CHAPTER IX. 


DIVIDED CIRCUIT. 

72. A circuit consisting of two branches or multiple circuits 
1 and 2 may be supplied, over a line or circuit 3, with an impressed 
e.m.f., e0. 

Let, in such a circuit, shown diagrammatically in Fig 31, 
rv Lv Cl and r2, L2, Cz — resistance, inductance, and capacity, 
respectively, of the two branch circuits 1 and 2; r0, L0, C0 = 

Co 


Fig. 31. Divided circuit. 

resistance, inductance, and capacity of the undivided part of the 
circuit, 3. Furthermore let e = potential difference at terminals 
of branch circuits 1 and 2, it and i2 respectively = currents in 
branch circuits 1 and 2, and i3 = current in undivided part of 
circuit, 3. 

Then ia = il + i2 

and e.m.f. at the terminals of circuit 1 is 


of circuit 2 is 


e = 


di 


121 


(2) 


(3) 


122 TRANSIENT PHENOMENA 

and of circuit 3 is 

(4) 

Instead of the inductances, L, and capacities, C, it is usually 
preferable, even in direct-current circuits, to introduce the 

reactances, x = 2 nfL = inductive reactance, xc = = con- 

2 7T/G 

densive reactance, referred to a standard frequency, such as 
/ = 60 cycles per second. Instead of the time t, then, an angle 

0 = 2 nft (5) 

is introduced, and then we have 

di x di dd di 

^ * (6) 


i/iift - 2 Kfo f Id0 - Xc/i dd, 


t 
since 


Hereby resistance, inductance, and capacity are expressed in 
the same units, ohms. 

Time is expressed by an angle 6 so that 360 degrees correspond 
to sV of a second, and the time effects thus are directly com- 
parable with the phenomena on a 60-cycle circuit. 

A better conception of the size or magnitude of inductance 
and capacity is secured. Since inductance and capacity are 
mostly observed and of importance in alternating-current cir- 
cuits, a reactor having an inductive reactance of x ohms and 
i amperes conveys to the engineer a more definite meaning as 
regards size: it has a volt-ampere capacity of tfx, that is, the 
approximate size of a transformer of half this capacity, or of a 

^2x 

— -watt transformer. A reactor having an inductance of L 

henrys and i amperes, however, conveys very little meaning to 


DIVIDED CIRCUIT 123 

the engineer who is mainly familiar with the effect of inductance 
in alternating-current circuits. 

Substituting therefore (5) and (6) in equations (2), (3), (4), 
gives the e.m.f. in circuit 1 as 

dL 
e = rli1 + xl -r1 + a 

in circuit 2 as 

dL ' C 
* = r** + **-fi + *ctJi,M', (8) 

in circuit 3 as 

e = e a. r { 4. x -h. _j_ x I { ^. /Q\ 

0 03 ' 0 J/j ' CQ I 3 J v*^/ 

tZC7 «/ 

hence, the potential differences at the condenser terminals are 

/di< 
i,dd = e-r1i1- xt— S (10) 

e2= <J *i dd = e- r2i2 -*,-^> (11) 

and e3 = xco I i3dd = e0- e - r0i3 - x0 -^ • (12) 

Differentiating equations (7), (8), and (9), to eliminate the 
integral, gives as differential equations of the divided circuit: 

d?ii dil . de 
d in din de 


cPt' . di~ . deQ de 

and + r*--- 


Subtracting (14) from (13) gives 

d\\ di, .\ / d?i2 di2 


124 TRANSIENT PHENOMENA 

Multiplying (15) by 2, and adding thereto (13) and (14), gives, 
by substituting (1), i3 = it + t'2, 

(] i fi'l 

(2 x0 + x,) -^ + (2 r, + r^+ (2 xco + xji, + 

(2 *0 + x,) J| + (2 r0 + r2) ^ + (2 *Co + xc>'2 = 2 ^ . (17) 

These two differential equations (16) and (17) are integrated 
by the functions 

and - (18) 

i2 = i2' + A2e~ae, 

where if and i2 are the permanent values of current, and 
i" = A^~a9 and i2" = A2e~ae are the transient current terms. 
Substituting (18) in (16) and (17) gives 

