﻿70 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  tJie 
  Theory/ 
  of 
  

  

  24. 
  If 
  a 
  source 
  of 
  alternating 
  e.m.f. 
  = 
  E 
  where 
  

  

  E 
  = 
  El 
  sin 
  {at 
  + 
  hi) 
  + 
  E3 
  sin 
  (3<w^ 
  + 
  ^3) 
  + 
  &c., 
  

  

  =^i 
  + 
  ^3 
  + 
  ^5 
  + 
  &c. 
  (vectors) 
  

  

  be 
  included 
  in 
  the 
  armature 
  circuit, 
  [and 
  if 
  the 
  armature 
  

   rotate 
  in 
  synchronism 
  with 
  this 
  e.m.f., 
  we 
  have 
  the 
  case 
  

   of 
  the 
  synchronous 
  A.C. 
  motor. 
  

  

  In 
  this 
  case 
  the 
  armature 
  and 
  field 
  currents 
  x 
  and 
  f 
  are 
  

   connected 
  by 
  the 
  equations 
  (see 
  § 
  1) 
  

  

  r.T 
  4- 
  -77 
  (Lt 
  + 
  m^ 
  cos 
  (ot) 
  = 
  XEq 
  sin 
  \ 
  (qoot 
  + 
  Jig) 
  

  

  pi~^~Tf 
  O^i 
  + 
  '^''^'^ 
  ^^^ 
  *"0 
  = 
  v- 
  

  

  Assuming 
  as 
  in 
  § 
  2 
  that 
  

  

  X 
  = 
  ai 
  -f 
  as 
  + 
  as 
  + 
  &c., 
  

  

  and 
  proceeding 
  exactly 
  as 
  in 
  § 
  5, 
  we 
  obtain 
  the 
  infinite 
  

   series 
  of 
  equations 
  

  

  ^lai 
  + 
  ag 
  

  

  = 
  — 
  

  

  «o- 
  

  

  -Ki 
  

  

  ai-f-T2«2 
  + 
  a3 
  

  

  = 
  

  

  

  

  "2 
  + 
  ^33-3 
  + 
  ^4 
  

  

  = 
  _ 
  

  

  'f^s 
  

  

  

  a3 
  + 
  T4a4 
  + 
  a5 
  

  

  = 
  

  

  

  

  ^4 
  + 
  ^5^5 
  + 
  "6 
  

  

  , 
  = 
  — 
  

  

  ■fC5 
  

  

  

  &C., 
  ( 
  

  

  fee, 
  

  

  

  

  in 
  which 
  

  

  

  

  

  ao= 
  the 
  vector 
  to 
  the 
  point 
  2??//c, 
  

  

  v-^ 
  

  

  as 
  

  

  before 
  

  

  and 
  Kq— 
  L'^eq, 
  

  

  ^ 
  qmo) 
  ^ 
  

  

  

  

  

  where 
  Xeq 
  is 
  the 
  applied 
  e.m.f., 
  and 
  the 
  t 
  and 
  t 
  operators 
  have 
  

   the 
  same 
  values 
  as 
  in 
  § 
  5. 
  

   Solving 
  for 
  ai 
  we 
  find 
  that 
  

  

  Pjai 
  = 
  —n2(ao 
  + 
  '^i)i--n4(/C3)i 
  — 
  116(^5)1— 
  &c., 
  

  

  where 
  Pi, 
  Ilg, 
  II4, 
  &c., 
  are 
  the 
  infinite 
  determinant 
  operators 
  

   whose 
  leading 
  terms 
  are 
  ti, 
  t^, 
  T4, 
  Tg, 
  &c., 
  respectively 
  as 
  

   in 
  § 
  6. 
  

  

  