 632912 630252 627615 615 620 625 378225 384400 390625 0.10645 0.10819 0.10994 487805 483871 480000 480 482 484 230400 232324 234256 0.064846 0.065386 0.065932 625000 622407 619835 630 635 640 396900 403225 409600 0.11171 0.11349 0.11528 476191 472441 468750 486 488 490 236196 238144 240100 0.066478 0.067025 0.067576 617284 614754 612245 645 650 655 416025 422500 429025 0.11709 0.11891 0.12075 465116 461539 458015 492 494 496 242064 244036 246016 0.068128 0.068685 0.069242 609756 607287 604839 660 665 670 435600 442225 448900 0.12260 0.12447 0.12634 454545 451128 447761 498 500 505 248004 250000 255025 0.069800 0.070363 0.071778 602410 600000 594059 675 680 685 455625 462400 469225 0.12824 0.13014 0.13206 444444 441176 437956 TABLE I Continued 507 X X 2 LXC n X X 2 LXC n 690 695 700 476100 483025 490000 0.13400 0.13595 0.13791 434783 431655 428571 870 875 880 756900 765625 774400 0.21303 0.21549 0.21795 344828 342857 340909 705 710 715 497025 504100 511225 0.13989 0.14188 0.14389 425532 422535 419580 885 890 895 783225 792100 801025 0.22044 0.22294 0.22545 338983 337079 335195 720 725 730 518400 525625 532900 0.14590 0.14794 0.14998 416667 413793 410959 900 905 910 810000 819025 828100 0.22797 0.23052 0.23307 333333 331492 329670 735 740 745 540225 547600 555025 0.15205 0.15412 0.15621 408163 405405 402685 915 920 925 837225 846400 855625 0.23564 0.23822 0.24082 327869 326087 324324 750 755 760 562500 570025 577600 0.15832 0.16043 0.16257 400000 397351 394737 930 935 940 864900 874225 883600 0.24343 0.24605 0.24869 322581 320856 319149 765 770 775 585225 592900 600625 0.16471 0.16687 0.16905 392157 389610 387097 945 950 955 893025 902500 912025 0.25134 0.25401 0.25669 317460 315790 314136 780 785 790 608400 616225 624100 0.17123 0.17344 0.17565 384615 382166 379747 960 965 970 921600 931225 940900 0.25938 0.26209 0.26482 312500 310881 309278 795 800 805 632025 640000 648025 0.17788 0.18013 0.18239 377359 375000 372671 975 980 985 950625 960400 970225 0.26755 0.27030 0.27307 307693 306122 304568 810 815 820 656100 664225 672400 0.18466 0.18695 0.18925 370370 368098 365854 990 995 1000 980100 990025 1000000 0.27585 0.27864 0.28145 303030 301508 300000 825 830 835 680625 688900 697225 0.19156 0.19389 0.19624 363636 361446 359282 1005 1010 1015 1010025 1020100 1030225 0.28427 0.28711 0.28996 298507 297030 295567 840 845 850 705600 714025 722500 0.19859 0.20096 0.20335 357143 355030 352941 1020 1025 1030 1040400 1050625 1060900 0.29282 0.29569 0.29859 294118 292683 291262 855 860 865 731025 739600 748225 0.20575 0.20816 0.21059 350877 348837 346821 1035 1040 1045 1071225 1081600 1092025 0.30149 0.30442 0.30734 289855 288462 287081 508 TABLE I Continued \ X 2 LXC n x X 2 LXC n 1050 1102500 0.31030 285714 1150 1322500 0.37222 260870 1055 1113025 0.31325 284360 1155 1334025 0.37545 259740 1060 1123600 0.31624 283019 1160 1345600 0.37872 258621 1065 1134225 0.31922 281690 1165 1357225 0.38198 257511 1070 1144900 0.32223 280374 1170 1368900 0.38528 256410 1075 1155625 0.32524 279069 1175 1380625 