106  The  Theory  of  Phasemeters. 
is  required.  The  fixed  phase  relation  of  the  voltages  will  be 
unaffected  by  the  variation  o£  the  load-currents,  and  thus  the 
instrument  will  indicate  the  true  power-factor  of  the  particular 
current  chosen.  Indeed,  most  actual  phasemeters  are  con- 
structed and  used  in  precisely  this  manner;  and  the  error  on 
unbalanced  loads  is  given  by  a  formula  like  (24)  with  Qx 
substituted  for  00.  Reference  to  (30)  and  (31)  will  show  that 
the  error  made  in  assuming  that  the  reading  indicates  the 
average  power-factor  of  the  load  is  essentially  the  same  as  in 
the  case  considered,  and  to  which  Table  III.  applies. 
In  conclusion,  the  main  results  of  the  foregoing  investiga- 
tion may  be  thus  summarised  : — 
Summary. 
1.  Phasemeters  for  multi-phase  circuits  are  all  equally 
accurate  on  balanced  loads  provided  they  have  been  correctly 
calibrated  and  possess  no  faults  due  to  purely  mechanical 
causes.  Their  accuracy  is  not  affected  by  variations  in  wave- 
form or  in  current-frequency.  The  calibration  of  the  scale 
is  affected  by  the  number  of  coils  used  in  the  instrument,  by 
the  ratios  of  the  ampere-turns  used  with  these  coils,  by  the 
distribution  of  the  windings,  and  by  the  magnetic  nature  and 
properties  of  the  magnetic  circuits,  especially  if  these  contain 
iron  ;  but  the  accuracy  of  the  indications  is  not  dependent 
upon  any  of  these  considerations. 
2.  Phasemeters  can  be  simply  and  accurately  calibrated 
for  balanced  loads  by  means  of  a  direct-current  method  of 
test. 
3.  The  error  of  phasemeters  on  unbalanced  circuits  is 
generally  serious  for  loads  which  are  badly  out  of  balance. 
The  error,  like  that  of  a  wattmeter,  increases  rapidly  as  the 
power-factor  of  the  load  diminishes.  It  can  only  be  reduced 
at  the  expense  of  complication  in  the  instrument,  by  in- 
creasing the  number  of  coils  used  in  the  fixed  and  moving 
systems,  and  by  arranging  the  coils  and  magnetic  circuits  to 
be  symmetrical  in  regard  to  one  another.  If  the  true  power- 
factor  of  the  load  is  cos  (/>,  the  reading  of  the  instrument  is 
cos  cj)  +  6  sin  cf), 
where  6  is  the  phase  error  due  not  to  the  instrument  but  to 
the  unbalanced  load,  and  is  the  product  of  two  factors  one  of 
which  is  the  maximum  value  of  6  determined  by  the  amount 
the  load  is  out  of  balance,  and  the  other  may  have  any  value 
between  +1  and  —1  and  is  the  factor  which  determines 
whether  the  instrument  reads  high  or  low.  The  maximum 
value  of  6  is  as  follows  : — 
