D.  C.  COMMUTATING  MACHINES  205 

the  current  entering  over  the  brush  shifts  from  segment  to  seg- 
ment in  direct  proportion  to  the  motion  of  the  gap  between  ad- 
jacent segments  across  the  brush,  that  is,  if  the  current  density 
is  uniform  all  over  the  contact  surface  of  the  brush.  This  means 
that  the  current  i  in  the  short-circuited  coil  varies  from  +  io  to 
—  iQ  as  a  linear  function  of  the  time.  In  this  case  it  can  be  rep- 
resented by 

.       .  to-2t 

^  =  ^o — r— J 
to 

thus, 

di  =       2ip 
dt~         to  * 

Substituting  this  value  in  the  general  differential  equation 
gives,  after  some  transformation, 

-?-*<>  +  r(to  -  20  -  2L  =  0; 
or, 

e  =  i    I  — 

which  gives  at  the  beginning  of  commutation,  t  =  0, 


at  the  end  of  commutation,  t  =  tQ, 


that  is,  even  with  high-resistance  brushes,  for  perfect  com- 
mutation, voltage  commutation  is  necessary,  and  the  e.m.f.  e 
impressed  upon  the  commutated  coil  must  increase  during  com- 
mutation from  ei  to  62,  by  the  above  equation.  This  e.m.f.  is 
proportional  to  the  current  iQ>  but  is  independent  of  the  brush 
resistance  r0. 

RESISTANCE  COMMUTATION 

67.  Herefrom  it  follows  that  resistance  commutation  cannot 
be  perfect,  but  that  at  the  contact  with  the  segment  that  leaves 
the  brush  the  current  density  must  be  higher  than  the  average. 
Let  g  =  ratio  of  actual  current  density  at  the  moment  of  leaving 
the  brush  to  average  current  density  of  brush  contact,  and  con- 


20G     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

sidering  only  the  end  of  commutation,  as  the  most  important 
moment,  we  have 

.       .  (2g  -l)U-2gt 

I     —    lQ  --  —  -  --    • 

to 

For  t  =  to  —  tl  this  gives 

^1 

i  =  -  io  +  2  g  -  io, 
to 

while  uniform  current  density  would  require 

^1 

i  =  —  io  +  2  -  i0. 
to 

The  general  differential  equation  of  resistance  commutation, 
e  =  0,  is 

di  rplo2    \      rQt0i0(2t-  t0) 


Substituting  in  this  equation  the  value  of  i  from  the  foregoing 
equation,  expanding  and  cancelling  to  —  t,  we  obtain 

2  r0*o2  (g  -  1)  +.  rtt0  (2  g  -  1)  -  2  grt2  -  2  gLt  =  0; 
hence, 

f  rt) 


g  ~  2  (r0*02  +  rtto  -  rt*  -  Lt) 
and  for  t  =  t0, 


2(rQtQ-L)  ~rQtQ-L' 

that  is,  g  is  always  >  1. 

The  smaller  L  and  the  larger  rQ)  the  smaller  is  g;  that  is,  the 
nearer  it  is  to  1,  the  condition  of  perfect  commutation,  and 
the  better  is  the  commutation. 

Sparkless  commutation  is  impossible  for  very  large  values  of 
g,  that  is,  when  L  approaches  roto,  or  when  r0  is  not  much  larger 

than  —  •     For  this  reason,  in  machines  in  which  L  cannot  be 
£o 

made  small,  r  is  sometimes  made  large  by  inserting  resistors 
in  the  leads  between  the  armature  and  the  commutator,  so-called 
''resistance"  or  "preventive"  leads  as  used  in  alternating-current 
commutator  motors.