Theory  of  Pliase meters.  95 
Symmetrical  Six- Circuit  Phasemeter. 
Now  the  most  likely  form  of  phasemeter  to  be  accurate  on 
three-phase  circuits  with  unbalanced  loads,  is  naturally  one  in 
which  each  system  of  coils  consists  of  three  similar  and 
symmetrically  arranged  coils,  having  a  common  line  of  sym- 
metry, and  with  their  planes  fixed  at  120  degrees  from 
each  other.  One  of  these  sets  of  coils  must  be  actuated  by 
the  load  currents,  and  the  other  set  by  the  line  voltages. 
With  such  an  instrument,  we  must  have  three  equations 
like  (1)  for  the  induction  densities  at  the  three  moving  coils, 
or  we  have : — 
B2=A1F12+A2F22  +  A3F32  ) 
B3  =  A1F13  +  AaF28  +  A8Fi 
where  in  the  quantities  F  the  first  suffix  refers  to  the  fixed 
coil,  assumed  to  take  one  of  the  load  currents,  and  the  second 
suffix  to  the  moving  coil.  Thus  F23  means  the  line  average 
effective  induction  density  due  to  a  unit  current  in  the  second 
fixed  coil  at  the  conductors  of  the  third  moving  coil  when 
the  deflexion  of  the  pointer  is  x.  Now  under  the  symmetrical 
conditions  assumed  we  have  : — 
F11=F22=F33=P"\ 
F23=F31  =  F12=Q  >;      ....'.     (11) 
F32=F13=F21=R^ 
but  Q  and  R  are  not  necessarily  the  same.     Hence  we  have  : 
}■■ 
IB^AxP+AaR-fAaQl 
b2=a1q+a2p+a3rI 
Ba=A,R  +  A-Q  +  A,P  ) 
(12) 
A^  +  AgQ-f 
Now  if  the  three  moving  coils  are  star  connected,  and  their 
three  free  ends  joined  through  three  equal  non-inductive 
resistances  to  the  three  mains,  the  currents  through  the  three 
coils  will  be  represented  by  V1?  V2,  V3,  the  voltages  between 
the  neutral  point  and  the  corresponding  mains.  The  condition 
for  balance  constituting  an  equation  for  x  is  : 
VjBi  +  VS  +  V3B3  =  0, 
or 
PS  ViAi  +  Q2  VA  +  R2  V,  A2  =  0. 
Now  referring  to  fig.  5,  and  remembering  that  the  voltage 
vectors  are  of  equal  magnitude,  and  are  symmetrically  spaced, 
