Self -Excitation in a Dynamo Machine. 473 



where a stands for the constant part of the electric resistances 

 (armature and main-circuit coils) in the circuit, and x for the 

 variable part ; that is to say, the resistance of the external 

 circuit, including under that term not only true ohmic resist- 

 ances, but also the effects of counter electromotive forces 

 opposed in the path of the current of the machine. 



We may now rewrite equation (6) in the following form 

 for the series- wound machine driven at a constant speed : — 



(a + 1) (a + x) = constant (7) 



Now at starting, when as yet the magnetization of the iron 

 is almost zero, the resistance of the iron part of the magnetic 

 circuit is small. Its value is not indeed quite zero, but de- 

 pends upon the prior history of the iron. It is certainly so 

 small as to be negligible compared with the value of a. 

 Hence there will be a particular maximum value of a, which 

 we may call x ly corresponding to the case when £ = 0. This 

 maximum value, which will be given by the equation 



#i= o, (8) 



a 



will be the critical external resistance of the machine at the 

 prescribed speed. If any greater resistance than Xi be inter- 

 calated in the circuit the machine will refuse to excite itself, 

 even though there is a small initial magnetization present. 

 So far as there is a weak magnetization present, the machine 

 will act as a feeble magneto-electric machine. 



Suppose, now, that x is so far decreased that the total elec- 

 tric resistance is reduced to half its limiting value, that is to 

 say, let x —\x Y — \a : then the current and the magnetization 

 will both run up until the magnetic resistance is equal to 

 double its initial value, and f will have the value f=a. The 

 permeability of the magnetic circuit, taken as a whole (iron, 

 copper, and air), will have been halved. This state of things 

 corresponds to the "diacritical point" in the excitation first 

 investigated by the author in 1884 ; and, if the Lamont- 

 Frolich formula were true, this point would correspond to 

 the state of semi-saturation of the magnetization. 



Again, suppose the variable part of the electric resistance 

 to be reduced to zero ; in other words, let the machine be 

 short-circuited. Then f will run up to the maximum value 

 it can have at that speed. Calling this value f 2 > we have 



^=—a—-« (9) 



It is further clear that if a fixed value be given to the 



