﻿the 
  Alternate 
  Current 
  Generator. 
  71 
  

  

  Reducing 
  to 
  the 
  continued 
  fraction 
  operators 
  of 
  § 
  7 
  we 
  

   obtain 
  

  

  a, 
  = 
  -g-(«„ 
  + 
  «0,-g^2^(«a)i-g^g^g^2;g:(«5)i-&c., 
  

  

  and 
  using 
  the 
  equations 
  

  

  «2 
  = 
  — 
  ^lai— 
  ao— 
  '^i 
  ; 
  % 
  = 
  — 
  T2a2 
  — 
  a-i 
  ; 
  &c., 
  

  

  the 
  successive 
  harmonics 
  of 
  the 
  armature 
  and 
  field 
  currents 
  

   can 
  be 
  obtained. 
  

  

  25. 
  If 
  in 
  the 
  last 
  example 
  the 
  e.m.f. 
  inserted 
  in 
  the 
  

   armature 
  circuit 
  be 
  sinusoidal 
  and 
  equal 
  to 
  E 
  sin 
  (wf-f 
  h) 
  = 
  e 
  

   (a 
  vector), 
  the 
  solution 
  will 
  (see 
  equations 
  in 
  last 
  paragraph) 
  

   obviously 
  be 
  identical 
  with 
  that 
  for 
  the 
  simple 
  generator 
  

   given 
  in 
  §§ 
  5 
  ef 
  seq., 
  when 
  in 
  the 
  latter 
  aQ-\-K 
  is 
  substituted 
  

   for 
  uq 
  where 
  

  

  2 
  9 
  

  

  K 
  =- 
  — 
  t'^e, 
  

   com 
  

  

  and 
  in 
  this 
  case 
  it 
  is 
  important 
  to 
  know 
  the 
  condition 
  which 
  

   determines 
  whether 
  the 
  machine 
  will 
  run 
  as 
  a 
  motor 
  and 
  

   develop 
  mechanical 
  power. 
  

  

  In 
  § 
  12 
  the 
  driving 
  torque 
  T 
  was 
  shown 
  to 
  be 
  given 
  by 
  

  

  T 
  = 
  — 
  ^{Saoai 
  + 
  Saia2 
  + 
  S«2a3 
  + 
  Sa3«4 
  4-&c.} 
  

  

  and 
  T 
  must 
  be 
  negative 
  for 
  a 
  motor. 
  

   Now, 
  as 
  

  

  1 
  L^^ 
  

  

  ai 
  = 
  - 
  ^ 
  {ao 
  + 
  K)i 
  = 
  («o 
  + 
  a:)i 
  (see 
  § 
  8) 
  

  

  Saoai 
  = 
  J 
  Sao 
  . 
  t^'ao+ 
  -— 
  Sa^ 
  . 
  l'^ 
  ^ 
  I 
  

  

  Si 
  1 
  com 
  J 
  

  

  = 
  _ 
  i 
  1^0^ 
  sin 
  h,+ 
  ^5o5 
  sin 
  (b. 
  + 
  A)}. 
  

   Again, 
  as 
  

  

  

  

  «2 
  

  

  1 
  

  

  A 
  

  

  -v.'^' 
  

  

  

  

  Sai«2 
  

  

  = 
  — 
  

  

  sin 
  ^2 
  - 
  

  

  

  2 
  

  

  similarly 
  

  

  

  

  sin 
  bo 
  . 
  

  

  ■ 
  2 
  . 
  

  

  

  S«2a3 
  = 
  2~~2~ 
  («o 
  + 
  f^) 
  7 
  &c., 
  &c. 
  ; 
  

  

  Si 
  0"2 
  53 
  

  

  