CHAPTER V. 

RESISTANCE, INDUCTANCE, AND CAPACITY IN SERIES. 
CONDENSER CHARGE AND DISCHARGE. 

29. If a continuous e.m.f . e is impressed upon a circuit contain- 
ing resistance, inductance, and capacity in series, the stationary 
condition of the circuit is zero current, i = o, and the poten- 
tial difference at the condenser equals the impressed e.m.f., 
et =• e, no permanent current exists, but only the transient 
current of charge or discharge of the condenser. 

The capacity C of a condenser is defined by the equation 

. de 


that is, the current into a condenser is proportional to the rate 
of increase of its e.m.f. and to the capacity. 
It is therefore 


and 

e-^-lidt (1) 


is the potential difference at the terminals of a condenser of 
capacity C with current i in the circuit to the condenser. 

Let then, in a circuit containing resistance, inductance, and 
capacity in series, e = impressed e.m.f., whether continuous, 
alternating, pulsating, etc.; i = current in the circuit at time t; 
r = resistance; L = inductance, and C = capacity; then the 
e.m.f. consumed by resistance r is 

n; 

the e.m.f. consumed by inductance L is 

di 


47 


48 TRANSIENT PHENOMENA 

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


hence, the impressed e.m.f. is 


and herefrom the potential difference at the condenser terminals 
is 

Ci = -L Cidt = e - ri - L -*• (3) 

(/ «/ at 

Equation (2) differentiated and rearranged gives 
„ d?i di 1 . de 


as the general differential equation of a circuit containing resist- 
ance, inductance, and capacity in series. 
30. If the impressed e.m.f. is constant, 

e = constant, 

de 

then 37 = 0, 

dt 

and equation (4) assumes the form, for continuous-current 
circuits, 


This equation is a linear relation between the dependent vari- 
able, i, and its differential quotients, and as such is integrated 
by an exponential function of the general form 

i = Ae-*. (6) 

(This exponential function also includes the trigonometric 
functions sine and cosine, which are exponential functions with 
imaginary exponent a.) 


CONDENSER CHARGE AND DISCHARGE 49 

Substituting (6) in (5) gives 


this must be an identity, irrespective of the value of t, to make 
(6) the integral of (5). That is, 

a?L-ar+- = Q. (7) 

A is still indefinite, and therefore determined by the terminal 
conditions of the problem. 
From (7) follows 


£. (8) 


hence the two roots, 

r — s 


and 


r + 


(9) 


where s = y/r2 - ^ . (10) 

Since there are two roots, at and a2, either of the two expres- 
ions (6), e~ait and e~a2t, and therefore also any combination of 
these two expressions, satisfies the differential equation (5). 

That is, the general integral equation, or solution of differential 
equation (5), is 

i = Ai^ + Ai^ . (11) 


Substituting (11) and (9) in equation (3) gives the potential 
difference at the condenser terminals as 


e — 


(12) 


50 TRANSIENT PHENOMENA 

31. Equations (11) and (12) contain two indeterminate con- 
stants, A! and A2, which are the integration constants of the 
differential equation of second order, (5), and determined by 
the terminal conditions, the current and the potential differ- 
ence at the condenser at the moment t = 0. 

Inversely, since in a circuit containing inductance and capac- 
ity two electric quantities must be given at the moment of 
start of the phenomenon, the current and the condenser poten- 
tial — representing the values of energy stored at the moment 
t = 0 as electromagnetic and as electrostatic energy, respec- 
tively — the equations must lead to two integration constants, 
that is, to a differential equation of second order. 

Let i = i0 = current and et = e0 = potential difference at 
condenser terminals at the moment t = 0; substituting in (11) 
and (12), 

t0 = A, + A2 

and e0 - 

hence, 


r — s . 
~~2~l« 


and 


r + s 


(13) 


s 

and therefore, substituting in (11) and (12), the current is 
r + s . r — s 

-^0 _rzf, 

~L > (14) 


_'+•, «o-«- -o-*« 


s s 

the condenser potential is 

r-t-s . r—s. 


