ALTERNATE CURRENT DYNAMO-ELECTRIC MACHINES. 
235 
of the first two terms; the difference of ordinates of these curves should at all 
times be equal to Lx. In particular, and this is the only point of moment, as we do 
not know how L may vary, when Lx vanishes filer clt = JE dt. 
If the effect of the current in the armature on the induction through the armature 
could be represented by a term which vanishes with the current in the armature, it 
is obvious that the curves A and B would cross at the epoch when x = 0. They do 
not. The difference of inductions as given by these curves at this epoch is 25 per 
cent, of the induction which would then traverse the armature coil if the machine 
were running on open circuit, that is to say, 25 per cent, of the ordinate at this epoch 
given by the A curve. If it is assumed that the average effect of the armature 
current upon the induction in the magnets is such as to lower this induction by 25 
per cent., the ordinates of the E curve would be decreased in like proportion, giving 
the curve E' of which C is the integral. The difference at any epoch between the 
curve C and the Ike curve is to be fully accounted for by the term (L;t)\ 
We may put it in this way. In this machine the armature current at the times 
when it has a value affects the field at the instant when the armature current is zero. 
The effect is produced by variations induced in the current in the field magnet¬ 
winding and in the solid iron of the magnets by the varying current in and the 
varying position of the armature. That these induced currents must exist is obvious, 
and it is easy enough to measure them in the copper coils. They have the effect 
of causing the armature reaction to produce an average effect upon the magnetism 
of the fields by partially annulling its periodic effect. If the current in the armature 
did not lag behind the electromotive force of the magnets E, there would be little or 
no diminution of the average magnetism of the magnets. We may correctly say that 
this diminution of the magnetism of the magnets is due to the self-induction of the 
armature causing a lag of current. The effect arises from the self-induction of the 
armature modified by currents induced in the magnets. 
The effect on the magnets of any current in the armature is greatest when the 
armature bobbins are opposite the pole-faces, it is nil or small when half-way between 
the pole-faces. We may therefore represent its effect at any instant approximately 
as proportional to the expression 
271-L/T sin 2irtlT + E cos 2rrtjT . 
- - --Cl 
(2ttL/T) 2 + R 2 
or 
_ 7tL/T 77-L/T cos 4tt^/T — E sin ivt/T 
(2ttL/T) 2 + E 2 ~ (2ttL/T) 2 + E 3 ' 
The constant term causes the fall in average magnetism, the periodic term causes 
currents in the magnets, and its effect on the magnetism is partly annulled thereby. 
The effect will vary as the square of the current if this is small. The effect of self- 
induction in diminishing the apparent electromotive foi'ce of the machine varies as 
2 h 2 
