LECTURE III. SINGLE-ENERGY TRANSIENTS IN CONTINUOUS- CURRENT CIRCUITS. 13. The simplest electrical transients are those in circuits in which energy can be stored in one form only, as in this case the change of stored energy can consist only of an increase or decrease ; but no surge or oscillation between several forms of energy can exist. Such circuits are most of the low- and medium-voltage circuits, — 220 volts, 600 volts, and 2200 volts. In them the capac- ity is small, due to the limited extent of the circuit, resulting from the low voltage, and at the low voltage the dielectric energy thus is negligible, that is, the circuit stores appreciable energy only by the magnetic field. A circuit of considerable capacity, but negligible inductance, if of high resistance, would also give one form of energy storage only, in the dielectric field. The usual high-voltage capacity circuit, as that of an electrostatic machine, while of very small inductance, also is of very small resistance, and the momentary discharge currents may be very consider- able, so that in spite of the very small inductance, considerable magnetic-energy storage may oc- cur; that is, the system is one storing energy in two forms, and oscillations appear, as in the dis- charge of the Leyden jar. Let, as represented in Fig. 10, a continuous voltage eo be im- pressed upon a wire coil of resistance r and inductance L (but A current Iq = — flows through the coil and % t 1 % io A C c c ..1. 1 Fig. 10. — Magnetic Single-energy Transient. negligible capacity), a magnetic field $0 10' Lio interlinks with the coil. Assuming now that the voltage eo is suddenly withdrawn, without changing 19 20 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the constants of the coil circuit, as for instance by short- circuiting the terminals of the coil, as indicated at A. With no voltage impressed upon the coil, and thus no power supplied to it, current i and magnetic flux $ of the coil must finally be zero. However, since the magnetic flux represents stored energy, it cannot instantly vanish, but the magnetic flux must gradually decrease from its initial value $0, by the dissipation of its stored energy in the resistance of the coil circuit as i-r. Plotting, there- fore, the magnetic flux of the coil as function of the time, in Fig. 11 A, the flux is constant and denoted by $0 up to the moment of *o I 1 ^^"^^-5 A K-L__ B io 1 1 C ^0 />:^^^ 0 ^ ■ ' Fig. 11. — Characteristics of Magnetic Single-energy Transient. time where the short circuit is applied, as indicated by the dotted line ^0. From ^0 on the magnetic flux decreases, as shown by curve . Since the magnetic flux is proportional to the current, the latter must follow a curve proportional to $, as shown in Fig. 115. The impressed voltage is shown in Fig. IIC as a dotted line; ii is Co up to to, and drops to 0 at ^o- However, since after ^0 a current I flows, an e.m.f. must exist in the circuit, proportional to the current. e = ri. SINGLE-ENERGY TRANSIENTS. 21 This is the e.m.f. induced by the decrease of magnetic flux $, and is therefore proportional to the rate of decrease of $, that is, to -J-. In the first moment of short circuit, the magnetic flux still has full value $o, and the current i thus also full value U. Hence, at the first moment of short circuit, the induced e.m.f. e must be equal to eo, that is, the magnetic flux $ must begin to decrease at such rate as to induce full voltage eo, as shown in Fig. IIC The three curves $, ^, and e are proportional to each other, and as e is proportional to the rate of change of $, $ must be propor- tional to its own rate of change, and thus also i and e. That is, the transients of magnetic flux, current, and voltage follow the law of proportionality, hence are simple exponential functions, as seen in Lecture I: (J) = $Qe-c(i-io) I = toe eo€-^('-^«). (1) $, I, and e decrease most rapidly at first, and then slower and slower, but can theoretically never become zero, though prac- tically they become negligible in a finite time. The voltage e is induced by the rate of change of the magnetism, and equals the decrease of the number of lines of magnetic force, divided by the time during which this decrease occurs, multiplied by the number of turns n of the coil. The induced voltage e times the time during which it is induced thus equals 7i times the decrease of the magnetic flux, and the total induced voltage, that is, the area of the induced-voltage curve. Fig. IIC, thus equals n times the total decrease of magnetic flux, that is, equals the initial current io times the inductance L: Zet = n$olO-« = Lio. (2) Whatever, therefore, may be the rate of decrease, or the shape of the curves of $, i, and e, the total area of the voltage curve must be the same, and equal to ?i$o = Liq. If then the current i would continue to decrease at its initial rate, as shown dotted in Fig. IIB (as could be caused, for instance, by a gradual increase of the resistance of the coil circuit), the induced voltage would retain its initial value eo up to the moment of time t = to -\- T, where the current has fallen to zero, as 22 ELECTRIC DISCHARGES, WAVES AND IMPULSES. shown dotted in Fig. llC The area of this new