CHAPTER V. FREE OSCILLATIONS. 28. The general equations of the electric circuit, (50) and (51), contain eight terms: four waves: two main waves and their reflected waves, and each wave consists of a sine term and a cosine term. The equations contain five constants, namely: the frequency constant, g; the wave length constant, &; the time attenuation constant, u\ the distance attenuation constant, h, and the time acceleration constant, s ; among these, the time attenuation, uy is a constant of the circuit, independent of the character of the wave. By the value of the acceleration constant, s, waves may be sub- divided into three classes, namely: s = 0, standing waves, as discussed in Chapter III; u > s > 0, traveling waves, as dis- cussed in Chapter IV; s = u, • alternating-current and e.m.f. waves, as discussed in Section III. The general equations contain eight integration constants C and C', which have to be determined by the terminal condi- tions of the problem. Upon the values of these integration constants C and C' largely depends the difference between the phenomena occurring in electric circuits, as those due to direct currents or pulsating currents, alternating currents, oscillating currents, inductive dis- charges, etc., and the study of the terminal conditions thus is of the foremost importance. 29. By free oscillations are understood the transient phe- nomena occurring in an electric circuit or part of the circuit to which neither electric energy is supplied by some outside source nor from which electric energy is abstracted. Free oscillations thus are the transient phenomena resulting from the dissipation of the energy stored in the electric field of the circuit, or inversely, the accumulation of the energy of the electric field; and their appearance therefore presupposes the possibility of energy storage in more than one form so as to allow 478 FREE OSCILLATIONS 479 an interchange or surge of energy between its different forms, electromagnetic and electrostatic energy. Free oscillations occur only in circuits containing both capacity C and inductance, L. The absence of energy supply or abstraction defines the free oscillations by the condition that the power p = ei at the two ends of the circuit or section of the circuit must be zero at all times, or the circuit must be closed upon itself. The latter condition, of a circuit closed upon itself, leads to a full-wave oscillation, that is, an oscillation in which the length of the circuit is a complete wave or a multiple thereof. With a cir- cuit of uniform constants as discussed here such a full-wave oscillation is hardly of any industrial importance. While the most important and serious case of an oscillation is that of a closed circuit, such a closed circuit never consists of a uniform conductor, but comprises sections of different constants • generat- ing system, transmission line and load, thus is a complex circuit comprising transition points between the sections, at which par- tial reflection occurs. The full-wave oscillation thus is that of a complex circuit, which will be discussed in the following chapters. Considering then the free oscillations of a circuit having two ends at which the power is zero, and representing the two ends of the electric circuit by I = 0 and I — 10, that is, counting the distance from one end of the circuit, the conditions of a free oscillation are I = 0, p = 0. I = *0, P = 0. Since p = ei, this means that at I = 0 and I = 10 either e or i must be zero, which gives four sets of terminal conditions: (1) e = 0 at I = 0; i = 0 at I = Z0. (2) i = 0 at I = 0; e = 0 at I = Z0. (3) e = 0 at I = 0; e = 0 at I = Z0. (4) i = 0 at I = 0; i = 0 at I = L. (197) Case (2) represents the same conditions as (1), merely with the distance I counting from the other end of the circuit — a line open at one end and grounded at the other end. Case (3) repre- 480 TRANSIENT PHENOMENA sents a circuit open at both ends, and case (4) a circuit grounded at both ends. 30. In either of the different cases, at the end of the circuit I = 0, either e = 0, or i = 0. Substituting I = 0 into the equations (50) and (51) gives eo = fi-<«- >'{[C/ (C/ + C2') - c, (C, + C2)] cos qt r ' (c1 ' -i- r f\ r (C1 -\- r v i • (L°2 V°3 ' U4 / °2 V°3 ' U4A - [ea' (C8 + C4) + c2 (C,7 + C/)] sin qt} (198) and i0 = s~(u~s)< { (Cj— C2) cos qt + (C/ — C/) sin qt} + e~(u+sn{(C3 -C4)cosqt+(C3' -C4')smqt}. (199) If neither g nor s equals zero, for e0 = 0, c/ (C/ + <72') - ct (Ct + Ca) = 0 I and c/ (C, + C2) + ct (C/ + Ca7) =0; J hence, \' \ _ '/ i (200) and for i0 = 0, n - r r - r 1 tf» '• V. j ^4:~^3J c'.c' c/-c" J (201) Substituting in (50) and (51), i = e~(u~s^ {Cj [e~hl cos (g£ — kl) ± s+hl cos (g^ + kl)] + C/ [s~w sin (g^ - kl) ± s+hl sin (qt + kl)]} + €-(«+•)« JC3 [£+w cos (gi - kl) ± e~hl cos (g^ + kl)] + C/ [£+w sin (qt - kl) ± e~hl sin (qt + kl)]} (202) FREE OSCILLATIONS 481 and e = f-<«->< { (c/C/- C/7J [e~u cos (?*-&/) T £+wcos (qt + kl)] - (c/C4 + c.C/) [e-w sin (# - &) =F £+wsin(^ + kl)]\ Ca/(7a/_ C2c3) [€+« cos (#-&) =F e~wcos (qt + kl)] 2'03 + c203')[e+hlsm(qt-kl) ~hl sm (qt + kl)]}, (203) where the upper sign refers to e = 0, the lower sign to i = 0 for I = 0. 31. In a free oscillation, either e or i must be zero at the other end of the oscillating circuit, or at I = 10. Substituting, therefore, / = Z0 in equations (202) and (203), and resolving and arranging the terms by functions of t} the respective coefficients of must equal zero, either in equation (202), if i = 0 at I = 10, or in equation (203), if 6 = 0 at I = Z0, provided that, as assumed above, neither s nor q vanishes. This gives, for i = 0 at / = Z0, from equation (202), C. (e~M° ± e+hl°) cos kL - C/(e~M° =F e+ °) sin kL = 0,1 [ (204) C, (e~Mo q= £+w°) sin kl0 + (7/(e-^ ± e~hl°) cos /cZ0 = 0, J and analogously for C3 and C3'. In equations (204), either Cv C/, (73, Cg' vanish, and then the whole oscillation vanishes, or, by eliminating Cv and C/ from equations (204), we get C0g2 ^ + (£-o T €+o)a sin2 ^ = 0. hence, osW0 = 0 and - 482 TRANSIENT PHENOMENA hence, for the upper sign, or if e = 0 for I = 0, h = 0 and cos klQ = 0, thus: (206) and for the lower sign, or if i = 0 for I = 0, 1 (207) «L = mi. 1 In the same manner it follows, for e = 0 at I = 1Q, from equa- tion (203), if e = 0 for Z = 0, thus: and if i = 0 for I = 0, thus: h = 0, sin kl0 = 0 kl — nn. (208) 0 and cos kl0 = 0, (2 n + 1) TT (209) From equations (206) to (209) it thus follows that h = 0, that is, the free oscillation of a uniform circuit is a standing wave. Also (2 n (210) if e = 0 at one, i = 0 at the other end of the circuit, and kl, - nn (211) if either e = 0 at both ends of the circuit or i = 0 at both ends of the circuit. 32. From (210) it follows that or an odd multiple thereof; that is, the longest wave which can exist in the circuit is that which makes the circuit a quarter- FREE OSCILLATIONS 483 wave length. Besides this fundamental wave, all its odd multi- ples can exist. Such an oscillation may be called a quarter-wave oscillation. The oscillation of a circuit which is open at one end, grounded at the other end, is a quarter-wave oscillation, which can contain only the odd harmonics of the fundamental wave of oscillation. From (211) it follows that or a multiple thereof; that is, the longest wave which can exist in such a circuit is that wave which makes the circuit a half- wave length. Besides this fundamental wave, all its multiples, odd as well as even, can exist. Such an oscillation may be called a half-wave oscillation. The oscillation of a circuit which is open at both ends, or grounded at both ends, is a half-wave oscillation, and a half-wave oscillation can also contain the even harmonics of the funda- mental wave of oscillation, and therefore also a constant term forn = Oin (211). It is interesting to note that in the half-wave oscillation of a circuit we have a case of a circuit in which higher even harmonics exist, and the e.m.f. and current wave, therefore, are not sym- metrical. From h = 0 follows, by equation (56), s=0, if k2 > LCm\ ] and (212) 9 = 0, if k2< LCm\ J The smallest value of k which can exist from equation (210) is and, as discussed in paragraph (15), this value in high-potential high-power circuits usually is very much larger than LCm2, so that the case q = 0 is realized only in extremely long circuits, as long-distance telephone or submarine cable, but not in trans- mission lines, and the first case, s = 0, therefore, is of most importance. 