﻿the 
  Alternate 
  Current 
  Generator. 
  

  

  73 
  

  

  In 
  fig. 
  -4, 
  let 
  Oao, 
  taken 
  in 
  the 
  axis 
  of 
  Y 
  be 
  equal 
  (as 
  in 
  § 
  13) 
  

   to 
  twice 
  the 
  steady 
  exciting 
  current 
  o£ 
  the 
  machine. 
  Draw 
  

  

  Fio-. 
  4. 
  

  

  the 
  vector 
  OE' 
  to 
  represent 
  in 
  amplitude 
  and 
  phase 
  2/ft)»i 
  

   times 
  the 
  applied 
  e.m.f. 
  ; 
  that 
  is, 
  i£ 
  the 
  latter 
  

  

  = 
  e' 
  sin 
  (wt 
  + 
  It), 
  OE' 
  = 
  2e^/(om, 
  

  

  and 
  the 
  angle 
  from 
  OX 
  to 
  OE^ 
  measured 
  in 
  the 
  positive 
  

   direction 
  is 
  = 
  h. 
  

  

  Rotating 
  OE' 
  forward 
  through 
  90° 
  gives 
  us 
  /c 
  of 
  § 
  25, 
  

   and 
  completing 
  the 
  parallelogram 
  ccqOkj 
  its 
  diagonal 
  is 
  ixq-\-k 
  

   in 
  the 
  line 
  OY'. 
  

  

  Knowing 
  the 
  motor 
  circuits, 
  we 
  can 
  determine 
  Si, 
  bi, 
  

   <^-2^ 
  ^2, 
  H^ 
  ^35 
  &c., 
  and 
  then 
  construct 
  for 
  ai, 
  ag, 
  sl^, 
  oc^, 
  &c., 
  

   exactly 
  as 
  in 
  § 
  13, 
  except 
  that 
  in 
  this 
  construction 
  the 
  vector 
  

   ccq 
  + 
  k 
  takes 
  the 
  place 
  of 
  ccq 
  in 
  § 
  13 
  (see 
  § 
  24} 
  . 
  

  

  Xow 
  the 
  mechanical 
  torque 
  developed 
  hy 
  the 
  machine 
  is 
  

   (see 
  § 
  12) 
  

  

  71X 
  

  

  = 
  ^{SaoBi 
  + 
  Saiag 
  + 
  Sagao 
  + 
  cfcc.} 
  

  

  = 
  ^into 
  the 
  sum 
  of 
  the 
  areas, 
  attending 
  to 
  signs, 
  

   of 
  the 
  triangles 
  aoO^-i? 
  a^Oas, 
  aoOaj, 
  &c. 
  

  

  But 
  the 
  triangles 
  aiOa^? 
  «20%5 
  ^•S^o'-A', 
  &c., 
  are 
  all 
  essen- 
  

   tially 
  negative 
  [their 
  sum 
  is 
  = 
  — 
  ^Bao 
  + 
  /c^ 
  of 
  § 
  25], 
  so 
  that 
  

   if 
  the 
  machine 
  is 
  to 
  develop 
  mechanical 
  power 
  and 
  run 
  as 
  a 
  

   motor, 
  the 
  phase 
  of 
  OE' 
  must 
  be 
  such 
  that 
  the 
  area 
  of 
  the 
  

   triangle 
  aQOai 
  is 
  positive 
  (as 
  it 
  is 
  in 
  fig. 
  4) 
  and 
  numerically 
  

   greater 
  than 
  the 
  sum 
  of 
  a^Oas, 
  a20a3, 
  &c. 
  

  

  