94  Dr.  W.  E.  Sumpner  on  the 
Poicer-f actor  for  Unbalanced  Loads. 
It  has  been  assumed  throughout  the  foregoing  theory  that 
the  load  currents  are  balanced  ;  a  most  desirable  condition  in 
practice,  but  one  rarely  attained.  So  far  as  the  writer  is 
aware,  the  behaviour  of  phasemeters  on  unbalanced  loads 
has  not  previously  been  investigated.  There  is  a  fairly 
general  impression  that  they  are  inaccurate  under  these 
circumstances,  and  for  this  phasemeters  have  been  blamed  ; 
though  the  fault  rather  lies  with  those  who  assume  that  an 
instrument,  which  is  only  affected  by  a  few  of  the  con- 
ditions of  a  circuit,  can  be  reasonably  expected  to  measure 
the  average  of  such  conditions. 
Now,  in  the  first  place,  there  is  not  even  any  generally 
accepted  definition  of  the  power-factor  of  a  three-phase 
circuit  when  the  load  currents  are  unbalanced.  There  are 
really  three  circuits,  consisting  of  each  line  current  and  the 
voltage  between  this  line  and  the  neutral  point.  There  are 
thus  three  power-factors,  one  for  each  circuit.  Fortunately 
the  voltages,  being  fixed  by  the  generator,  can  always  be 
regarded  as  equal  and  in  symmetrical  phase  relation;  other- 
wise the  definition  of  the  average  power-factor  of  the  load 
would  be  most  complicated.  But  wre  shall  take  the  following 
definition  of  cos  </>  for  unbalanced  loads  : — 
_  Ax  cos  0!  4-  A2  cos  cj)2  +  A3  cos  j>3  . 
COS0~  A1  +  A2  +  A3  ;    *         ^ 
where  A1?  A2,  A3  are  the  magnitudes  of  the  three  load 
currents,  and  <£]?  cf>2,  4>s  are  the  phase  angles  between  these 
currents  and  the  corresponding  voltages  to  the  neutral  point. 
This  formula  for  cos  <£  merely  amounts  to  defining  the  volt- 
amperes  of  an  unbalanced  three-phase  circuit  as  being  the 
product  of  the  voltage  to  the  neutral 
point  and  the  sum  of  the  three- 
line  currents.  The  relations  of  the 
different  quantities  are  indicated  in 
fig.  5,  in  which  the  voltage  vectors 
are  supposed  to  be  of  equal  mag- 
nitude and  symmetrically  spaced.  V," 
If  this  were  not  the  case,  we  could 
still  determine  a  neutral  point 
such  that  the  voltages  between  it 
and  each  line  would  be  equal  ;  and 
the  definition  (9)  would  still  hold 
good  provided  the  phase  angles  <j>1} 
</>25  </>3  were  reckoned  from  the  corresponding  voltages  to  the 
neutral  point  so  determined. 