484 TRANSIENT PHENOMENA Substituting, therefore, h = 0 and s = 0 into the equation (52) gives *-y-h-. and m (213) and substituting into equations (202) and (203) of the free oscilla- tion gives i = e~ut{A1[cos (qt — kl) ± cos (qt + kl)] + A2 [sin (qt - kl) ± sin (qt + kl)]} (214) and e = -£-* {(mA2- qAj) [cos (qt-kl) =F cos (qt + kl)] ^ qA2) [sin ($-fcQ =F sin (qt + . (215) where: A, = C, + C, and A = C/ + <73'. ' Since k and therefore # are large quantities, m can be neglected compared with q, and k = hence and the equation (215) assumes, with sufficient approximation, the form t {A, [cos (^ - kl) =F cos (g^ + kl)] -A2 [sin (g/ - kl) =F sin (^+ kl)]}, (216) where the upper sign in (214) and (216) corresponds to e = 0 at I = 0, the lower sign to i = 0 at I = 0, as is obvious from the equations. FREE OSCILLATIONS 485 Substituting A, = A cos Y and A2 = A sin 7- (217) into (214) and (216) gives the equations of the free oscillation, thus : and e = — i = A£~W'{COS (qt-kl-f) =F cos (qt + kl - -£~^{cos (qt-kl-f) T cos (qt + kl - 7-)}. (218) With the upper sign, or for e = 0 at I = 0, this gives i = 2 A£~^ cos kl cos (qt — r) and e= -2AV/|e-" sin kl sin (qt — f) . (219) With the lower sign, or for i = 0 at I = 0, this gives i = 2 As~w' sin &/ sin ($ - 7-) and e = — 2 A\f -e- "* cosklcos (qt - (220) 33. While the free oscillation of a circuit is a standing wave, the general standing wave, as represented by equations (139) and (140), with four integration constants Av A,', Av A2', is not necessarily a free oscillation. To be a free oscillation, the power ei, that is, either e or i, must be zero at two points of the circuit, the ends of the circuit or sec- tion of circuit which oscillates. At a point ^ of the circuit at which e = 0, the coefficients of cos qt and sin qt in equation (139) must vanish. This gives (A, + A2) cos kl, + (A/ - A/) sin ^ = Ol and - (A, - A,) sin kl, + (A/ + A/) cos ^ = 0. J Eliminating sin kl, and cos kl, from these two equations gives (A? - ,422) + (A* - A*) = 0, or (222) A* + A»-A> +A», 486 TRANSIENT PHENOMENA as the condition which must be fulfilled between the integration constants. The value Zx then follows from (221) as 4 ' 4_ 4 -t" 2 At a point Z2 of the circuit at which i = 0 the coefficients of cos qt and sin qt in equation (140) must vanish. This gives, in the same manner as above, (A* - A?) + (A* - A") = 0, that is, the same conditions as (221), and gives for 12 the value tfln + 4 A f I A t A.2 Aj -t- /12 From (223) and (224) it follows that I* = ~ lAr J (225) That is, the angles kll and kl2 differ by one quarter-wave length or an odd multiple thereof. Herefrom it then follows that if the integration constants of a standing wave fulfill the conditions At2 + A/2 = A* + A* = B\ (226) the circuit of this wave contains points lv distant from each other by a half-wave length, at which e = 0, and points Z2, distant from each other by a half-wave length, at which i = 0, and the points 12 are intermediate between the points lv that is, distant there- from by one quarter-wave length. Any section of the circuit, from a point ^ or 12 to any other point Zt or 12, then is a freely oscillating circuit. In the free oscillation of the circuit the circuit is bounded by one point Zt and one point 12; that is, the e.m.f. is zero at one end and the current zero at the other end of the circuit, case (1) or (2) of equation (197), and the circuit is then a quarter- wave or an odd multiple thereof, or the circuit is bounded by two points Zt or by two points Z2, and then the voltage is zero at both ends of the circuit in the former case, number (3) in equation (197), or FREE OSCILLATIONS 487 the current is zero at both ends of the circuit in the latter case, number (4) in equation (197), and in either case the circuit is one half-wave or a multiple thereof. Choosing one of the points ^ or 12 as starting point of the dis- tance, that is, substituting I — ^ or I — 12 respectively, instead of /, in the equations (139) and (140), with some transformation these equations convert into the equations (219) or (220). In other words, the equation (226), as relation between the integra- tion constants of a standing wave, is the necessary and sufficient condition that this standing wave be a free oscillation. 