CHAPTER VII. 

RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES IN 
ALTERNATING-CURRENT CIRCUIT. 

65. Let, at time t = 0 or 0 = 0, the e.m.f., 

e = E cos (0 - 00), (1) 

be impressed upon a circuit containing in series the resistance, r, 
the inductance, L, and the capacity, C. 

The inductive reactance is x = 2 TT/L 1 
and the condensive reactance is xc = > 

2 7T/C J 

where/ = frequency and 6 = 2 nft. (3) 

Then the e.m.f. consumed by resistance is ri\ 
the e.m.f. consumed by inductance, is 

di di 

Ldt = xJe' 

and the e.m.f. consumed by capacity is 

, (4) 


where i = instantaneous value of the current. 

di 


di C 

Hence, e = ri + x -- + xc I i dO, (5) 

da J 

di f* 

E cos (6 - 00) = ri + x — + xc li dO, (6) 

da J 


and hence, the difference. of potential at the condenser terminals 
is 

el = xc I idd = E cos (6 - 00) - ri - x — - (7) 

u cw 

88 


RESISTANCE, INDUCTANCE, AND CAPACITY 89 

Equation (6) differentiated gives 

E sin (0 - 0Q) + x-jj^ + r -^ + xei = 0. (8) 

The integral of this equation (8) is of the general form 

i = Ara9 + B cos (6 - o-). (9) 

Substituting (9) in (8), and rearranging, gives 
A£~ae\a2x— ar + xr\ + sin# \E cos On—rB cosv—B (x—xc) sin o-j 

( tj'( y \c,/j 

- cos 0 {£" sin 6Q — rB sin a- + J5 (x — xc) cos <r} = 0, 

and, since this must be an identity, 

a?x — ar + xc = 0, 
E cos #0 — rB cos a- — B (x — xc) sin o- = 0, , (10) 

# sin 00 -- rB sin o- + B (x — xc) cos a- = 0. 


Substituting 


— 4 


tan 7 = 


in equations (10) gives 


x - xc 


r ± s 


and 


= 00 + ? 

= indefinite, 


and the equation of current, (9), thus is 
i = -cos (6 - 60 -7) + A/~ 


(ID 


(12) 


(13) 


90 TRANSIENT PHENOMENA 

and, substituting (12) in (7), and rearranging, the potential 
difference at the condenser terminals is 


. (14) 


The two integration constants Al and A2 are given by the 
terminal conditions of the problem. 
Let, at the moment of start, 


0-0, 

i = iQ = instantaneous value of current and 

e1 = e0 = instantaneous value of condenser potential 
difference. 

Substituting in (13) and (14), 

ET 


(15) 


(#o + 7) + A, + A, 
zn 


and 


• Q 

Therefore 


Aj + Aj^^- -cos (60 + 7) 

and 

n0 + 2e0 E , 
Aj-A2=- - + -- rcos(<90 + 7)-2^csm(i9o+7) , 

S SZ0 


(16) 


or, 

r— s . 


' e 


and 


S SZ 

r + s . 

2 %^"eo -fi 
- -cos 


(17) 


RESISTANCE, INDUCTANCE, AND CAPACITY 


91 


Substituting (17) in (13) and (14) gives the integral equations 
of the problem. 
The current is 


Tfl 


Tjl f _ T~S Q p 


p 

^ 

\_ J 


and the potential difference at the condenser terminals is 


E 


where 


and 


r 

r-±£ 


Cos 


r+s 


(x - 


x — xc 
tan 7 . = - — > 


= Vr2 - 4 x xr 


(19) 


(ID 


The expressions of i and et consist of three terms each : 

(1) The permanent term, which is the only one remaining 
after some time; 

(2) A transient term depending upon the constants of the 
circuit, r, s, xci z0, x, the impressed e.m.f., E, and its phase 00 at 
the moment of starting, but independent of the conditions 
existing in the circuit before the start; and 


92 TRANSIENT PHENOMENA 

(3) A term depending, besides upon the constants of the 
circuit, upon the instantaneous values of current and potential 
difference, iQ and e0, at the moment of starting the circuit, and 
thereby upon the electrical conditions of the circuit before 
impressing the e.m.f., e. This term disappears if the circuit is 
dead before the start. 


