Theory  of  Pluuemeters.  87 
the  position  of  balance  will  be  the  same  as  in  the  correspond- 
ing alternating  current  test,  when  the  phase  of  the  current 
through,  the  moving  coil  differs  from  that  of  the  current  Ai 
by  0. 
From  (7)  we  have 
tan*=4vFA? (8) 
from  which  <£  and  cos  <j>  can  be  found.  If  cos  </>  so  found  is 
to  represent  the  power-factor  of  the  alternate  current  load,  all 
that  is  necessary  is  to  select  for  the  voltage  V  that  between 
the  main  conductor  carrying  the  current  Ax  and  the  neutral 
point  of  the  system,  and  to  apply  it  through  a  non-inductive 
resistance  to  the  moving  coil.  If,  however,  instead  of  this 
resistance  a  condenser  be  used,  the  voltage  selected  should  be 
that  between  the  lines  carrying  A2  and  A3,  since  the  effect  of 
the  condenser  will  be  to  cause  the  moving  coil  current  to  be  in 
quadrature  with  the  voltage  producing  it.  The  shape  of  the 
curve  connecting  the  deflexion  x  with  0  or  cos  <£  will  depend 
on  the  nature  of  the  functions  F  in  (6),  and  these  are  deter- 
mined by  the  structure  of  the  instrument.  With  actual 
phasemeters  the  coils  are  arranged  with  the  object  of  making 
the  quantities  F  of  equal  magnitude,  and  as  nearly  as  possible 
sinuous  f  mictions  of  the  deflexion  x  ;  and  the  angle  between 
the  planes  of  any  two  coils  is  made  equal  to  the  angle 
representing  the  phase  difference  of  the  currents  passing 
through  the  coils.  Such  assumptions  are  equivalent  to 
putting  : 
F^/sin*  ;     F2=/sin  (a?- 120) ;     F3=/sin  (#  +  120); 
and  if  we  substitute  these  expressions  in  (6)  and  reduce,  we 
easily  find  sin  ($  +  *)  =  0, 
or  the  deflexion  x  in  degrees  is  equal  to  <j>  when  measured 
from  an  appropriate  zero. 
But  the  accuracy  of  the  instrument  as  a  phasemeter  in 
no  way  depends  upon  the  fulfilment  of  these  conditions  of 
construction.  The  introduction  of  iron  into  the  magnetic 
circuits,  and  the  use  of  unsymmetrical  winding  in  the  coils, 
will  affect  the  shape  of  the  curve  connecting  x  and  </>,  but 
will  not  prevent  as  from  being  an  accurate  indication  of  <f>. 
Nor  is  it  even  desirable  that  x  should  be  proportional  to  <£. 
It  would  be  much  more  useful  to  arrange  to  make  x  fairly 
proportional  to  cos  <f>.  Phasemeters  are  wanted  to  measure 
not  (f>,  but  cos  </>,  or  the  power-factor,  and  this  for  load 
currents  which  in  practice  are  always  lagging.  By  making 
x  proportional  to  (/>  the  scale-readings  For  different  power- 
factors  are    widely   separated   where    least    wanted,   that    is 
