LECTURE IX. OSCILLATIONS OF THE COMPOUND CIRCUIT. 38. The most interesting and most important application of the travehng wave is that of the stationary oscillation of a com- pound circuit, as industrial circuits are never uniform, but consist of sections of different characteristics, as the generating system, transformer, line, load, etc. Oscillograms of such circuits have been shown in the previous lecture. If we have a circuit consisting of sections 1, 2, 3 . . . , of the respective lengths (in velocity measure) Xi, X2, X3 . . . , this entire circuit, when left to itself, gradually dissipates its stored energy by a transient. As function of the time, this transient must decrease at the same rate Uq throughout the entire circuit. Thus the time decrement of all the sections must be Every section, however, has a power-dissipation constant, Ui, U2, U3 . . . , which represents the rate at which the stored energy of the section would be dissipated by the losses of power in the section, t , t , t ... But since as part of the whole circuit each section must die down at the same rate e~"o', in addition to its power-dissipation decrement e""'^, e~"2< , . . ^ each section must still have a second time decrement, e~^"o~"i^V' e~^"''~"2)< . . . This latter does not represent power dissipation, and thus represents power transfer. That is, §1 = Wo "~ Ui, S2 = Uo — II2, (1) It thus follows that in a compound circuit, if Uo is the average exponential time decrement of the complete circuit, or the average 108 OSCILLATIONS OF THE COMPOUND CIRCUIT. 109 power-dissipation constant of the circuit, and u that of any section, this section must have a second exponential time decrement, s = Uq — u, (2) which represents power transfer from the section to other sections, or, if s is negative, power received from other sections. The oscil- lation of every individual section thus is a traveling wave, with a power-transfer constant s. As Uo is the average dissipation constant, that is, an average of the power-dissipation constants u of all the sections, and s = uq — u the power-transfer constant, some of the s must be positive, some negative. In any section in which the power-dissipation constant u is less than the average Uq of the entire circuit, the power-transfer con- stant s is positive; that is, the wave, passing over this section, in- creases in intensity, builds up, or in other words, gathers energy, which it carries away from this section into other sections. In any section in which the power-dissipation constant u is greater than the average Uq of the entire circuit, the power-transfer con- stant s is negative; that is, the wave, passing over this section, decreases in intensity and thus in energy, or in other words, leaves some of its energy in this section, that is, supplies energy to the section, which energy it brought from the other sections. By the power-transfer constant s, sections of low energy dissi- pation supply power to sections of high energy dissipation. 39. Let for instance in Fig, 43 be represented a circuit consist- ing of step-up transformer, transmission line, and load. (The load, consisting of step-down transformer and its secondary cir- cuit, may for convenience be considered as one circuit section.) Assume now that the circuit is disconnected from the power sup- ply by low-tension switches, at A. This leaves transformer, line, and load as a compound oscillating circuit, consisting of four sections : the high-tension coil of the step-up transformer, the two lines, and the load. Let then Xi = length of line, X2 = length of transformer circuit, and X3 = length of load circuit, in velocity measure.* If then * If Zi = length of circuit section in any measure, and Lo = inductance, Co = capacity per unit of length U, then the length of the circuit in velocity measure is Xi = aoh, where o-q = v LoCo. Thus, if L = inductance, C = capacity per transformer coil, n = number of transformer coils, for the transformer the unit of length is the coil; hence the 110 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Ui = 900 = power- dissipation constant of the line, W2 = 100 = power-dissipation constant of transformer, and u^ = 1600 = power- dissipation constant of the load, and the respective lengths of the circuit sections are Xi = 1.5 X 10-3; X2 = 1 X 10-3; X3 = 0.5 X 10-^, it is: Line. Transformer. Line. Load. Sum. Length: X= 1.5X10-^ IXIQ-^ I.SXIQ-^ .5X10-^ 4.5X10-3 Power-dissipa- tion constant: u = 900 