LECTURE VII. LINE OSCILLATIONS. 28. In a circuit containing inductance and capacity, the tran- sient consists of a periodic component, by which the stored energy surges between magnetic — and dielectric — , and a transient component, by which the total stored energy decreases. Considering only the periodic component, the maximum value of magnetic energy must equal the maximum value of dielectric '^'^e^gy- Li„^ Ce, 0 "^^0 (1) where Iq = maximum value of transient current, 60 = maximum value of transient voltage. This gives the relation between Bq and Iq, ^^ = Jl ,^ = 1, (2) where Zq is called the natural impedance or surge impedance, 2/0 the natural or surge admittance of the circuit. As the maximum of current must coincide with the zero of voltage, and inversely, if the one is represented by the cosine function, the other is the sine function; hence the periodic com- ponents of the transient are ii = Iq cos (0 — 7) ei = eo sm (0 — 7) ) where 0 = 2 Tft, (4) and is the frequency of oscillation. The dissipative or " transient " component is M = €-"', (6) 72 LINE OSCILLATIONS. T6 where u 2 U ^ C; hence the total expression of transient current and voltage is ^ = ^oe~ "^ cos (0 — 7) e = eoe~ ^^ sin (0 — 7) (7) (8) 7, eo, and io follow from the initial values e' and i' of the transient, 2bt t = 0 or (t> = 0: hence ^ = ^o cos 7 e' = —eo sin 7 tan 7 (9) (10) The preceding equations of the double-energy transient apply to the circuit in which capacity and inductance are massed, as, for instance, the discharge or charge of a condenser through an in- ductive circuit. Obviously, no material difference can exist, whether the capacity and the inductance are separately massed, or whether they are intermixed, a piece of inductance and piece of capacity alternating, or uniformly distributed, as in the transmission line, cable, etc. Thus, the same equations apply to any point of the transmission line. j J - ..1- _ A B Fig. 34. However, if (8) are the equations of current and voltage at a point A of a line, shown diagrammatically in Fig. 34, at any other point B, at distance I from the point A, the same equations will apply, but the phase angle 7, and the maximum values eo and Iq, may be different. Thus, if I = c^e -"* cos ((f) ut -t) ) -t) ! (11) 74 ELECTRIC DISCHARGES, WAVES AND IMPULSES. are the current and voltage at the point A, this oscillation will appear at a point B, at distance I from ^, at a moment of time later than at A by the time of propagation U from A to B, if the oscillation is traveling from A to B; that is, in the equation (11), instead of t the time {t — ti) enters. Or, if the oscillation travels from 5 to A, it is earlier at B by the time ti; that is, instead of the time t, the value (t + ^i) enters the equation (11). In general, the oscillation at A will appear at B, and the oscillation at B will appear at A, after the time ti; that is, both expressions of (11), with {t — ^i) and with {t + ^i), will occur. The general form of the line oscillation thus is given by substi- tuting {t =F ^i) instead of t into the equations (11), where ^i is the time of propagation over the distance I. li V = velocity of propagation of the electric field, which in air, as with a transmission line, is approximately V = 3X W, (12) and in a medium of permeability fx and permittivity (specific capacity) k is v= y=-y (13) and we denote then and if we denote a = -, (14) h = at', (15) 2 tt/^i = CO = 2 Trfal, (16) we get, substituting t T k for t and 0 =F co for (/> into the equation (11), the equations of the line oscillation: i = ce-"' COS ((/) =F CO - 7) ) . e = 2;oCe-"' sin (0 =F co — 7) j In these equations, 4> = 27rft ^ is the time angle, and >- (18) CO = 2 irfal ) is the space angle, and c = C(,e=^"'i is the maximum value of current, ZfjC the maximum value of voltage at the point I. LINE OSCILLATIONS. 