34. A single term of a free oscillation of a circuit, with the dis- tance counted from one end of the circuit, that is, one point of zero power, thus is represented by equations (219) or (220), respectively. Reversing the sign of Z, that is, counting the distance in the opposite direction, and substituting B = ± 2 A y - , these equations assume a more convenient form, thus: for and and for and e = 0 at I = 0, e = Be'"* sin kl sin (qt - y) — s~ut cos kl cos (qt — y), j i = 0 at I = 0, e = Be~"* cos kl cos (gtf — y) (227) (228) :~^sin &Zsin (qt - y). Introducing again the velocity of propagation as unit distance, J — /T-7 1 (229) from equation (66) and (229) we get: kl = X Vq2 + m2 488 TRANSIENT PHENOMENA hence, if m is small compared with q} Id = qX, (230) and substituting (229) in (230) gives k = VW (233) Denoting the length of the circuit in a quarter-wave oscillation by and the length -of the circuit in a half-wave oscillation by (234) (235) the wave length of the fundamental or lowest frequency of oscillation is >10 = 4 ^ = 2 Ja; (236) or the length of the fundamental wave, with the velocity of prop- agation as distance unit, in a quarter-wave oscillation is (237) and in a half-wave oscillation is *, = 2 Z0 VW. FREE OSCILLATIONS 489 Substituting (237) into (232) and (233) for a quarter-wave oscillation gives and 2 7T q = (2 n + 1) — , (238) and for a half-wave oscillation gives and Writing now 27T (239) (240) that is, representing a complete cycle of the fundamental fre- quency, or complete wave in time, by 6 = 2 TT, and a complete wave in space by t = 2 TT, from (239) and (240) we have kl (241) where n may be any integer number with a half-wave oscillation, but only an odd number with a quarter-wave oscillation. 35. Substituting (241) into (227) and (228) gives as the complete expression of a free oscillation the following equation A. Quarter- wave oscillation. (a) e = 0 at I = 0 (or r = 0) e = and «£n sin (2n+ I)T sin [(2 n + 1) 0 - r (242) 490 TRANSIENT PHENOMENA (b) i = 0 at Z = 0 (or T = 0) 00 = £~ut Bn cos (2 n + 1) T cos [(2 n + 1) 0 - rn] and i = \/T£~Ut i>#n sin (2 n + 1) r sin [(2 n + 1)6- Tn]. T *-» o 5. Half-wave oscillation. (a) e = 0 at I = 0 (or r = 0) n J5n sin TIT sin (nd — yn) (243) and /(7 °° i = V Ts~ut Sn -^n cos nr cos (r&0 - /-J; T L o (6) t = 0 at I = 0 (or r = 0) (244; and e = s~ » n cos nr cos 0 t i/5 -v *:; • ,* ^ * == V 7 e 2,n#r> sm nr sin (w0 - jj, * L Y where (245) ko 2^VW (240) ^0 = 4 Z0 v L(7 in a quarter-wave t (237) = 2 Z0 v LC in a half-wave oscillation, J and MV £~w< = e ~ .-^T. (246) ^0 is the wave length, and thus — the frequency, of the funda- mental wave, with the velocity of propagation as distance unit. It is interesting to note that the time decrement of the free oscillation, e~ut, is the same for all frequencies and wave lengths, FREE OSCILLATIONS 491 and that the relative intensity of the different harmonic compo- nents of the oscillation, and thereby the wave shape of the oscillation, remains unchanged during the decay of the oscillation. This result, analogous to that found in the chapter on traveling waves, obviously is based on the assumption that the constants of the circuit do not change with the frequency. This, however, is not perfectly true. At very high frequencies r increases, due to unequal current distribution in the conductor, as discussed in Section III, L slightly decreases hereby, g increases by the energy losses resulting from brush discharges and from electro- static radiation, etc., so that, in general, at very high frequency an increase of y and ^, and therewith of u, may be expected; Li C that is, very high harmonics would die out with greater rapidity, which would result in smoothing out the wave shape with increas- ing decay, making it more nearly approach the fundamental and its lower harmonics. 