100 900 1600 uX= 1.35 .1 1.35 .8 3.6 hence, Wo= average u = — — = 800, and: Power-transfer constant: s = Uo-u= -100 +700 -100 -800 The transformer thus dissipates power at the rate U2 = 100, while it sends out power into the other sections at the rate of S2 = 700, or seven times as much as it dissipates. That is, it sup- plies seven-eighths of its stored energy to other sections. The load dissipates power at the rate uz = 1600, and receives power at the rate —s = 800; that is, half of the power which it dissipates is supplied from the other sections, in this case the transformer. The transmission line dissipates power at the rate iii = 900, that is, only a little faster than the average power dissipation of the entire circuit, Uo = 800 ; and the line thus receives power only at the rate —s= 100, that is, receives only one-ninth of its power from the transformer; the other eight-ninths come from its stored energy. The traveling wave passing along the circuit section thus increases or decreases in its power at the rate e^^^^; that is, in the line: p = 79ie"2oox^ the energy of the wave decreases slowly; in the transformer: p = p2€+^''°°^, the energy of the wave increases rapidly; length li = n, and the length in velocity measure, X = aou = n ^ LC. Or, if L = inductance, C = capacity of the entire transformer, its length in velocity measure is \ = ^ LC. Thus, the reduction to velocity measure of distance is very simple. Oscillations of the compound circuit. Ill in the load: p = pse~^^^^^, the energy of the wave decreases rapidly. Here the coefficients of pi, p2, ps must be such that the wave at the beginning of one section has the same value as at the end of the preceding section. In general, two traveling waves run around the circuit in opposite direction. Each of the two waves reaches its maximum intensity in this circuit at the point where it leaves the transformer and enters the line, since in the transformer it increases, while in the line it again decreases, in intensity. Fig. 54. — Energy Distribution in Compound Oscillation of Closed Circuit; High Line Loss. Assuming that the maximum value of the one wave is 6, that of the opposite wave 4 megawatts, the power values of the two waves then are plotted in the upper part of Fig. 54, and their difference, that is, the resultant flow of power, in the lower part of Fig, 54. As seen from the latter, there are two power nodes, or points over which no power flows, one in the transformer and one in the load, and the power flows from the transformer over the line into the load; the transformer acts as generator of the power, and of this 112 ELECTRIC DISCHARGES, WAVES AND IMPULSES. power a fraction is consumed in the line, the rest suppUed to the load. 40. The diagram of this transient power transfer of the system thus is very similar to that of the permanent power transmis- sion by alternating currents: a source of power, a partial con- sumption in the line, and the rest of the power consumed in the load. However, this transient power-transfer diagram does not repre- sent the entire power which is being consumed in the circuit, as power is also supplied from the stored energy of the circuit; and the case may thus arise — which cannot exist in a permanent power transmission — that the power dissipation of the line is less than corresponds to its stored energy, and the line also supplies power to the load, that is, acts as generator, and in this case the power would not be a maximum at the transformer terminals, but would still further increase in the line, reaching its maxi- mum at the load terminals. This obviously is possible only with transient power, where the line has a store of energy from which it can draw in supplying power. In permanent condition the line could not add to the power, but must consume, that is, the permanent power-transmission diagram must always be like Fig. 54. Not so, as seen, with the transient of the stationary oscillation. Assume, for instance, that we reduce the power dissipation in the hne by doubling the conductor section, that is, reducing the resistance to one-half. As L thereby also slightly decreases, C increases, and g possibly changes, the change brought about in I (r q\ . the constant '^ = 9(7+7;) is not necessarily a reduction to one- half, but depends upon the dimensions of the line. Assuming therefore, that the power-dissipation constant of the line is by the doubling of the line section reduced from Ui = 900 to Ui = 500, this gives the constants: hence, Line. Transformer. X= 1.5X10-3 1X10-3 u= 500 100 u\= .75 .1 Line. 1.5X10-3 500 .75 Load. .5X10-3 1600 .8 Sum. 4.5X10-3 2.4 2mX _„„ U(i = average u = -~— = 533, and: s= +33 -f-433 +33 -1067 OSCILLATIONS OF THE COMPOUND CIRCUIT. 