~ 2L 


(15) 


CONDENSER CHARGE AND DISCHARGE 51 

For no condenser charge, or i0 = 0, e0 = 0, we have 


e 

1 s 
and 


substituting in (11) and (12), we get the charging current as 

ii^/ _ r±» * ) 

2L -•" j. (16) 

The condenser potential as 


r+sH) 

2L • 

J ) 


For a condenser discharge or i'0 = 0, e = e0, we have 


and 

hence, the discharging current is 

( _ r~s r+s 

S ( 

The condenser potential is 


N 

' 
/ 


that is, in condenser discharge and in condenser charge the 
currents are the same, but opposite in direction, and the con- 
denser potential rises in one case in the same way as it falls in 
the other. 

32. As example is shown, in Fig. 11, the charge of a con- 
denser .of C = 10 mf. capacity by an impressed e.m.f. of 


52 


TRANSIENT PHENOMENA 


e = 1000 volts through a circuit of r = 250 ohms resistance 
and L = 100 mh. inductance; hence, s = 150 ohms, and the 
charging current is 

6.667 fe-500' - £-; 


i = D.DD/ [e " - e~* ""j amperes. 
The condenser potential is 

el = 1000 {1 - 1.333 e~5<Mt + 0.333 £-2000<j volts. 


12 16 20 24 28 32 36 40 


Fig. 11. Charging a condenser through a circuit having resistance and induc 
tance. Constant potential. Logarithmic charge. 

33. The equations (14) to (19) contain the square root, 


'4L 


hence, they apply in their present form only when 

4L 


If r2 = -— - , these equations become indeterminate, or = — > 
0 0 

and if r2 < — , s is imaginary, and the equations assume a 
C 

complex imaginary form. In either case they have to be 
rearranged to assume a form suitable for application. 
Three cases have thus to be distinguished : 

(a) r2 > — -, in which the equations of the circuit can be 
o 

used in their present form. Since the functions are exponen- 
tial or logarithmic, this is called the logarithmic case. 


CONDENSER CHARGE AND DISCHARGE 53 

(6) r2 = — is called the critical case, marking the transi- 
tion between (a) and (c), but belonging to neither. 

(c) r2 < — . In this case trigonometric functions appear; it 

is called the trigonometric case, or oscillation. 
34. In the logarithmic case, 


4L<W, 

that is, with high resistance, or high capacity, or low induc- 
tance, equations (14) to (19) apply. 

*•-« t ^ r+s t 

The term e 2L is always greater than e 2L , since the 
former has a lower coefficient in the exponent, and the differ- 
ence of these terms, in the equations of condenser charge and 
discharge, is always positive. That is, the current rises from 
zero at t = 0, reaches a maximum and then falls again to 

zero at £ = oo, but it never reverses. The maximum of the 

/? 

current is less than i = - • 

s 

The exponential term in equations (17) and (19) also never 
reverses. That is, the condenser potential gradually changes, 
without ever reversing or exceeding the impressed e.m.f. in the 
charge or the starting potential in the discharge. 

, in the case r2 > — — , no abnormal voltage is pro- 


duced in the circuit, and the transient term is of short duration, 
so that a condenser charge or discharge under these conditions 
is relatively harmless. 

In charging or discharging a condenser, or in general a circuit 
containing capacity, the insertion of a resistance in series in the 

circuit of such value that r2 > — therefore eliminates the 

C 

danger from abnormal electrostatic or electromagnetic stresses. 

In general, the higher the resistance of a circuit, compared 
with inductance and capacity, the more the transient term is 
suppressed. 


54 TRANSIENT PHENOMENA 

35. In a circuit containing resistance and capacity but no 
inductance, L = 0, we have, substituting in (5), 

rf + jL; = 0, (20) 

or, transposing, 


which is integrated by _t_ 

i - ce rc, (21) 

where c = integration constant. 

Equation (21) gives for t = 0, i = c; that is, the current at 
the moment of closing the circuit must have a finite value, or 
must jump instantly from zero to c. This is not possible, but 
so also it is not possible to produce a circuit without any induc- 
tance whatever. 

