D.  C.  COMMUTATING  MACHINES  201 

is,  in  the  armature  during  commutation  an  e.m.f.  is  generated 
by  its  rotation  through  a  magnetic  field.  This  magnetic  field 
may  be  the  magnetic  field  of  armature  reaction,  or  the  reverse 
magnetic  field  of  a  commutating  pole,  or  the  fringe  of  the  main 
field  of  the  machine,  into  which  the  brushes  are  shifted.  In 
this  case  the  commutation  depends  upon  the  inductance  and  the 
resistance  of  the  armature  coil  and  the  e.m.f.  generated  therein 
by  the  main  magnetic  field,  and  if  this  magnetic  field  is  a  corn- 
mutating  field,  is  called  voltage  commutation. 

In  either  case  the  resistance  of  the  brushes  and  their  contact 
may  either  be  negligible,  as  usually  the  case  with  copper  brushes, 
or  it  may  be  of  the  same  or  a  higher  magnitude  than  the  internal 
resistance  of  the  armature  coil  A.  The  latter  is  usually  the  case 
with  carbon  or  graphite  brushes. 

In  the  former  case  the  resistance  of  the  short-circuit  of  arma- 
ture coil  A  under  commutation  is  approximately  constant;  in 
the  latter  case  it  varies  from  infinity  in  the  moment  of  beginning 
commutation  down  to  minimum,  and  then  up  again  to  infinity  at 
the  end  of  commutation. 

65.  (a)  Negligible  resistance  of  brush  and  brush  contact. 

This  is  more  or  less  approximately  the  case  with  copper  brushes. 

Let  iQ  =  current, 

L  =  inductance, 

r  —  resistance  of  armature  coil, 

to  =  -£•  =  time  of  commutation, 

and  —  e  =  e.m.f.  generated  in  the  armature  coil  by  its  rotation 
through  the  magnetic  field,  where  e  is  negative  for  the  magnetic 
field  of  armature  reaction  and  positive  for  the  commutating  field. 
Denoting  the  current  in  the  coil  A  at  time  t  after  beginning 
of  commutation  by  i,  the  e.m.f.  of  self-inductance  is 

_  di 


Thus  the  total  e.m.f.  acting  in  coil  A, 

di 

—  e  -\-  ei  =  —  e  —  L  -77* 
at 

and  the  current  is 

e      Ldi 
r      r  dt 


202     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Transposing,  this  expression  becomes 

rdt         di 


the  integral  of  which  is 

-  j-  =  loge  (^  +  i)  -  loge  c, 


where  loge  c  =  integration  constant. 
Since  at  t  =  0,  i  =  io,  we  have 


loge  c  =  log       +  i0  , 
therefore 

(g  \  In 

-  +  i,J  ,  and  ^  =  (-  + 

and,  at  the  end  of  commutation,  or,  t  =  to, 


For  perfect  commutation, 

that  is,  the  current  at  the  end  of  commutation  must  have  reversed 
and  reached  its  full  value  in  opposite  direction. 

Substituting  in  this  last  equation  the  value  i\  from  the  pre- 
ceding equation,  and  transforming,  we  have 


taking  the  logarithms  of  both  terms, 


L   o 

-t-> 

and,  solving  the  exponential  equation  for  e,  we  obtain