D.  C.  COMMUTATING  MACHINES 


203 


It  is  evident  that  the  inequality  e  >  i<>r  must  be  true,  otherwise 

perfect  commutation  is  not  possible. 

If 


we  have 


that  is,  the  current  never  reverses,  but  merely  dies  out  more  or 
less,  and  in  the  moment  when  the  gap  G  of  the  armature  coil 
leaves  the  brush  B  the  current  therein  has  to  rise  suddenly  to 
full  intensity  in  opposite  direction.  This  being  impossible,  due 
to  the  inductance  of  the  coil,  the  current  forms  an  arc  from  the 
brush  across  the  commutator  surface  for  a  length  of  time  depend- 
ing upon  the  inductance  of  the  armature  coil. 

Therefore,  with  low-resistance  brushes,  resistance  commutation 
is  not  permissible  except  with  machines  of  extremely  low  arma- 


FIG.  108. — Brush  commutating  coil  A. 

ture  inductance,  that  is,  armature  inductance  so  low  that  the 

magnetic  energy  -7^—,  which  appears  as  "spark"  in  this  case,  is 
& 

harmless. 

Voltage  commutation  is  feasible  with  low-resistance  brushes, 
but  requires  a  commutating  e.m.f.  e  proportional  to  current  z'o; 
that  is,  requires  shifting  of  brushes  proportionally  to  the  load,  or 
a  commutating  pole. 

In  the  preceding,  the  e.m.f.  e.has  been  assumed  constant  dur- 
ing the  commutation.  In  reality  it  varies  somewhat,  usually 
increasing  with  the  approach  of  the  commutated  coil  to  a  denser 
field.  It  is  not  possible  to  consider  this  variation  in  general,  and 
e  is  thus  to  be  considered  the  average  value  during  commutation. 

66.  (b)  High-resistance  brush  contact. 

Fig.  108  represents  a  brush  B  commutating  armature  coil  A. 


204     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

Let  r0  =  contact  resistance  of  the  brush,  that  is,  resistance 
from  the  brush  to  the  commutator  surface  over  the  total  bearing 
surface  of  the  brushes.  The  resistance  of  the  commutated  cir- 
cuit is  thus  internal  resistance  of  the  armature  coil  r  plus  the 
resistance  from  C  to  B  plus  the  resistance  from  B  to  D. 

Thus,  if  to  =  time  of  commutation,  at  the  time  t  after  the  be- 
ginning of  the  commutation,  the  resistance  from  C  to  B  is  — 

and  from  B  to  D  is -;  thus,  the  total  resistance  of  the  corn- 
to  —  * 

mutated  coil  is 

T-»  ,      to?*0     i         'o7*0  to  TQ 

R  =  r  +  —  +  -. :  =  r  +     (          v 

t         to  —  t  t  (to  —  t) 

If  i0  =  current  in  coil  A  before  commutation,  the  total  cur- 
rent into  the  armature  from  brush  B  is  2  i0.  Thus,  if  i  —  current 
in  commutated  coil,  the  current  from  B  to  D  =  iQ  +  i,  the  cur- 
rent from  B  to  C  =  io  —  i. 

Hence,  the  difference  of  potential  from  D  to  C  is 


The  e.m.f.  acting  in  coil  A  is 

Ldi 


and  herefrom  the  difference  of  potential  from  D  to  C  is 

L--  >• 

hence, 

di        .  tofo     , .         .>.        tofo  / .         .\ 

dt  to  —  t  t 

or,  transposing, 

Ldi  .        toToio  (2 1  —  to)          tozroi 

dt  t  (to  —  t)  t  (to  —  t) 

T  di  .  /  roto2     \        Totoio  C2>t  —  to) 


'    t(tQ-t)  * 

The  further  solution  of  this  general  problem  becomes  difficult, 
but  even  without  integrating  this  differential  equation  a  number 
of  important  conclusions  can  be  derived. 

Obviously  the  commutation  is  correct  and  thus  sparkless,  if