The Theory of Automatic Regulators. 
273 
results, i. e. excess regulation occurs, which produces voltage fluctuations 
which are unbearable in lighting plants. 
The limiting speed of a sluggish regulator may be determined as follows : 
We assume at present that the coefficient of self-induction of the field to 
be regulated is constant, and that the pressure increases linearly with the 
exciting current. These assumj)tions are not quite correct except for small 
ranges of the regulation, but the errors do not materially affect the result. 
The exponential curves of the exciting currents may then be considered also 
to represent the pressure curves of the machine, with an abcissa axis displaced 
by a constant value y. We further assume that the speed of the regulator 
and the variations of the exciting current are uniform. 
Let Ve — exciting pressure, 
I =- maximum exciting current, 
al = the exciting current when the regulation is complete, 
O < a < 1, 
= resistance of the exciting winding, 
— ^ = total resistance of the excitinu^ circuit, including: the rei^ulator, 
L = coefficient of self-induction of the exciting circuit, 
t = time in seconds counted from the beginning of the switching 
period, 
= current at time t for total resistance — - 
Y V 
We have I = ^/ and al = a - . /-^\ 
• • • V y 
The rising excitnig current curve (starting from zero) follows 
exponential law, as a heating curve, viz. : 
la 
al 
Differentiating, we obtain 
1 - ^ J J (2) 
dia _ Ye - ^ m 
dia Ye I 
For t = 0, ~ jj ~ t ~ constant (4) 
This holds, no matter what the value of a may be, and means that all 
the curves have the same tangent in their origin, for constant values of 
V„ L, and R,. 
In a similar manner we consider the falling curves, starting from the 
maximum possible value. The current is, however, not switched off, so that 
an equation similar to that for the cooling curve cannot be used here. The 
