LECTURE VI. DOUBLE-ENERGY TRANSIENTS. 24. In a circuit in which energy can be stored in one form only, the change in the stored energy which can take place as the result of a change of the circuit conditions is an increase or decrease. The transient can be separated from the permanent condition, and then always is the representation of a gradual decrease of energy. Even if the stored energy after the change of circuit conditions is greater than before, and during the transition period an increase of energy occurs, the representation still is by a decrease of the transient. This transient then is the difference between the energy storage in the permanent condition and the energy storage during the transition period. If the law of proportionality between current, voltage, magnetic flux, etc., apphes, the single-energy transient is a simple exponential function : _ j_ y = 2/oe ^°, (1) where 2/0 = initial value of the transient, and To = duration of the transient, that is, the time which the transient voltage, current, etc., would last if maintained at its initial value. The duration To is the ratio of the energy-storage coefficient to the power-dissipation coefficient. Thus, if energy is stored by the current i, as magnetic field. To = -, (2) r where L = inductance = coefficient of energy storage by the cur- rent, r = resistance = coefficient of power dissipation by the current. If the energy is stored by the voltage e, as dielectric field, the duration of the transient would be TV - -, (3) g 59 60 ELECTRIC DISCHARGES, WAVES AND IMPULSES. where C = capacity = coefficient of energy storage by the volt- age, in the dielectric field, and g = conductance = coefficient of power consumption by the voltage, as leakage conductance by the voltage, corona, dielectric hysteresis, etc. Thus the transient of the spontaneous discharge of a condenser would be represented by e = eoe~'^\ (4) Similar single-energy transients may occur in other systems. For instance, the transient by which a water jet approaches con- stant velocity when falling under gravitation through a resisting medium would have the duration T = '-^ (5> 9 where Vq = limiting velocity, g = acceleration of gravity, and would be given by v = Vo[l-e~'^). (6) In a system in which energy can be stored in two different forms, as for instance as magnetic and as dielectric energy in a circuit containing inductance and capacity, in addition to the gradual decrease of stored energy similar to that represented by the single-energy transient, a transfer of energy can occur between its two different forms. Thus, if i = transient current, e = transient voltage (that is, the difference between the respective currents and voltages exist- ing in the circuit as result of the previous circuit condition, and the values which should exist as result of the change of circuit conditions), then the total stored energy is 2^2^ (7) While the total energy W decreases by dissipation, Wm may be converted into Wd, or inversely. Such an energy transfer may be periodic, that is, magnetic energy may change to dielectric and then back again; or unidirectional, that is, magnetic energy may change to dielectric (or inversely, dielectric to magnetic), but never change back again; but the DOUBLE-ENERGY TRANSIENTS. 61 energy is dissipated before this. This latter case occurs when the dissipation of energy is very rapid, the resistance (or conductance) high, and therefore gives transients, which rarely are of industrial importance, as they are of short duration and of low power. It therefore is sufficient to consider the oscillating double-energy transient, that is, the case in which the energy changes periodically between its two forms, during its gradual dissipation. This may be done by considering separately the periodic trans- fer, or pulsation of the energy between its two forms, and the gradual dissipation of energy. A. Pulsatio7i of energy. 25. The magnetic energy is a maximum at the moment when the dielectric energy is zero, and when all the energy, therefore, is magnetic ; and the magnetic energy is then where Iq = maximum value of transient current. The dielectric energy is a maximum at the moment when the magnetic energy is zero, and all the energy therefore dielectric, and is then cv 2 ' where eo = maximum value of transient voltage. As it is the same stored energy which alternately appears as magnetic and as dielectric energy, it obviously is This gives a relation between the maximum value of transient current and the maximum value of transient voltage: '" \l^- (9) v/ U V c -^ therefore is of the nature of an impedance z^^ and is called the natural impedance, or the surge im'pedance, of the circuit ; and Ic its reciprocal, \ j = Vq, is the natural admittance, or the surge admittance, of the circuit. 