/-/2/j / sl/\ * 

\Jj t/o (JLlci 


+ A^-a9 (a\ - ar1 + xc) - A2e~a0 (a?x2 - ar2 + xc) = 0 (19) 

and 

(Pi ' di' - 


(2 r0 + r,) + (2'^ + xc)i 


- a (2 r0 + r,) + (2 zco + xCl)} + A^-^ {a2 (2 x0 + xa) 
- a (2r0 + r2) + (2 xco + xc)} -2^- (20) 

73. For 0 = oo, the exponential terms eliminate, and there 
remain the differential equations of the permanent terms 
i/ and t/, thus 


0 (21) 

and 

cPt ' di r 

(2^o + x,) -^- + (2r0 + r,) ^- + (2 xco + ajj i/ + (2 a0 + x2) 

^ +(2r +r)-^+(2z+zH'-2^. C22) 

2 ^ ro -t r2; k ^co+ *C2; ^2 - * 


DIVIDED CIRCUIT 125 

The solution of these equations (21) and (22) is the usual 
equation of electrical engineering, giving t/ and i2 as sine waves 
if the e.m.f., e0, is a sine wave; giving ^'1/ and i2 as constant 
quantities if e0 is constant and xco and either xCi or xct or both 
vanish, and giving i{ and i2 = 0 if either xco or both xCl and 
xCt differ from zero. 

Subtracting (21) and (22) from (19) and (20) leaves as dif- 
ferential equations of the transient terms i" and i"t 

e~ae {A, (a\ - or, + xci) '- A2 (a?x2 - ar2 + xj} -= 0 (23) 
and 

e-«* {A, [a2 (2 x0 + «,)•-? (2 ro + 'i) + (2 zco + xci)] + A2 

[a2 (2 z0 + x2) - a (2r0 + r2) + (2 xco +xC2)]} = 0. (24) 

Introducing a new constant 5, these equations give, from (23), 
A1 = B (a?x2 - ar2 + xc) 


and 


(25) 


then substituting (25) in (24) gives 

(a?x2 - ar2 + xc) [a2 (2 x0 + xj - a (2 r0 + r,) + (2 ^Co + xj] 
+ (a2^, - ort + xci)[a2(2 x0 + x2) - a (2 r0 + r2) + (2 xco 
+ *c2)] = 0, (26) 

while B remains indeterminate as integration constant. 

Quartic equation (26) gives four values of a, which may be all 
real, or two real and two conjugate imaginary, or two pairs of 
conjugate imaginary roots. 

Rearranged, equation (26) gives 


a4 (XQX, + x0x2 + x,x2) - a3 {r0 (xl + x2) + r1 (x, -f x2) 
+ r2 (x0 + x^) } + a2 { (r/! + r/a + r/,) + xCo (x1 + x2) 
+ xci (x0+ x2)+ xcz (XQ+ x,)}- a {xeo(rl+ r2)+ xCi (r0-f ra) 
• + ^c, (ro + ri) } + ( VCl + XA* + ^c^c.) - 0. (27) 

Let o17 a2, a3, a4 be the four roots of this quartic equation (27) ; 


126 

then 

and 

i2 = i 


B 


TRANSIENT PHENOMENA 


B 


- a/2 + xc) 


xc 


B4 (a*x2- a4r2+ 
^ + B a2x - a 


(28) 


(29) 


where the integration constants Bv B2, B3 and B4 are deter- 
mined by the terminal conditions: the currents and condenser 
potentials at zero time, 6 = 0. 

The quartic equation (27) usually has to be solved by approxi- 
mation. 

74. Special Cases: Continuous-current divided circuit, with 
resistance and inductance but no capacity, e0 = constant. 


Fig. 32. Divided continuous-current circuit without capacity. 