75 Resolving the trigonometric expressions of equation (17) into functions of single angles, we get as equations of current and of voltage products of the transient e~"', and of a combination of the trigonometric expressions : cos cf) cos 0)^ sin 0 cos CO, cos cf) sin CO, sin cf) sin co. (19) Line oscillations thus can be expressed in two different forms, either as functions of the sum and difference of time angle cf) and distance angle co: (0 =t co), as in (17); or as products of functions of cf) and functions of co, as in (19). The latter expression usually is more convenient to introduce the terminal conditions in station- ary waves, as oscillations and surges; the former is often more convenient to show the relation to traveling waves. In Figs. 35 and 36 are shown oscillograms of such line oscilla- tions. Fig. 35 gives the oscillation produced by switching 28 miles of 100-kv. line by high-tension switches onto a 2500-kw. step-up transformer in a substation at the end of a 153-mile three- phase line; Fig. 36 the oscillation of the same system caused by switching on the low-tension side of the step-up transformer. 29. As seen, the phase of current i and voltage e changes pro- gressively along the line I, so that at some distance Iq current and voltage are 360 degrees displaced from their values at the starting point, that is, are again in the same phase. This distance Zo is called the wave length, and is the distance which the electric field travels during one period to — -j. of the frequency of oscillation. As current and voltage vary in phase progressively along the line, the effect of inductance and of capacity, as represented by the inductance voltage and capacity current, varies progressively, and the resultant effect of inductance and capacity, that is, the effective inductance and the effective capacity of the circuit, thus are not the sum of the inductances and capacities of all the line elements, but the resultant of the inductances and capacities of all the line elements combined in all phases. That is, the effective inductance and capacity are derived by multipljdng the total 2 inductance and total capacity by avg/cos/, that is, by - • 76 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 00 ^ I s o 3 02 f=4 LINE OSCILLATIONS. 77 00 . .2 ^ 03 •3 O O fl bC -^^ 78 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Instead of L and C, thus enter into the equation of the double- energy oscillation of the line the values — and — ^ IT IT In the same manner, instead of the total resistance r and the total conductance g, the values — and —^ appear. TT IT The values of Zq, yo, u, 4>, and co are not changed hereby. The frequency /, however, changes from the value correspond- ing to the circuit of massed capacity, / = 7= , to the value 2 7r vLC 4:VLC Thus the frequency of oscillation of a transmission line is / = — 7= = T-^ ■ (20) where a = VlC. (21) If h is the length of the line, or of that piece of the line over which the oscillation extends, and we denote by Lo,Co,ro,go (22) the inductance, capacity, resistance, and conductance per unit length of line, then that is, the rate of decrease of the transient is independent of the length of the line, and merely depends on the line constants per unit length. It then is (T = ll = period, (31) LINE OSCILLATIONS. 81 thus is lo = vto = -.J (32) and, substituting (32) into (31), gives a = (To, (33) or 1 1 0-0 VLoCc (34) This gives a very important relation between inductance Lo and capacity Co per unit length, and the velocity of propagation. It allows the calculation of the capacity from the inductance, Co = ^^, (35) and inversely. As in complex overhead structures the capacity usually is difficult to calculate, while the inductance is easily de- rived, equation (35) is useful in calculating the capacity by means of the inductance. This equation (35) also allows the calculation of the mutual capacity, and thereby the static induction between circuits, from the mutual magnetic inductance. The reverse equation, Lo = ^^. (36) is useful in calculating the inductance of cables from their meas- ured capacity, and the velocity of propagation equation (13). 31. If Zi is the length of a line, and its two ends are of different electrical character, as the one open, the other short-circuited, and thereby i = 0 at one end, e = 0 at the other end, the oscilla- tion of this line is a quarter-wave or an odd multiple thereof. The longest wave which may exist in this circuit has the wave length Zo = 4 k, and therefore the period ^0 = co^o = 4 aoh, that is, the frequency /o = -. — 7- . This is called the fundamental wave 4 aoti of oscillation. In addition thereto, all its odd multiples can exist as higher harmonics, of the respective wave lengths „ , _ .. and the frequencies (2 k — l)/o, where k = 1,2,3 .. . 