GIFT OF 
 
 ^^*"*^ 
 
OSCILLATING -CURRENT 
 CIRCUITS 
 
 AN EXTENSION OF THE THEORY OF GENERALIZED 
 
 ANGULAR VELOCITIES, WITH APPLICATIONS TO 
 
 THE COUPLED CIRCUIT AND THE ARTIFICIAL 
 
 TRANSMISSION LINE 
 
 BY 
 
 V. BUSH 
 
 ABSTRACT 
 
 OF 
 A THESIS 
 
 SUBMITTED TO THE FACULTY OF THE 
 MASSACHUSETTS INSTITUTE OF TECH- 
 NOLOGY IN PART FULFILMENT OF 
 THE REQUIREMENTS FOR THE DEGREE 
 OF DOCTOR OF ENGINEERING 
 
 JUNE, 1916 
 
OSCILLATING-CURRENT 
 CIRCUITS 
 
 AN EXTENSION OF THE THEORY OF GENERALIZED 
 
 ANGULAR VELOCITIES, WITH APPLICATIONS TO 
 
 THE COUPLED CIRCUIT AND THE ARTIFICIAL 
 
 TRANSMISSION LINE 
 
 BY 
 
 V. BUSH 
 
 ABSTRACT 
 
 OF 
 
 A THESIS 
 
 SUBMITTED TO THE FACULTY OF THE 
 MASSACHUSETTS INSTITUTE OF TECH- 
 NOLOGY IN PART FULFILMENT OF 
 THE REQUIREMENTS FOR THE DEGREE 
 OF DOCTOR OF ENGINEERING 
 
 JUNE, 1916 
 
CONTENTS. 
 
 PAGE 
 
 LIST OF SYMBOLS EMPLOYED . . 4 
 
 INTRODUCTION 5 
 
 THE COUPLED CIRCUIT 8 
 
 APPLICATION TO THE ARTIFICIAL LINE 1 1 
 
 SUGGESTIONS FOR A CONTINUATION OF THE WORK 14 
 
 SUMMARY 14 
 
 333554 
 
LIST OF SYMBOLS EMPLOYED IN THESIS. 
 
 i The instantaneous oscillating current in a branch of a network am- 
 peres 
 
 n A generalized angular velocity of oscillation hyperbolic radians per 
 second Z 
 
 Z Generalized impedance ohms Z 
 
 E Initial potential volts 
 
 Naperian base 2.718 .... 
 
 twi 2 Roots of the equation Z =o, hyperbolic radians per second Z 
 
 C Total capacitance farads 
 
 L Total self inductance henries 
 
 M Total mutual inductance henries 
 
 R Total resistance ohms 
 
 Constants of the primary and secondary of a coupled circuit are dis- 
 tinguished by subscripts 
 
 a, /3, 7, 5, 77 Coefficients of the equation Z = o for the coupled circuit 
 
 q Correction to be applied to the absolute values of the free angular 
 velocities of a resistanceless coupled circuit to obtain the absolute 
 values of the angular velocities of the complete circuit. numeric 
 
 p A correction to be added and subtracted to - to obtain the decrements 
 
 4 
 of the complete coupled circuit. hyp. rad. per sec. 
 
 s, t Sum and difference respectively of the squares of the angular velocities 
 of the resistanceless coupled circuit. 
 
 /hyp. rarl \2 
 
 >. rad.V 
 sec. / 
 
 \ sec 
 
 j The pure imaginary, V i 
 
 A A generalized amplitude of current oscillation amperes Z 
 m Number of sections of an artificial line 
 h Auxiliary constant numeric 
 
 .Z This sign appended to the units of an equation denotes that the expres- 
 sion contains, in general, complex quantities 
 
OSCILLATING-CURRENT CIRCUITS. 
 
 INTRODUCTION. 
 
 Heaviside,* and since then several others,! have shown that for the free 
 oscillations of a network the generalized impedance, formed from the con- 
 stants of the network and the complex angular velocity of oscillation, is 
 zero for any complete circuit. This principle enables the frequencies and 
 decrements of the free oscillations of a network to be readily found. There 
 is a similar principle which enables the finding of the amplitudes of free 
 oscillation at the several frequencies, which is also in Heaviside, derived 
 from a series of theorems concerning the distribution of energy during 
 subsidence. It is the purpose of the thesis, of which this is an abstract, to 
 demonstrate the application of this second principle to practical engineering 
 problems. 
 