Equations (18) and (19) contain the term s = Vr2 — 4 x xc 
= V/7*2 — 4 — ; hence apply only when r2 > 4 x xc, but become 

indeterminate if. r2=4xxc} and imaginary if r2< 4 x xc; in the 
latter cases they have to be rearranged so as to appear in real 
form, in manner similar to that in Chapter V. 

56. In the critical case, r2 = 4 xxc and s = 0, equation (18), 
rearranged, assumes the form 

TJ1 Tfl H £ 

i = - cos (0 - 00-7) + -£ 

-cos(00+7)-zcsin>(00 + fy) - -— cos(00+7)[ 

_2 s ) 


However, developing in a series, and canceling all but the 
first term as infinitely small, we have 


- 
e 2x -e 


- > 

X 


hence the current is 

EJ ^ _JL 

2x 


( I-f ~\ ft ) 

\ \- cos (00+ 7) - xc sin (00 + 7) - - cos (00 + 7) ( 
(\_2 Jx 


(20) 


RESISTANCE, INDUCTANCE, AND CAPACITY 


93 


and in the same manner the potential difference at condenser 
terminals is 


Ex, 


E -™ 


[T^ ~l fi 

- cos (00 + 7) - xcr sin (00 + 7) I - - 2 xc sin (0 + 7) 


(21) 


Here again three terms exist, namely: a permanent term, a 
transient term depending only on E and 00, and a transient 
term depending on iQ and e0. 

57. In the trigonometric or oscillatory case, r2 < 4 a; xc, s be- 
comes imaginary, and equations (18) and (19) therefore contain 
complex imaginary exponents, which have to be eliminated, 
since the complex imaginary form of the equation obviously 
is only apparent, the phenomenon being real. 

Substituting 

q = V4 x xc - r2 = js (22) 

in equations (13) and (14), and also substituting the trigono- 
metric expressions 


and 


and separating the imaginary and the real terms, gives 


(23) 


E 


(A, 4- A2) cos -^ 


- —9 

: 


j (A, - A,) sin 6 


94 TRANSIENT PHENOMENA 

and 


then substituting herein the equations (16) and (22) the imagi 
nary disappears, and we have the current, 


-0 
2x 


JL a [2xc • 

!e°^K°sin2lei' (24) 

and the potential difference at the condenser terminals, 


Here the three component terms are seen also. 

58. As examples are shown in Figs. 20 and 21, the starting 
of the current i, its permanent term i1 ', and the two transient 
terms i1 and iv and their difference, for the constants E = 1000 
volts = maximum value of impressed e.m.f.; r = 200 ohms 
= resistance ; x = 75 ohms = inductive reactance, and xc = 75 
ohms = condensive reactance. We have 

4 x xc = 22,500 

and r2 = 40,000; 

therefore 

r2 > 4 x xc, 


RESISTANCE, INDUCTANCE, AND CAPACITY 


95 


that is, the start is logarithmic, and z0 = 200, s = 132, and 
7 = 0. 


20 


60 80 100 120 140 160 180 200 
Degrees 


Fig. 20. Starting of an alternating-current circuit, having capacity, inductance 
and resistance in series. Logarithmic start. 

In Fig. 20 the circuit is closed at the moment 00 = 0, that 
is, at the maximum value of the impressed e.m.f., giving from 
the equations (18) and (19), since i0 = 0, e0 = 0, 


and 


i = 5 {cos 6 - 1.26 s-2-22' + 0.26 £-°'452' } 
el = 375 {sin0 + 0.57 (e-»-»«_fi-o.462*)}p 


0 20 40 


100 120 140 160 180 200 
Degrees 


Fig. 21. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Logarithmic start. 

In Fig. 21 the circuit is closed at the moment 00 = 90°, that 
is, at the zero value of the impressed e.m.f., giving the equa- 
tions 

i = 5 {sinfl + 0.57 Or2'22' - fi-o-«")} 
and 

e, = - 375 {cosfl + 0.26 *-'•»•- 1.26 e~°-™°)}. 


96 


TRANSIENT PHENOMENA 


There exists no value of #0 which does not give rise to a 
transient term. 


