﻿with 
  Alternate 
  Current 
  Measuring 
  Instruments. 
  147 
  

  

  P.D. 
  Phase 
  Relations. 
  — 
  From 
  equations 
  (18) 
  and 
  (19) 
  we 
  

   obtain 
  

  

  ** 
  (cos 
  <j> 
  s 
  + 
  j- 
  c 
  J 
  - 
  r 
  p 
  (sin 
  <f> 
  s 
  + 
  ^J 
  

  

  tan 
  "V>= 
  ?= 
  — 
  r 
  -r 
  — 
  r 
  — 
  A, 
  

  

  E 
  + 
  A, 
  |r 
  p 
  (cos 
  cf> 
  s 
  + 
  ^) 
  + 
  x 
  v 
  (sin 
  0, 
  + 
  -£ 
  J 
  j- 
  

  

  # 
  s 
  cos 
  s 
  —r 
  s 
  sin 
  <£ 
  s 
  A 
  

  

  tan 
  vr 
  s 
  = 
  -^ 
  r~r 
  — 
  i 
  ^ 
  — 
  rvA 
  s 
  . 
  

  

  7 
  Jii 
  — 
  A 
  g 
  {r 
  g 
  cos9 
  s 
  + 
  ^s 
  sin 
  S 
  } 
  

  

  If, 
  as 
  should 
  be 
  the 
  case 
  with 
  voltage 
  transformers, 
  the 
  

   resistance 
  and 
  inductive 
  drop 
  in 
  the 
  transformer 
  are 
  small 
  

   compared 
  with 
  E, 
  we 
  have 
  

  

  Yp-^s=P= 
  jf 
  • 
  (23) 
  

  

  as 
  is 
  geometrically 
  obvious. 
  

  

  For 
  the 
  phase 
  -difference 
  to 
  be 
  zero 
  it 
  follows 
  that 
  either 
  

   the 
  resistance 
  and 
  inductance 
  must 
  be 
  zero, 
  or 
  that 
  

  

  , 
  , 
  __ 
  A 
  s 
  2 
  rx 
  + 
  (AeXp 
  - 
  A 
  m 
  r 
  p 
  ) 
  <y 
  A 
  s 
  \x 
  2 
  + 
  r 
  2 
  ) 
  - 
  (A 
  c 
  x 
  p 
  - 
  A 
  m 
  r 
  P 
  ) 
  2 
  ( 
  ^., 
  

   tan 
  * 
  s 
  A 
  s 
  r 
  2 
  -(A 
  C 
  ^-A 
  w 
  ^)' 
  " 
  " 
  V* 
  } 
  

  

  In 
  this 
  expression 
  if 
  As 
  is 
  large 
  compared 
  with 
  A 
  c 
  or 
  A 
  m 
  

  

  tan<f)5= 
  - 
  or 
  the 
  ratio 
  of 
  the 
  reactance 
  to 
  the 
  resistance 
  of 
  

   r 
  r 
  J 
  

  

  the 
  instrument 
  should 
  be 
  the 
  same 
  as 
  for 
  the 
  transformer, 
  and 
  

   the 
  instrument 
  should 
  therefore 
  be 
  fairly 
  inductive. 
  

   With 
  a 
  non-inductive 
  instrument 
  

  

  E~ 
  

  

  Q 
  _ 
  A 
  ?' 
  6 
  ~r~ 
  A 
  e'^ 
  ^m' 
  p 
  (*>*)) 
  

  

  and 
  if 
  the 
  instrument 
  takes 
  no 
  appreciable 
  current 
  compared 
  

   with 
  A 
  c 
  or 
  A 
  m 
  , 
  

  

  O 
  __ 
  Ae^'p 
  An 
  r 
  p 
  __ 
  (26) 
  

  

  To 
  find 
  the 
  variation 
  of 
  phase-displacement 
  with 
  voltage 
  we 
  

   have 
  evidently 
  from 
  (23) 
  

  

  and 
  this 
  is 
  evidently 
  zero 
  for 
  an 
  air-core 
  transformer. 
  

  

  Relation 
  between 
  the 
  Secondary 
  Current 
  and 
  the 
  Magnetizing 
  

   and 
  Core-Loss 
  Currents. 
  — 
  If 
  I 
  is 
  the 
  length 
  of 
  the 
  magnetic 
  

  

  L2 
  

  

  