VOLTAGE REGULATION OF THE ALTERNATOR. 139 



turns, of the amount that the armature current is helping the field winding at the 

 given instant. Let this be represented by m, so that 



m = iTcosat (iii) 



or, substituting the value of i from equation (ii) we have 



m = I Tcos ut sin (ut 0) 

 or 



m = I Tcos ut sin at cos I Tcos 2 at sin (iv) 



The average value of m is the average amount in ampere-turns that the field wind- 

 ing is helped by the armature current, and the average value of m is equal to the sum 

 of the average values of the two terms of the right-hand member of equation (iv), but 

 the average value of the first term is zero and the average value of the second term is 

 1 T'sin 0, inasmuch as the average value of cos 2 uf is equal to . Therefore 



average value of m = 1 7"sin 

 or, substituting the effective value /I/ 1/2, we have 



average value of m = IT sin 6 ( v) 



V/2 



The meaning of this equation is as follows : (a) When 6 is zero the armature current 

 neither helps nor opposes the passage of flux, on the average ; (b] when 6 is posi- 

 tive, that is when the current lags behind the voltage in phase, the armature current 

 opposes the passage of flux, on the average ; and (<r) when 6 is negative, that is 

 when the current is ahead of the voltage in phase, the armature current helps the pas- 

 sage of flux, on the average. 



The constancy of the magnetizing action of the armature currents in a two-phase 

 armature, when the receiving circuits are balanced, is easily shown. Let m, equation 

 (iv), be the instantaneous value of the magnetizing action of phase A of a two-phase 

 armature. The instantaneous value of the magnetizing action m' of phase B is 

 obtained by substituting ut 90 for at in equation (iv) which gives : 



m f = I T sin ut cos at cos 6 IT sin 2 at sin 0. (vi) 



Therefore the total magnetizing action of the two phases is 



m -f- m' = I T( sin 2 at -f- cos 2 at) sin 

 or 



m-\-m' = l T^sin fl 

 which is constant. 



66. Power rating of alternators. The current delivered by an 

 alternator generates heat in the armature which heat, together 

 with the heat generated by eddy currents and hysteresis, causes 

 the temperature of the armature to rise until it gives off heat as 

 fast as heat is generated in it. Therefore, to increase the current 

 output of an alternator causes an increased rise of temperature of 

 the machine. This heating effect usually determines the current 

 rating of an alternator, and the power rating of an alternator is 