0.38857 255319 1080 1166400 0.32828 277778 1180 1392400 0.39189 254237 1085 1177225 0.33132 276498 1185 1404225 0.39521 253165 1090 1188100 0.33439 275229 1190 1416100 0.39856 252101 1095 1199025 0.33746 273973 1195 1428025 0.40191 251046 1100 1210000 0.34055 272727 1200 1440000 0.40529 250000 1105 1221025 0.34365 271493 1205 1452025 0.40867 248963 1110 1232100 0.34677 270270 1210 1464100 0.41207 247934 1115 1243225 0.34990 269058 1215 1476225 0.41548 246914 1120 1254400 0.35305 267857 1220 1488400 0.41891 245902 1125 1265625 0.35620 266667 1225 1500625 0.42234 244898 1130 1276900 0.35938 265487 1230 1512900 0.42581 243902 1135 1288225 0.36256 264317 1235 1525225 0.42927 242915 1140 , 1299600 0.36577 263158 1240 1537600 0.43276 241935 1145 1311025 0.36898 262009 1245 1550025 0.43625 240964 TABLE II 509 Table H Radiation Resistance in Ohms of Flat-top Antenna X = natural wavelength of antenna unloaded, X = wavelength when loaded with inductance at base, _ length of flat horizontal part of antenna. total length of antenna X/Xo Radiation resistance in ohms for y equal 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 .1 .2 36.60 28.00 21.80 33.30 26.00 20.20 29.70 23.90 18.80 25.50 20.00 15.80 20.30 16.00 12.40 14.70 11.90 9.00 9.70 7.60 6.00 4.90 3.80 2.90 1.200 1.070 0.940 .3 .4 .5 18.20 15.10 12.80 16.90 14.00 11.70 15.10 12.20 10.40 12.60 10.50 9.00 11.20 8.60 7.30 7.20 6.10 5.20 4.90 4.00 3.30 2.40 2.00 1.70 0.810 0.700 0.600 .6 .7 .8 11.00 9.50 8.3d 10.00 8.60 7.70 9.00 7.60 6.70 7.80 6.70 6.00 6.30 5.40 4.70 4.40 3.70 3.20 2.80 2.50 2.20 1.40 1.20 1.10 0.500 0.400 0.330 1.9 2.0 2.2 7.40 6.50 5.20 6.80 6.10 5.00 6.20 5.50 4.60 5.30 4.80 3.90 4.20 3.80 3.00 2.90 2.70 2.20 1.90 1.70 1.40 0.90 0.75 0.57 0.240 0.180 0.160 2.4 2.6 2.8 4.40 3.80 3.30 4.20 3.50 3.00 3.80 3.10 2.60 3.20 2.70 2.30 2.50 2.10 1.80 1.80 1.50 1.30 1.20 1.00 0.86 0.48 0.42 0.37 0.140 0.120 0.100 3.0 3.2 3.4 2.80 2.50 2.20 2.50 2.30 2.00 2.20 2.00 1.80 1.90 1.70 1.60 1.50 1.30 1.10 1.10 0.92 0.84 0.74 0.64 0.55 0.33 0.29 0.25 0.090 0.080 0.072 3.6 3.8 4.0 2.00 1.75 1.62 1.90 1.70 1.50 1.60 1.40 1.30 1.40 1.30 1.10 1.00 0.94 0.88 0.77 0.71 0.66 0.47 0.39 0.31 0.22 0.19 0.16 0.066 0.060 0.055 4.5 5.0 5.5 1.30 1.00 0.78 1.21 0.92 0.73 1.05 0.80 0.65 0.89 0.68 0.56 0.75 0.63 0.53 0.54 0.42 0.36 0.26 0.22 0.19 0.12 0.09 0.08 0.042 0.032 0.025 6.0 6.5 7.0 0.61 0.48 0.38 0.54 0.45 0.36 0.49 0.41 0.33 0.44 0.38 0.32 0.43 0.35 0.28 0.29 0.25 0-320 0.16 0.14 0.12 0.07 0.07 0.06 0.019 0.015 0.013 7.5 8.0 8.5 0.32 0.28 0.26 0.31 0.27 0.25 0.29 0.25 0.23 0.28 0.23 0.21 0.25 0.22 0.19 0.19 0.17 0.15 0.11 0.10 0.09 0.06 0.05 0.05 0.013 0.012 0.012 9.0 9.5 10.0 0.25 0.24 0.22 0.22 0.20 0.18 0.20 0.19 0.17 o!is 0.17 0.15 0.16 0.15 0.13 0.13 0.12 0.11 0.08 0.08 0.07 0.05 0.05 0.04 0.012 0.011 0.011 10.5 11.0 11.5 0.21 0.20 0.19 0.16 0.14 0.13 0.15 0.13 0.12 0.14 0.12 0.11 0.12 0.11 0.10 0.10 0.09 0.08 0.07 0.06 0.06 0.04 0.04 0.04 0.010 0.010 0.009 12.0 12.5 13.0 0.18 0.16 0.15 0.12 0.11 0.10 0.11 0.10 0.09 0.10 0.09 0.09 0.09 