In such a circuit, shown diagrammatically in Fig. 32, equations 
(7), (8), and (9) are greatly simplified by the absence of the 
integral, and we have 

e~ '& + *&> (30) 


and eQ = e -\ 

(30) and (31) combined give 


di. 
1- 


. 3 

'°d0 


- 


(31) 
(32) 

(33) 


DIVIDED CIRCUIT 127 

Substituting (1), i3 = i1 + iv in (32), multiplying it by 2 and 
adding thereto (30) and (31), gives 

2 e0= (2 r0+ rt) t1 + (2 r0+ r2) *,+ (2 z0 + xt) ^ 


Equations (33) and (34) are integrated by 
and F (35) 

Substituting (35) in (33) and (34) gives 
(VV - r2V) + e-^Ufa- ax,) - A2(r2 - ax2)l = 0 

and 

2 e0= (2 r0 + rt) i/ + (2 r0 + ra) i/ + e~ai{Al [(2 r0 + rf) 

- a (2 z0 + x,)] + A2 [(2 r0 + r2)- a (2 x0 + *,)]}. , 

These equations resolve into the equations of permanent 
state, thus 

and (2 r0 + r,) i/ + (2 r0 + ra) t/ = 2 e0. J 

Hence, t/ = 60-^ 

•(36) 
and t'2' = e0-j? 

where r2 = r0rt + r/2 + rtr2, (37) 

and the transient equations having the coefficients 

A (r nv\- A (r -- nr "I — 0 

Ai yt *f*v A2 vr2 ax2/ 
and 

At [(2 r0 + rt) - a (2 x0 + xj] + A2 [(2 r0 + r2) 
- a (2 z0 + xa)] = 0. 


128 TRANSIENT PHENOMENA 

Herefrom it follows that 


and 
and 


A2 = B (r1 - axj, 


a2 


x,)] 


- a[r0 (xl + x2) 


= 0, 


x2) 


5 - indefinite. 
Substituting the abbreviations, 


(38) 


(39) 
(40) 


and 


\ + V2 + rjr2 = r2, 
r0 (^i + sa) + ri (^o + ^2) + r2 (*0 + ^i) = ZG (ri + r2) 


^2 (r 


gives (39) 
hence two roots, 

and 
where 


= 0; 


2X2 


°2 = "r^ 


- 4 rV. 


(4D 


(42) 


(43) 


(44) 


The two roots of equation (42), ax and a2, are always real, since 
in (f 

s4 > 4 rV, 


as seen by substituting (41) therein. 
The final integral equations thus are 


and 


2x2 


*+<f 


-2 


(45) 


DIVIDED CIRCUIT 


129 


B^ and B2 are determined by the terminal conditions, as the 
currents ^ and t'2 at the start, 6 = 0. 
Let, at zero time, or 0 = 0, 


and 


then, substituting in (45), we have 


(46) 


4- 

l 1 


and 


(47) 


and herefrom calculate Bl and £2. 

75. For instance, in a continuous-current circuit, let the 
impressed e.m.f., e0 == 120 volts; the resistance of the undivided 
part of the circuit, r0 — 20 ohms; the reactance, x0 = 20 ohms; 
the resistance of one of the branches, rl = 20 ohms; the reactance, 
xl = 40 ohms, and the resistance of the other branch, r2 = 
5 ohms, the reactance, x2 = 200 ohms. 

Thus one of the branches is of low resistance and high react- 
ance, the other of high resistance and moderate reactance. 

The permanent values of the currents, (r2 = 600), are 


and 


t/ = 1 amp. 


i 


i2 = 4 amp. 

(a) Assuming now the resistance r0 suddenly decreased from 
ro = 20 ohms to r0 = 15 ohms, we have the permanent values 
of current as 

and 


t/ = 1.265 amp. 
if = 5.06 amp. 


The previous values of currents, and thus the values of currents 
at the moment of start, 0 = 0, are 

i* = 1 amp. 1 

and r 

i2° = 4 amp. J 


130 TRANSIENT PHENOMENA 

therefrom follow the equations of currents, by substitution in 
the preceding, 


i, = 1.265 + 0.455 £-<>'<™3e - 0.720 e~0'^6 1 
and 

i2 - 5.06 - 1.038 r0'0633* _ 0.022 e" °'586'. J 

(6) Assuming now the resistance r0 suddenly raised again 
from r0 = 15 ohms to r0 = 20 ohms, leaving everything else 
the same, we have 

ij0 = 1.265 amp. ~] 
and 

t'2° = 5.06 amp.; J 

and then 


i, = 1 - 0.528 e-0'06970 + 0.793 r0'674' 1 

and 

ia = 4 + 1.018 s-0-069™ + 0.042 £-°'6740 . J 

(c) Assuming now the resistance r0 suddenly raised from 
r0 = 20 ohms to r0 = 25 ohms, gives 