113 That is, the power-transfer constant of the hne has become posi- tive, Si = 33, and the hne now assists the transformer in supplying power to the load. Assuming again the values of the two travel- ing waves, where they leave the transformer (which now are not the maximum values, since the waves still further increase in intensity in passing over the lines), as 6 and 4 megawatts respec- tively, the power diagram of the two waves, and the power dia- gram of their resultant, are given in Fig. 55. \AAAAf Traiisuiission Line 500 Step up Transformer u = ioo Transmission Line U = 500 Load u Fig. 55. U„= 533 Energy Distribution in Compound Oscillation of Closed Circuit; Low Line Loss. In a closed circuit, as here discussed, the relative intensity of the two component waves of opposite direction is not definite, but depends on the circuit condition at the starting moment of the transient. In an oscillation of an open compound circuit, the relative intensities of the two component waves are fixed by the condition that at the open ends of the circuit the power transfer must be zero. As illustration may be considered a circuit comprising the high- potential coil of the step-up transformer, and the two lines, which are assumed as open at the step-down end, as illustrated diagram- matically in Fig. 56. 114 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Choosing the same lengths and the same power-dissipation constants as in the previous illustrations, this gives: Line. 1.5X10-3 900 1.35 2 2/ A Transformer. 1X10-3 100 .1 iience, Wo = average u = —— = 700, and: S = —2^0 +600 Line. 1.5X10-3 900 1.35 ■200 Sum. 4X10-3 2.8 Transformer Line Fig. 56. The diagram of the power of the two waves of opposite direc- tions, and of the resultant power, is shown in Fig. 57, assuming 6 megawatts as the maximum power of each wave, which is reached at the point where it leaves the transformer. Transmission Line Transformer Transmission Line H ^TRfMrifOT^p 1- U =900 U =100 yv U=' Fig. 57. — Energy Distribution in Compound Oscillation of Open Circuit. In this case the two waves must be of the same intensity, so as to give 0 as resultant at the open ends of the line. A power node then appears in the center of the transformer. 41. A stationary oscillation of a compound circuit consists of two traveling waves, traversing the circuit in opposite direction, and transferring power betwceu the circuit sections in such a manner OSCILLATIONS OF THE COMPOUND CIRCUIT. 115 as to give the same rate of energy dissipation in all circuit sections. As the result of this power transfer, the stored energy of the sj^stem must be uniformly distributed throughout the entire circuit, and if it is not so in the beginning of the transient, local traveling waves redistribute the energy throughout the oscillat- ing circuit, as stated before. Such local oscillations are usually of very high frequency, but sometimes come within the range of the oscillograph, as in Fig. 47. During the oscillation of the complex circuit, every circuit element d\ (in velocity measure), or every wave length or equal part of the wave length, therefore contains the same amount of stored energy. That is, if eo = maximum voltage, ^o = maximum current, and Xo = wave length, the average energy " ° must be constant throughout the entire circuit. Since, however, in velocity measure, Xo is constant and equal to the period Tq through- out all the sections of the circuit, the product of maximum voltage and of maximum current, eoio, thus must be constant throughout the entire circuit. The same applies to an ordinary traveling wave or impulse. Since it is the same energy which moves along the circuit at a constant rate, the energy contents for equal sections of the circuit must be the same except for the factor e~ ^ ut^ j^y which the energy decreases with the time, and thus with the distance traversed during this time. Maximum voltage eo and maximum current Zo, however, are related to each other by the condition_ f = ^» = sj^' (3) ^0 'Co and as the relation of Lo and Co is different in the different sections, and that very much so, Zo, and with it the ratio of maximum voltage to maximum current, differ for the different