0.08 0.08 0.07 0.07 0.06 0.05 0.05 0.05 0.03 0.03 0.03 0.009 0.008 0.008 13.5 14.0 14.5 0.14 0.12 0.11 0.09 0.08 0.08 0.08 0.07 0.07 0.08 0.07 0.06 0.07 0.06 0.06 0.06 0.05 0.05 0.04 0.04 0.04 0.03 0.02 0.02 0.007 0.007 0.006 15.0 15.5 16.0 0.10 0.08 0.06 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.006 0.005 0.005 510 TABLE III RELATION OF UNITS Table III. For the Conversion of Units Containing the Practical Units Together With Their Values in Terms of the Two Sets of c.g.s. Units, Where c = 3 X 10 10 cm. /sec. TTnit rf C. g. s. units Electromag- netic Electrostatic Quantity Current 1 Coulomb = 1 Ampere = io- j = ID" 1 = IO- 1 X c = 3 X IO 9 IQ-i X c = 3 X IO 9 Potential 1 Volt 10 8 = IO 8 -5- c = H X 10~ 2 Resistance Capacity 1 Obm 1 Farad = 10 9 = io- 9 = IO 9 -5- c 2 = K X IO- 11 ID" 9 X v* = 9 X IO 11 Inductance Energy Power 1 Henry = 1 Joule 1 Watt 10 9 = IO 7 - IO 7 = IO 9 -i- v 2 = MX IO- 11 IO 7 ergs IO 7 ergs INDEX Abraham, M., 436, 486 Absolute units, conversion table of, 510 Adams, E. P., 324 Addition of complex quantities, 46 Additivity of solutions, 13, 489 Ampere, 510 Amplitude at optimum resonance, 167, 192, 221, 224, 237 in two resistanceless coupled circuits, 86, 90, 92 in two resistive circuits, 138 of current, 61, 64, 211, 252 Angular velocity, 45, 77, 99 velocity undamped, 178 Anisotropic media, 350 Antenna circuit replaced, 176, 240 field due to horizontal part of, 452, 455 field due to vertical part of, 440 not a doublet, 436 power radiated from horizontal part of, 457, 463, 472 power radiated from vertical part of, 444 radiation characteristics of, 435 table of radiation resistance of, 509 total radiation resistance of, 473 Antitangents, caution regarding sign of, 49 Apparent resistance, reactance and impedance, 159, 160 Appendix, 489 Arbitrary constants, 13, 15, 490, 492 incidence, 394 Argand's method, 43 Artificial lines, 285 Attenuation constant, 293, 295, 326 factor, 411 of high frequency waves on wires, 332 Austin, L. W., 482 Average current, 38 Avoidance of interference, 194 Axes, 359 B Backward equivalences, 216, 229 Bedell, F., 156 Bjerkness, V., 73 Blondlot, 334 Bound electrons, 349 Buzzer excitation, 27, 30, 40 Campbell, G. A., 285, 288 Capacity, 3 coupling, 219 distributed, 3, 324 Capacity-inductance product (table) 502 Capacity per unit of length, 332 Carson, J. R., 287 Chaffee, E. Leon, 86 Chain of circuits, 210, 226 Charge compared with discharge, 22 energy during, 32, 40 intrinsic, 4, 349 Charging of a condenser, 20 Circuit containing R, L, C, and a sinusoidal e.m.f., 51 free oscillation of a single, 9 Circuits, chain of, 210 Circular motion, uniform, 45 C. G. S. units in terms of practical units, 510 Clarendon type for vectors, 347 Coefficient of coupling, 78, 178 of reflection, 292, 403 Coefficients of radiation resistance, 474 Cohen, L., 73 Coil with distributed capacity, 340 511 512 INDEX Compensator, electric, 286, 320 Complementary function, 157 Complete product, 371 solution for condenser discharge, 17 \ Complex attenuation constant, 