236 THE MAGNETIC CIRCUIT [ART. 68 



In reality there is a common flux which links with the coil 

 under consideration and with the other coils undergoing com- 

 mutation at the same time. This flux is changing with the 

 time, and consequently it induces additional voltages in the 

 coil CD. The induced e.m.f. depends upon the relative 

 position of CD and the other coils (whether in the same slot 

 or in the adjacent slots) and upon the rate of change of the cur- 

 rent in each coil of the group. It would be too complicated for 

 practical purposes to take all these factors into account with 

 any degree of accuracy. Therefore, Hob art makes a further 

 assumption, namely, that the current in all the coils, which are 

 short-circuited at the same time, varies at the same rate and that 

 the whole leakage flux is linked with all the coils of the group (Fig. 

 57). 



Let s be the average number of coils simultaneously short- 

 circuited under a set of brushes (the actual number varies from 

 instant to instant). Consider a group of mutually influencing 

 conductors, such as are shown at C or at D. One-half of the 

 conductors in the same group are short-circuited by the positive 

 brushes, the other half by the adjacent negative brushes. The 

 total number of coils in each group is 2s, and since by assumption 

 the current in all of them varies at the same rate, and all of 

 the flux is linked with all of the coils, the equivalent inductance 

 of the coil A B is 2s times larger than if this coil were alone. 

 Thus for a multiple-wound armature 



L^ = 2L PP .2s= 4sq 2 ((P^k + (P a 'l a + $(P e 'l e ) X KT 8 henrys. (157) 



On the basis of Mr. Hobart's tests and until more accurate data 

 are available, the following average values of the unit permeances 

 may be used: (Pi =4 and (P a f = (P e ' = 0.8 perms per centimeter. 



The frequency / which enters into formula (156) is calculated 

 as follows : The time between the positions 1 and 2 of the brushes 

 corresponds to one-half of one cycle, because during this interval 

 the current changes from +/i to 1\. Let v be the peripheral 

 velocity of the commutator, in meters per sec., let b be the thick- 

 ness of the brushes, and b' the thickness of the mica insulation 

 between the commutator segments, both in millimeters. The 

 time between the positions 1 and 2 of the brushes is (b -b')/WQOv 

 seconds. Hence 



/=500v/ (&-&') cy./sec ...... (158) 



