Chapter 5 SYNCHRONOUS MACHINES

142

5.1. INTRODUCTION

The electrical and electromechanical behavior of most synchronous machines can be

predicted from the equations that describe the three-phase salient-pole synchronous

machine. In particular, these equations can be used directly to predict the performance

of synchronous motors, hydro, steam, combustion, or wind turbine driven synchronous

generators, and, with only slight modifi cations, reluctance motors.

The rotor of a synchronous machine is equipped with a fi eld winding and one or

more damper windings and, in general, each of the rotor windings has different electri￾cal characteristics. Moreover, the rotor of a salient-pole synchronous machine is mag￾netically unsymmetrical. Due to these rotor asymmetries, a change of variables for the

rotor variables offers no advantage. However, a change of variables is benefi cial for

the stator variables. In most cases, the stator variables are transformed to a reference

frame fi xed in the rotor (Park ’ s equations) [1] ; however, the stator variables may also

be expressed in the arbitrary reference frame, which is convenient for some computer

simulations.

In this chapter, the voltage and electromagnetic torque equations are fi rst estab￾lished in machine variables. Reference-frame theory set forth in Chapter 3 is then used

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.

SYNCHRONOUS MACHINES

5

VOLTAGE EQUATIONS IN MACHINE VARIABLES 143

to establish the machine equations with the stator variables in the rotor reference frame.

The equations that describe the steady-state behavior are then derived using the theory

established in Chapter 3 . The machine equations are arranged convenient for computer

simulation wherein a method for accounting for saturation is given. Computer traces

are given to illustrate the dynamic behavior of a synchronous machine during motor

and generator operation and a low-power reluctance motor during load changes and

variable frequency operation.

Nearly all of the electric power used throughout the world is generated by syn￾chronous generators driven either by hydro, steam, or wind turbines or combustion

engines. Just as the induction motor is the workhorse when it comes to converting

energy from electrical to mechanical, the synchronous machine is the principal means

of converting energy from mechanical to electrical. In the power system or electric grid

environment, the analysis of the synchronous generator is often carried out assuming

positive currents out of the machine. Although this is very convenient for the power

systems engineer, it tends to be somewhat confusing for beginning machine analysts

and inconvenient for engineers working in the electric drives area. In an effort to make

this chapter helpful in both environments, positive stator currents are assumed into the

machine as done in the analysis of the induction machine, and then in Section 5.10 , the

sense of the stator currents is reversed, and high-power synchronous generators that

would be used in a power system are considered. The changes in the machine equations

necessary to accommodate positive current out of the machine are described. Computer

traces are then given to illustrate the dynamic behavior of typical hydro and steam

turbine-driven generators during sudden changes in input torque and during and fol￾lowing a three-phase fault at the terminals. These dynamic responses, which are calcu￾lated using the detailed set of nonlinear differential equations, are compared with those

predicted by an approximate method of calculating the transient torque–angle charac￾teristics, which was widely used before the advent of modern computers and which still

offer an unequalled means of visualizing the transient behavior of synchronous genera￾tors in a power system.

5.2. VOLTAGE EQUATIONS IN MACHINE VARIABLES

A two-pole, three-phase, wye-connected, salient-pole synchronous machine is shown

in Figure 5.2-1 . The stator windings are identical sinusoidally distributed windings,

displaced 120°, with N s equivalent turns and resistance r s . The rotor is equipped with

a fi eld winding and three damper windings. The fi eld winding ( fd winding) has N fd

equivalent turns with resistance r fd . One damper winding has the same magnetic axis

as the fi eld winding. This winding, the kd winding, has N kd equivalent turns with resis￾tance r kd . The magnetic axis of the second and third damper windings, the kq 1 and kq 2

windings, is displaced 90° ahead of the magnetic axis of the fd and kd windings. The

kq 1 and kq 2 windings have Nkq 1 and Nkq 2 equivalent turns, respectively, with resistances

rkq 1 and rkq 2 . It is assumed that all rotor windings are sinusoidally distributed.

In Figure 5.2-1 , the magnetic axes of the stator windings are denoted by the as,

bs , and cs axes. This notation was also used for the stator windings of the induction

144 SYNCHRONOUS MACHINES

machine. The quadrature axis ( q -axis) and direct axis ( d -axis) are introduced in Figure

5.2-1 . The q -axis is the magnetic axis of the kq 1 and kq 2 windings, while the d -axis is

the magnetic axis of the fd and kd windings. The use of the q- and d -axes was in exis￾tence prior to Park ’ s work [1] , and as mentioned in Chapter 3 , Park used the notation

of f q , f d , and f0 in his transformation. Perhaps he made this choice of notation since, in

effect, this transformation referred the stator variables to the rotor where the traditional

q -and d -axes are located.

We have used f qs , f ds , and f0s , and fqr′ , fdr′ , and f ′0r to denote transformed induction

machine variables without introducing the connotation of a q- or d -axis. Instead, the

q- and d -axes have been reserved to denote the rotor magnetic axes of the synchronous

machine where they have an established physical meaning quite independent of any

transformation. For this reason, one may argue that the q and d subscripts should not

be used to denote the transformation to the arbitrary reference frame. Indeed, this line

of reasoning has merit; however, since the transformation to the arbitrary reference

Figure 5.2-1. Two-pole, three-phase, wye-connected salient-pole synchronous machine.