293, 295 impedance, 158 mutual impedance, 207 reflection coefficient, 292, 293 Complex quantities, addition of, 46 division of, 48 evolution of, 49 geometry of, 42 involution of, 49 multiplication of, 47 Condenser, charging of, 20 discharge, 11, 17, 18 discharge, energy expended in, 37 discharge in primary, 86, 138 energy supplied to a perfect, 34 power supplied to a perfect, 33 Condensive and non-condensive flow, 366 Conductivity, 370 Conservation of electricity, 3 Constants, determination of arbi- trary, 15 Construction of antenna, 483 Continuity of tangential components of E and H, 368 Conversion table of units, 510 Cosine, series for, 44 Coulomb, 348, 510 Counter e.m.f., 7 Coupled circuits forced, 156 periods of, 79 power in, 171 two, 73 under impressed e.m.f., 156, 204 wavelengths of, 81 Coupling by capacity, 219 by resistance, 223 coefficient of, 78, 178 critical, deficient and sufficient, 167 nearly perfect, 84 Coupling, negligible, 83 perfect, 84 Crehore, A. C, 156 Critical case, 14, 18 coupling, 167 Crystalline media, 350 Cubic equation reduced, 105, 107 Cunningham 285 Curl, 358 Curl curl A, 377 Curl, denned, 363 equations examined, 364 Current amplitude in smooth line, 329 average and mean-square, 38 density, 368 doublet, 422 interruption, energy and power during, 40 resonance condition, 63 Curves of radiation resistance, 476. 477, 478, 479, 480 D Damping constant, 26, 99, 113, 133 factor, 23 Decrement, determination of, 70, 148 logarithmic, 23, 27 of energy, 36 per undamped period, 66 Decrements, resonance curves for various, 67 Deficient coupling, 167 Demoivre's formula, 44 Density of energy, 375 Design of compensator. 320 of filter, 318 Detector in shunt, 240 resistance, 201 Dielectric constant, 348 effect of, 348 Difference equation, 289 Direct coupled system, 74 Discharge compared with charge, 22 energy during, 32 of a condenser, 11, 17, 18 Discontinuity of induction, 357 INDEX 513 Displacement assumption, 367 current, 368 Distributed capacity, 3 in coils, 340 Divergence of a curl equals zero, 365 of a vector, 353 of a vector product, 373 surface, 355 Division of complex quantities, 48 Domalip, 73 Dorsey, 385 Double periodicity, 99 Doublet, 421, 422, 429 power radiated by a, 432, 433 radiation resistance of a, 433 Doubly- periodic system, 80 Drude, P., 73 Duane, Wm., 334 E Efficiency of transfer, 173 Electric induction, 349 intensity, 347, 348 Electricity conservation of, 3 Electric waves, 347 due to a doublet, 421 in an imperfect conductor, 408 on wires, 324 Electromagnetic units, 510 Electromotive force, 5, 359, 363 counter, 7 induced by buzzer excitation, 30 Electrons, free and bound, 349 Electrostatic units, 510 Elimination among Maxwell's equa- tions, 378 Energy and power, general notions, 32 in buzzer excitation, 40 and radiation, 373 during charge or discharge, 32 electric and magnetic, in plane wave, 388 log. dec. of, 36 lost in resistance during charge, 40 33 Energy of electromagnetic field, 370 supplied to a perfect condenser, 34 to a resistance, 36 to a