82 ELECTRIC DISCHARGES, WAVES AND IMPULSES. If then 0 denotes the time angle and co the distance angle of the fundamental wave, that is, 0 = 2 tt represents a complete cycle and CO = 2 TT a complete wave length of the fundamental wave, the time and distance angles of the higher harmonics are 3 0, 3 CO, 5 0, 5 CO, 7 0, 7 CO, etc. A complex oscillation, comprising waves of all possible fre- quencies, thus would have the form tti cos (0 T CO — 7i) + as cos 3 (0 =F co — 73) + a5 cos 5 (0 =F CO - 75) + . . . , (37) and the length ^1 of the line then is represented by the angle CO = i^, and the oscillation called a quarter-wave oscillation. If the two ends of the line l\ have the same electrical charac- teristics, that is, e = 0 at both ends, or i = 0, the longest possible wave has the length ?o = 2 Zi, and the frequency h aolo 2 aoli or any multiple (odd or even) thereof. If then 0 and co again represent the time and the distance angles of the fundamental wave, its harmonics have the respective time and distance angles 2 0, 2 CO, 3 0, 3 CO, 4 0, 4 CO, etc. A complex oscillation then has the form ai cos (0 =F CO — 7i) + a2 cos 2 (0 =F co — 72) + as cos 3 (0 =F CO - 73) + . . . , (38) and the length k of the line is represented by angle coi = t, and the oscillation is called a half-wave oscillation. The half-wave oscillation thus contains even as well as odd harmonics, and thereby may have a wave shape, in which one half wave differs from the other. Equations (37) and (38) are of the form of equation (17), but LINE OSCILLATIONS. 83 usually are more conveniently resolved into the form of equa- tion (19). At extremely high frequencies (2 k —l)f, that is, for very large values of k, the successive harmonics are so close together that a very small variation of the line constants causes them to overlap, and as the line constants are not perfectly constant, but may vary slightly with the voltage, current, etc., it follows that at very high frequencies the line responds to any frequency, has no definite frequency of oscillation, but oscillations can exist of any frequency, provided this frequency is sufficiently high. Thus in long trans- mission lines, resonance phenomena can occur only with moderate frequencies, but not with frequencies of hundred thousands or millions of cycles. 32. The line constants Vq, go, Lq, Cq are given per unit length, as per cm., mile, 1000 feet, etc. The most convenient unit of length, when dealing with tran- sients in circuits of distributed capacity, is the velocity unit v. That is, choosing as unit of length the distance of propagation in unit time, or 3 X 10^° cm. in overhead circuits, this gives v = 1, and therefore "" = ^^«^» -' ^ (39) LnCn — 1, '1 or Oo — 7^ , i^o — TT • That is, the capacity per unit of length, in velocity measure, is inversely proportional to the inductance. In this velocity unit of length, distances will be represented by X. Using this unit of length, o-q disappears from the equations of the transient. This velocity unit of length becomes specially useful if the transient extends over different circuit sections, of different con- stants and therefore different wave lengths, as for instance an overhead line, the underground cable, in which the wave length is about one-half what it is in the overhead line (k = 4) and coiled windings, as the high-potential winding of a transformer, in which the wave length usually is relatively short. In the velocity measure of length, the wave length becomes the same throughout all these circuit sections, and the investigation is thereby greatly simplified. 