-2 


-4 


E =1000 tolts 
200-phiii 
133.3 otii 


20 


80 100 120 140 


180 200 


Fig. 22. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Critical start. 

In Fig. 22 the start of a circuit is shown, with the inductive 
reactance increased so as to give the critical condition, 

r2 = 4zzc, 

but otherwise the constants are the same as in Figs. 20 and 21, 
that is, E = 1000 volts; r = 200 ohms; x = 133.3 ohms, and 
xc = 75 ohms; 

therefore z0 = 208.3, 

fro o 

tan 7 = ^ - = 0.2915, or 7 = 16°, 
ZOO 

assuming that the circuit is started at the moment 00 = 0, or 
at the maximum value of impressed e.m.f. 
Then (20) and (21) give 

i = 4.78 cos (6 - 16°) + r°'75' (2.7 0 - 4.6) 
and 

e,= 358 sin (6 - 16°) - £-°-75'(410# - 99). 

Here also no value of 00 exists at which the transient term 
disappears. 

69. The most important is the oscillating case, r2 < 4 x xc, 
since it is the most common in electrical circuits, as underground 
cable systems and overhead high potential circuits, and also is 
practically the only one in which excessive currents or excessive 
voltages, and thereby dangerous phenomena, may occur. 


RESISTANCE, INDUCTANCE, AND CAPACITY 


97 


If the condensive reactance xc is high compared with the 
resistance r and the inductive reactance x, the equations of 
start for the circuit from dead condition, that is, t'0 = 0 and 
e0 = 0, are found by substitution into the general equations 
(24) and (25), which give the current as 


E( . , 

{ sm (0 — 

Xc f 


2xTsin00cosV/-c0 
L r x 


(26) 


and the potential difference at the condenser terminals as 


cos# cos V -H 
where 


cos 


xc 


sin , (27) 


xc, and 7 = - 90°. 


(28) 


In this case an oscillating term always exists whatever the 
value of 00, that is, the point of the wave, where the circuit is 
started. 

The frequency of oscillation therefore is 


/o 

or, approximately, 


2x" 


_ 
4X2 


(29) 


where/ = fundamental frequency. 

Substituting x = 2nfL and zc = — -r-, we have 


CL 


or, approximately, 

/o 


(30) 


98 TRANSIENT PHENOMENA 

60. The oscillating start, or, in general, change of circuit 
conditions, is the most important, since in circuits containing 
capacity the transient effect is almost always oscillating. 

The most common examples of capacity are distributed 
capacity in transmission lines, cables, etc., and capacity in the 
form of electrostatic condensers for neutralizing lagging currents, 
for constant potential-constant current transformation, etc. 

(a) In transmission lines or cables the charging current is a 
fraction of full-load current i0, and the e.m.f. of self-inductance 
consumed by the line reactance is a fraction of the impressed 
e.m.f. e0. Since, however, the charging current is (approximately) 

p 

= — and the e.m.f. of self-inductance = xi0, we have 

Xc 


eo 

< i0, xi 0 

Xc 


hence, multiplying, 

x 

— < 1 and x < xc. 

Xc 

The resistance r is of the same magnitude as x\ thus 

4xxc >r\ 

For instance, with 10 per cent resistance drop, 30 per cent 
reactance voltage, and 20 per cent charging current in the line, 
assuming half the resistance and reactance as in series with the 
capacity (that is, representing the distributed capacity of the 
line by one condenser shunted across its center) and denoting 

?-*• 

where e0 = impressed voltage, i0 = full-load current, we have 


x = 0.5 X 0.3 p = 0.15 p, 

r = 0.5 X 0.1 p = 0.05 p, 
and 

r -5- s -s- zc = 1 -f- 3 -r- 100, 
and 

4 x xr + r2 = 1200 -4- 1. 


RESISTANCE, INDUCTANCE, AND CAPACITY 99 

In this case, to make the start non-oscillating, we must have 

x < — — - r, or x < 0.000125 p, which is not possible; t>r r > 

which can be done only by starting the circuit through a very 
large non-inductive resistance (of such size as to cut the starting 

current down to less than — of full-load current). Even in 

this case, however, oscillations would appear by a change of 
load, etc., after the start of the circuit. 

