Theory  of  Phasemeters.  105 
to  the  way  the  load  is  out  of  balance,  between  the  values 
cos  (£-|-e2isin  </>, 
so  that  for  a  load  2  per  cent,  out  of  balance  the  reading  of 
the  power-factor  cannot  vary  from  the  true  \alue  by  more 
than  +'028  as  extreme  limits.  The  above  formula  applies  to 
an  instrument  having  three  coils  in  one  system  and  a  single 
coil  in  the  other  and  the  error  is  independent  of  the  structure 
of  the  instrument.  The  error  is  largely  controlled  by  the  value 
of  tan  <j>,  where  <fi  is  the  angle  which  we  have  assumed  to 
represent  both  the  power-factor,  and  the  phase-difference  of  the 
moving-coil  current,  in  reference  to  the  current  in  one  of  the 
fixed  coils.  As  any  voltage  of  the  multi-phase  system  may  be 
chosen  for  the  moving  coil,  provided  the  instrument  has  been 
correspondingly  calibrated,  it  might  at  first  sight  appear 
possible  to  select  such  a  voltage  for  the  moving  system  as  to 
make  the  value  of  tan  <f>  small  for  the  particular  power-factors 
the  instrument  is  most  required  to  read,  and  thus  render  the 
error  under  practical  conditions  negligible  on  unbalanced 
loads.  A  careful  examination  will,  however,  show  that  this 
is  not  the  case. 
As  already  shown,  an  instrument  having  three  current- 
coils  and  three  voltage-coils,  all  symmetrically  arranged,  will 
read  correctly  whether  the  load-currents  are  balanced  or  not. 
But  such  perfect  symmetry  is  easier  to  assume  in  a  mathe- 
matical investigation  than  to  ensure  in  the  structure  of  so 
complicated  an  instrument.  If  one  of  the  moving  coils  is 
dispensed  with,  as  in  the  other  symmetrical  case  considered, 
the  instrument  is  simpler  to  construct  and  its  symmetry 
easier  to  secure.  Such  a  five-circuit  instrument  is  not  quite 
accurate  on  unbalanced  loads ;  but  (19)  shows  that  the  error 
is  only  one -third  as  great  as  in  the  four-circuit  instrument  to 
which  Table  III.  applies  ;  and  it  will  be  seen  that  for  all 
load-currents  such  as  are  likely  to  occur  in  practice,  the  error 
is  small  enough  in  this  case  to  be  neglected.  The  six-circuit 
instrument  consists  essentially  of  three  single-phase  instru- 
ments combined  into  one.  The  best  solution  of  the  phase- 
meter  problem  would  no  doubt  consist  of  three  single-phase 
instruments  used  one  on  each  circuit.  There  are  difficulties, 
not  yet  overcome,  in  constructing  good  instruments  for  single- 
phase  circuits.  But  such  instruments,  even  if  they  existed, 
would  at  present  be  used  mostly  on  multi-phase  circuits,  and 
on  such  circuits  no  difficulty  arises.  All  that  is  needed  is  for 
the  instrument  to  have  one  of  its  systems  of  coils  arranged 
multi-phase  for  two  or  more  of  the  voltages,  and  to  have  its 
other  system  single-phase  for  the  current  whose  power-factor 
