176 PRINCIPLES OF ELECTRICAL DESIGN 



ally similar to Fig. 56, except that the coil connecting segments 

 A and B has moved nearer to the edge of the commutation zone. 

 The distance, in inches, still to be travelled before the removal of 

 the short-circuit is w, which is supposed to be only a small per- 

 centage of the total brush arc W. 



Let R be the resistance, in ohms, of the coil connecting the 

 commutator segments A and B. 



R c the contact resistance of the brushes per square inch of 

 area. 



l c the total length, in inches, of the set of brushes measured 

 parallel to the axis of rotation. 



A = the average current density, in amperes per square inch, 

 over the brush-contact surface. It will also be, approxi- 

 mately, the density over the surface Sb of the segment 

 B (Fig. 67). 



A, = the maximum permissible current density over the sur- 

 face of the brush tip of width w. 



Summing up the e.m.fs. and potential differences in the path 

 of the short-circuit (the resistance of the material in the body of 

 the brush being neglected), we can write, for the value of the 

 volts developed in the short-circuited coil at the instant con- 

 sidered, 



e = iR + Afl c - & W R C 

 But 



i = I c i a 



wherein the meaning of the symbols will be evident from an in- 

 spection of Fig. 67. This last equation may be written, 



i = I c - bwlcW 

 Substituting in the previous equation, we get, 



e = I C R - & w l c wR - R c (&w - A) 



If, now, we imagine w to become smaller and smaller as it ap- 

 proaches zero value, the second term on the right-hand side of 

 the equation becomes of relatively less and less importance, and 

 we may therefore write, 



e = I C R - R c (& w - A) 



which gives an approximate value for the permissible e.m.f. in 

 the short-circuited coil at the end of the commutation period. 

 If preferred, this equation may be put in the form 



e = I C R - RMk - 1) (87) 