Therefore equation (21) does not apply for very small values 
of time, t, but for very small t the inductance, L, of the circuit, 
however small, determines the current. 

The potential difference at the condenser terminals from (3) is 

el = e — ri 
hence t 

et = e - res r° (22) 

The integration constant c cannot be determined from equation 
(21) at t = 0, since the current i makes a jump at this moment. 

But from (22) it follows that if at the moment t = 0, el = e0, 
e0 = e - re, 

e-eQ 

hence, c = - -, 

r 

and herefrom the equations of the non-inductive condenser 

circuit, 

_t_ 

f _(6-e>"'<7 (23) 

r 

and ' 

' 


As seen, these equations do not depend upon the current iQ in 
the circuit at the moment before t = 0. 


CONDENSER CHARGE AND DISCHARGE 65 

36. These equations do not apply for very small values of t, 
but in this case the inductance, L, has to be considered, that is, 
equations (14) to (19) used. 
For L = 0 the second term in (14) becomes indefinite, as it 

«, 

contains e , and therefore has to be evaluated as follows: 
For L = 0, we have 

s = r. 


and 

r — 


0 


and, developed by the binomial theorem, dropping all but the 
first term, 


r — s = • 

2L 


and 

r-s 1 

2L rC' 
r -f s _r_ 

2L ~~L 

Substituting these values in equations (14) and (15) gives the 
current as 


r r 

and the potential difference at the condenser as 

_j_ 

€l = e - (e - e0) e * ', (26) 

that is, in the equation of the current, the term 


56 TRANSIENT PHENOMENA 

has to be added to equation (23) . This term makes the transition 
from the circuit conditions before t = 0 to those after t = 0, 
and is of extremely short duration. 

For instance, choosing the same constants as in § 32, namely : 
e = 1000 volts; r = 250 ohms; C = 10 mf., but choosing the 
inductance as low as possible, L = 5 mh., gives the equations 
of condenser charge, i.e., for iQ = 0 and e0 = 0, 


and 

e, = 1000 {1 - e-400'}. 

The second term in the equation of the current, f-50-000', has 
decreased already to 1 per cent after t = 17.3 X 10~6 seconds, 
while the first term, £-400'; has during this time decreased only 
by 0.7 per cent, that is, it has not yet appreciably decreased. 

37. In the critical case, 


• c 

and s = 0, . 

r 


A, = - A 


s 
Hence, substituting in equation (14) and rearranging, 


O 7" o 7" 


^=(e-eQ-r-i^-^(^ ^ -)• (27) 

The last term of this equation, 


s s 

'-L _,~"f o 


D s 'a' 


CONDENSER CHARGE AND DISCHARGE 57 

that is, becomes indeterminate for s = 0, and therefore is 
evaluated by differentiation, 


ds 
Substituting (28) in (27) gives the equation of current, 

^'- (29) 

The condenser potential is found, by substituting in (15), to be 


(30) 


The last term of this equation (30) is 


(31) 
For s = 0, the first term of this equation (31), by substituting 

(28), becomes ^-~ , the second term = 1, and substituting in (30), 
2 Li 

this gives the condenser potential as 

^'. (32) 


Herefrom it follows that for the condenser charge, iQ = 0 and 
e0 = 0, 


= Le 
and 

e. = e ?!-(! + 


2L 


58 TRANSIENT PHENOMENA 

for the condenser discharge, i0 = 0 and e = 0, 


and 


38. As an example are shown, in Fig. 12, the charging current 
and the potential difference at the terminals of the condenser, 


5—1000 


36 40 


Fig. 12. Charging a condenser through a circuit having resistance and induc- 
tance. Constant potential. Critical charge. 

in a circuit having the constants', e = 1000 volts; C = 10 mf.; 
L = 100 mh., and such resistance as to give the critical start, 
that is, 