resistanceless inductance, 35 transmission and absorption, 415 Equivalences for three circuits, 229 for two circuits, 217 Equivalent resistance, reactance and impedance, 216, 217, 229 Erg, 510 in and 772 defined, 178 Evolution of complex quantities, 49 Excitation by current interruption, 27 Exponential, series for, 44 Exponentials, integration by use of, 49 Extinction coefficient, 411 Farad, 510 Faraday, 348 Field due to doublet, 429 due to horizontal part of an antenna, 452 due to vertical part of an antenna, 440 intensities on reflection, 397, 403 Filter action, 296 design, 318 Filters, 285 Flat-top antenna, radiation resis- tance of, 478, 509 power contributed by, 473 Flux of induction, 350 Forced solution, 161 Forward equivalences, 216, 229 Fourth order differential equation, 94 Free electrons, 349 oscillation of single circuit, 9 oscillation of two coupled re- sistanceless circuits, 73, 86 oscillation of two coupled resis- tive circuits, 94, 138 514 INDEX Frequency for different wavelengths etc. (Table), 502 Fresnel's equations, 407 G Galizine, B., 73 Gaussian system of units, 359 Gauss's theorem, 350, 352, 353 Geitler, J. von, 73 Geometry of complex quantities, 42 Grand maxima of current, 238 of relative power, 265, 277, 282 Graphic method for wavelengths, 81 H Hagen, 419 Harmonic plane wave, 388 wave in imperfect conductor, 409 Heaviside, 324 Henry, 510 Hertz, 423, 482 High frequencies, line that attenu- ates, 301 High-frequency waves on wires, 332 Homogeneous isotropic medium, 378 linear differential equation, 12, 490 Horizontal plane, field in, 484 Hubbard, J. C., 340 I Impedance, 158 apparent, 160 complex mutual, 207 equivalent, 216 input, 292 pure mutual, 213 surge, 292, 310, 314, 317 Imperfect conductor, electric waves in an, 408 Index of refraction for electric waves, 385, 411 of imperfect conductors, 411 Inductance, discharge of primary, 91, 150 Inductance, mutual, 75 per unit of length, 333 resistanceless, power supplied to, 35 self, 6 Induction, electric, 349 flux of, 350 magnetic, 357 Inductivity, 348 Input impedance, 292 Insulating medium, 379 Insulator, reflection at surface of an, 399 refraction at surface of an, 399 Integral effect in secondary, 143, 146, 159 Integration by use of exponentials, 49 Intensity before a reflector, 397 electric, 347 in horizontal plane, 484, 485 magnetic, 357 Interference, avoidance of, 194 ratio of, 194 Intrinsic charge, 4, 349 Inverse square law, 348 Involution of complex quantities, 49 Isochronism, 83 quasi, 108 Isotropic medium, 349 Jones, D. E., 423 Joule, 510 K Kelvin, Lord, 324 Kennelly, A. E., 285 Key, 238, 253, 256 Kirchhoff, 324 Kirchhoff's current law, 1 e.m.f. law, 5, 8 Kolacek, 73 L X C vs. \, X 2 ,and n (Table), 502 Laplace, 358 INDEX 515 Large conductivity, waves in medium of, 413 Law of reflection, 396, 401 of refraction, 401 Lines, artificial, 285 resistanceless, 296 resistive, 309 Linkage, positive, 359 Logarithmic decrement 23, 27 of energy, 36 per undamped period, 66 Loose-coupled system, 136 Low frequencies, line that attenu- ates, 300 M Macku, B., 73 Magnetic intensity, 357 Magnetically coupled, 76 coupled system, relations in, 82 Magnetomotive force, 359, 360 Martin, J., 69 Maximum efficiency, 173 Max. max. current, 167 and detector resistance, 201 Maxwell, J. C., 156, 348, 363, 364, 368 Maxwell's displacement assumption. 