D.  C.  COMMUTATING  MACHINES 


191 


ture  in  centimeters  per  second,  lp  =  pitch  of  armature  slot  (that 
is,  width  of  one  slot  and  one  tooth  at  armature  surface),  the 

S 
frequency  is  /i  =  y-.     Or,  if  /  =  frequency   of   machine,  n  — 

number  of  armature  slots  per  pair  of  poles,  /i  =  nf. 

For  instance,/  =  33.3,  n  =  51,  thus/i  =  1700. 

Under  the  assumption,  width 
of  slots  equals  width  of  teeth 
=  2  X  width  of  air  gap,  the  dis- 
tribution of  magnetic  flux  at  the 
pole  face  is  plotted  in  Fig.  103. 

The  drop  of  density  opposite 
each  slot  consists  of  two  curved 
branches  equal  to  those  in  Fig. 
92,  that  is,  calculated  by 


•B' 


-3 

n 

FIG.  103.—  I 

< 

« 

i 

slots  on  flu 

Iffect  of 

B 


distribution. 


V  + 1*2 

The  average  flux  is  7525;  that  is,  by  cutting  half  the  armature 
surface  away  by  slots  of  a  width  equal  to  twice  the  length  of  air 
gap,  the  total  flux  under  the  field  pole  is  reduced  only  in  the 
proportion  8000  to  7525,  or  about  6  per  cent. 

The  flux  B  pulsating  between  8000  and  5700  is  equivalent  to 
a  uniform  flux  B\  =  7525  superposed  with  an  alternating  flux 


FIG.  104. — Effect  of  slots  on  flux  distribution. 

BO,  shown  in  Fig.  104,  with  a  maximum  of  475  and  a  minimum 
of  1825.  This  alternating  flux  BQ  can,  as  regards  production  of 
eddy  currents,  be  replaced  by  the  equivalent  sine  wave  B0o,  that 
is,  a  sine  wave  having  the  same  effective  value  (or  square  root  of 
mean  square).  The  effective  value  is  718. 

The  pulsation  of  magnetic  flux  farther  in  the  interior  of  the 
field-pole  face  can  be  approximated  by  drawing  curves  equi- 


192     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

distant  from  BQ.  Thus  the  curves  #0.5,  BI>  ^1.5,  #2,  #2.5,  and  B3 
are  drawn  equidistant  from  B0  in  the  relative  distances  0.5,  1, 
1.5,  2,  2.5,  and  3  (where  la  =  1  is  the  length  of  air  gap).  They 
give  the  effective  values: 

BQ  BQ.S  BI  BI.Z  BZ  Bz.s  B3 

718          373  184  119  91  69  57 

That  is,  the  pulsation  of  magnetic  flux  rapidly  disappears 
toward  the  interior  of  the  magnet  pole,  and  still  more  rapidly 
the  energy  loss  by  eddy  currents,  which  is  proportional  to  the 
square  of  the  magnetic  density. 

54.  In  calculating  the  effect  of  eddy  currents,  the  magnetizing 
effect  of  eddy  currents  may  be  neglected  (which  tends  to  reduce 
the  pulsation  of  magnetism);  this  gives  the  upper  limit  of  loss 

Let  B  =  effective  density  of  the  alternating  magnetic  flux, 
S  =  peripheral   speed    of    armature    in    centimeters    per 

second,  and 
I  =  length  of  pole  face  along  armature. 

The  e.m.f.  generated  in  the  pole  face  is  then 

e  =  SIB  X  10-8, 
and  the  current  in  a  strip  of  thickness  Al  and  1  cm.  width, 

eAl      SIBAl  10~8       SBAl  10~8 

Ai  =  —r  =  -     — jr—    -  =  -  —i 

pl>  pi  p 

where 

p  =  resistivity  of  the  material. 

Thus  the  effect  of  eddy  currents  in  this  strip  is 

'  SHB2Al  10-16 

Ap  =  eAi  =  -  —i 

or  per  cubic  centimeter, 

S*B*  10~16 

P  =        ~~P~ 

that  is,  proportional  to  the  square  of  the  effective  value  of  mag- 
netic pulsation,  the  square  of  peripheral  speed,  and  inversely 
proportional  to  the  resistivity. 
Thus,  assuming  for  instance, 
S  =  2000, 

p  =  20  X  10~6,  for  cast  steel, 
p  =  100  X  10~6,  for  cast  iron, 
we  have  in  the  above  example,