D.  C.  COMMUTATING  MACHINES 


179 


acting  upon  the  air  gap  between  armature  and  field  pole,  la  = 
length  of  air  gap,  from  iron  to  iron,  the  density  under  the  magnet 
pole,  that  is,  in  the  range  BC  of  Fig.  90,  is 


At  a  point  having  the  distance  lx  from  the  end  of  the  field  pole 
on  the  armature  surface,  the  distance  from  the  next  field  pole 
is  ld  =  Vk2  +  lx2,  and  the  density  thus,  approximately, 


B  C 

FIG.  92. — Distribution  of  mganetic  flux  under  a  single  pole. 


Herefrom  the  distribution  of  magnetic  flux  is  calculated  and 
plotted  in  Fig.  92,  for  a  single  pole  BC,  along  the  armature  sur- 
face A,  for  the  length  of  air  gap  la  =  1,  and  such  a  m.m.f.  as  to 


L 


FIG.  93. — Distribution  of  magnetic  force  and  flux  at  no  load. 

give  Bo  =  8000  under  the  field  pole;  that  is,  for  /0  =  6400  or 
HQ  =  8000. 

Around  the  surface  of  the  direct-current  machine  armature, 
alternate  poles  follow  each  other.  Thus  the  m.m.f.  is  constant 
only  under  each  field  pole,  but  decreases  in  the  space  between 
the  field  poles,  from  C  to  E  in  Fig.  93,  from  full  value  at  C  to 
full  value  in  opposite  direction  at  E.  The  point  D  midway 


180     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

between  C  and  E}  at  which  the  m.m.f.  of  the  field  equals  zero, 
is  called  the  "neutral."  The  distribution  of  m.m.f.  of  field 
excitation  is  thus  given  by  the  line  F  in  Fig.  91.  The  distribu- 
tion of  magnetic  flux  as  shown  in  Fig.  91  by  BQ  is  derived  by 
the  formula 

4irF 


B 


10  I 


where 


This  distribution  of  magnetic  flux  applies  only  to  the  no-load 
condition.  Under  load,  that  is,  if  the  armature  carries  current, 
the  distribution  of  flux  is  changed  by  the  m.m.f.  of  the  armature 
current,  or  armature  reaction. 


J 


FIG.  94. — Distribution  of  flux  with  current  in  the  armature. 

44.  Assuming  the  brushes  set  at  the  middle  points  between 
adjacent  poles,  D  and  G,  Fig.  94,  the  m.m.f.  of  the  armature  is 
maximum  at  the  point  connected  with  the  commutator  brushes, 
in  this  case  at  the  points  D  and  G}  and  gradually  decreases  from 
full  value  at  D  to  equal  but  opposite  value  at  G,  as  shown  by 
the  line  -Fa  in  Fig.  94,  while  the  line  FQ  gives  the  field  m.m.f. 
or  impressed  m.m.f. 

If  n  =  number  of  turns  in  series  between  brushes  per  pole, 
i  =  current  per  turn,  the  armature  reaction  is  Fa  =  ni  ampere- 
turns.  Adding  Fa  and  F0  gives  the  resultant  m.m.f.  F,  and  there- 
from the  magnetic  distribution: 


B  = 


The  latter  is  shown  as  line 


10  ld 
in  Fig.  94.