Handbook of Electrical Engineering

GENERALISED THEORY OF ELECTRICAL MACHINES 489

Armature time constantTa

X 2

ωRa

Q-axis sub-transient open-circuit time constantTqo′′=

1

ωRkq

(Xkq+Xmq)

Q-axis sub-transient short-circuit time constantTq′′=

1

ωRkq

(

Xkq+

XmqXa

Xmq+Xa

)

Q-axis damper leakage time constantTkq=

1

ωRkq

Xkq

Negative phase sequence reactance

X 2 =

√

X′′d·Xq′′ or

X′′d+X′′q

2

or

2 Xd′′X′′q

X′′d+X′′q

Zero phase sequence reactanceXohas a value lower thanXd′′and is a complex function of

the slot pitching of the stator windings and the leakage reactance present in their end windings,

see Reference 7, Chapter XII.

i) Operational impedances in thed-axis.

The equation for the operational impedance that relates thed-axis flux linkages to the stator current

idand the rotor excitationvfis,

d=

Xd(p)

ω

id+

G(p)

ω

vf ( 20. 19 )

where,Xd(p)=

( 1 +Td′′p)( 1 +Td′′p)

( 1 +Tdo′′p)( 1 +Tdo′′p)

Xd

and,G(p)=

( 1 +Tkdp)

( 1 +Tdo′′p)( 1 +Tdo′′p)

Xmd

Rf

j) Operational impedance in theq-axis.

The equation that relates theq-axis flux linkages to the stator currentiqis,

q=

Xq(p)

ω

iq ( 20. 20 )

Where,Xq(p)=

( 1 +Tq′′p)

( 1 +Tqo′′p)

Xq

The process of obtaining expressions for the derived reactances, operational impedances and
time constants was based on the notion that only one damper winding exists on each axis. Krause
in Reference 5 applied the process to a synchronous machine that has two damper windings on the
q-axis. This would be advantageous when studies are being performed with large solid pole machines
such as steam power plant generators, which are nowadays rated between 100 and 660 MW. Very
similar functions are formed for theq-axis as are formed for thed-axis. To represent three windings
on the d-axis would require a formidable amount of algebraic manipulation, from which the benefits
may only be small and there will then be the problem of obtaining the extra parameters from either
design data or factory tests.