CHAPTER VIII. LOW FREQUENCY SURGES IN HIGH POTENTIAL SYSTEMS. 64. In electric circuits of considerable capacity, that is, in extended high potential systems, as long distance transmission lines and underground cable systems, occasionally destructive high potential low frequency surges occur; that is, oscillations of the whole system, of the same character as in the case of localized capacity and inductance discussed in the preceding chapter. While a system of distributed capacity has an infinite number of frequencies, which usually are the odd multiples of a funda- mental frequency of oscillation, in those cases where the fundamental frequency predominates and the effect of the higher frequencies is negligible, the oscillation can be approxi- mated by the equations of oscillation given in Chapters V and VII, which are far simpler than the equations of an oscillation of a system of distributed capacity. Such low frequency surges comprise the total system, not only the transmission lines but also the step-up transformers, gen- erators, etc., and in an underground cable system in such an oscillation the capacity and inductance are indeed localized to a certain extent, the one in the cables, the other in the generating system. In an underground cable system, therefore, of the infinite series of frequencies of oscillations which theoretically exist, only the fundamental frequency and those very high harmonics which represent local oscillations of sections of cables can be pronounced, and the first higher harmonics of the fundamental frequency must be practically absent. That is, oscillations of an underground cable system are either (a) Low frequency high power surges of the whole system, of a frequency of a few hundred cycles, frequently of destructive character, or, (6) Very high frequency low power oscillations, local in character, so called "static," probably of frequencies of hundred 105 106 TRANSIENT PHENOMENA thousands of cycles, rarely directly destructive, but indirectly harmful in their weakening action on the insulation and the possibility of their starting a low frequency surge. The former ones only are considered in the present chapter. Their causes may be manifold, — changes of circuit conditions, as starting, opening a short circuit, existence of a flaring arc on the system, etc. In the circuit from the generating system to the capacity of the transmission line or the underground cables, we have always r2 < —j-; that is, the phenomenon is always oscillatory, and (_/ equations (24) and (25), Chapter VII, apply, and for the current we have ~n> (1) and for the condenser potential we have c Z X (2) 65. These equations (1) and (2) can be essentially simplified by neglecting terms of secondary magnitude. xc is in high potential transmission lines or cables always very large compared with r and x. The full-load resistance and reactance voltage may vary from less than 5 per cent to about 20 per cent of the impressed e.m.f., the charging current of the line from 5 per cent to about 20 per cent of full-load current, at normal voltage and frequency. In this case, xc is from 25 to more than 400 times as large as r or x, and r and x thus negligible compared with xc. HIGH POTENTIAL SYSTEMS It is then, in close approximation : q = 2 Vx xc = - = -90o. 107 (3) Substituting these values in equations (1) and (2) gives the current as i= -- sin(0-#0) + e~2^ \\\ -- sin 00~| cosy - 0 xc ( |_ xc J x and the potential difference at the condenser as l = E cos (0 - 00) + [e0 - J^ cos 0J cos T2re0 + 4xa:ct0 . _ 4VxZ 4 E XX, in 00) Jsin y/^-c (2 r cos 00 + 4 x sin These equations consist of three terms: x, . _ x, / I p " -I- £ "' • el ~ 61 Kl 1 ; . ^ ' (a ft \ I — Sin (C7 — UQ), (5) (6) 108 TRANSIENT PHENOMENA E -~e ( = £ 2* ) cos 6, cos 00 cos V - 0 + ''cos 6> - Ee or, by dropping terms of secondary order, E - ~e — Ee 2x cos 6 n cos V/ — and: or, by dropping terms of secondary