242 ALTERNATING CURRENTS 



shall determine the relative outputs, for the same mean temperature 

 rise, when the armature is used as a converter from single-, two-, 

 and three-phase currents respectively to continuous current. 



The points of connection to the slip-rings being fixed relatively 

 to the armature winding, while the brushes are constantly changing 

 their position with respect to it, it is evident that the distribution 

 of currents in the armature at any instant will depend (among 

 other things) on the position of the brushes relatively to the slip- 

 ring attachments. Since the effect is one depending on relative 

 position, we shall find it convenient to suppose that the armature is 

 stationary, while the brushes are carried round by the revolving field. 



In Fig. 150, the circle is intended to represent the closed winding 

 of a stationary two-pole armature, E and S being the fixed points 

 in the winding to which the two slip-rings of the single-phase 

 converter are attached, while the brushes BI, B 2 are carried round 

 the winding with an angular velocity^?, such thatp = 2?r X frequency 

 of the single-phase currents. Let the instantaneous value of the 

 single-phase current be given by 



! = I TO cos pt = I m cos 9 



where we write, for the sake of brevity, S for pt. The current reaches 

 its maximum positive value at t = 0, i.e. = 0. In the figure, we 

 assume the positive direction of the current to be from left to right, 

 as shown by the arrows. Now, since by supposition the power factor 

 is unity, the maximum value of the current must occur when the 

 p d. reaches its maximum value, i.e. when BiB 2 is horizontal. The 



I 



value of the continuous current of the converter is, by 3, ~ x r.m.s. 



v2 



value of alternating current = ^ X ^ = I TO , so that the con- 

 tinuous current is half the maximum alternating current. When 

 t = 0, therefore, the brushes are horizontal, the alternating current 

 has the value I m , and of this %I m passes into the external circuit, 

 while the remaining I m flows into the armature, driving it as 

 a motor. 



Consider any point P in the winding at an angular distance 

 POE = ^ from E. It is evident from considerations of symmetry 

 that the cycle of changes in the numerical value of the current in 

 the winding at P during the first half-revolution of BiB 2 will be 

 repeated during the next half-revolution. So far as the heating at 

 P is concerned, therefore, it will be sufficient to study the changes 

 in the current at P during one half-revolution of BiB 2 , i.e. while 

 BI moves, in a counter-clockwise direction, from E to S, 



Now, it is evident that as the brush BI sweeps past the point 