 The principle may be stated as follows: If Z is the generalized impedance 
 of a branch of the network initially containing a store of energy, corre- 
 sponding to the initial voltage E, and if n is the complex angular velocity 
 of oscillation, so that Z= /(), then the first order term in the Taylor ex- 
 
 j rj 
 
 pansion of Z, namely, n , will be of the nature of an impulsive impedance ; 
 
 d n 
 
 and the oscillatory current will be of the form : 
 
 E nt , 
 
 amperes Z 
 
 --^ dz 
 
 n 
 dn 
 
 where the summation extends over the roots i, n^, - - of the equation Z =o. 
 
 It will be convenient to call the expression n the "threshold im- 
 
 d n 
 pedance."* 
 
 The equation, as given, applies to the current in the branch initially 
 charged, where the generalized and threshold impedances are formed for 
 that branch. 
 
 The discussion of the application of this principle to various typical net- 
 works has indicated the truth of the following additional propositions 
 which will be found useful in attacking particular problems: 
 
 * Heaviside. Electrical Papers, Electromagnetic Theory, Vol. II. 
 t Campbell, Proc. AIEE, 1911; 
 
 Kennelly, Proc. IRE, 1915; 
 
 Eccles & Makower, Electrician, 1915. 
 
(1) In determining the amplitude of oscillation at some point of the net- 
 work distant from the branch initially charged, the generalized impedances 
 of the elements combine in the manner of simple resistances. Upon com- 
 bining with the generalized impedance of an element, each term of a cur- 
 rent or voltage expression is combined with the generalized impedance of 
 the element formed for the free angular velocity of the term considered. 
 
 (2) When several stores of energy are simultaneously discharged they 
 may be considered separately and the results added. 
 
 (3) In order to ensure that the correct free angular velocities be ob- 
 tained, the generalized impedance should be formed for the branch under 
 examination; as in special cases certain free angular velocities may be 
 absent in particular branches of the network. 
 
 (4) The threshold impedance is formed always from the generalized 
 impedance which considers the initially charged element as the main 
 branch. 
 
 (5) The sudden application of a steady electromotive force may be 
 treated as the inverse of the discharge from the final state attained. 
 
 (6) The sudden application of an alternating electromotive force may be 
 treated in similar manner, the unbalanced stores of energy being in this 
 case the differences between the initial stores of energy in the network, and 
 the energies at the same points of the network corresponding in the steady 
 state to the point of the voltage wave at which it was suddenly applied. 
 
 The method of applying the threshold impedance is shown by various 
 examples. One of these, the series circuit containing resistance, inductance, 
 and capacitance is included here for illustration. 
 
 Resistance 
 f\ o/ims. 
 
 Inductance 
 L_ henries. 
 
 C farads. 
 
 P/g. / . 5imf)le Series Osci/latin<j Circuit. 
 

 7 
 
 In this circuit (see fig. i) the generalized impedance will be: 
 
 Z = R+Ln+ ohms Z 
 
 Cn 
 
 Equaling to zero and solving for n, we obtain the free angular velocities: 
 
 L 
 
 LC 
 
 _ R . // R\* r 
 
 71% % / I - I - 
 
 *L V \ 2 L/ LC 
 
 hyp- ra d. 
 
 sec. 
 
 The threshold impedance is: 
 
 n =Ln ohms Z 
 
 dn Cn 
 
 If now we consider the condenser as discharging through the circuit from 
 an initial voltage E, the current will be: 
 
 n=nz E ni , 
 
 i= ^ e amperes Z 
 
 n^x n ^ 
 H dn 
 
 or 
 
 E nit i E n%t / 
 
 t = e -j- e amperes Z. 
 
 i _ i 
 
 which, with the values of n\ and n* given above, is the complete oscillatory 
 solution. This expression may be reduced to the usual form by inserting 
 the values of i and n^. There will, of course, be three cases according as 
 the quantity under the radical is positive, zero, or negative. For the third 
 case the expression becomes upon reducing: 
 
 amperes 
 LC \2L' 
 
 which is the solution obtained by the usual methods. The solutions for 
 the other cases may be obtained by similar reductions. 
 
 It will be noted that this method of solving the circuit is much more con- 
 cise and direct than is the method of determining the constants of integra- 
 tion in the differential equation solution, in accordance with the boundary 
 conditions. It is also convenient to retain all three cases in the single 
 expression. 
 