sections of the circuit. If then ei and ii are maximum voltage and maximum current respectively of one section, and Zi = \/ -^ is the ''natural imped- ance " of this section, and 62, ^2, and ^2 = y tt are the correspond- ing values for another section, it is 6^12 = eiiij (4) 116 ELECTRIC DISCHARGES, WAVES AND IMPULSES. and since substituting into (4) gives or and or ^2 , . T ^ ^2 €2 = = i2Z2, ei = ■■ iiZi, 12% = ■ ii%, ii V Z2 y c,L2 el = !l!, Zl Zl ' (5) (6) (7) That is, in the same oscillating circuit, the maximum voltages eo in the different sections are proportional to, and the maximum currents Iq inversely proportional to, the square root of the natural impedances Zq of the sections, that is, to the fourth root of the ratios of inductance to capacity y^ • At every transition point between successive sections traversed by a traveling wave, as those of an oscillating system, a trans- formation of voltage and of current occurs, by a transformation ratio which is the square root of the ratio of the natural imped- ances, ^0 = V PT , of the two respective sections. ^ Co When passing from a section of high capacity and low induc- tance, that is, low impedance Zq, to a section of low capacity and high inductance, that is, high impedance Zq, as when passing from a transmission line into a transformer, or from a cable into a trans- mission line, the voltage thus is transformed up, and the current transformed down, and inversely, Avith a wave passing in opposite direction. A low-voltage high-current wave in a transmission line thus becomes a high-voltage low-current wave in a transformer, and inversely, and thus, while it may be harmless in the line, may become destructive in the transformer, etc. OSCILLATIONS OF THE COMPOUND CIRCUIT. 117 42. At the transition point between two successive sections, the current and voltage respectively must be the same in the two sections. Since the maximum values of current and voltage respectively are different in the two sections, the phase angles of the waves of the two sections must be different at the transition point; that is, a change of phase angle occurs at the transition point. This is illustrated in Fig. 58. Let Zq = 200 in the first section (transmission line), Zo = 800 in the second section (transformer). The transformation ratio between the sections then is V ^t^'^^; that is, the maximum voltage of the second section is twice, and the maximum current half, that of the first section, and the waves of current and of voltage in the two sections thus may be as illustrated for the voltage in Fig. 58, by 6162. Fig. 58. — Effect of Transition Point on Traveling Wave. If then e' and i' are the values of voltage and current respec- tively at the transition point between two sections 1 and 2, and ei and ii the maximum voltage and maximum current respec- tively of the first, 62 and 22 of the second, section, the voltage phase and current phase at the transition point are, respectively: For the wave of the first section: — = cos 7i and — = cos 5i. ei ii For the wave of the second section: e' i' — = cos 72 and — = cos §2. 62 ^2 (9) 118 ELECTRIC DISCHARGES, WAVES AND IMPULSES. Dividing the two pairs cos 72 of equations of (9) gives = ^ = i M hence, multipHed, C0S71 cos ^2 _ 2i _ . /Z2 io V zi cos bi 12 (10) or or cos 72 y cos §2 _ cos 7i cos 5i ' cos 72 _ cos bi cos 7i cos ^2 cos 7i cos 5i = cos 72 cos 62; (11) that is, the ratio of the cosines of the current phases at the tran- sition point is the reciprocal of the ratio of the cosines of the voltage phases at this point. Since at the transition point between two sections the voltage and current change, from ei, ^l to e^, 4, by the transformation ratio v/ — , this change can also be represented as a partial reflection. That is, the current ^l can be considered as consisting of a compo- nent 1*2, which passes over the transition point, is '' transmitted " current, and a component ii = ii — 2*2, which is " reflected " current, etc. The greater then the change of circuit constants at the transition point, the greater is the difference between the currents and voltages of the two sections; that is, the more of current and voltage are reflected, the less transmitted, and if the change of constants is very great, as when entering from a trans- mission line a reactance of very low capacity, almost all the current is reflected, and very little passes into and through the reactance, but a high voltage is produced in the reactance.