CONDITIONS OF STABILITY OF SYNCHRONOUS MOTORS. 139 



This gives us as a first condition that the quantity under the 

 square root must be positive, or, in the limit, zero. That is 



(72^2 _ /2 t ,2 _ 2rl*w -f 2V) 2 is greater than or equal to 

 - e 2 - 2rw)* + 4 V] 



Since 



/2 = r 2 + 3 



this reduces to 



7 2 e 2 J 2 is greater than or equal to rV + 2I*rwe* -f /V 2 

 or, taking the square root of both sides 



leE is greater than or equal to re 2 4- I*w . . (15) 



The limiting values of e, for any given output, w, are therefore 

 given by the equation 



re 2 - leE + Ifw = ..... (16) 

 provided, at the same time, the condition 



pjp _ /2 6 2 _ 2 r pw + 2/SfV is greater than or equal to zero (17) 



is satisfied, since this is necessary in order that i z may be 

 positive. 



It is not difficult to show that (17) is satisfied by all values of 

 w and e which satisfy (15), for, by (15), we have 



is less than or equal to leE re 2 

 therefore 

 72^2 _ 72^2 _ 2r /2 w + 22 6 2 is grea ter than 7 2 ^ 2 - /V - 2r(IeE- re 1 } 



greater than 7 2 ^ 2 - 2rIeE+ rV - 

 greater than (Ztf-re) 2 



Thus, when (15) is satisfied, so also is (17). It will, therefore, be 

 sufficient to see that condition (15) is always satisfied. 

 Provided, then, that 



re 1 lEe -f Pw is equal to or less than zero 



the current will be real, and we shall have a possible working 

 condition of the plant. 