36. The equations of a free oscillation of a circuit, as quarter- wave or half -wave, (242) to (245), still contain the pairs of inte- gration constants Bn and yn, representing, respectively, the intensity and the phase of the nth harmonic. These pairs of integration constants are determined by the ter- minal conditions of time ; that is, they depend upon the amount and the distribution of the stored energy of the circuit at the starting moment of the oscillation, or, in other words, on the distribution of current and e.m.f. at t = 0. The e.m.f., e0, and the current, i0, at time t = 0, can be ex- pressed as an infinite series of trigonometric functions of the distance Z; that is, the distance angle T, or a Fourier series of such character as also to fulfill the terminal conditions in space, as dis- cussed above, that is, e = 0, and i = 0, respectively, at the ends of the circuit. The voltage and current distribution in the circuit, at the starting moment of the oscillation, t = 0, or, 6 = 0, can be represented by the Fourier series, thus: cos nr "*" ttn sm and i0 = JTn (bn cos nr + 6n' sin nr), o (247) 492 TRANSIENT PHENOMENA where an 1 f27r - / en cos nrdr = 2 avg [e0 cos TIT-JO , 71 ^0 sin WT dr = 2 avg [e0 sin nr]02 *, (248) and analogously for b. The expression avg [Fgj denotes the average value of the function F between the limits at and a2. Since these integrals extend over the complete wave 2 TT, the wave thus has to be extended by utilizing the terminal conditions regarding T, but the wave is symmetrical with regard to I = 0 and with regard to I = 10, and this feature in the case of a quarter- wave oscillation excludes the existence of odd values of n in equations (247) and (248). 37. Substituting in equations (242) to (245), t = 0, ' 0 = 0, and then equating with (247), gives, from (242), 00 oo eQ — 2^n Bn sin (2 n + 1) T sin fn = V n [an cos (2 n + I)T o o f a/ sin (2 n + 1) r] and i0 = \ j Vn 5n cos (2n + 1) T cos ?-n = V« [6n cos (2 n + 1) T ^o o + &/ sin (2 n + 1) r]; hence, <*n = 0, 6n' = 0, m Bn sin ?-n = an' and v -5n cos j-n= bn. * iv FREE OSCILLATIONS Equation (242) gives the constants an = 0; bn' = 0, 493 (249) tan^n =-~ y^; 71 in the same manner equation (243) gives the constants a>n = 0; bn = 0, Bn (250) Equation (244) gives the same values as (242), and (245) the same values as (243). Examples. 38. As first example may be considered the discharge of a transmission line : A circuit of length Z0 is charged to a uniform voltage E, while there is no current in the circuit. This circuit then is grounded at one end, while the other end remains insulated. Let the distance be counted from the grounded end, and the time from the moment of grounding, and introducing the deno- tations (235). The terminal conditions then are: (a) T = 0 e = 0, T = — 0. (6) at 6 = 0 e = 0 for T = 0; e = E for T^ 0, i = 0 for T 7^ 0; i = indefinite for r = 0. 494 TRANSIENT PHENOMENA The distribution of e.m.f., ew and current, i0, in the circuit, at the starting moment 6 = 0, can be expressed by the Fourier series (247), and from (248), On' = 2 avg [E sin (2 TT+ l)r] = 71 TT L) TC and and from (249), hence, and tan -„ = and substituting (252) into (242), 4 E sin (2 n + 1) r cos (2 n + and I = cos(2n+ n L From (240) it follows that i d = 2x gives the period, and the frequency, i = 4 (251) (252) (253) and T = 2 re ^ves the wave length, of the fundamental wave, or oscillation of lowest frequency and greatest wave length. FREE OSCILLATIONS 495 Choosing the same line constants as in paragraph 16, namely : Z0 = 120 miles; r = 0.41 ohm per mile; L = 1.95X10"3 henry per mile; g = .25 X 10~6 mho per mile, and C = 0.0162 X 10~6 farad per mile, we have u = 113, and 147 £-°-0485' and the fundamental frequency of oscillation is fl = 371 cycles per second. If now the e.m.f. to which the line is charged is E = 40,000 volts, substituting these values in equations (253) gives e = 51,000 £-°-0485