﻿the 
  Alternate 
  Current 
  Generator. 
  57 
  

  

  In 
  OY 
  take 
  Oao 
  = 
  2Vp 
  = 
  2x 
  exciting 
  current. 
  Produce 
  

  

  10 
  through 
  to 
  ai 
  so 
  that 
  Oai 
  = 
  -^. 
  In 
  02 
  take 
  

  

  0«2 
  = 
  Oai/o-2. 
  Produce 
  30 
  through 
  to 
  ag 
  so 
  that 
  

  

  Oa, 
  = 
  — 
  ^. 
  and 
  so 
  on 
  where 
  5i, 
  o-g, 
  53, 
  0-4, 
  &c., 
  are 
  the 
  

  

  sz 
  

   quantities 
  determined 
  in 
  § 
  8. 
  

  

  Then 
  the 
  vectors 
  to 
  ai, 
  ag, 
  as, 
  &c., 
  represent 
  completely 
  in 
  

   amplitude 
  and 
  phase 
  the 
  different 
  harmonics 
  of 
  the 
  armature 
  

   current, 
  the 
  subscribed 
  numbers 
  indicating 
  the 
  orders 
  of 
  the 
  

   harmonics 
  ; 
  and 
  those 
  to 
  a.^^ 
  a^, 
  cxq, 
  &c., 
  represent 
  completely 
  

   in 
  the 
  same 
  way 
  the 
  different 
  harmonics 
  o£ 
  the 
  induced 
  

   alternating 
  field 
  current. 
  

  

  Again 
  (see 
  § 
  10), 
  if 
  we 
  rotate 
  the 
  vector 
  ^ 
  drawn 
  to 
  the 
  

   middle 
  point 
  of 
  ao«2 
  backwards 
  through 
  a 
  right 
  angle, 
  we 
  

  

  obtain 
  the 
  vector 
  OEi 
  that 
  represents 
  — 
  into 
  the 
  first 
  

  

  ^ 
  met) 
  

  

  harmonic 
  of 
  the 
  total 
  e.m.f. 
  E 
  generated 
  in 
  the 
  armature 
  ; 
  

   and 
  if 
  we 
  rotate 
  backwards 
  through 
  7r/2 
  the 
  vector 
  to 
  the 
  

   middle 
  point 
  of 
  a2«4 
  ^^ 
  obtain 
  the 
  vector 
  OE3 
  that 
  re- 
  

   presents 
  :^ 
  — 
  into 
  the 
  third 
  harmonic 
  of 
  E 
  ; 
  and 
  similarly 
  

  

  for 
  the 
  other 
  harmonics 
  of 
  E. 
  

  

  In 
  the 
  same 
  way, 
  by 
  rotating 
  backwards 
  through 
  7r/2 
  the 
  

   vector 
  to 
  the 
  middle 
  point 
  of 
  aias, 
  we 
  obtain 
  the 
  vector 
  OHg 
  

  

  that 
  represents 
  7^ 
  — 
  into 
  the 
  fundamental 
  harmonic 
  of 
  the 
  

   ^ 
  zoom 
  

  

  E.M.F. 
  H 
  induced 
  in 
  the 
  field 
  circuit 
  ; 
  and 
  so 
  on 
  for 
  the 
  

   other 
  harmonics 
  of 
  H. 
  

  

  Again, 
  as 
  Sa/S 
  is 
  = 
  twice 
  the 
  area 
  of 
  the 
  triangle 
  whose 
  

   sides 
  are 
  a 
  and 
  ^ 
  and 
  is 
  positive 
  if 
  y8 
  follows 
  a 
  in 
  rotation 
  

   order 
  in 
  the 
  diagram, 
  the 
  mean 
  torque 
  exerted 
  on 
  the 
  

   generator 
  by 
  the 
  driver 
  (see 
  § 
  12) 
  is 
  equal 
  to 
  ^m 
  into 
  the 
  

   sum 
  of 
  the 
  areas 
  of 
  the 
  triangles 
  ^QOai, 
  aiOag, 
  a20a3, 
  Si^Ooi4^, 
  &c., 
  

   these 
  triangles, 
  in 
  the 
  case 
  of 
  any 
  generator^ 
  being 
  all 
  taken 
  

   as 
  positive. 
  

  

  14. 
  When, 
  for 
  any 
  generator, 
  the 
  t, 
  r 
  operators 
  have 
  been 
  

   calculated 
  for 
  a 
  particular 
  load 
  (see 
  § 
  5), 
  a 
  geometrical 
  

   solution 
  can 
  easily 
  be 
  obtained 
  to 
  a 
  high 
  degree 
  of 
  accuracy 
  

   by 
  aid 
  of 
  a 
  ruler, 
  scale, 
  slide-rule, 
  and 
  protractor. 
  

  

  Thus 
  if 
  we 
  neglect 
  the 
  harmonics 
  ag, 
  ag, 
  u^q, 
  &c., 
  then, 
  

   drawing 
  any 
  vector 
  from 
  the 
  origin 
  to 
  represent 
  Si^ 
  we 
  can 
  

   construct 
  for 
  ct^ 
  as 
  aQ= 
  —tjSij 
  (see 
  § 
  6). 
  From 
  uq 
  we 
  can 
  con- 
  

   struct 
  for 
  T6a6 
  and 
  as 
  SL^ 
  + 
  '^e'^e-^^j 
  — 
  O 
  the 
  triangle 
  of 
  vectors 
  

  

  