8 
 
 If we wish the oscillatory voltage across, for instance, the reactor in this 
 circuit, we may obtain it by multiplying the oscillatory current by the 
 generalized impedance of the reactor, and treat the current terms separately, 
 thus: 
 
 nit , ELnz n 2 t , 
 
 - e + volts Z 
 
 i i 
 
 Cn\ Cn 2 
 
 and this expression may also be reduced by inserting the values of n\ and n^. 
 
 THE COUPLED CIRCUIT. 
 
 The coupled circuit has been thoroughly solved by the method of differ- 
 ential equations.* These solutions have been discussed from the point of 
 view of the applications of this circuit, particularly to radio work. Many 
 approximate solutions have been obtained for the case of the free discharge 
 of the primary condenser, either by neglecting the effects of resistance, or 
 the reaction of the secondary upon the primary, or in some similar way. 
 The complete exact solution has been generally avoided, principally because 
 of its complication. The resistance operator method, or the method of 
 generalized angular velocities, has also been applied to this circuit in as far 
 as the frequencies and decrements are concerned.! This method gives the 
 same equation for the determination of the free angular velocities as does 
 the differential equation solution, namely: 
 
 numerc 
 
 where n is the complex angular velocity, and the constants are those shown 
 on fig. 2. 
 
 Fi<j.i. JndvctiVely Counted Circuit. 
 
 * Bjerkness, Wied. Ann. 55, 1895; Oberbeck, Wied. Ann. 55, 1895; Domalys & Kolacek, 
 Wied. Ann. 57, 1896; Wien, Wied. Ann. 61, 1897; Rayleigh, Theory of Sound; Braun, Phys. 
 Za. 3, 1901; Drude, Ann. d. Phys. 13, 1904; Jones, Phil. Mag. 1907; Cohen, Bui. Bu. Stds. 
 S, 1909; Pierce, Proc. Am. Ac. A. & Sc. 46, 1911; Fleming, Proc. Phys. Soc. 1913- 
 
 t Eccles, Phys. Soc. Proc. 24, 1912. Kennelly, Proc. IRE, 1915. 
 
The solution of this fourth degree equation is laborious, and may be 
 avoided in the following manner. The equation may be written in the form : 
 
 /hyp. rad.X 4 , 
 
 and if we treat the same circuit without resistance we obtain the easily 
 solved equation: 
 
 /hyp- rad.V 
 
 I -^- - ) 
 
 \ sec. / 
 
 The roots of this last equation will differ but little in absolute value from 
 the absolute values of the roots of the complete equation. If (i -\-q) and 
 (i q) are the correction factors to be applied to the absolute values of the 
 resistanceless roots in order to obtain the absolute values of the complete 
 roots, we may find an expression for q by means of the algebraic relations 
 between the roots and coefficients of the above equations; and in deriving 
 this relation the square of q may be neglected. This expression is: 
 
 P-yt-ctS 
 
 - - - numeric 
 
 where s is the sum and / the difference of the squares of the roots of the 
 resistanceless equation. 
 
 In a similar manner the relation may be derived: 
 
 27 as+aqt 
 4t8qs 
 
 where ( -+p } and ( - p } are the decrements in the solution of the com- 
 \4 / \4 / 
 
 plete circuit. In this manner the frequencies and decrements of the oscil- 
 lations in the coupled circuit may be obtained without the necessity of 
 solving the fourth degree equation. 
 
 This method, tested on a typical circuit with constants: 
 
 Ci = io~ 9 farads 
 C 2 = io- 10 farads 
 RI =1000 ohms 
 RZ =2000 ohms 
 LI =0.025 henries 
 L 2 =0.040 henries 
 M =0.020 henries 
 
 gave by exact solution: 
 
 17638.9=117192683. 
 
10 
 
 and by the approximate method : 
 
 - 57361. o; 664750. 
 -17639.0^192680. 
 
 The amplitudes of oscillation may be readily found by the use of the 
 threshold impedance. If we consider the discharge of the primary con- 
 denser, so that the primary is the main branch, the threshold impedance is: 
 
 n 
 
 ohms Z 
 
 Inserting into this expression the four roots MI, w 2 , w 3 , w 4 of the equation Z = o, 
 gives four particular values of the threshold impedance. Dividing the 
 initial primary condenser voltage by each of these values gives the four 
 complex amplitudes for the primary current expression: 
 
 t nit , n%t . . n z t . . w 4 / / 
 
 ^l = Al +A 2 e +A 3 e + A t e , amperes L 
 
 This expression may be readily reduced to trigonometric form, when the 
 imaginary portions of the expression cancel out. 
 
