﻿with 
  Alternate 
  Current 
  Measuring 
  Instruments, 
  145 
  

  

  have 
  no 
  effect 
  upon 
  the 
  ratio. 
  This 
  agrees 
  with 
  Mr. 
  Campbell's 
  

   theoretical 
  and 
  experimental 
  conclusions. 
  

  

  Phase 
  Difference. 
  — 
  Reverting 
  to 
  equations 
  (11) 
  and 
  (12), 
  

   we 
  have 
  

  

  _ 
  A«sin<fts 
  + 
  A 
  w 
  

   tan 
  ft,- 
  A.cosfc 
  + 
  A*' 
  

  

  Hence 
  

  

  -x 
  cos(£ 
  s 
  — 
  x-sin^ 
  

   tan 
  = 
  tan 
  (*,,-*,) 
  = 
  V" 
  — 
  J" 
  ' 
  ( 
  17 
  > 
  

  

  1+ 
  -^-cos0 
  s 
  + 
  x-sin^ 
  

  

  which 
  is 
  zero 
  for 
  A 
  m 
  and 
  A 
  c 
  both 
  zero 
  or 
  for 
  tan 
  <j) 
  s 
  =s 
  — 
  as 
  

  

  is 
  obvious. 
  c 
  

  

  Hence 
  for 
  zero 
  phase-displacement 
  the 
  instrument 
  should 
  

  

  be 
  so 
  inductive 
  that 
  the 
  ratio 
  of 
  its 
  reactance 
  to 
  resistance 
  

  

  equals 
  the 
  ratio 
  of 
  the 
  magnetizing 
  to 
  the 
  core-loss 
  currents 
  

  

  of 
  the 
  transformer. 
  

  

  A, 
  Ac 
  

  

  As 
  the 
  relation 
  between 
  -~^ 
  and 
  -r- 
  is 
  not 
  in 
  general 
  linear 
  

  

  -a- 
  8 
  As 
  

  

  it 
  follows 
  that 
  the 
  phase-displacement 
  cannot 
  be 
  constant, 
  but 
  

   it 
  is 
  fairly 
  evident 
  that 
  the 
  best 
  constancy 
  would 
  be 
  approxi- 
  

   mately 
  obtained 
  when 
  the 
  actual 
  displacement 
  is 
  smallest, 
  

   i. 
  e. 
  with 
  an 
  inductive 
  instrument. 
  

  

  With 
  variation 
  of 
  frequency 
  it 
  is 
  again 
  obvious 
  that 
  with 
  a 
  

   fully 
  inductive 
  instrument, 
  without 
  eddy 
  currents 
  in 
  the 
  

   transformer, 
  the 
  phase-displacement 
  should 
  be 
  unaffected 
  

   by 
  frequency. 
  The 
  less 
  inductive 
  the 
  instrument 
  the 
  

   more 
  rapidly 
  should 
  the 
  phase-displacement 
  decrease 
  with 
  

   frequency, 
  owing 
  to 
  the 
  drop 
  in 
  the 
  induction 
  density. 
  

  

  P.D. 
  Relations. 
  — 
  Writing 
  the 
  secondary 
  current 
  in 
  the 
  

   form 
  A 
  s 
  (cos 
  fa+j 
  s 
  ^ 
  n 
  4>s) 
  and 
  the 
  primary 
  

  

  A 
  s 
  cos 
  (j) 
  s 
  + 
  A 
  c 
  +j(A 
  s 
  sin 
  <j>, 
  + 
  A 
  m 
  ), 
  

  

  let 
  r 
  p 
  and 
  r 
  s 
  be 
  the 
  primary 
  and 
  secondary 
  resistances 
  and 
  

   x 
  v 
  and 
  x, 
  the 
  primary 
  and 
  secondary 
  reactances 
  for 
  coils 
  of 
  

   a 
  single 
  turn 
  each. 
  

   Then 
  

  

  V 
  p 
  = 
  E 
  + 
  { 
  A, 
  cos 
  <£ 
  s 
  + 
  A 
  c 
  +j(A 
  8 
  sin 
  cp 
  8 
  + 
  A 
  m 
  ) 
  } 
  (r 
  p 
  —jap) 
  

  

  — 
  Y 
  8 
  = 
  E 
  — 
  A*(cos 
  (f> 
  8 
  +j 
  sin 
  <f) 
  8 
  ) 
  (r 
  8 
  —jx 
  s 
  ), 
  

   Phil 
  Mag. 
  S. 
  6. 
  Vol. 
  16. 
  No. 
  91. 
  July 
  1908. 
  L 
  

  

  