Theory  of  Phasemeters.  93 
other  instrument,  the  coils  were  wound  in  six  slots  stamped 
symmetrically  in  the  periphery  of  the  inner  stator  S;.  In 
either  case,  the  curve  found  between  x  and  <f>  was  of  the  same 
general  character  as  that  shown  in  fig.  3,  except  that,  as  the 
windings  were  in  these  instruments  60  degrees  apart,  it  was 
found  that  the  flat  parts  of  the  curve  covered  a  greater 
angular  range,  and  that  the  angular  period  of  the  steps  was  60° 
instead  of  30°.  In  each  instrument  the  direct  and  alternating 
current  tests  were  found  to  be  in  close  agreement,  and 
the  direct-current  calibration  of  one  of  them  was  found  to 
coincide  with  a  large  number  of  alternating-current  tests  taken 
on  the  same  instrument  more  than  eight  months  previously. 
It  therefore  appears  clear,  both  from  theory  and  experiment, 
that  the  accuracy  of  these  instruments  on  balanced  loads  in 
no  way  depends  upon  the  structure  of  the  coils,  or  upon  the 
presence  or  absence  of  iron,  or  upon  the  mode  of  variation 
of  the  currents.  The  theory  given  of  the  four-circuit  instru- 
ment described  will  be  seen  to  be  equally  applicable 
whichever  of  the  two  systems  of  coils  is  fixed,  so  that  if  the 
fixed  system  consists  of  a  single  coil  and  the  movable  system 
consists  of  three  relatively  fixed  coils,  the  same  theory  applies. 
Three- Circuit  and  Monophase  Instruments. 
If  the  three-coil  system  is  reduced  to  a  two-coil  system,  as 
in  most  actual  phasemeters,  only  a  slight  modification  of  the 
theory  is  required,  and  the  instrument  can  still  be  tested  by 
direct-current  methods.  All  that  is  needed  is  to  put  Fx  =  0 
in  equations  1,  2,  3,  and  6.  Equations  4,  5,  7,  and  8  still 
hold  good,  and  in  the  direct-current  calibration  there  are 
only  two  currents  to  consider,  and  these  may  have  any 
values.  For  a  single-phase  instrument,  which  merely  differs 
from  a  multi-phase  instrument  in  that  its  two-coil  system  is 
parallel  connected,  and  the  branch  circuits  made  of  different 
inductive  properties,  a  similar  theory  applies.  For  a  fixed 
frequency  and  wave  form  there  will  be  a  constant  ratio 
between  P  and  Q,  the  two  alternating  currents  in  the  two- 
coil  system,  and  a  constant  phase  difference  a  between  these 
currents.  If,  then,  <£  is  the  phase  difference  between  the 
alternating  current  in  the  single-coil  system  and  that  through 
P,  the  corresponding  direct  currents  through  the  coils  P  and 
Q  which  will  produce  the  same  deflexion  will  be 
A2  =  Pcos<£, 
A3  =  Qcos  ((£  —  a). 
If  therefore  a  and  the  ratio  P  :  Q  are  known,  it  is  possible 
to  determine  <fi  for  any  ratio  A2  :  A3    of   the    currents  pro- 
ducing the  observed  deflexion. 