(6) When using electrostatic condensers for producing watt- 
less leading currents, the resistance in series with the condensers 
is made as low as possible, for reasons of efficiency. 

Even with the extreme value of 10 per cent resistance, or 

r 4- xc = I -f- 10, the non-oscillating condition is x < — r, or 

0.23 per cent, which is not feasible. 
In general, if 

x consumes 12 4 9 16 per cent of the con- 
denser potential 
difference, 

r must consume > 20 28.3 40 60 80 per cent of the con- 
denser potential 
difference. 

That is, a very high non-inductive resistance is required to 
avoid oscillations. 

The frequency of oscillation is approximately /0 = y — / 

that is, is lower than the impressed frequency if xc < x (or the 
permanent current lags), and higher than the impressed fre- 
quency if xc > x (or the permanent current leads). In trans- 
mission lines and cables the latter is always the case. 

Since in a transmission line— is approximately the charging 

Xc 

X 

current, as fraction of full-load current, and ~- half the line 

e.m.f. of self-inductance, or reactance voltage, as fraction of 
impressed voltage, the following is approximately true : 


100 


TRANSIENT PHENOMENA 


The frequency of oscillation of a transmission line is the 
impressed frequency divided by the square root of the product 
of charging current and of half the reactance voltage of the line, 
given respectively as fractions of full-load current and of im- 
pressed voltage. For instance, 10 per cent charging current, 
20 per cent reactance voltage, gives an oscillation frequency 


vui x o.i 


10 f. 


Fig. 23. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Oscillating start of transmission line. 

61. In Figs. 23 and 24 is given as example the start 
of current in a circuit having the constants, E = 35,000 
cos (6 — 00); r = 5 ohms; x = 10 ohms, and xc = 1000 ohms. 

In Fig. 23 for 00= 0°, or approximately maximum oscilla- 
tion, 

i = - 35 {sin 0 - 10 e- °25 ' sin 10 6} 
and 

el = 35,000 {cos 6 - e~ °25 • [cos 10 0 + 0.025 sin 10 0]} . 

In Fig. 24 for #0 = 90°, or approximately minimum oscilla- 
tion, 

i = 35 {cos 0 - r °25 9 cos 10 6} \ 

and 

e, = 35,000 { sin 0 + 0.1 <r ° 25 ° sin 10 0 } . ) 

As seen, the frequency is 10 times the fundamental, and in 
starting the potential difference nearly doubles. 


RESISTANCE, INDUCTANCE, AND CAPACITY 


101 


As further example, Fig. 25 shows the start of a circuit of a 
frequency of oscillation of the same magnitude as the funda- 
mental, in resonance condition, x = xc, and of high resistance. 


60 
40 
20 
0 

-20 

-40 

-60 

60 

40 

20 

0 

-20 
-40 


\s 


4 


\ 


= 35000 volts 
-5-ohW- 
10 oh' ma1 


-\ 


100 ohms 

gd-p- 


Fig. 24. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Oscillating start of transmission line. 


The circuit constants are E = 1500 volts; r = 30 ohms; 
x = 20 ohms; xc = 20 ohms, and 00 = — 7; which give 
3 = 26.46; z0 = 30; 7 = 0, and 00 = 0. 


Fig. 25. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Oscillating start. High resistance. 

Substituting in equations (24) and (25) gives 
i = 50 {cos 0 - r °-75 0 [cos 0.661 Q - 1.14 sin 0.661 6]} ] 

= 1000 {sin e - 1.51 e- ° 75 • sin 0.661 6}. 


102 


TRANSIENT PHENOMENA 


As example of an oscillation of long wave, Fig. 26 represents 
the start of a circuit having the constants E = 1500 volts; 
r = 10 ohms; x = 62.5 ohms; xc = 10 ohms, and 00 = — 7; 
which give q = 49; 20 = 53.4; 7 = 79°, and 00 = - 79°. 

