﻿the 
  Alternate 
  Current 
  Generator., 
  69 
  

  

  The 
  first 
  two 
  of 
  equations 
  VII. 
  may 
  now 
  be 
  written 
  ^ 
  

  

  dt 
  

  

  P 
  i-J 
  . 
  ...^r,|:/^. 
  .....,..._ 
  ^7 
  

  

  T] 
  being 
  divided 
  by 
  1-\-k 
  as 
  the 
  constant 
  term 
  in 
  f 
  ' 
  is 
  equal 
  

   to 
  the 
  constant 
  term 
  in 
  f 
  (J" 
  having 
  no 
  constant 
  term). 
  

  

  Now 
  X 
  and 
  ^' 
  determined 
  from 
  these 
  equations 
  will 
  be 
  

   very 
  approximately 
  the 
  same 
  as 
  x 
  and 
  f 
  determined 
  from 
  

   the 
  equations 
  in 
  § 
  1 
  for 
  the 
  alternator 
  without 
  dampers 
  ; 
  

   for 
  a.Q 
  the 
  given 
  vector 
  is 
  the 
  same 
  for 
  both, 
  as 
  are 
  also 
  all 
  

   the 
  t 
  operators. 
  

  

  The 
  T 
  operators 
  differ 
  in 
  that 
  for 
  p, 
  pj{1-\-k) 
  is 
  substituted, 
  

   but 
  in 
  § 
  8 
  and 
  in 
  note 
  § 
  22 
  it 
  is 
  shown 
  that 
  the 
  r 
  operators 
  

   for 
  generators 
  as 
  ordinarily 
  constructed 
  are 
  practically 
  inde- 
  

   pendent 
  of 
  the 
  value 
  of 
  p 
  the 
  field 
  resistance. 
  

  

  Hence 
  we 
  see 
  that 
  f 
  ' 
  is 
  the 
  alternating 
  field 
  current 
  if 
  the 
  

   dampers 
  are 
  absent, 
  f 
  its 
  value 
  when 
  the 
  dampers 
  are 
  attached 
  

   and 
  these 
  currents 
  are 
  connected 
  b}^ 
  the 
  relation 
  

  

  In 
  addition 
  as 
  2zt,'^ 
  = 
  Kp^^, 
  

  

  Hence 
  if 
  H^ 
  be 
  the 
  copper 
  loss 
  in 
  the 
  field 
  coils 
  due 
  to 
  

   induced 
  alternating 
  current 
  when 
  the 
  generator 
  is 
  without 
  

   dampers, 
  H 
  the 
  same 
  when 
  dampers 
  are 
  attached, 
  and 
  h 
  the 
  

   loss 
  in 
  the 
  dampers 
  when 
  attached, 
  

  

  TX 
  H 
  , 
  kSl 
  TX 
  7 
  H 
  

  

  H 
  = 
  ,— 
  — 
  -7-0, 
  h=——^, 
  H 
  + 
  /i 
  

  

  (l 
  + 
  /f)-' 
  (I 
  + 
  a:)^' 
  1 
  + 
  fc' 
  

  

  If 
  we 
  assume 
  that 
  the 
  mean 
  length 
  of 
  a 
  field 
  turn 
  is 
  equal 
  

   to 
  the 
  length 
  of 
  a 
  damper 
  turn, 
  it 
  is 
  easy 
  to 
  show 
  that 
  /c 
  is 
  

   the 
  ratio 
  of 
  the 
  volume 
  of 
  copper 
  in 
  the 
  dampers 
  to 
  the 
  

   volume 
  of 
  copper 
  in 
  the 
  field 
  windings 
  when 
  there 
  is 
  no 
  

   resistance 
  external 
  to 
  the 
  windings 
  in 
  the 
  field 
  circuit. 
  

   If 
  there 
  is 
  resistance 
  external 
  to 
  the 
  windings 
  in 
  the 
  field 
  

   circuit, 
  K 
  is 
  greater 
  than 
  the 
  above 
  volume 
  ratio. 
  

  

  The 
  magnetic 
  flux 
  in 
  the 
  field 
  coils 
  being 
  equal 
  to 
  

  

  ff{v^' 
  'hnx 
  cos 
  cot} 
  

  

  is 
  practically 
  unaffected 
  by 
  the 
  presence 
  of 
  dampers, 
  so 
  that 
  

   the 
  iron 
  loss 
  in 
  the 
  field-magnets 
  remains 
  the 
  same. 
  

  

  