V — = 200 ohms. 
V C 


In this case, 


and 


i = 10,000 t£~looot 
e, = 1000 {!-(! + 10000 £"1000'}. 
39. In the trigonometric or oscillating case, 


The term under the square root (10) is negative, that is, the 
square root, s, is imaginary, and al and a2 are complex imaginary 
quantities, so that the equations (11) and (12) appear in imagi- 
nary form. They obviously can be reduced to real terms, 


CONDENSER CHARGE AND DISCHARGE 


59 


since the phenomenon is real. Since an exponential function 
with imaginary exponents is a trigonometric function, and 
inversely, the solution of the equation thus leads to trigono- 
metric functions, that is, the phenomenon is periodic or oscil- 
lating. 
Substituting s = jq, we have 


(33) 


and 


Substituting (34) in (11) and (12), and rearranging, 


^ = 


2L 


(34) 


(35) 


(36) 


Between the exponential function and the trigonometric 
functions exist the relations 


and 


= cos v + j sin v 


y _ 


(37) 


Substituting (37) in (35), and rearranging, gives 


- (A, + A,) cos^^ + j (A, - A2) sin -^« j. 

Substituting the two new integration constants, 

B, = AV + A2 ] 

and (38) 

j^-7(At-A,),J 

gives 

(39) 


60 TRANSIENT PHENOMENA 

In the same manner, substituting (37) in (36), rearranging, 
and substituting (38), gives 


.t-.-™+**. (40 

{ Zi Zi Ll Zi A Ll } 

Bl and B2 are now the two integration constants, determined 
by the terminal conditions. That is, for t = 0, let i = i0 = cur- 
rent and e^ = e0 = potential difference at condenser terminals, 
and substituting these values in (39) and (40) gives 


and 

rB, -I- qB, 


hence, 
and 


(41) 


Substituting (41) in (39) and (40) gives the general equations 
of condenser oscillation: 
the current is 


and the potential difference at condenser terminals is 

r (e — ^O i 

0' o 0 

' (43) 

Herefrom follow the equations of condenser charge and dis- 
charge, as special case : 
For condenser charge, iQ = 0; e0 = 0, we have 

A * (44) 


CONDENSER CHARGE AND DISCHARGE 


and 


and for condenser discharge, i0 = 0, e = 0, we have 


and 


<V 


T . 

-* 


61 

(45) 

(46) 

(47) 


40. As an example is shown the oscillation of condenser 
charge in a circuit having the constants, e = 1000 volts; L = 
100 mh., and C = 10 mf. 


Fig. 13. Charging a condenser through a circuit having resistance and induc- 
tance. Constant potential. Oscillating charge. 

(a) In Fig. 13, r = 100 ohms, hence, q = 173 and the current is 

i = 11.55 e-500t sin 866 Z; 
the condenser potential is 

e, = 1000 { 1 - e- 50° ' (cos 866 t + 0.577 sin 866 0 } . 

(b) In Fig. 14, r = 40 ohms, hence, q = 196 and the current 
is 

i = 10.2 £-200' sin 980*; 

the condenser potential is 

e, - 1000 { 1 - e" 20° ' (cos 980 t + 0.21 sin 980 0 } . 


62 


TRANSIENT PHENOMENA 


41. Since the equations of current and potential difference 
(42) to (47) contain trigonometric functions, the phenomena 
are periodic or waves, similar to alternating currents. They 

r 

differ from the latter by containing an exponential factor e 2 L , 
which steadily decreases with increase of t. That is, the sue- 


16UUI — 

f 

^ 

f 

N, 

c = 

« 1QOO volts 

L = 

= 1 

X)mh 

1 

X 

\ 

T = 

40 oh, 

as 

C = 

= 

OE 

af. 

\ 

^ 

~-v, 

. 

