CHAPTER II. 

CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOMENA. 

6. As an example of a system of periodic transient phenomena, 
used for the control of electric circuits, may be considered an 
automatic potential regulator operating in the field circuit of 
the exciter of an alternating current system. 

Let, r0 = 40 ohms = resistance and L = 400 henrys = 
inductance of the exciter field circuit. 

A resistor, having a resistance, rl = 24 ohms, is inserted in 
series to r0, L in the exciter field, and a potential magnet, con- 
trolled by the alternating current system, is arranged so as to 
short circuit resistance, rv if the alternating potential is below, 
to throw resistance rl into circuit again, if the potential is 
above normal. 

With a single resistance step, rv in the one position of the 
regulator, with rx short circuited, and only r0 as exciter field 
winding resistance, the alternating potential would be above 
normal, that is, the regulator cannot remain in this position, 
but as soon after short circuiting resistance rl as the potential 
has risen sufficiently, the regulator must change its position 
and cut resistance T\ into the circuit, increasing the exciter field 
circuit resistance to r0 + rr This resistance now is too high, 
would lower the alternating potential too much, and the regula- 
tor thus cuts resistance rl out again. That is, the regulator 
continuously oscillates between the two positions, corresponding 
to the exciter field circuit resistances r0 and (r0 + rt) respec- 
tively, at a period depending on the momentum of the moving 
mass, the force of the magnets, etc., that is, approximately 
constant. The time of contact in each of the two positions, 
however, varies: when requiring a high field excitation, the 
regulator remains a longer time in position r0, hence a shorter 
time in position (r0 + rt), before the rising potential throws it 
over into the next position; while at light load, requiring low 
field excitation, the duration of the period of high resistance, 

223 


224 


TRANSIENT PHENOMENA 


(TO _|_ rj} is greater, and that of the period of low resistance, r0, 
less. 

7. Let, ^ = the duration of the short circuit of resistance rx; 
t2 = the time during which resistance rx is in circuit, and t0 = 

t, + tr 

During each period t0, the resistance of the exciter field, 
therefore, is r0 for the time tv and (r0 + rj for the time ty 

Furthermore, let, i1 = the current during time tv and i2 = 
the current during time tr 

During each of the two periods, let the time be counted 
anew from zero, that is, the transient current il exists during the 
time 0 < t < tv through the resistance r0, the transient 
current, iv during the time 0 < t < t2, through the resistance 
<>o + rj. 

This gives the terminal conditions : 


and 


(1) 


that is, the starting point of the current, iv is the end value of 
the current, iv and inversely. 

If now, e = voltage impressed upon the exciter field circuit, 
the differential equations are : 


and 


e = (T, 


di. 


(2) 


or, 


dil 


rt -, 

ro 


dt. 


(3) 


CIRCUIT CONTROL 


225 


Integrated, 


and 


ro + 


(4) 


Substituting the terminal conditions (1) in equations (4), 
gives for the integration constants cl and c2 the equations, 

e e 


+ 'I 


and 


herefrom, 


and 


ct = - 


(ro 


»"o 
- T 

- L 


erl-e 


(5) 


Substituting (5) in (4), 


and 


1 + 


_ rp+ri 
L 


(6) 


If, e = 250 volts; tQ = 0.2 sec., or 5 complete cycles per sec.; 
\ = 0.15, and t2 = 0.05 sec.; then 

i, = 6.25 {1 - 0.128 r°-li\ 


and 


i = 3.91 1 + 0.391 fi 


-°-16' 


(7) 


226 TRANSIENT PHENOMENA 

8. The mean value of current in the circuit is 

1 ( Ck 
I'-T-rr J VB 

h + t2 '^o 

This integrated gives, 


(8) 


r0 + rlt 


(9) 


and, if 
and 


(10) 


are the two extreme values of permanent current, corresponding 
respectively to the resistances r0 and (r0 + rj, we have 


(11) 


that is, the current, i, varies between i/ and i2' as linear function 
of the durations of contact, ^ and ty 

The maximum variation of current during the periodic change 
is given by the ratio of maximum current and minimum current; 
or, 


and is 


where, 


and 


r0(l -e-" 


r0(l - 


(12) 


(13) 


(14) 


CIRCUIT CONTROL 227 

Substituting 

x2 x3 
1-. =,__ + __+..., (15) 

by using only term of first order; 

gives (16) 

q = 1; J 

that is, the primary terms eliminate, and the difference between 
it and i2 is due to terms of secondary order only, hence very 
small. 

Substituting 

1 -".•"-*-!_; (17) 

that is, using also terms of second order, gives 

(18) 

or, approximately, 

q =1 +r r^2 rs > (19) 

and, substituting (14), 

g-i + LJ1*+O ; (2°) 

that is, the percentage variation of current is 


Equation (21) is a maximum for 


and, then, is 

9-1 = ; (23) 


228 TRANSIENT PHENOMENA 

or, in the above example, (rl = 24; L = 400; tQ = 0.2); 

q - 1 = 0.003; 

that is, 0.3 per cent. 

The time t0 of a cycle, which gives 1 per cent variation of 
current, q — 1 = 0.01, is 

t0 = ^ (? - 1), (24) 

' 1 

= f sec. 

The pulsation of current, 0.3 per cent respectively 1 per cent, 
thus is very small compared with the pulsation of the resistance, 
rt = 24 ohms, which is 46 per cent of the average resistance 

r0 -f ~ = 52 ohms.