Substituting in equations (24) and (25) gives 


i = 28 f cos 0 - e 


and 


[cos 0.390 - 0.2 sin 0.390]} 


e, = 280 {sin 6 - 2.55 r °-°8 « sin 0.396 0 


62. While in the preceding examples, Figs. 23 to 26, con- 
stants of transmission lines have been used, as will be shown 
in the following chapters, in the case of a transmission line 


V 


9 

xt. 


62.5 ohms 


Fig. 26. Starting of an alternating-current circuit having capacity, inductance 
and resistance in series. Oscillating start of long period. 

with distributed capacity and inductance, the oscillation does 
not consist of one definite frequency but an infinite series of 
frequencies, and the preceding discussion thus approximates 
only the fundamental frequency of the system. This, however, 
is the frequency which usually predominates in a high power 
low frequency surge of the system. 

In an underground cable system the preceding discussion 
applies more closely, since in such a system capacity and induc- 
tance are more nearly localized : the capacity is in the under- 
ground cables, which are of low inductance, and the inductance 
is in the generating system, which has practically no capacity. 

In an underground cable system the tendency therefore is 


RESISTANCE, INDUCTANCE, AND CAPACITY 103 

either towards a local, very high frequency oscillation, or travel- 
ing wave, of very limited power, in a part of the cables, or a low 
frequency high power surge, frequently of destructive magnitude, 
of the joint capacity of the cables, against the inductance of the 
generating system. 

63. The physical meaning of the transient terms can best be 
understood by reviewing their origin. 

In a circuit containing resistance and inductance only, but a 
single transient term appears of exponential nature. In such a 
circuit at any moment, and thus at the moment of start, the 
current should have a certain definite value, depending on 
the constants of the circuit. In the moment of start, however, the 
current may have a different value, depending on the preceding 
condition, as for instance the value zero if the circuit has been 
open before. The current thus adjusts itself from the initial 
value to the permanent value on an exponential curve, which 
disappears if the initial value happens to coincide with the final 
value, as for instance if the circuit is closed at the moment of 
the e.m.f. wave, when the permanent current should be zero. 
The approach of current to the permanent value is retarded by 
the inductance, accelerated by the resistance of the circuit. 

In a circuit containing inductance and capacity, at any 
moment the current has a certain value and the condenser a 
certain charge « that is, potential difference. In the moment of 
start, current intensity and condenser charge have definite 
values, depending on the previous condition, as zero, if the 
circuit was open, and thus two transient terms must appear, 
depending upon the adjustment of current and of condenser 
e.m.f. to their permanent values. 

Since at the moment when the current is zero the condenser 
e.m.f. is maximum, and inversely, in a circuit containing induc- 
tance and capacity, a change of circuit conditions always results 
in the appearance of a transient term. 

If the circuit is closed at the moment when the condenser 
e.m.f. should be zero, that is, about the maximum value of cur- 
rent, the transient term of current cannot exceed in amplitude its 
final value, since its maximum or initial value equals the value 
which the current should have at this moment. If, however, 
the circuit is closed at the moment where the current should be 
zero and the condenser e.m.f. maximum, the condenser being 


104 TRANSIENT PHENOMENA 

without charge acts in the first moment like a short circuit, that 
is, the current begins at a value corresponding to the impressed 
e.m.f. divided by the line impedance. Thus if we neglect the 
resistance and if the condenser reactance equals n2 times line 
reactance, the current starts at n2 times its final rate; thus it 
would, in a half wave, give n2 times the full charge of the con- 
denser, or in other words, charge the condenser in - of the time 
of a half wave. That is, the period of the starting current is 

- and the amplitude n times that of the final current. How- 
n l 

ever, as soon as the condenser is charged, in - of a period of 

Ti 

the impressed e.m.f., the magnetic field of the charging current 
produces a return current, discharging the condenser again at 
the same rate. 

Thus the normal condition of start is an oscillation of such a 
frequency as to give the full condenser charge at a rate which 
when continued up to full frequency would give an amplitude 
equal to the impressed e.m.f. divided by the line reactance. 
The effect of the line resistance is to consume e.m.f. and thus 
dampen the oscillation, until the resistance consumes during the 
condenser charge as much energy as the magnetic field would 
store up, and then the oscillation disappears and the start becomes 
exponential. 

Analytically the double transient term appears as the result 
of the two roots of a quadratic equation, as seen above.