V.  Synchronous  Reactance 

14.  In  general,  both  effects,  armature  self-inductance  and 
armature  reaction,  can  be  combined  by  the  term  "  synchronous 
reactance." 


FIG.    55. — Diagram  showing  effect 
of  synchronous  reactance. 


FIG.  56. — Diagram  of  generator 
e.m.f  s.  showing  affect  of  synchronous 
reactance  with  non-reactive  load. 


In  a  polyphase  machine,  the  synchronous  reactance  is  different, 
and  lower,  with  one  phase  only  loaded,  as  "  single-phase  synchro- 
nous reactance,"  than  with  all  phases  uniformly  loaded,  as  "  poly- 
phase synchronous  reactance."  The  resultant  armature  reac- 
tion of  all  phases  of  the  polyphase  machine  is  higher  than  that 
with  the  same  current  in  one  phase  only,  and  so  also  the  self- 


SYNCHRONOUS  MACHINES 


137 


inductive  flux,  as  resultant  flux  of  several  phases,  and  thus  rep- 
resents a  higher  synchronous  reactance. 

Let  r  =  effective  resistance, 

XQ  =  synchronous  reactance  of  armature,  as  discussed  in 
Section  II. 

Let  E  =  terminal  voltage, 
/  =  current, 

0  =  angle  of  lag  of  the  current  behind '  the  terminal  vol- 
tage. 

It  is  in  vector  diagram,  Fig.  55. 

OE  =  E  =  terminal  voltage  assumed  as  zero  vector.     01  = 


FIG.  57. — Diagram  of  generator 
e.m.fs.  showing  effect  of  synchronous 
reactance  with  lagging  reactive  load. 
6  =  60  degrees. 


FIG.  58. — Diagram  of  generator 
e.m.fs.  Showing  effect  of  synchro- 
nous reactance  with  leading  reactive 
load  6  =  —  60  degrees. 


I  =  current  lagging  by  the  angle  EOI  =  0  behind  the  terminal 
voltage. 

OE\  =  Ir  is  the  e.m.f.  consumed  by  resistance,  in  phase 
with  01  j  and  OE'o  =  Ix0  the  e.m.f.  consumed  by  the  synchronous 
reactance,  90  degrees  ahead  of  the_current  OI. 

OE'i  and  OE'Q  combined  give  OE'  =  E'  the  e.m.f.  consumed 
by  the  synchronous  impedance. 

Combining  OE'i,  OE'o,  OE  gives  the  nominal  generated  e.m.f. 
OEo  =  EQ,  corresponding  to  the  field  excitation  FQ. 

In  Figs.  56,  57,  58,  are  shown  the  diagrams  for  6  =  0  or  non- 
inductive  load,  6  =  60  degrees  lag  or  inductive  load,  and  &  — 
—  60  degrees  or  anti-inductive  load. 

Resolving  all  e.m.fs.  into  components  in  phase  and  in  quad- 
rature with  the  current,  or  into  power  and  reactive  components, 
in  symbolic  expression  we  have: 


138     ELEMENTS  OF  ELECTRICAL  ENGINEERING 

the  terminal  voltage  E  =  E  cos  6  +  jE  sin  6 ; 
the  e.m.f.  consumed  by  resistance,  E\  =  ir; 
the  e.m.f.  consumed  by  synchronous  reactance,  E'0  =  +  jixQ, 
and  the  nominal  generated  e.m.f., 

E0  =  E  +  E\  +  E'Q  =  (E  cos  0  +  ir)  +  j  (E  sin  19  +  ix0) ; 

or,     since 

.  „  »    ,     ,      ,  /      power  current  \ 

cos  6  =  p  =  power-factor  of  the  load  (  =  -.   —. — ) 

\       total  current  / 

and 

q  =  \/l  —  p2  =  sin  0  =  inductance  factor  of  the  load 

(wattless  current\ 
total  current'    /  ' 
it  is 

Eo  =  (Ep  +»  +  j  (Eq  +  ix0), 


or,  in  absolute  values, 


Eo  =  V(Ep  +  ir)2  +  (Eq  +  ^0)2; 
hence, 

E  =  VE02  -  i2  (x0p  -  rq)2  -  i  (rp  -f  x^q). 

The  power  delivered  by  the  alternator  into  the  external  cir- 
cuit is 

P  =  iEp; 

that  is,  the  current  times  the  power  component  of  the  terminal 
voltage. 

The  electric  power  produced  in  the  alternator  armature  is 

Po  =  i(Ep  +  ir)-, 

that  is,  the  current  times  the  power  component  of  the  nomi- 
nal generated  e.m.f.,  or,  what  is  the  same  thing,  the  current  times 
the  power  component  of  the  real  generated  e.m.f.