i, = 0.828 - 0.374 e-Q'9™e + 0.546 £-°'7640 1 
and 

i2 = 3.312 + 0.649 rQ'm39 + 0.039 r0-764'. j 

(^) Assuming now the resistance r0 lowered again from r0 = 
25 ohms to r0 = 20 ohms, gives 

i, = 1 + 0.342 r0-069™ - 0.514 r0'6740 ^ 

and L 

ia = 4 - 0.660 g-0-06970 _ 0.028 £-°-674'. J 

76. In Fig. 33 are shown the variations of currents il and iv 
resultant from a sudden variation of the resistance r0 from 20 
to 15, back to 20, to 25, and back again to 20 ohms. As seen, 
the readjustment of current iv that is, the current in the induc- 
tive branch of the circuit, to its permanent condition, is very 
slow and gradual. Current iv however, not only changes very 
rapidly with a change of r0, but overreaches greatly; that is, a 
decrease of r0 causes il to increase rapidly to a temporary value 
far in excess of the permanent increase, and then gradually it 


DIVIDED CIRCUIT 


131 


falls back to its normal, and inversely with an increase of r0. 
Hence, any change of the main current is greatly exaggerated 
in the temporary component of current i\; a permanent change 
of about 20 per cent in the total current results in a practically 
instantaneous change of the branch current iv by about 50 per 
cent in the present instance. 

Thus, where any effect should be produced by a change of 
current, or of voltage, as a control of the circuit effected thereby, 
the action is made far more sensitive and quicker by shunting 
the operating circuit iv of as low inductance as possible, across 


t=-0 20 40 


Fig. 33. Current in divided continuous-current circuit resulting from sudden 
variations in resistance. 

a high inductance of as low resistance as possible. The sudden 
and temporary excess of the change of current il takes care of 
the increased friction of rest in setting the operating mechanism 
in motion, and gives a quicker reaction than a mechanism 
operated directly by the main current. 

This arrangement has been proposed for the operation of arc 
lamps of high arc voltage from constant potential circuits. 
The operating magnet, being in the circuit iv more or less 
anticipates the change of arc resistance by temporarily over- 
reaching. 

77. The temporary increase of the voltage, e, across the 
branch circuit, iv corresponding to the temporary excess current 
of this circuit, may, however, result in harmful effects, as de- 
struction of measuring instruments by the temporary excess 
voltage. 


132 TRANSIENT PHEXOMEXA 

Let, for instance, in a circuit of impressed continuous e.m.f., 
^ = 600 volts, as an electric railway circuit, the resistance of 
the circuit equal 25 ohms, the inductive reactance 44 ohms. 
This gives a permanent current of i7 = 24 amperes. 

Let now a small part of the circuit, of resistance ra = 1 ohm, 
but including most of the reactance x, = 40 ohms — as a motor 
series field winding — be shunted by a voltmeter, and rl = 1000 
ohms = resistance, xx = 40 ohms = reactance of the volt- 
meter circuit. 

In permanent condition the voltmeter reads ,V X 600 = 24 
volts, but any change of circuit condition, as a sudden decrease 
or increase of supply voltage ev results in the appearance of a 
temporary term which may greatly increase the voltage impressed 
upon the voltmeter. 

In this divided circuit, the constants are: undivided part of 
the circuit, rt = 24 ohms; xt = 4 ohms; first branch, voltmeter 
(practically non-inductive), rt = 1000 ohms, x4 = 40 ohms; 
second branch, motor field, highly inductive, r, = 1 ohm, x, = 
40 ohms. 

(a) Assuming now the impressed e.m.f., ev suddenly dropped 
from e% = 600 volts to e% = 540 volts, that is, by 10 per cent, 
gives the equations 


•ad 


i, = 0.0216 - 0.0806 £-••"" + 0.0830 £-*-lf| 
t, = 21.6 + 2.407 £-••«*• - 0.007 £-*Jf . 


(5) Assuming now the voltage, ev suddenly raised again from 
et = 540 volts to c. = 600 volts, gives the equations 

i, = 0.024 + 0.0806 £-••«•- 0.0830 £- 
and 

i, = 24 - 2.407 e-*402* + 0.007 


.0830 £-s-lfl 

r*'1' . 