367 equations, 358 Mean-square current, 38, 61 of sine, 62 secondary current, 143, 146, 153 Muirhead, 285 Multiplication of complex quantities, 47 of vectors, 371 Mutual impedance, complex, 206, 207 pure, 213 inductance, 75 power, 457, 465, 472 N Negative roots, 105 Negligible coupling, 83, 121, 136 Non-crystalline media, 349 Non-reflection, condition for, 304 Normal incidence, 391 Numerical cases, 112, 122, 180, 188, 197, 257, 279, 306, 320, 322 Oberbeck, 73, 156 Ohm, 510 Ohm's law, 5 Optimum combinations for a chain of circuits, 237 resonance, 165, 186, 191, 275 simultaneous adjustments, 247 Oscillation, free, of single circuit, 9 Partial resonance, 161, 162, 163, 177, 180, 232 Particular integral, 157 solution, 52 Perfect coupling, 84 Period and wavelength, 60, 73 during charging, 26 of single circuit, 23, 24, 25, 73 Period undamped, 25, 64, 100 Periods of two couple circuits, 73, 79 Permeability, 357 Persistent waves, 176 Phase change at reflection, 330, 419 Phase lag per section, 293, 295 Pickard, G. W., 502 Pierce, G. W., 4, 81, 156, 176 Planck, M., 423 Plane field, 380 polarized wave, 388 wave, 379 equation, 380, 409 harmonic, 388 reflection of, 391, 394 solution, 377, 381, 383 Poisson, 358 Poisson's equation, 355 Polarized wave, plane, 388 Positive linkage, 359 Potential difference on wires, 341 fall of, 5 Power and energy, general notions, 32 in buzzer excitation, 40 516 INDEX Power in coupled circuits, 171, 258, 265, 277, 282 Power, maximum, transferred at maximum efficiency, 174 radiated by doublet, 432 from flat-top, 457, 463 from vertical part, 444, 450 . mutually, 457, 465, 472 relative, 258, 265, 277, 282 supplied to a condenser, 33 to a resistance, 36 to an inductance, 35 Power transferred at maximum effi- ciency, 173 Poynting, J. H., 375 Poynting's vector, 370, 375, 415 Pupin, 285 Pure mutual impedance, 213 Q Quasi isochronism, 108 R Radiation characteristics of an an- tenna, 435 resistance of a doublet, 434 of an antenna, 477, 509 of a straight vertical antenna, 481 of vertical part, 451 Radiotelegraphic receiving station, 176 with detector in shunt, 240 Ratio of interference, 194 of units, 359, 510 quantities, 255 .yleigh, Lord, 73 ^sactance, 53, 158 apparent, 160 equivalent, 216 Reciprocity theorem, 204, 217, 228 Recurrent sections, 256 Reflection coefficient, 292, 293, 403, 419 from an imperfect conductor, 416 from a perfect conductor, 391 Reflection, law of, 396, 401 on smooth line, 330 repeated, 291 vitreous, 399 Refraction, law of, 401 vitreous, 399 Refractive index, 385 Relative power, 258, 265, 277, 282 Relaxation time, 408 Repeated reflection, 291 Resistance, apparent, 160 coupling, 223 equivalent, 216 Resistance, power and energy sup- plied to, 36 radiation, of