COMMUTATION 



147 



is constant over the surface of contact, the current entering the 



S 



brush through any surface of width S is 21 c X TTT* To cal- 

 culate the volts e that must be developed in the coil of resistance 

 R when the distance yet to be travelled before the end of com- 

 mutation is w, consider the sum of the potential differences of 

 the local circuit AabB which is closed through the material in 

 the brush. This leads to the equation 



e = iR + i b Rb - i a R a (67) 



where R a and Rb are contact resistances depending upon the 

 areas of the surfaces through which the current enters the brush. 

 Under the conditions shown in Fig. 56, the contact surfaces S a 

 and Sb are equal, and the currents i a and ib are therefore also 

 equal. It follows that the voltage drops i a R a and ibRb are equal 

 and cancel out from equation (67). The same is true in the 

 later stages of commutation when S a is no longer equal to Sb 

 but to the portion w of the brush which remains in contact with 

 the segment A. The relations between the currents and the 

 surface resistances are then obtained by expressing these quan- 

 tities in terms of the contact surface, thus: 



i a = w X ki 



ii = S b X ^ 



D . Ny/ JU 



-tl/6 rt /N I" 2 



06 



where &i and k z are constants, and the voltage drop i a R a is seen 

 to be still equal to the drop ibRb. It follows that the only e.m.f. 

 to be developed in the short-circuited coil when uniform current 

 distribution is required will be e = iR. 



The instantaneous value (i) of the current in the coil under- 

 going commutation can be expressed in terms of the brush width 

 W and the distance (w) through which the coil still has to travel 

 before completion of commutation, because, 



w 



I = l c 2l c X TIT 



/W - 2w\ 

 lc \ W I 



and 



W 

 W -2w 



(68) 



