﻿the 
  Alternate 
  Current 
  Generator. 
  53 
  

  

  similarly 
  

  

  «4 
  = 
  —^ 
  — 
  ?>mU(Dt+ 
  ^ 
  4-bi 
  + 
  A2 
  + 
  ^3 
  ^- 
  a) 
  ; 
  

  

  a5 
  = 
  — 
  sinfoo;^- 
  ^ 
  + 
  ^i 
  + 
  A 
  + 
  ^s 
  + 
  A 
  + 
  b^) 
  ; 
  

  

  &C., 
  &C. 
  

  

  And 
  substituting 
  these 
  values 
  in 
  

  

  X 
  = 
  ai 
  + 
  ag 
  + 
  ag 
  + 
  &c., 
  ? 
  = 
  - 
  + 
  «2 
  + 
  ^4 
  + 
  &c., 
  

  

  r 
  

  

  we 
  obtain 
  the 
  armature 
  and 
  field 
  currents 
  in 
  the 
  usual 
  

   trigonometrical 
  form 
  o£ 
  expression. 
  

  

  It 
  is 
  worth 
  while 
  drawing 
  attention 
  to 
  the 
  fact 
  that 
  the 
  

   period 
  of 
  the 
  alternating 
  current 
  induced 
  in 
  the 
  field 
  circuit 
  

   is 
  half 
  that 
  of 
  the 
  armature 
  current, 
  and 
  that 
  it 
  contains 
  all 
  

   harmonics, 
  both 
  odd 
  and 
  even, 
  relative 
  to 
  its 
  own 
  funda- 
  

   mental, 
  and 
  so 
  its 
  wave-form 
  will 
  in 
  general 
  be 
  unsymmetrical 
  

   with 
  respect 
  to 
  the 
  time-axis. 
  

  

  10. 
  The 
  total 
  e.m.f. 
  E 
  generated 
  in 
  the 
  armature 
  circuit 
  

  

  being 
  equal 
  to 
  — 
  ;T7(wif 
  cos 
  (ot\ 
  

  

  E 
  = 
  - 
  |^-^X(«g_i 
  + 
  «^+l)^, 
  (see 
  § 
  4) 
  

  

  TT 
  

  

  Also 
  as 
  «g_i 
  + 
  ^f^a^H-aj^4-l 
  = 
  

  

  TT 
  

  

  „ 
  wm 
  ^^ 
  ^ 
  

   -E 
  = 
  ^ 
  t 
  Xqt^2.q^ 
  

  

  And 
  in 
  either 
  of 
  these 
  formulae 
  the 
  trigonometrical 
  ex- 
  

   pressions 
  in 
  § 
  9 
  for 
  the 
  vectors 
  can 
  be 
  substituted. 
  

  

  In 
  the 
  first, 
  however, 
  it 
  must 
  be 
  noted 
  that 
  both 
  «^_i 
  and 
  

   a^+l 
  are 
  to 
  be 
  taken 
  of 
  order 
  q 
  (odd). 
  Thus 
  the 
  funda- 
  

   mental 
  harmonic 
  of 
  E 
  is 
  

  

  (omrj 
  

   P 
  

  

  < 
  sin 
  cot-\ 
  sin 
  (cot 
  -f 
  bj 
  H-zSg) 
  r 
  

  

  