274 
Transactions of the Royal Society of South Africa. 
final value after a switching operation will be — ^ , where a depends on the 
CtJAi g 
position of the regulator when the regulation is complete. Before this value 
If 
is reached the current has a higher value, viz., i'" = a~ Ce — -^t, 
where C is some integration constant, found as follows 
For t = 0, and ami, the current i'a — 
for ^ = 00 and a any value, i'a=^ " ]^ ! whence 
C = (1 - a) and 
i'a = a ^ (I — a) e " 
f 
= al -f (1 — a) l£ " 
(5) 
t 
Differentiating this equation, we get " = — ^ - ^- t ato, and 
dt a t° 
n , di'a 1 — a I 
for t^o, ~ = X ^ (6) 
dt a t^ 
from which follows that these curves have not a common tangent. 
To every value of a corresponds a definite position of the regulator. For 
an increase of « (or a decrease in the regulating resistance) the exciting 
current follows a rising curve. When fi decreases, the current drops along 
a corresponding falling characteristic. When an indirectly acting regulator 
is employed the sensitiveness of the relay or contact voltmeter must be 
limited : or, for a given sensitiveness of the relay or contact voltmeter the 
number of steps of the regulator cannot be any if hunting is to be prevented. 
For a given load the variation of the pressure by shifting the regulator one 
step must be smaller than that which will actuate the relay. Suppose the 
latter acts when the pressure rises 1 per cent, above or drops 1 per cent, 
below the normal value, then the pressure variation caused by moving the 
regulator one step (for a given load) must be less than 2 per cent. When 
this is the case excess regulation and hunting is caused only by too great a 
speed of the switch lever. To determine this limiting speed, consider at first 
the rising exciting current curves. 
Let the whole regulation take place (from a = 0 to a = 1) in T 
seconds, whereby it must be remembered that from a = 0 (resistance = 
infinity) to a = , say J, it takes as long as from a = ^ to a = This 
means that the first curve, for a = , say i, does not commence in the origin 