Tbe voltage, e, across the voltmeter, or on circuit 1, is 
e = r^ + *i^r = 1000 1/ =F 77.9 e~9jm§ ± 6.2 r**9 , 

where 


DIVIDED CIRCUIT 


133 


Hence, in case (a), drop of impressed voltage, ev by 10 per cent, 

e = 21.6 - 77.9 £-••""+ 6.2 
and in (6), rise of impressed voltage, 

e = 24.0 + 77.9 «-•"""- 6.2 

This voltage, e, in the two cases, is plotted in Fig. 34. As 
seen, during the transition of the voltmeter reading from 21.6 
to 24.0 volts, the voltage momentarily rises to 95.7 volts, or 


Fig. 


jve apparatus in series with cdrcuit of high 
mniirtann* 


four times its permanent value, and during the decrease of 
permanent voltage from 24.0 to 21.6 volts the voltmeter momen- 
tarily reverses, going to 50.1 volts in reverse direction. 

In a high voltage direct-current circuit, a voltmeter shunted 
across a low resistance, if this resistance is highly inductive, is in 
danger of destruction by any sudden change of voltage or current 
in the circuit, even if the permanent value of the voltage is well 
within the safe range of the voltmeter. 


CAPACITY SHUNTING A PART OF A OONTINUOU^CURRENT 

CIRCUIT. 

78. A circuit of resistance rt and inductive reactance xl is 
shunted by the condensive reactance xc, and supplied over the 
resistance r0 and the inductive reactance x. by a continuous 
impressed e.m.f., ^, as shown diagrammatically in Fig. 35. 


134 TRANSIENT PHENOMENA 

In the undivided circuit, 


di. di 


In the inductive branch, 


In the condenser branch, 
e = xc I i2 dd. 


(48) 


(49) 


(50) 


L0, 


Fig. 35. Suppression of pulsations in direct-current circuits by series induc- 
tance and shunted capacity. 

Eliminating e gives, from (48) and (49), 

fl'l {11 


and from (49) and (50), 


(52) 


Differentiating (52), to eliminate the integral, 


(53) 


DIVIDED CIRCUIT 135 


Substituting (53) in (51), and rearranging, 


1 ( di 

eo = fro + ri) h + - frori + Vo + 3e*i) -Z 


i + r^o) ^ + XO*I-J£JT \ > 


a differential equation of third order. 
This resolves into the permanent term 

<>0 = 0*0 + O \', 

hence, i,' = — ^— , (55) 

TQ H- rt 

and a transient term 

t," - 4«-»; (56) 

that is, 

^ - V + A£~^ = — ^— + Ae-ae. (57) 

r0 + rt 

Equation (57) substituted in (54) gives as equation of a, 

*c fro + ri) ~ a frori + «A + %) + °2 fro^i + ri^o) - a'^i = °> 
or 

^L+i1) = 0 (58) 


while A. remains indefinite as integration constant. 

Equation (58) has three roots, av a2, and a3, which either are 
all three real, when the phenomenon is logarithmic, or, one 
real and two imaginary, when the phenomenon is oscillating. 

The integral equation for the current in branch 1 is 


*e • (59) 
the current in branch 2 is by (53) 


(60) 


136 TRANSIENT PHENOMENA 

and the potential difference at the condenser is 

/dfL 
i2dd=rlil + z,^1 


"'. (61) 

In the case of an oscillatory change, equations (59), (60), and 
(61) appear in complex imaginary form, and therefore have to 
be reduced to trigonometric functions. 

The three integration constants, Av A2, and A3, are deter- 
mined by the three terminal conditions, at 6 = 0, il = if, 
iz = if, e = e°. 

79. As numerical example may be considered a circuit having 
the constants, e0 = 110 volts; r0 = 1 ohm; x0 = 10 ohms; 
rl = 10 ohms; x^ = 100 ohms, and xc = 10 ohms. 

In other words, a continuous e.m.f. of 110 volts supplies, 
over a line of r0 = 1 ohm resistance, a circuit of rl = 10 ohms 
resistance. An inductive reactance x0 = 10 ohms is inserted 
into the line, and an inductive reactance xl = 100 ohms in the 
load circuit, and the latter shunted by a condensive reactance of 
xc = 10 ohms. 

