154 ELEMENTS OF ELECTRICAL ENGINEERING. 



Let V be the angle between the mean position of the armature and its actual posi- 

 tion at a given instant, and let T be the unbalanced torque acting on the armature at 

 this instant. Our problem is to find the relation between ^ and 71 This relation, 

 when -i/; is small, is 



T= 61> (i) 



where b is a constant. Therefore, from the laws of harmonic motion, we have 



in which K is the moment of inertia of the rotating part of the machine, and t is the 

 period of the hunting oscillations. 



Derivation of equation (i) . We shall derive equation (i) for a special case, 

 namely, for the case in which the moments of inertia of the armatures (or rotating 

 parts) of generator and synchronous motor are equal, and for the particular phase 

 angle 180, see Figs. 125 and 128. In this case, namely, when = 180, a small 

 change of P f is accompanied by an equal and opposite change of P ff y so that equal 

 unbalanced torques act at each instant on the armatures of machines A and B, and 

 their moments of inertia being equal the ranges of the oscillations of the armatures of 

 both machines are equal. That is, the armature of machine A is as much ahead 

 of its mean position at each instant as the armature of machine B is behind its mean 

 position at the same instant and vice versa. Therefore the change of the phase angle 

 <j> is equal to 2^i/, 2i/> being the angular displacement of one armature referred to the 

 other and p being the number of pairs of field magnet poles in each machine. 



Differentiating equation (15) with respect to (j>, writing 2/i/> for d$, and after 

 the differentiation is performed, putting = 180, we have 



***** . sin a 



Now dP rf equals ZTTH T where n is the speed of the machine and T is the 

 unbalanced torque. Therefore 



pbAB 

 T= -- g -- sm 



The value of b, equation (i), is therefore 



* AB 



Substituting this value of b in equation (ii) and solving for / 2 , we have 



X* 



pAB sin 6 

 or, since 



wZ 

 sin = - and u -= 27rn 



V& + x 2 



tve have 



pABL 