62 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The maximum transient voltage can thus be calculated from the maximum transient current: eo = ^0 y g = ^'o^Jo, (10) and inversely, fc ^0 = eo y Y = eo2/o. (11) This relation is very important, as frequently in double-energy transients one of the quantities eo or io is given, and it is impor- tant to determine the other. For instance, if a line is short-circuited, and the short-circuit current io suddenly broken, the maximum voltage which can be induced by the dissipation of the stored magnetic energy of the short-circuit current is eo = IoZq. If one conductor of an ungrounded cable system is grounded, the maximum momentary current which may flow to ground is iQ = e^yo, where eo = voltage between cable conductor and ground. If lightning strikes a line, and the maximum voltage which it may produce on the line, as limited by the disruptive strength of the line insulation against momentary voltages, is e^, the maximum discharge current in the line is limited to Iq = eoyo. If L is high but C low, as in the high-potential winding of a high-voltage transformer (which winding can be considered as a circuit of distributed capacity, inductance, and resistance), Zq is high and 2/0 low. That is, a high transient voltage can produce only moderate transient currents, but even a small transient cur- rent produces high voltages. Thus reactances, and other reactive apparatus, as transformers, stop the passage of large oscillating currents, but do so by the production of high oscillating voltages. Inversely, if L is low and C high, as in an underground cable, Zq is low but yo high, and even moderate oscillating voltages pro- duce large oscillating currents, but even large oscillating currents produce only moderate voltages. Thus underground cables are little liable to the production of high oscillating voltages. This is fortunate, as the dielectric strength of a cable is necessarily relatively much lower than that of a transmission line, due to the close proximity of the conductors in the former. A cable, therefore, when receiving the moderate or small oscillating cur- rents which may originate in a transformer, gives only very low DOUBLE-ENERGY TRANSIENTS. 63 oscillating voltages, that is, acts as a short circuit for the trans- former oscillation, and therefore protects the latter. Inversely, if the large oscillating current of a cable enters a reactive device, as a current transformer, it produces enormous voltages therein. Thus, cable oscillations are more liable to be destructive to the reactive apparatus, transformers, etc., connected with the cable, than to the cable itself. A transmission line is intermediate in the values of Zq and t/o between the cable and the reactive apparatus, thus acting like a reactive apparatus to the former, like a cable toward the latter. Thus, the transformer is protected by the transmission line in oscillations originating in the transformer, but endangered by the transmission line in oscillations originating in the transmission line. __ The simple consideration of the relative values of ^o = y 7^ in the different parts of an electric system thus gives considerable information on the relative danger and protective action of the parts on each other, and shows the reason why some elements, as current transformers, are far more liable to destruction than others ; but also shows that disruptive effects of transient voltages, observed in one apparatus, may not and very frequently do not originate in the damaged apparatus, but originate in another part of the system, in which they were relatively harmless, and become dangerous only when entering the former apparatus. 26. If there is a periodic transfer between magnetic and dielec- tric energy, the transient current i and the transient voltage e successively increase, decrease, and become zero. The current thus may be represented by i = iocos ((f) — y), (12) where io is the maximum value of current, discussed above, and = 2 7rft, (13) where / = the frequency of this transfer (which is still undeter- mined), and y the phase angle at the starting moment of the transient; that is, ^l = ^o cos 7 = initial transient current. (14) As the current iissi maximum at the moment when the magnetic energy is a maximum and the dielectric energy zero, the voltage e 64 ELECTRIC DISCHARGES, WAVES AND IMPULSES. must be zero when the current is a maximum, and inversely; and if the current is represented by the cosine function, the voltage thus is represented by the sine function, that is, e = eo sin (0 — 7), (15) where ei = — 60 sin 7 = initial value of transient voltage. (16) The frequency / is still unknown, but from the law of propor- tionality it follows that there must be a frequency, that is, the suc- cessive conversions between the two forms of energy must occur in equal time intervals, for this reason: If magnetic energy converts to dielectric and back again, at some moment the proportion be- tween the two forms of energy must be the same again as at the starting moment, but both reduced in the same proportion by the power dissipation. From this moment on, the same cycle then must repeat with proportional, but proportionately lowered values. Fig. 31. — CD10017. — Oscillogram of Stationary Oscillation of Varying Frequency: Compound Circuit of Step-up Transformer and 28 Miles of 100,000-volt Transmission Line. If, however, the law of proportionality does not exist, the oscil- lation may not be of constant frequency. Thus in Fig. 31 is shown an oscillogram of the voltage oscillation of the compound circuit consisting of 28 miles of 100,000-volt transmission line and the 2500-kw. high-potential step-up transformer winding, caused by switching transformer and 28-mile line by low-tension switches off a substation at the end of a 153-mile transmission line, at 88 kv. With decreasing voltage, the magnetic density in the transformer DOUBLE-ENERGY TRANSIENTS. 