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 eo 
storing energy in two forms, and ^ 

oscillations appear, as in the dis- ' ~ 

charge of the Leyden jar. Fig 10._Magnetie Single.energy 

Let, as represented in Fig. 10, Transient, 

a continuous voltage e0 be im- 
pressed upon a wire coil of resistance r and inductance L (but 


negligible capacity). 


A current iQ = — flows through the coil and 

a magnetic field $0 10~8 = - - interlinks with the coil. Assuming 

now that the voltage e0 is suddenly withdrawn, without changing 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 3>o, 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 


Fig. 11. — Characteristics of Magnetic Single-energy Transient. 


time where the short circuit is applied, as indicated by the dotted 
line t0. From t0 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. IIB. 
The impressed voltage is shown in Fig. 1 1C as a dotted line; it is 
CQ up to t0, and drops to 0 at t0. However, since after t0 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 

d<& 

-j- . In the first moment of short circuit, the magnetic flux $ still 

has full value 3>0, and the current i thus also full value iQ. Hence, 
at the first moment of short circuit, the induced e.m.f. e must be 
equal to eQ, that is, the magnetic flux $ must begin to decrease at 
such rate as to induce full voltage e0, as shown in Fig. 11C. 

The three curves <£, i, and e are proportional to each other, and 
as e is proportional to the rate of change of 3>, <£ 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: 


(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 n times the 
decrease of the magnetic flux, and the total induced voltage, 
that is, the area of the induced-voltage curve, Fig. 11C, thus 
equals n times the total decrease of magnetic flux, that is, equals 
the initial current i0 times the inductance L: 


Zet = w£010-8 = LiQ. (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 w£0 = Li0. 

If then the current i would continue to decrease at its initial 
rate, as shown dotted in Fig. 115 (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 e0 up to the moment 
of time t = tQ + T, where the current has fallen to zero, as 


22 ELECTRIC DISCHARGES, WAVES AND IMPULSES. 

shown dotted in Fig. 11C. The area of this new voltage curve 
would be e0T, 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 


= ri.r, 

and, combining (2) and (3), i0 cancels, and we get the value of T: 

: • .:' :V : T-\- >•'••; • (4) 

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 T = - 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. 


by the numerical values of the exponential function, y = e 
given in Table III. 

TABLE III. 

Exponential Transient of Initial Value 1 and Duration 1. 
y = e~x. 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 ^, 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, 


r 

r 


and the equations of the single-energy magnetic transient, (1), 
thus may be written in the forms: 


I = lot~ c (t ~ 'o) = IQ€ 

e = e0e~c('~'o) = e0e 


- 7 # - to 

= ^Qe L , 

- y (t - *0) 

^r /?„£ L 


(5) 


Usually, the starting moment of the transient is chosen as the 
zero of time, Zo = 0, and equations (5) then assume the simpler 
form: 


24 ELECTRIC DISCHARGES, WAVES AND IMPULSES, 


(6) 


The same equations may be derived directly by the integration 
of the differential equation: 


where L -=- is the inductance voltage, ri the resistance voltage. 

and their sum equals zero, as the coil is short-circuited. 
Equation (7) transposed gives 


hence 


logi =- 


i = Ce~~L\ 


and, as for t = 0: i = to, it is: C = i0; hence 


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 IQ = field-exciting current, n = number of field turns 
per pole, r = resistance, and L = inductance of the field-exciting 

circuit, it is 

iQ2r = 500, 

hence 

500 


SINGLE-ENERGY TRANSIENTS. 25 

The magnetic flux is $0 = 8 X 106, and with 4 n total turns 
the total number of magnetic interlinkages thus is 

4 n$0 = 32 n X 106, 
hence the inductance 

L0~8 .32 n 


T 
L = 


^o 


, 
henrys. 


The field excitation is 

ra'o = 6000 ampere turns, 

6000 


hence 
hence 
and 


n = 


, .32 X 6000 , 

L = - —r henrys, 


*<r 
L 1920 


0 OA 
- 3'84 sec' 


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 


I 


Fig. 12. — Magnetic Single-energy Transient. 

coil is not short-circuited ' upon itself, but its circuit closed by a 
resistance r1 '. Such would, for instance, be the case in Fig. 12, 
when opening the switch S. 


26 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 i0 at the moment to 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. 


Fig. 13. — Characteristics of Magnetic Single-energy Transient. 

The impressed voltage eQ is withdrawn at the moment t0, 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, to, the current is still iQ, and the induced voltage 
thus must be 

eo = io (r + r'), 

while the impressed voltage, before to, was 

eQ = ior; 

hence the induced voltage eo' is greater than the impressed volt- 
age eo, 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 


e0 


SINGLE-ENERGY TRANSIENTS. 27 

The duration of the transient now is 

T L 

T = 7+-r" 

that is, shorter in the same proportion as the resistance, and 
thereby the induced voltage is higher. 

If rf = oo , that is, no resistance is in shunt to the coil, but the 
circuit is simply opened, if the opening were instantaneous, it 
would be : e0f = 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. 

16. 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. e\ produces a current i\ in a circuit, and an e.m.f. ez 
produces in the same circuit a current i2, then the e.m.f. e\ + ez 
produces the current i\ -\- 1%, as is obvious. 

If now the voltage e\ + ez, and thus also the current ii + iZj con- 
sists of a permanent term, e\ and ii, and a transient term, e2 and iz, 
the transient terms ez, iz follow the same curves, when combined 
with the permanent terms e\, i\, 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. 


28 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. 


B 


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 eQ be impressed upon the coil. At 
the moment of time, tQ, 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 i0, 3>0, e0 are reached. We may 
then, as discussed above, separate the transient from the perma- 
nent term, and consider that at the time U the coil has a permanent 
current i0, permanent flux <i>0, permanent voltage e0, and in addi- 


SINGLE-ENERGY TRANSIENTS. 29 

tion thereto a transient current —i0j a transient flux — <£0, and a 
transient voltage — eQ. 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 i0, <J>0, e0. This is shown 
in Fig. 14. 

A shows the permanent flux <£0, and the transient flux — <J>0, 
which are assumed, up to the time tQ, 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 3>, which is the tran- 
sient from zero flux to the permanent flux 3>0. 

In the same manner B shows the construction of the actual 
current change i by the addition of the permanent current iQ and 
the transient current i', which starts from —iQ at to. 

C then shows the voltage relation: eQ the permanent voltage, e' 
the transient voltage which starts from — e0 at t0, and e the re- 
sultant or effective voltage in the coil, derived by adding eQ and e'.