voltage curve would be CqT, and since it is the same as that of the curve e, as seen above, it follows that the area of the voltage curve e is (3) Se^ = eoT, I = rioT, I and, combining (2) and (3), I'o cancels, and we get the value of T: T = -. (4) r That is, the initial decrease of current, and therefore of mag- netic flux and of induced voltage, is such that if the decrease continued at the same rate, the current, flux, and voltage would become zero after the time T = —- r The total induced voltage, that is, voltage times time, and therefore also the total current and magnetic flux during the transient, are such that, when maintained at their initial value, they would last for the time T = —• Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient we may determine the time in which the transient decreases to half, or to one-tenth, etc., of its initial value. It is preferable, however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is, the duration of a transient is considered as the time T = -' r This time T has frequently been called the " time constant " of the circuit. The higher the inductance L, the longer the transient lasts, obviously, since the stored energy which the transient dissipates is proportional to L. The higher the resistance r, the shorter is the duration of the transient, since in the higher resistance the stored energy is more rapidly dissipated. Using the time constant 7" = - as unit of length for the abscissa, and the initial value as unit of the ordinates, all exponential transients have the same shape, and can thereby be constructed SINGLE-ENERGY TRANSIENTS. 23 by the numerical values of the exponential function, y = (T', given in Table III. TABLE III. Exponential Transient of Initial Value 1 and Duration 1. y = e-^. e = 2.71828. X y X y 0 1.000 1.0 0.368 0.05 0.951 1.2 0.301 0.1 0.905 1.4 0.247 0.15 0.860 1.6 0.202 0.2 0.819 1.8 0.165 0.25 0.779 2.0 0.135 0.3 0.741 2.5 0.082 0.35 0.705 3.0 0.050 0.4 0.670 3.5 0.030 0.45 0.638 4.0 0.018 0.5 0.607 4.5 0.011 0.6 0.549 5.0 0.007 0.7 0.497 6.0 0.002 0.8 0.449 7.0 0.001 0.9 0.407 8.0 0.000 1.0 0.368 As seen in Lecture I, the coefficient of the exponent of the single-energy transient, c, is equal to yp, where T is the projection of the tangent at the starting moment of the transient, as shown in Fig. 11, and by equation (4) was found equal to -. That is, 1 and the equations of the single-energy magnetic transient, (1), thus may be written in the forms: cE> = $oe~'^^'~'"^ = *o€ I'oe- " ^' - ^"^ = i'oe e = eoe""^^'"'"^ e^e t-tn t-tn T t-tp T = $ = ioe eoe -I,. -£<'- -k) 5 -£"- -k) (5) Usually, the starting moment of the transient is chosen as the zero of time, ^o = 0, and equations (5) then assume the simpler form : 24 ELECTRIC DISCHARGES, WAVES AND IMPULSES. ^ = $oe - ct = $oe T = $oe L I = ioe- - ct = loe t T t = loe rt 'L rt e = e^e-''^ = eoe ^ = e^e ^. (6) The same equations may be derived directly by the integration of the differential equation: Lf + n = 0, (7) di where L -7- is the inductance voltage, n the resistance voltage, and their sum equals zero, as the coil is short-circuited. Equation (7) transposed gives di r — = — J dt, I L hence T logi = — J t -\- logC, Ce-''\ and, as for t = 0: i = io, it is: C = to; hence t = iQe -£' 14. Usually single-energy transients last an appreciable time, and thereby become of engineering importance, only in highly inductive circuits, as motor fields, magnets, etc. To get an idea on the duration of such magnetic transients, consider a motor field: A 4-polar motor has 8 ml. (megalines) of magnetic flux per pole, produced by 6000 ampere turns m.m.f. per pole, and dissi- pates normally 500 watts in the field excitation. That is, if io = field-exciting current, n = number of field turns per pole, r = resistance, and L = inductance of the field-exciting circuit, it is ZoV = 500, hence ^500 ^ io' ' SINGLE-ENERGY TRANSIENTS. 25 The magnetic flux is $o = 8 X 10^, and with 4 7z total turns the total number of magnetic interlinkages thus is 4 no = 32 n X 10^ hence the inductance r 4n$olO-5 .32 n, L = : = — : — henrys. ^o to The field excitation is m'o = 6000 ampere turns, hence hence and n = 6000 ^"0 r .32 X 6000 ^ L = r^ henrys, ^0 7 = ^ = ^5 = 3.84 sec. r 500 That is, the stored magnetic energy could maintain full field excitation for nearly 4 seconds. It is interesting to note that the duration of the field discharge does not depend on the voltage, current, or size of the machine, but merely on, first, the magnetic flux and m.m.f., — which determine the stored magnetic energy, — and, second, on the excitation power, which determines the rate of energy dissipation. 