Chapter 7 machine equations in operational impedances and time constants

271

7.1. INTRODUCTION

In Chapter 5 , we assumed that the electrical characteristics of the rotor of a synchronous

machine could be portrayed by two windings in each axis. This type of a representation

is suffi cient for most applications; however, there are instances where a more refi ned

model may be necessary. For example, when representing solid iron rotor machines, it

may be necessary to use three or more rotor windings in each axis so that transient

dynamics are accurately represented. This may also be required to accurately capture

switching dynamics when modeling machine/rectifi er systems.

R.H. Park [1] , in his original paper, did not specify the number of rotor circuits.

Instead, he expressed the stator fl ux linkages in terms of operational impedances and

a transfer function relating stator fl ux linkages to fi eld voltage. In other words, Park

recognized that, in general, the rotor of a synchronous machine appears as a distributed

parameter system when viewed from the stator. The fact that an accurate, equivalent

lumped parameter circuit representation of the rotor of a synchronous machine might

require two, three, or four damper windings was more or less of academic interest until

digital computers became available. Prior to the 1970s, the damper windings were

seldom considered in stability studies; however, as the capability of computers increased,

it became desirable to represent the machine in more detail.

Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk,

Scott Sudhoff, and Steven Pekarek.

© 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc.

MACHINE EQUATIONS IN

OPERATIONAL IMPEDANCES

AND TIME CONSTANTS

7

272 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS

The standard short-circuit test,which involves monitoring the stator short-circuit

currents, provides information from which the parameters of the fi eld winding and one

damper winding in the d -axis can be determined. The parameters for the q -axis damper

winding are calculated from design data. Due to the need for more accurate parameters,

frequency–response data are now being used as means of measuring the operational

impedances from which the parameters can be obtained for any number of rotor wind￾ings in both axes.

In this chapter, the operational impedances as set forth by Park [1] are described.

The standard and derived synchronous machine time constants are defi ned and their

relationship to the operational impedances established. Finally, a method of approxi￾mating the measured operational impedances by lumped parameter rotor circuits is

presented.

7.2. P ARK ’ S EQUATIONS IN OPERATIONAL FORM

R.H. Park [1] published the original qd 0-voltage equations in the form

v ri p qs

r s qsr r

b

ds

r

b

qs

r =− + +

ω

ω

ψ

ω

ψ (7.2-1)

v ri p ds

r s dsr r

b

qs

r

b

ds

r =− − +

ω

ω

ψ

ω

ψ (7.2-2)

v ri p s ss

b

00 0 =− + s ω

ψ (7.2-3)

where

ψqs

r q qsr = −X pi ( ) (7.2-4)

ψds

r d dsr fd

r =− + X pi G pv () () ′ (7.2-5)

ψ0 0 s ls s = −X i (7.2-6)

In these equations, positive stator current is assumed out of the machine, the operator

X q ( p ) is referred to as the q -axis operational impedance, X d ( p ) is the d -axis operational

impedance, and G ( p ) is a dimensionless transfer function relating stator fl ux linkages

per second to fi eld voltage.

With the equations written in this form, the rotor of a synchronous machine can be

considered as either a distributed or lumped parameter system. Over the years, the elec￾trical characteristics of the rotor have often been approximated by three lumped param￾eter circuits, one fi eld winding and two damper windings, one in each axis. Although this

type of representation is generally adequate for salient-pole machines, it does not suffi ce

for a solid iron rotor machine. It now appears that for dynamic and transient stability

considerations, at least two and perhaps three damper windings should be used in the

q -axis for solid rotor machines with a fi eld and two damper windings in the d -axis [2] .