order, Vx cos x (7) (8) (9) (10) Thus the £ota£ current is approximately e0 — E cos 60 . and the difference of potential at the condenser is — r 9 C I et=E cos (6 - 00) + £ j (e0 - £ cos «„) cos \/ ^ (11) HIGH POTENTIAL SYSTEMS 109 Of the three terms: i* ', e/; in \ e/'; i"r , e/", the first obviously represents the stationary condition of charging current and con- denser potential, since the two other terms disappear for t = oo . The second term, i", e^', represents that component of oscilla- tion which depends upon the phase of impressed e.m.f., or the point of the impressed e.m.f. wave, at which the oscillation begins, while the third term, inf , e/", represents the component of oscillation which depends upon the instantaneous values of current and e.m.f. respectively, at the moment at which the oscillation begins, s c is the decrement of the oscillation. 66. The frequency of oscillation is where / is the impressed frequency. That is, the frequency of oscillation equals the impressed frequency times the square root of the ratio of condensive reactance and inductive reactance of the circuit, or is the impressed frequency divided by the square root of inductance voltage and capacity current, as fraction of impressed voltage and full-load current. Since the frequency of oscillation is that is, is independent of the frequency of the impressed e.m.f. Substituting 6 = 2 xfi, xc = —^ and x = 2 rfL & 7T/U in equations (8), (10), and (11), we have t sin - — y VCL (12) 1C - — t t i" = V/F#£ 2L cos<90sin VCL t e/' =- Ee cos ^ cos--—; 110 TRANSIENT PHENOMENA r . , . ,„ t 1'" = e VCL ==• + i0 V 7: sin — — [ ; C< T ° V /7 A //nr r > -~t ( i =-27zfCEsin(d - 60)+ £ 2L U0 cos (13) r 2L e, = E cos (d - 00)+ £ I (e0 - £ cos 00) cos 7^=- X , « ) (14) The oscillating terms of these equations are independent of the impressed frequency. That is, the oscillating currents and potential differences, caused by a change of circuit conditions (as starting, change of load, or opening circuit), are independent of the impressed frequency, and thus also of the wave shape of the impressed e.m.f., or its higher harmonics (except as regards terms of secondary order). The first component of oscillation, equation (12), depends not only upon the line constants and the impressed e.m.f., but principally upon the phase, or the point of the impressed e.m.f. wave, at which the oscillation starts; however, it does not depend upon the previous condition of the circuit. Therefore this component of oscillation is the same as the oscillation produced in starting the transmission line, that is, connecting it, unexcited, to the generator terminals. There exists no point of the impressed e.m.f. wave where no oscillation occurs (while, when starting a circuit containing resistance and inductance only, at the point of the impressed e.m.f. wave where the final current passes zero the stationary condition is instantly reached). With capacity in circuit, any change of circuit conditions involves an electric oscillation. HIGH POTENTIAL SYSTEMS 111 The maximum intensities of the starting oscillation occur near the value #0 = 0, and are and Since E 2x& sin i/l£ xxf e/' = - # K cos \/_£ X (15) 7 = sin (0 - 00) is the stationary value of charging current, it follows that the maximum intensity which the oscillating current, produced in starting a transmission line, may reach is y — • times the sta- tionary charging current, or the initial current bears to the stationary value the same ratio as the frequency of oscillation to the impressed frequency. The maximum oscillating e.m.f. generated in starting a trans- mission line is of the same value as the impressed e.m.f. Thus the maximum value of potential difference occurring in a trans- mission line at starting is less than twice the impressed e.m.f. and no excessive voltages can be generated in starting a circuit. The minimum values of the starting oscillation occur near #0 = 90°, and are, from equations (7), ;„ _ •" _ 2X COS\/^£ and -V/-#£. "c (16) that is, the oscillating current is of the same intensity as the charging current, and the maximum rush of current thus is less than twice the stationary value. The potential difference in the circuit rises only little above the impressed e.m.f. The second component of the oscillation, equation (13), does not depend upon the point of .the impressed e.m.f. wave at 112 TRANSIENT PHENOMENA which the oscillation starts, 00, nor upon the impressed e.m.f. as a whole, E, but, besides upon the constants of the circuit, it depends only upon the instantaneous values of current and of potential difference in the circuit at the moment when the oscillation starts, iQ and eQ. Thus, if i0 = 0, e0 = 0, or in starting a transmission line, unexcited, by connecting it to the impressed e.m.f., this term disappears. It is this component which may cause excessive potential differences. Two cases shall more fully be discussed, namely : (a) Opening the circuit of a transmission line under load, and (6) rupturing a short-circuit on the transmission line. 67. (a) If iQ is the instantaneous value of full-load current, e0 the instantaneous value of difference of potential at the condenser, n0 is small compared with e0, and \/~xxc iQ is of the same magnitude as e0. Writing and substituting in equations (10), we have **' cos 0 + d and e»> = Ve/ + ia2xxet sin U/ ^ 0 + d (17) e that is, the amplitude of oscillation isV/%2 +- — for the current, and \/eo2 + i*x xc for the e.m.f. Thus the generated e.m.f. can be larger than the impressed e.m.f., but is, as a rule, still of the same magnitude, except when xc is very large. In the expressions of the total current and potential difference at condenser, in equations (11), (e0 — E cos 00) is the difference between the potential difference at the condenser and the impressed e.m.f., at the instant of starting of the oscillation, or the voltage consumed by the line impedance, and this is small HIGH POTENTIAL SYSTEMS 113 if the current is not excessive. Thus, neglecting the terms with (e0 — E cos 0Q)j equations (11) assume the form t - - sin (d - 60) + v'^'cos \/-c V 0 and e1= E cos (d - 60) + i0 -JL0 ce '' siny (18) that is, the oscillation of current is of the amplitude of full-load current, and the oscillation of condenser potential difference is of the amplitude i x xc is the ratio of inductance voltage to condenser current, in fractions of full-load voltage and current. We have, therefore, L Thus in circuits of very high inductance L and relatively low capacity C, i0Vx xc may be much higher than the impressed e.m.f., and a serious rise of potential occur when opening the circuit under load, while in low inductance cables of high capacity i0Vxxc is moderate; that is, the inductance, by tending to maintain the current, generates an e.m.f., producing a rise in potential, while capacity exerts a cushioning effect. Low inductance and high capacity thus are of advantage when breaking full-load current in a circuit. 68. (6) If a transmission line containing resistance, induc- tance, and capacity is short-circuited, and the short-circuit suddenly opened at time t = 0, we have, for t < 0, and where and ET I =-COS (d - 0Q - y), tan y = -; (19) 114 TRANSIENT PHENOMENA thus, at time t = 0, _ E z (20) Substituting these values of e0 and i0 in equations (9) gives and E_ z r_ 2x 7 a 2x ^- O Xc or, neglecting terms of secondary magnitude, and .