Then, substituting in equation (58), 

a3 - 0.2 a2 + 1.11 a - 0.11 = 0. 

This cubic equation gives by approximation one root, at = 0.1, 
and, divided by (a — 0.1), leaves the quadratic equation 

a2 - 0.1 a + 1.1 = 0, 

which gives the complex imaginary roots a2 = 0.05 — 1.047 / 
and a3 = 0.05 + 1.047 j; then from the equation of current, 
by substituting trigonometric functions for the exponential 
functions with imaginary exponent, we get the equation for the 
load current as 

i, = if + A^-*'10 + £-°-05' (Bl cos 1.047 6 + B2 sin 1.047 0), 
the condenser potential as 

e=W if + e-0'050 {(5^ + 104.7 B2) cos 1.047 d - (104.7 Bl 
- 5 £2) sin 1.047/9}, 


DIVIDED CIRCUIT 137 

and the condenser current as 

ia = 10.9 r0-05' {B, cos 1.047 0 + B2 sin 1.047 6}. 

At e0 = 110 volts impressed, the permanent current is if = 10 
amp., the permanent condenser potential is ef = 100 volts, and 
the permanent condenser current is if = 0. 

Assuming now the voltage, e0, suddenly dropped by 10 per 
cent, from e0 = 110 volts to e0 = 99 volts, gives the permanent 
current as if = 9 amp. At the moment of drop of voltage, 
0 = 0, we have, however, il = if = 10 amp.; e = e' = 100 
volts, and t'2 = 0; hence, substituting these numerical values 
into the above equations of iv e, iv gives the three integration 
constants : 

A, = 1; Bi = 0, and B2 = 0.0955; 
therefore the load current is 

i, = 9 + r°'ie + 0.0955 r0'05' sin 1.047 0, 
the condenser current is 

i2 = 1.05 r0'05' sin 1.0470, 
and the condenser, or load, voltage is 

e = 90 + e-0'06* (10 cos 1.047 0 + 0.48 sin 1.047 0). 
Without the condenser, the equation of current would be 
i=9 + r°'ie. 

In this combination of circuits with shunted condensive 
reactance xc, at the moment of the voltage drop, or 0 = 0, the 
rate of change of the load current is, approximately, 

^ = [_ O.lr0'1' + 0.0955 X 1.047r°-05e cos 1.047 0]0 = 0, 
do 

while without the condenser it would be 

| = [_ o.l r «•"]. = -0.1. 

80. By shunting the circuit with capacity, the current in the 
circuit does not instantly begin to change with a change or 
fluctuation of impressed e.m.f. 


138 


TRANSIENT PHENOMENA 


In Fig. 36 is plotted, with 6 as abscissas, the change of the 
current, ilt in per cent, resulting from an instantaneous change 
of impressed e.m.f., eQ, of 10 per cent, with condenser in shunt 
to the load circuit, and without condenser. 

As seen, at 0 = 172°, both currents, it with the condenser 
and i without condenser, have dropped by the same amount, 


2.6 
2.4 
2.2 
2.0 
1.8 
1.6 
1.4 

1.0 

o.s 

0.6 
0.4 
0.2 
0 

j 

upply: 

co- 

= 110 volts 

/I 

1 0 = 

= 1 ohm 
= 10 ohms 

X 

/ 

/ 

Lc 

id i 

nd 

series i'ndu'ctan 
l\ == 10 ohms 

ce 

/ 

/ 

Xi i 100 oh 

tns 

/ 

/ 

SI 

nut 

(_•<! c 

apa 

e-ily 
-- 1C 

oh 

IIS 

/ 

/ 

^ 

X 

/ 

V 

^ 

/ 

/ 

/ 

'</ 

A 

/ 

/ 

'^£> 

j 

// 

/ 

*i\ 

• 

/ 

^ 

' 

/ 

/ 

/ 

— —• 

*^ 

^ 

0.4 


1.2 


1.6 


2.0 


2.4 


2.8 


Fig. 36. Suppression of pulsations in direct-current circuits by series induc- 
tance and shunted capacity. Effect of 10 per cent drop of voltage. 