84 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Substituting o-q = 1 in equations (30) and (31) gives ^0 = Xo, / = 1 To' 4> = 2Tft = 2'Kt CO = 2irf\-- 2 7rX Xo (40) and the natural impedance of the line then becomes, in velocity measure, Zq =v/ Co L - ^ - ^ Co 2/0 (41) where eo = maximum voltage, io = maximum current. That is, the natural impedance is the inductance, and the natural admittance is the capacity, per velocity unit of length, and is the main characteristic constant of the line. The equations of the current and voltage of the line oscillation then consist, by (19), of trigonometric terms cos (f) cos CO, sin 0 cos CO, cos (f) sin CO, sin 4> sin co, multiplied with the transient, e~"*, and would thus, in the most general case, be given by an expression of the form i = e~ "' J ai cos cos co + c/ cos sin co + di sin = 0, and by the values of i and e Sit all times t (or cf) respectively) at the ends of the circuit, that is, for CO = 0 and co = ^• For instance, if: (a) The circuit is open at one end co = 0, that is, the current is zero at all times at this end. That is, ?; = 0 for CO = 0; the equations of i then must not contain the terms with cos o), cos 2 CO, etc., as these would not be zero for co = 0. That is, it must be a, = 0, 6i = 0, ) a, = 0, h2 = 0, I (43) as = 0, 63 = 0, etc. ) The equation of i contains only the terms with sin co, sin 2 co, etc. Since, however, the voltage e is a maximum where the current i is zero, and inversely, at the point where the current is zero, the voltage must be a maximum; that is, the equations of the voltage must contain only the terms with cos co, cos 2 co, etc. Thus it must be c/ = 0, d/ = 0, ) c/ = 0, rf/ = 0, [ (44) C3' = 0, ds' = 0, etc. ) Substituting (43) and (44) into (42) gives i = e~" * f Ci cos 4> + c^i sin 0 j sin co, e = e-"* J a/ cos 0 + bi sin 0 j cos co (45) and the higher harmonics hereof. (b) If in addition to (a), the open circuit at one end co = 0, the line is short-circuited at the other end co = ^, the voltage e must be zero at this latter end. Cos co, cos 3 co, cos 5 co, etc., become zero for co = ^, but cos 2 co, cos 4 co, etc., are not zero for CO = ^, and the latter functions thus cannot appear in the expres- sion of e. 86 ELECTRIC DISCHARGES, WAVES AND IMPULSES. That is, the voltage e can contain no even harmonics. If, however, the voltage contains no even harmonics, the current produced by this voltage also can contain no even harmonics. That is, it must be C2 = 0, d2 = 0, a2' = 0, 62' = 0, ) C4 = 0, d^ = 0, a^' = 0, W = 0, > (46) Ce = 0, de = 0, ae' = 0, be' = 0, etc. ) The complete expression of the stationary oscillation in a circuit open at the end co = 0 and short-circuited at the end <^ = 9 thus would be i = ^-ut J (^^^ (.Qg (f, -\- di sin 0) sin co + (cs cos 3 0 + c/s sin 3 0) sin 3 CO + . . . ( , g _ g-uM (c^/ cos 0 + 6/ sin 0) cos co + (as' cos 3 0 + 63' sin 3 0) cos 3 CO + ... 1 . (47) (c) Assuming now as instance that, in such a stationary oscilla- tion as given by equation (47), the current in the circuit is zero at the starting moment of the transient for 0 = 0. Then the equation of the current can contain no terms with cos 0, as these would not be zero for 0 = 0. That is, it must be ci = 0, ) cs = 0, [ (48) Co = 0, etc. ) At the moment, however, when the current is zero, the voltage of the stationary oscillation must be a maximum. As i = 0 for 0 = 0, at this moment the voltage e must be a maximum, that is, the voltage wave can contain no terms with sin 0, sin 3 0, etc. This means W = 0, ) b/ = 0, (49) W = 0, etc. ) Substituting (48) and (49) into equation (47) gives i = e~"' 5 rii sin 0 sin 00 + d^ sin 3 0 sin 3 co + ^5 sin 5 0 sin 5 co + ■ ■ ■ I, e = €~"' \ai cos 0 cos co+as' cos 3 0 cos 3 co+a5' cos 5 0 cos 5 co + . . . J. (50) LINE OSCILLATIONS. 87 In these equations (50), d and a' are the maximum values of current and of voltage respectively, of the different harmonic waves. Between the maximum values of current, to, and of volt- age, Bq, of a stationary oscillation exists, however, the relation ^0 - ^ i /^ where Zq is the natural impedance or surge impedance. That is a' = dzo, (51) and substituting (51) into (50) gives i = ^-^t \ di sin 0 sin co + (is sin 3 (/> sin 3 co + t?5 sin 5 (^ sin 5 co + • ■ • i, e = Zq e~"' 5 d\ cos 0 cos co + c^s cos 3 0 cos 3 co + ^5 cos 5 0 cos 5 co (52) {d) If then the distribution of voltage e along the circuit is given at the moment of start of the transient, for instance, the voltage is constant and equals ei throughout the entire circuit at the starting moment <^ = 0 of the transient, this gives the relation, by substituting in (52), ei = Zq e~"^ \ di cos co + ^3 cos 3 co + (^5 cos 5 co + . . . | , (53) for all values of co. Herefrom then calculate the values of di, ds, d^, etc., in the manner as discussed in '' Engineering Mathematics," Chapter III.