65 decreases, and as at lower magnetic densities the permeability of the iron is higher, with the decrease of voltage the permeability of the iron and thereby the inductance of the electric circuit inter- linked with it increases, and, resulting from this increased magnetic energy storage coefficient L, there follows a slower period of oscil- lation, that is, a decrease of frequency, as seen on the oscillogram, from 55 cycles to 20 cycles per second. If the energy transfer is not a simple sine wave, it can be repre- sented by a series of sine waves, and in this case the above equa- tions (12) and (15) would still apply, but the calculation of the frequency / would give a number of values which represent the different component sine waves. The dielectric field of a condenser, or its " charge,'' is capacity times voltage: Ce. It is, however, the product of the current flowing into the condenser, and the time during which this current flows into it, that is, it equals i t. Applying the law Ce = it (17) to the oscillating energy transfer: the voltage at the condenser changes during a half-cycle from —eo to +eo, and the condenser charge thus is 2eoC; 2 the current has a maximum value io, thus an average value -io, IT and as it flows into the condenser during one-half cycle of the frequency /, that is, during the time ^, it is 2eoC = ~iQ ^—., TV ZJ which is the expression of the condenser equation (17) applied to the oscillating energy transfer. Transposed, this equation gives and substituting equation (10) into (18), and canceling with io, gives 66 ELECTRIC DISCHARGES, WAVES AND IMPULSES. as the expression of the frequency of the oscillation, where a = VlC (20) is a convenient abbreviation of the square root. The transfer of energy between magnetic and dielectric thus occurs with a definite frequency / = ^ — , and the oscillation thus is a sine wave without distortion, as long as the law of proportion- ality applies. When this fails, the wave may be distorted, as seen on the oscillogram Fig. 31. The equations of the periodic part of the transient can now be written down by substituting (13), (19), (14), and (16) into (12) and (15): i = ^o cos (0 — t) = io cos y cos 4> + io sin y sin 0 and by (11): t to . t ^l cos ei — sm - a eo (T i = ii cos 2/0^1 sin-, (21) (7 (T and in the same manner: ei cos - + Zoii sin -, (22) (7 (7 where ei is the initial value of transient voltage, ii the initial value of transient current. B. Power dissipation. 27. In Fig. 32 are plotted as A the periodic component of the oscillating current i, and as B the voltage e, as C the stored mag- Li'^ . . Ce^ netic energy — , and as D the stored dielectric energy -^ • As seen, the stored magnetic energy pulsates, with double frequency, 2/, between zero and a maximum, equal to the total stored energy. The average value of the stored magnetic energy thus is one-half of the total stored energy, and the dissipation of magnetic energy thus occurs at half the rate at which it would occur if all the energy were magnetic energy; that is, the transient resulting from the power dissipation of the magnetic energy lasts twice as long as it would if all the stored energy were magnetic, or in other words, if the transient were a single (magnetic) energy DOUBLE-ENERGY TRANSIENTS. 67 transient. In the latter case, the duration of the transient would be To--, and with only half the energy magnetic, the duration thus is twice as longj or T,=2T, = —, (23) r and hereby the factor h = e T^ multiplies with the values of current and voltage (21) and (22). Fig. 32. — Relat The same appl on of Magnetic and Dielectric Energy of Transient. ies to the dielectric energy. If all the energy were dielectric, it would be dissipated by a transient of the dura- tion: rp f ^ . i 0 = —f 68 ELECTRIC DISCHARGES, WAVES AND IMPULSES. as only half the energy is dielectric, the dissipation is half as rapid, that is, the dielectric transient has the duration T2 = 2 To' = — , (24) 9 and therefore adds the factor _t_ /c = e ^'2 to the equations (21) and (22). While these equations (21) and (22) constitute the periodic part of the phenomenon, the part which represents the dissipa- tion of power is given by the factor hk = e T^e T,^^ '\t,^tJ^ (25) The duration of the double-energy transient, T, thus is given by 2 \To ^ To' (26) and this is the harmonic mean of the duration of the single-energy magnetic and the single-energy dielectric transient. It is, by substituting for To and To', T = l{i+iy'^' (27) where u is the abbreviation for the reciprocal of the duration of the double-energy transient. Usually, the dissipation exponent of the double-energy transient -K2+8 IS given as r 2L* This is correct only if g = 0, that is, the conductance, which rep- resents the power dissipation resultant from the voltage (by leak- age, dielectric induction and dielectric hysteresis, corona, etc.), is negligible. Such is the case in most power circuits and trans- mission lines, except at the highest voltages, where corona appears. It is not always the case in underground cables, high-potential DOUBLE-ENERGY TRANSIENTS. 69 transformers, etc., and is not the case in telegraph or telephone lines, etc. It is very nearly the case if the capacity is due to elec- trostatic condensers, but not if the capacity is that of electrolytic condensers, aluminum cells, etc. Combining now the power-dissipation equation (25) as factor with the equations of periodic energy transfer, (21) and (22), gives the complete equations of the double-energy transient of the circuit containing inductance and capacity: \ . t . t I = e~^^ < Zi cos 2/0^1 sm - (a a A i , ■ - i I e = e-^^