2.6 per cent. But at 0 = 57.3°, ^ has dropped only J per cent, 
and i nearly 1 per cent, and at 0 = 24°, il has not yet dropped 
at all, while i has dropped by 0.38 per cent. 

That is, without condenser, all pulsations of the impressed 
e.m.f., eQ, appear in the load circuit as pulsations of the current, 
i, of a magnitude reduced the more the shorter the duration of 
the pulsation. After 0 = 60°, or t = 0.00275 seconds, the 
pulsation of the current has reached 10 per cent of the pulsation 
of impressed e.m.f. 

With a condenser in shunt to the load circuit, the pulsation 
of current in the load circuit is still zero after 0 = 24°, or after 
0.001 seconds, and reaches 1.25 per cent of the pulsation of 
impressed e.m.f., e0, after 6 = 60°, or t = 0.00275 seconds. 

A pulsation of the impressed e.m.f., e0, of a frequency higher 
than 250 cycles, practically cannot penetrate to the load circuit, 
that is, does not appear at all in the load current it regardless 
of how much a pulsation of the impressed e.m.f., e0, it is, and a 


DIVIDED CIRCUIT 139 

pulsation of impressed e.m.f., e0, of a frequency of 120 cycles re- 
appears in the load current iv reduced to 1 per cent of its value. 

In cases where from a source of e.m.f., e0, which contains a 
slight high frequency pulsation — as the pulsation corresponding 
to the commutator segments of a commutating machine — a 
current is desired showing no pulsation whatever, as for instance 
for the operation of a telephone exchange, a very high inductive 
reactance in series with the circuit, and a condensive reactance 
in shunt therewith, entirely eliminates all high frequency pulsa- 
tions from the current, passing only harmless low frequency 
pulsations at a greatly reduced amplitude. 

81. As a further example is shown in Fig. 37 the pulsation 
of a non-inductive circuit, xl = 0, of the resistance rl = 4 ohms, 
shunted by a condensive reactance xc = 10 ohms, and supplied 
over a line of resistance r0 = 1 ohm and inductive reactance 
x0 = 10 ohms, by an impressed e.m.f., e0 = 110 volts. 

Due to xl = 0 equation (58) reduces to 


1 0 

or, substituting numerical values, 

a2 - 2.6 a + 1.25 = 0 
and at ='0.637, a2 = 1.963; 

that is, both roots are real, or the phenomenon is logarithmic. 
We now have 

l\-ty + A/-«-«7'+A4r'-B«', 

ia = - 0.255 A^-0'63™- 0.785 A^-1'"*' , 

and e - r,i, =- 4 (i,' + Af-*'"1' + A/-1'963'). 

The load current is 

i/ = 22 amp. 

A reduction of the impressed e.m.f., e0, by 10 per cent, or from 
110 to 99 volts, gives the integration constants Al = 3.26 and 
A2 = - 1.06; hence, 

i, = 19.8 + 3.26 e-0'637' - 1.06 r1'**9 , 


and e = 4 ir 


140 TRANSIENT PHENOMENA 

Without a condenser, the equation of current would be 
i = 19.8 + 2.2 £-°-50. 

In Fig. 37 is shown, with 6 as abscissas, the drop of current 
il and i, in per cent. 

Although here the change is logarithmic, while in the former 
paragraph it was trigonometric, the result is the same — a very 
great reduction, by the condenser, of the drop of current imme- 
diately after the change of e.m.f. However, in the present case 


Fig. 37. Suppression of pulsations in nori -inductive direct-current circuits by 
series inductance and shunted capacity. Effect of 10 per cent drop of 
voltage. 

the change of the circuit is far more rapid than in the preceding 
case, due to the far lower inductive reactance of the present case. 
For instance, after 6 = 0.1, the drop of current, with condenser, 
is 0.045 per cent, without condenser, 0.5 per cent. At 6 = 0.2, 
the drop of current is 0.23 and 0.95 per cent respectively. For 
longer times or larger values of 6, the difference produced by the 
condenser becomes less and less. 

This effect of a condenser across a direct-current circuit, of 
suppressing high frequency pulsations from reaching the circuit, 
requires a very large capacity.