 An examination of the generalized impedance of the several elements of 
 the circuit gives for the ratio between the primary and secondary ampli- 
 tudes: v 
 
 Mn . / 
 
 - numeric Z. 
 
 The four values of this ratio applied to the primary amplitudes give the 
 corresponding secondary amplitudes* 
 
 The results above were checked by means of oscillograms taken upon 
 a typical coupled .circuit. The constants chosen 
 
 ^i =I -937 ohms 
 
 Rz =2.531 ohms 
 
 LI =7.52 x io~ 3 henries 
 
 Lt =7.63 x i o~ 3 henries 
 
 M =3475 x io~ 3 henries 
 
 C\ =13.51 microfarads 
 
 2 =24.62 microfarads 
 
 gave frequencies of oscillation 609.5 an d 339- 2 which were within con- 
 venient range for the oscillograph. The computed points checked the os- 
 cillograms within the errors of measurement. A solution was also made by 
 differential equations as a check. 
 
II 
 
 APPLICATION TO THE ARTIFICIAL LINE. 
 
 There has been much difficulty encountered in the analysis of smooth line 
 transients. The -cable has been comparatively easily handled,* but the 
 analysis of the aerial line has given in general results too complicated for 
 engineering use. For experimental analysis for steady state phenomena 
 the lumped artificial line has proved invaluable,! but there has been much 
 doubt as to just how far such a line of a given number of sections could be 
 trusted for transient effects, t 
 
 The method of generalized angular velocities is applied in the thesis to 
 the analysis of the oscillations of the artificial line under certain typical con- 
 ditions. The distant-end current on a grounded artificial line, when a 
 steady voltage is suddenly applied at the home end, is considered for the 
 artificial cable, and the artificial aerial line. The IT line is used, but the 
 formulas apply also to the T line with small changes. 
 
 The purpose of this analysis is to determine, for specific cases, the number 
 of sections requisite in an artificial line, in order that it may represent its 
 corresponding smooth line, not only for the steady state, but also for certain 
 transient effects, to a sufficient degree of approximation for engineering 
 investigations. 
 
 The method used is simply to analyze artificial lines with various num- 
 bers of sections, considered simply as networks with concentrated constants. 
 The results of these successive solutions are grouped, and from them is 
 derived the solution for the general case of m sections. 
 
 The general solutions for the application of a steady electromotive force 
 to the grounded artificial line obtained in this way follow: 
 
 For the cable of m sections containing resistance and capacitance only: 
 
 [ m*hit m*fe* "~| 
 
 <* I+ !^=l ( RC -^Z2e RC + . . . amperes 
 
 RL 22 J 
 
 where 
 
 *# is the received current 
 E the applied steady voltage 
 R the total resistance 
 Cthe total capacitance. 
 
 * Kelvin, Proc. Roy. Soc. 1855; 
 
 Poincar6, EC. Elect. 40, 1904; 
 
 Malcolm, Electrician 1911; 12. 
 fPupin, Trans. AIEE 1890, 1900; Trans. Am. Math. Soc. 1900; 
 
 Kennelly, Proc. Am. Acad. Arts & Sci., 44, 1908 ; 
 . Huxley. Thesis M. I. T., 1914. 
 J Cunningham and Davis, Proc. AIEE 1911, 1912; 
 
 Ricker. Thesis M. I. T. 1915. 
 
12 
 
 The values of h are found as roots of the auxiliary equation: 
 
 =0 
 
 2 ! 
 
 numeric 
 
 where there are terms if m is even, and terms if m is odd. 
 
 A curve for obtaining these roots is presented for convenient use in prac- 
 tical cases. , 
 
 For the aerial line containing resistance, capacitance and inductance, 
 the corresponding equation is: 
 
 r 
 
 <-![- 
 
 
 LC 
 
 hm2 ^ 
 
 2L 2L 
 
 I 
 
 J m'fe ( R\* 
 
 V ~7C " (71 ^ + ' ' J 
 
 LC \* ^/ amperes 
 
 As the line is subdivided an oscillatory term appears in this equation for 
 each section introduced. 
 
