Page 494 - Handbook of Electrical Engineering

484 HANDBOOK OF ELECTRICAL ENGINEERING

√
the 0.5 and 1.0 constant become 1/2 see References 10, 13 and 14. The most commonly used
constants are k = 2/3and k i = 1.0. Harris et al, Reference 13, Chapter 3, discuss this subject at
length, in relation to power invariance and the choice of base parameters for per-unit systems.
Bimbhra, Reference 10, also discusses transformations in considerable detail.
From (20.1) the emf induced in a winding is,
dψ
e =
dt
The voltage (v) applied to the winding must always balance this emf (e) and the resistive
volt-drop (IR) of the winding conductor carrying the current, hence:-

dψ
v = RI +
dt
Where dψ/dt will in some windings be a combination of transformer induced and rotation-
ally induced emfs. The flux linkages ψ will be the sum of its own linkages due to its own currents
and all the linkages from windings sharing the same magnetic circuit. For the synchronous gener-
ator which has three stator windings and three rotor windings, as described in sub-section 20.2 a)
to g), the set of voltage equations are:-

     
v a R a 0 0 0 0 0 i a
   0 R a 0 0 0 0   i 
 v b     b 
 0 0 R a 0 0
     
 v c  0   i c 
 0 0 0 0
  =    
 v f  R fd 0   i f 
     
 0
 v kd  0 0 0 R kd 0   i kd 
0 0 0 0 0
v kq R kq i kq
   
L aa M ab M ac M af M akd M akq i a
   i 
M ba L bb M bc M bf M bkd
 M bkq   b 
   
 M ca M cb L cc M cf M ckd M ckq   i c 
(20.4)
  
+ p 
 M fa M fb M fc L fdfd M fkd M fkq   i f 
   
 M kda M kdb M kdc M kdf L kdkd M kdkq   i kd 
M kqa M kqb M kqc M kqf M kqkd L kqkq i kq
d
Where, p is the differential operator .
dt
Equation (20.4) has the matrix form, [v] = [R][i] + p[L][i].
The mutual inductances M ij in the triangle above the leading diagonal are equal to those
M ji in the lower triangle and represent the mutual inductance between winding i and winding j.
Where i and j take the suffices a, b, c through to k q . For a salient pole synchronous generator
or motor some of the mutual and self-inductances are sinusoidal functions of the rotor position θ.