1 

\ 

\ 

k 

* 

— 

***** 

1 

\ 

^ 

\ 

^r 

, 

IE 

— 

De 

gre 

es 

XX 

3 or 

8 

0 

1& 

°\ 

2A 

fy 

3i 

20 

7400 

480' 

\5( 

^) 

& 

(0 

72 

^ 

-« 

1 — ~, 

[_ 

\ 

1 

s 

^ 

S 

I/ 

J 

\ 

1 

\ 

/ 

\ 

/ 

Fig. 14. Charging a condenser through a circuit having resistance and induc- 
tance. Constant potential. Oscillating charge. 

cessive half waves of current and of condenser potential pro- 
gressively decrease in amplitude. Such alternating waves of 
progressively decreasing amplitude are called oscillating waves. 

Since equations (42) to (47) are periodic, the time t can be 
represented by an angle 6, so that one complete period is denoted 
by 2 n or one complete revolution, 

f) * 4 O -ft (A Q\ 

0 = —t = 2rft. (48) 


hence, the frequency of oscillation is 


or, substituting 


gives the frequency of oscillation as 


(49) 


(50) 


CONDENSER CHARGE AND DISCHARGE 


68 


This frequency decreases with increasing resistance r, and 

becomes zero for ( — —\ = — , that is, r2 = — , or the critical 
\2 LI L/C C 

case, where the phenomenon ceases to be oscillating. 

If the resistance is small, so that the second term in equa- 
tion (50) can be neglected, the frequency of oscillation is 


Substituting 0 for t by equation (48) 


in equations (42) and (43) gives the general equationSj 


_U 


2q 
r(e- e0)- 


= e - 


and 


1 (e - e0) cos 
d = 2nft 

f-^-. 


sin 0 


JL /JlV 
LC V2L/ ' 


(52) 


,(53) 
(48) 


(50) 


42. If the resistance r can be neglected, that is, if r2 is small 
compared with — , the following equations are approximately 


exact: 

and 


r _ __ _ _ 
J ' 2 n VW 


(54) 


(55) 


64 TRANSIENT PHENOMENA 

Introducing now x = 2 TT/L = inductive reactance and 

x' = ^- = capacity reactance, and substituting (55), we 

2 TT/ C 

have 


and 

0 

hence, xf = x, 

that is, the frequency of oscillation of a circuit containing 
inductance and capacity, but negligible resistance, is that 

frequency / which makes the condensive reactance xf = — — — 

2 7T/C 

equal the inductive reactance x = 2 nfL : 


(56) 

Then (54), 

q = 2x, (57) 

and the general equations (52) and (53) are 


i = s 2ar ] i0cos^ + - - sin/9 [ ; (58) 

4 ic 


-e0) cos 0 + r g~c*~2a"Bin 0 1 • (59) 
^ x 


x- (56) 

and by (48) and (55) : 


CONDENSER CHARGE AND DISCHARGE 


65 


43. Due to the factor e ' , successive half waves of oscilla- 
tion decrease the more in amplitude, the greater the resistance r. 
The ratio of the amplitude of successive half waves, or the 


decrement of the oscillation, is A = e 2L \ where tl = duration 

of one half wave or one half cycle, = -— . 

2/ 


A 

a.o 


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

Fig. 15. Decrement of Oscillation. 


Hence, from (50), 


and 


Denoting the critical resistance as 


2 _ 


4L 


r> :: c' 


we have 


or, 


A «e 


-s 


-^-i 


(60) 
(61) 


(62) 


66 TRANSIENT PHENOMENA 

that is, the decrement of the oscillating wave, or the decay of 
the oscillation, is a function only of the ratio of the resistance 
of the circuit to its critical resistance, that is, the minimum 
resistance which makes the phenomenon non-oscillatory. 

In Fig. 15 are shown the numerical values of the decrement A, 

for different ratios of actual to critical resistance — • 

ri 

As seen, for r > 0.21 rv or a resistance of the circuit of more 
than 21 per cent of its critical resistance, the decrement A is 
below 50 per cent, or the second half wave less than half the first 
one, etc. ; that is, very little oscillation is left. 

Where resistance is inserted into a circuit to eliminate the 
danger from oscillations, one-fifth of the critical resistance, or 

r = 0.4 y—, seems sufficient to practically dampen out the 
G 

oscillation.