﻿^2 
  Prof. 
  T. 
  R. 
  Lyle 
  on 
  the 
  Theory 
  of 
  

  

  18. 
  Returning 
  to 
  the 
  alternator, 
  if 
  n 
  be 
  the 
  number 
  of 
  

   armature 
  turns, 
  v 
  the 
  number 
  of 
  field 
  turns, 
  and 
  

  

  X 
  = 
  ai4-a3 
  + 
  a5 
  + 
  &c., 
  f 
  = 
  ^ 
  + 
  «2 
  + 
  «4-r 
  &c., 
  

  

  the 
  armature 
  and 
  field 
  currents 
  respectively, 
  the 
  magnetizing 
  

   current-turns 
  M.^ 
  producing 
  flux 
  across 
  the 
  air-gap 
  and 
  

   through 
  the 
  armature 
  in 
  a 
  direction 
  axial 
  to 
  its 
  windings 
  

   are 
  given 
  by 
  

  

  Mj. 
  = 
  nx 
  + 
  yf 
  cos 
  wt, 
  

  

  and 
  the 
  current-turns 
  M^ 
  producing 
  flux 
  across 
  the 
  air-gap 
  

   and 
  through 
  the 
  armature 
  in 
  a 
  direction 
  parallel 
  to 
  the 
  

   planes 
  of 
  the 
  windings, 
  and 
  behind 
  that 
  of 
  M^, 
  by 
  90^, 
  are 
  

   given 
  by 
  

  

  My 
  = 
  vf 
  sin 
  0)^, 
  

  

  or 
  in 
  vector 
  notation 
  (see 
  § 
  4) 
  

  

  which 
  produce 
  the 
  armature 
  fluxes 
  A^, 
  and 
  A^ 
  given 
  by 
  

  

  TT 
  

   ]/ 
  

  

  where 
  G^ 
  = 
  g^L~ 
  ^ 
  and 
  q 
  any 
  odd 
  number. 
  

  

  [Note 
  that 
  the 
  directions 
  of 
  A^ 
  and 
  A^ 
  are 
  fixed 
  in 
  the 
  

   armature.] 
  

  

  Now 
  if 
  magnetic 
  leakage 
  be 
  otherwise 
  taken 
  account 
  of, 
  

   the 
  flux 
  in 
  the 
  stator 
  must 
  be 
  continuous 
  with 
  that 
  in 
  the 
  

   rotor 
  so 
  that 
  the 
  flux 
  F 
  looped 
  on 
  the 
  field-windings 
  at 
  any 
  

   instant 
  is 
  given 
  by 
  

  

  F 
  = 
  A^ 
  cos 
  G)i 
  + 
  Ay 
  sin 
  wt, 
  

   which 
  by 
  means 
  of 
  the 
  relations 
  in 
  § 
  4 
  can 
  be 
  reduced 
  to 
  

  

  F 
  ^ 
  s[vG^«^ 
  + 
  |(G^_,a^_i 
  + 
  a^+ia^+i)J 
  

   where 
  7:* 
  is 
  even 
  and 
  2G^ 
  = 
  G^_i 
  + 
  G^_j_i. 
  

  

  