15. Assume now that in the moment where the transient be- gins the resistance of the coil in Fig. 10 is increased, that is, the Fig. 12. — Magnetic Single-energy Transient. coil is not short-circuited upon itself, but its circuit closed by a resistance r'. Such would, for instance, be the case in Fig. 12, when opening the switch S. ^6 ELECTRIC DISCHARGES, WAVES AND IMPULSES, The transients of magnetic flux, current, and voltage are shown as A, B, and C in Fig. 13. The magnetic flux and therewith the current decrease from the initial values $o and ^o at the moment ^o of opening the switch >S, on curves which must be steeper than those in Fig. 11, since the current passes through a greater resistance, r + r', and thereby dissipates the stored magnetic energy at a greater rate. *o A ^\$ B io \ / C ^0 t. Fig. 13. — Characteristics of Magnetic Single-energy Transient. The impressed voltage eo is withdrawn at the moment ^o, and a voltage thus induced from this moment onward, of such value as to produce the current i through the resistance r -\- r\ In the first moment, U, the current is still ^o, and the induced voltage thus must be eo' = io (r + /), while the impressed voltage, before to, was Co = lor; hence the induced voltage eo' is greater than the impressed volt- age Co, in the same ratio as the resistance of the discharge circuit r + r' is greater than the resistance of the coil r through which the impressed voltage sends the current eo^ ^ r + r\ eo r SINGLE-ENERGY TRANSIENTS. 27 The duration of the transient now is r -\- r that is, shorter in the same proportion as the resistance, and thereby the induced voltage is higher. If r' = 00 , that is, no resistance is in shunt to the coil, but the circuit is simply opened, if the opening were instantaneous, it would be : e^ =co ; that is, an infinite voltage would be induced. That is, the insulation of the coil would be punctured and the circuit closed in this manner. The more rapid, thus, the opening of an inductive circuit, the higher is the induced voltage, and the greater the danger of break- down. Hence it is not safe to have too rapid circuit-opening devices on inductive circuits. To some extent the circuit protects itself by an arc following the blades of the circuit-opening switch, and thereby retarding the cir- cuit opening. The more rapid the mechanical opening of the switch, the higher the induced voltage, and further, therefore, the arc follows the switch blades and maintains the circuit. 1 6. Similar transients as discussed above occur when closing a circuit upon an impressed voltage, or changing the voltage, or the current, or the resistance or inductance of the circuit. A discus- sion of the infinite variety of possible combinations obviously would be impossible. However, they can all be reduced to the same simple case discussed above, by considering that several currents, voltages, magnetic fluxes, etc., in the same circuit add algebraically, without interfering with each other (assuming, as done here, that magnetic saturation is not approached). If an e.m.f. ei produces a current ^l in a circuit, and an e.m.f. 62 produces in the same circuit a current 2*2, then the e.m.f. ei + 62 produces the current ii + ^2, as is obvious. If now the voltage ei + ^2, and thus also the current ii + ii, con- sists of a permanent term, ei and ^l, and a transient term, 62 and ^2, the transient terms 62, ^"2 follow the same curves, when combined with the permanent terms ei, ii, as they would when alone in the circuit (the case above discussed). Thus, the preceding discus- sion applies to all magnetic transients, by separating the transient from the permanent term, investigating it separately, and then adding it to the permanent term. 5i« ELECTRIC DISCHARGES, WAVES AND IMPULSES. The same reasoning also applies to the transient resulting from several forms of energy storage (provided that the law of propor- tionality of i, e, $, etc., applies), and makes it possible, in inves- tigating the phenomena during the transition period of energy readjustment, to separate the permanent and the transient term, and discuss them separately. ._ j. -^^ A " i ^^^—^^^ [ B ''\^-^-^ \ ^,'' C '° i ^^^'"^ — j ! Fig. 14. — Single-energy Starting Transient of Magnetic Circuit. For instance, in the coil shown in Fig. 10, let the short circuit A be opened, that is, the voltage eo be impressed upon the coil. At the moment of time, U^ when this is done, current i, magnetic flux $, and voltage e on the coil are zero. In final condition, after the transient has passed, the values io, o, eo are reached. We may then, as discussed above, separate the transient from the perma- nent term, and consider that at the time ^o the coil has a permanent current io, permanent flux fpo, permanent voltage Co, and in addi- SINGLE-ENERGY TRANSIENTS. 29 tion thereto a transient current —io, a transient flux — $o, and a transient voltage —eo. These transients are the same as in Fig. 11 (only with reversed direction). Thus the same curves result, and to them are added the permanent values 2*0, $0, eo. This is shown in Fig. 14. A shows the permanent flux ^0, and the transient flux — $0, which are assumed, up to the time ^o, to give the resultant zero flux. The transient flux dies out by the curve $', in accordance with Fig. 11. $' added to $0 gives the curve ^, which is the tran- sient from zero flux to the permanent flux $o- In the same manner B shows the construction of the actual current change i by the addition of the permanent current ^o and the transient current i\ which starts from —io at ^o- C then shows the voltage relation: eo the permanent voltage, e' the transient voltage which starts from —eo at ^0, and e the re- sultant or effective voltage in the coil, derived by adding eo and e'.