doublet, 434 of flat-top antenna, 473, 509 of straight vertical antenna, 481 of vertical part of antenna, 451 Resistanceless coupled circuits, 73, 86 line, 296 Resonance combinations, 248 curve, equation to, 65 in simple circuit, 60 partial, 161, 232, 240 relations, 240, 243 relations restricted, 232 Resonant fundamental system on wires, 338 Restricted resonance relations, 232 Restrictions, 243 Retardation angle per section, 294, 296, 305, 308 per unit length, 326 by resistanceless line, 305 R. M. S. current and e.m.f., 38 Roots, negative, 105 of fourth' degree equation, 99 Rosa, 385 Rubens, 419 Riidenberg, 434 Rule of signs, 103 S Saunders, 334 Scalar and vector product, 371 INDEX 517 Secondary, e.m.f. induced in, 31 Semiconductors, 408, 411 Sharpness of resonance, 194 Shepherd, G. M. B., 285 Signs, rule of, 103 Sine, series for, 45 Sinusoidal e.m.f. impressed, 51, 156 Solenoidal vector, 367 Solution of differential equations, 494, 495, 500 Spherical waves, 423 Stationary waves in insulator, 393 on wires, 335, 337, 342, 344 Steady state, 58 Stoppage condenser, 242, 284 Sufficient coupling, 167 Surface divergence, 355 Surge impedance, 292, 310, 314, 317 of smooth line, 328 Systems of recurrent sections, 256 r-case, 118, 119 Taylor, 285 Telegraph and telephone lines, arti- ficial, 285 equation, 409 Terminal conditions, 290 impedance, 303 Three circuits, chain of, 226 Thomson doublet, 421 Thomson, Sir J. J., 421 Sir Wm., 11, 24, 324 Thomson's formula, 24 Time between maxima, 25 Time-lag independent of frequency, 308 Transformation into periodic form, 53, 98 of e.m.f. equation, 363 of m.m.f. equation, 360 Transformer coupling, 73, 214 Transient term made zero, 57 Transverse wave, 386 Trowbridge, John, 334 Two circuits with transformer coup- ling, 214 coupled circuits, 73, 94 Two-way equivalences, 229 Types of artificial line, 298 U w-case, 118 Undamped angular velocity, 178 period, 64, 100, 113, 129 log. dec. per, 66 wavelength, 67, 179, 258 Units, conversion table of, 510 Gaussian, 359 ratio of, 359, 385, 510 Varley, 285 Vector, exponential expression for, 44 product, 371 trigonometric expression for, 44 Velocity of electric waves, 383, 384, 411 of high frequency waves on wires, 332, 334 of light and ratio of units, 385 on wires, 330 Vertical part of antenna, 440, 444, 450, 451 Vitreous reflection and refraction, 397 Volt, 510 W Watt, 510 Wave, electric, 347 Wave equation, 377, 378 Wavelength, definition of, 60 '^ graphic method, 81 square vs. added capacity," 339 vs. X and L X C (Table), 502 undamped, 67, 179, 258 Wave, plane harmonic, 388 transverse, 386 Waves, on wires, 324 persistent, 176 Wien, M., 73 Wires, waves on, 324 FOURTEEN DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. -^IfcX REC'D LD ', > _'^ " FEB 23*65 -3PM DEC* 7'1955tl- 12lvn'62j^ REC'D LD JUN 1 2 1962 4lW63ft~ ~. . HP JjAAl l n ioc^ H '^ i u lyoj n#** R.C'Q L.D MAY 1 9 '64 -EM SMar-ftSLO LD 21-100m-2,'55 (B139s22)476 General Library University of California Berkeley T L P5 THE UNIVERSITY OF CALIFORNIA LIBRARY