232 ALTERNATING CURRENTS 



e- a (P + w )^ j a (V ~ w )^ 



Since sin a = -7- -- ---- - _ , and sin 2 = ^---- -- ----- , 



X/9M- 0? + t^A 2 v^ 5 + (p - w) 2 A a 



the expression for the current may be written in the form 



i = --- ri sm 0! cos [(P + w )^ ~~ ^i] + s ^ n 02 cos [(j? o>) 2 ]} 



In order to find the value of the total tangential pull T on the 

 coil, we have, as before, to form the product T = 2lib. Now, I may 

 be written in the form equation (1) of 136 



6 = ^B{cos (p tt))t cos (p + <*>}t} 



In determining the mean value T m of T, we need only consider 

 the products of terms of the same frequency, since the mean value of 

 the product of two terms of different frequency vanishes.* Now, 

 since 



cos [(p -\-u>}t- 0i] cos (p + w) = i{cos [2(p + w)zl - 0i] + cos 0i} 



the mean value of which is cos 0i, and since similarly the mean value 

 of cos [(p tt))t 62] cos (p w}t is cos 2 , it follows that 



r 

 T m = --- - v - ( sin 0i . cos 0i + sin 2 . cos 2 ) 



n 20 2 -sin20i) 



138. Effect of Varying Resistance of Rotor 



Circuits 



Eemembering ( 136) that when T m is negative the pull is from 

 left to right, i.e. in the direction of motion, we see that the motor will 

 exert a driving torque if sin 20 2 > sin 20i, and an opposing torque if 

 sin 20 2 < sin 20 X . The angles 0i and 2 are given by the equations (2). 

 At standstill, 0i = 2 , and the torque vanishes which confirms our 



previous result. If, as is always the case, is a large quantity, then 



for a small value of w, i.e. at a low speed, both 0i and 2 will be very 

 large angles, and 20i, 20 2 will both exceed 90, 0i being the greater of 

 the two, so that sin 20 2 > sin 20i, as is evident from an inspection of 

 Fig. 145 ; the motor will, therefore, exert a driving torque. By plot- 

 ting sin 20 2 sin 20i as a function of the speed, we obtain a curve 



* Thus, cos [(p + u)t - 0,] cos (p -w) = i [cos (Ipt - 0,) + cos (2wt - 0,)], the 

 mean value of which is zero. 



