﻿with 
  Alternate 
  Current 
  Measuring 
  Instruments. 
  143 
  

  

  Fig. 
  2. 
  

  

  Then 
  we 
  have 
  

  

  A 
  p 
  4- 
  A 
  s 
  = 
  A 
  vectorially 
  ; 
  

  

  or 
  writing 
  the 
  currents 
  in 
  the 
  Steinmetz 
  notation, 
  

  

  A 
  5 
  = 
  — 
  A 
  s 
  cos 
  <j> 
  8 
  —jA 
  s 
  sin 
  <p 
  s 
  . 
  . 
  

   A 
  = 
  A 
  c 
  +;A 
  m 
  . 
  

  

  and 
  

  

  Hence 
  

  

  A_p=A 
  — 
  A 
  s 
  =A 
  s 
  cos 
  $ 
  8 
  + 
  A 
  c 
  + 
  (A 
  s 
  sin(j)s 
  + 
  A 
  m 
  ) 
  

  

  (11) 
  

  

  (12) 
  

  

  A 
  p 
  = 
  \/(A 
  s 
  cos0 
  s 
  + 
  A 
  c 
  ) 
  2 
  +(A,sin^ 
  s 
  + 
  A 
  TO 
  ) 
  2 
  

   = 
  VA 
  S 
  2 
  + 
  2A 
  S 
  (A 
  C 
  cos 
  <j) 
  s 
  + 
  A 
  m 
  sin 
  <j> 
  s 
  ) 
  + 
  A 
  2 
  . 
  

  

  If 
  A 
  is 
  small 
  in 
  comparison 
  with 
  A 
  s 
  as 
  it 
  should 
  always 
  

   be 
  in 
  practice, 
  we 
  may 
  write 
  

  

  A 
  p 
  -. 
  A 
  e 
  ± 
  t 
  A 
  m 
  

  

  A 
  s 
  _i 
  " 
  

  

  ~^- 
  cos 
  (j) 
  s 
  + 
  -^ 
  sin 
  <f> 
  s 
  . 
  

  

  (13) 
  

  

  I£ 
  R 
  c 
  is 
  the 
  ratio 
  o£ 
  the 
  primary 
  to 
  secondary 
  currents, 
  

   and 
  R* 
  the 
  ratio 
  of 
  secondary 
  to 
  primary 
  turns, 
  we 
  have 
  

  

  R 
  C 
  = 
  R, 
  (l+^cos0 
  s 
  +^sin<£ 
  s 
  \ 
  . 
  . 
  (14) 
  

  

  Hence 
  the 
  ratio 
  o£ 
  the 
  currents 
  can 
  only 
  be 
  equal 
  to 
  the 
  

   transformation 
  ratio 
  if 
  A 
  c 
  and 
  A 
  TO 
  are 
  both 
  zero 
  or 
  

  

  tan0 
  s 
  = 
  — 
  ~- 
  zs 
  —cot 
  a. 
  

  

  An 
  

  

  