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 arek= 2 /3andki= 1 .0. Harriset 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,

e=

dψ

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:-

v=RI+

dψ

dt

Where dψ/dtwill in some windings be a combination of transformer induced and rotation-

ally induced emfs. The flux linkagesψwill 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:-

        

va

vb

vc

vf

vkd

vkq

        

=

        

Ra 000 0 0

0 Ra 00 0 0

00 Ra 000

000 Rfd 00

0000 Rkd 0

0000 0Rkq

        

        

ia

ib

ic

if

ikd

ikq

        

+p

        

Laa Mab Mac Maf Makd Makq

Mba Lbb Mbc Mbf Mbkd Mbkq

Mca Mcb Lcc Mcf Mckd Mckq

Mfa Mfb Mfc Lfdfd Mfkd Mfkq

Mkda Mkdb Mkdc Mkdf Lkdkd Mkdkq

Mkqa Mkqb Mkqc Mkqf Mkqkd Lkqkq

        

        

ia

ib

ic

if

ikd

ikq

        

(20.4)

Where,pis the differential operator

d

dt

.

Equation (20.4) has the matrix form, [v]=[R][i]+p[L][i].

The mutual inductancesMijin the triangle above the leading diagonal are equal to those

Mjiin the lower triangle and represent the mutual inductance between winding i and windingj.

Whereiandj take the sufficesa,b,cthrough tokq. For a salient pole synchronous generator

or motor some of the mutual and self-inductances are sinusoidal functions of the rotor positionθ.