„, E - £ » V = — £ COS z cos (8, + y) sin 6; (21) that is, im is of the magnitude of short-circuit current, and e"' of higher magnitude than the impressed e.m.f., since z is small compared with \/xxc. The total values of current and condenser potential difference, from equation (11), are cos\/- X xx, and cos (0 - 00) - Ee \rxxc COS (00 + y) . cos \ cosV/- v X (22) HIGH POTENTIAL SYSTEMS 115 or approximately, since all terms are negligible compared with i'" and e/", and TJI T_Q i — i=-e 2x cos (00 + 7-) cos V- 2 ' X E\fxxc -£• fc £ cos (69 + 7-) sin y - (23) * • ju These values are a maximum, if the circuit is opened at the OQ = - r, that is, at the maximum value of the short- urrent, and are then moment 60 = - 7-, that is, al circuit current, and are then and (24) The amplitude of oscillation of the condenser potential dif ference is xx ; or, neglecting the line resistance, as rough approximation, x = z, CE x ' that is, the potential difference at the condenser is increased above the impressed e.m.f. in the proportion of the square root of the ratio of condensive reactance to inductive reactance, or inversely proportional to the square root of inductance voltage times capacity current, as fraction of the impressed voltage and the full-load current. Thus, in this case, the rise of voltage is excessive. The minimum intensity of the oscillation due to rupturing short-circuit occurs if the circuit is broken at the moment 116 TRANSIENT PHENOMENA 00 = 90° — r, that is, at the zero value of the short-circuit current. Then we have r ysiny— 0 * £ (25) that is, the potential difference at the condenser is less than twice the impressed e.m.f.; therefore is moderate. Hence, a short- circuit can be opened safely only at or near the zero value of the short-circuit current. The phenomenon ceases to be oscillating, and becomes an ordinary logarithmic discharge, if x/r2— 4 xxc is real, or r > 2 Vxx^. Some examples may illustrate the phenomena discussed in the preceding paragraphs. 69. Let, in a transmission line carrying 100 amperes at full load, under an impressed e.m.f. of 20,000 volts, the resistance drop = 8 per cent, the inductance voltage = 15 per cent of the impressed voltage, and the charging current = 8 per cent of full- load current. Assuming 1 per cent resistance drop in the step-up transformers, and a reactance voltage of 2i per cent, the resistance drop between the constant potential generator terminals and the middle of the transmission line is then 5 per cent, or r = 10 ohms, and the inductance voltage is. 10 per cent, or x = 20 ohms. The charging current of the line is 8 amperes, thus the condensive reactance xc = 2500 ohms. Then, assuming a sine wave of impressed e.m.f., we have E = 20,000 V2 = 28,280 volts; i' = - 11.3 sin (6 - 00); e{= 28,280 cos (0 - 00); i" = - 11.3 £-0'25'[sin 00 cos 11.2 6 - 11.2 cos 0, sin 11.2 6], and e/' = - 28,280 £-°-25' [cos 00 cos 1 1 .2 0 + (0.0222 cos 00 + 0.0283 sin 00) sin 11. 20] =* -28,280 £-°-25(? cos 00 cos 11.20. HIGH POTENTIAL SYSTEMS 117 Therefore the oscillations produced in starting the trans- mission line are i = - 11.3 [sin (0 - 00) + r0'25 e (sin 00 cos 11.2 0 - 11.2 cos 00 sin 11.20)] and e, = 28,280 { cos (0 - 00) - £-°'25' [cos 00 cos 11.2 d + (0.0222 cos 00 + 0.0283 sin 00) sin 1 1 .2 0] } ^ 28,280 [cos (0 - 00) - £-°-25' cos 00 cos 11.2 0]. 100 60 f \ s "•> K=" •>• — 28,2 lOo 20 o 80 v hru hm ilta i ' £-= " 25C Ooh ma 80 40 60 30 V \«i / \ ' / \ (>' \ / / \ / 'v ,- 40 "SO (_ \ / A^ N f o-w-o 1 1 \ ,' / \ 1 V - — »~--^ / \ \ / , \ / / \ \ t ^r • — ^y i -20 10 1 p \ ^ 7 ... --\ >- ^ I 1 1 \ \ I « i \, / 60 30 1 / I \ / on 1 / t ,' t / inn ; \ \ / ._ f -120 \ / v '' 20 70 40 50 Degrees Fig. 27. Starting of a transmission line. 90 100 m ^^ _ — - Zr=> •^^ • ^* ^ •*" 25 ofl on ~-x ^ — • -z>~* ^ — • ,~^- ^* s^ K = /' = »' »hOv .hm olts H /; \ ^ x^ ^^^" ,~v Z = x_ — • 20< jhiu lot, I 318 .10 ^15 i' / U< ^ / / "\ •;- 90 10 g 10 a a ""x^ ^ -^1 \ / ^-— » . --- --. \ ^ / \ ^ ^ ^X^*^ \ \ \ ~" 7- -. ^_ \ ^~ / ** \ "" — • ""/ 10 10 \ * ^^ r* ) 1 0 2 0 3 0 4 1 5 0 e 0 7 0 \ 0 I 0 1( w Fig. 28. Starting of a transmission line. Hence the maximum values for 00 = 0, are i -- -11.3 (sin0 - 11.2 £-°-25' sin 11.20) and e, = 28, 280 [cos 0-£-°-25' (cos 11.2 0 + 0.0222 sin 11.20)] ^ 28,280 (cos 0 - £-°-25' cos 11.2 0), and the minimum values, for 00 = 90°, are i -= 11.3 (cos 0 -