 This aerial line formula was checked by means of oscillograms taken 
 upon a typical artificial aerial power transmission line at Pierce Hall, 
 Harvard University.* Twelve sections were used, representing a line of 
 the following constants: 
 
 #000 A.W.G. aluminum stranded conductors 
 
 Overstrand diameter 0.47 inches 
 
 Interaxial distance 90.5 inches 
 
 Length 596.4 miles. 
 
 Total inductance 1.035 henries 
 
 Total capacitance 8.ioXio~ 6 farads 
 
 Total resistance 300. 1 ohms 
 
 The elements of this artificial line were grouped in such a manner that it 
 was arranged as a IT line of various numbers of sections. The arrival curve 
 of current computed and plotted to the scale of the oscillograms showed a 
 check to a reasonable degree of accuracy. 
 
 * Kennelly and Tabossi, Elec. World 1912. 
 
In order to determine the relation between the artificial and smooth 
 cables, the cable formula was plotted for the numerical case: 
 
 =200 ohms 
 
 C= io~ 6 farads 
 
 R=2O volts 
 
 for various numbers of sections. It was found that with these constants, 
 the arrival curve on a six section artificial cable coincided to a sufficient 
 degree of accuracy, for the purposes of engineering, with the smooth line 
 arrival curve as plotted from Kelvin's formula: 
 
 -C- 
 
 2 e 
 
 n-n 
 'RC 
 
 RC 
 
 amperes 
 
 The limiting value of the artificial aerial line formula as the number of 
 sections is indefinitely increased was also considered. From the fact that 
 the artificial cable formula approaches Kelvin's smooth line formula in the 
 limit, were derived the limits of the various values of m z h^ and h as m= o . 
 A certain approximation was also made because of the fact that in lines 
 
 encountered in practice 
 
 / R 
 ice, I - 
 
 \2 Li 
 
 L 
 may be neglected in comparison with - . 
 
 X/C 
 
 Applying these facts to the artificial aerial line formula gave the following 
 expression for the received current on a grounded smooth aerial line when a 
 steady voltage is suddenly applied at the home end : 
 
 Rt 
 
 Rt 
 
 Er 
 
 *-*-' 
 
 - 
 
 amperes 
 
 where F (/) is the discontinuous function represented in fig. 3. 
 
 HLC-+ 
 
 fig. 3. The 
 
 Fft) in tfte atriaL tine formula. 
 
14 
 
 The arrival curve plotted from this formula for the smooth line on which 
 the oscillograms were taken is shown in fig. 4. 
 
 -4-" ft /W 
 
 Time 
 
 Fig. 4. Arrival Current, Smooth Aerlat Line. 
 
 A comparison of this smooth line curve with the artificial line arrival 
 curves showed that the artificial line of four sections, or less, did not well 
 approximate the smooth line, for the transient due to the sudden application 
 of a steady voltage. The artificial line of twelve sections approximated 
 the smooth line fairly well; but a still greater number of sections would be 
 necessary, in order to enable the artificial line to be used for experimental 
 investigation with this type of transient. 
 
 SUGGESTION FOR A CONTINUATION OF THE WORK. 
 
 The method of generalized angular velocities, applied to the oscillations 
 of networks with concentrated constants, has proved to be valuable for 
 engineering purposes. It is believed that the same method may be profit- 
 ably applied to networks containing branches with distributed constants. 
 A starting point for such work would be found in Heaviside's application 
 of the resistance operator to the smooth line. 
 
 SUMMARY. 
 
 In addition to the theorem which determines the free angular velocities 
 of oscillation of a network, there is a theorem which will determine the 
 amplitudes. This theorem involves a "threshold impedance" which may 
 be formed for any circuit with concentrated constants, and which enables 
 the amplitudes of oscillation to be found from the initial potential of the 
 unbalanced energy. 
 
15 
 
 An application of this method to the coupled circuit gives an easily ap- 
 plied and convenient complete solution for the primary condenser dis- 
 charge. 
 
 Applied to the artificial line, it enables the lumped line of a given number 
 of sections to be compared with the represented smooth line for certain 
 transient effects. 
 
 The writer wishes to express his thanks to Prof. D. C. Jackson, Dr. A. E. 
 Kennelly, and other members of the Department of Electrical Engineering 
 who have assisted him in the preparation of the thesis. 
 
 R-6-i 6-400. 
 
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