θr

wr

as-axis

fd

fd′

bs-axis

as′

q-axis

kd′

kd

kq1 kq2

kq1′

kq2′

bs′

bs

cs′

cs-axis d-axis

as

cs

Nkd

Nfd

Nkq2

Nkq1

rs

ibs

ics

ikq1

ikq2 vkq2

vkq1

+

+

+

+

+

+ Ns

−

−

−

−

−

−

vas

vbs

vcs

ikd

ifd

rkd

vkd

ias vfd

rs

+

rfd

Ns

rs

Ns

VOLTAGE EQUATIONS IN MACHINE VARIABLES 145

frame is in essence a generalization of Park ’ s transformation, the q and d subscripts

have been selected for use in the transformation to the arbitrary reference primarily out

of respect for Park ’ s work, which is the basis of it all.

Although the damper windings are shown with provisions to apply a voltage, they

are, in fact, short-circuited windings that represent the paths for induced rotor currents.

Currents may fl ow in either cage-type windings similar to the squirrel-cage windings

of induction machines or in the actual iron of the rotor. In salient-pole machines at

least, the rotor is laminated, and the damper winding currents are confi ned, for the

most part, to the cage windings embedded in the rotor. In the high-speed, two- or four￾pole machines, the rotor is cylindrical, made of solid iron with a cage-type winding

embedded in the rotor. Here, currents can fl ow either in the cage winding or in the

solid iron.

The performance of nearly all types of synchronous machines may be adequately

described by straightforward modifi cations of the equations describing the performance

of the machine shown in Figure 5.2-1 . For example, the behavior of low-speed hydro

turbine generators, which are always salient-pole machines, is generally predicted suf￾fi ciently by one equivalent damper winding in the q -axis. Hence, the performance of

this type of machine may be described from the equations derived for the machine

shown in Figure 5.2-1 by eliminating all terms involving one of the kq windings. The

reluctance machine, which has no fi eld winding and generally only one damper winding

in the q -axis, may be described by eliminating the terms involving the fd winding and

one of the kq windings. In solid iron rotor, steam turbine generators, the magnetic

characteristics of the q- and d -axes are identical, or nearly so, hence the inductances

associated with the two axes are essentially the same. Also, it is necessary, in most

cases, to include all three damper windings in order to portray adequately the transient

characteristics of the stator variables and the electromagnetic torque of solid iron rotor

machines [2] .

The voltage equations in machine variables may be expressed in matrix form as

v ri abcs s abcs abcs = + pl (5.2-1)

v ri qdr r qdr qdr = + pl (5.2-2)

where

( )[ ] fabcs T as bs cs = fff (5.2-3)

()[ ] fqdr T kq kq fd kd = f f ff 1 2 (5.2-4)

In the previous equations, the s and r subscripts denote variables associated with

the stator and rotor windings, respectively. Both rs and rr are diagonal matrices,

in particular

rs sss = diag[ ] rrr (5.2-5)

rr kq kq fd kd = diag[ ] r r rr 1 2 (5.2-6)

146 SYNCHRONOUS MACHINES

The fl ux linkage equations for a linear magnetic system become

l

l

abcs

qdr

s sr

sr T r

abcs

qdr

⎡

⎣

⎢ ⎤

⎦

⎥ = ⎡

⎣

⎢ ⎤

⎦

⎥

⎡

⎣

⎢ ⎤

⎦

⎥

L L

L L

i

( ) i (5.2-7)

From the work in Chapters 1 and 2 , neglecting mutual leakage between stator windings,

we can write Ls as

Ls

LLL LL LL ls A B r A B r A B r

=

+− − − − ⎛

⎝

⎜ ⎞

⎠ cos cos cos 2 ⎟ −− +

1

2

2

3

1

2

2

3

θ θ π θ⎛ π

⎝

⎜ ⎞

⎠

⎟

−− − ⎛

⎝

⎜ ⎞

⎠

⎟ +− − ⎛

⎝

⎜ ⎞

⎠

⎟ − 1

2

2

3

2 2

3

1

2

LL LLL A B r ls A B r cos cos θ π θ π L L

LL LL L

AB r

AB r AB r

− +

−− + ⎛

⎝

⎜ ⎞

⎠

⎟ −− +

cos ( )

cos cos ( )

2

1

2

2

3

1

2

2

θ π

θ π θ π ls A B r +− + L L ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥ cos2 2

3

θ π

(5.2-8)

By a straightforward extension of the work in Chapters 1 and 2 , we can express the

self- and mutual inductances of the damper windings. The inductance matrices Lsr and

Lr may then be expressed as

Lsr

skq r skq r sfd r skd r

skq r

L L LL

= − L ⎛

⎝

1 2

1

2

3

cos cos sin sin

cos

θ θθθ

θ π ⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛ L LL skq r sfd r skd r 2

2

3

2

3

2

3

cos sin sin θ π θ π θ π

⎝

⎜ ⎞

⎠

⎟

+ ⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠ LLL skq r skq r sfd r 1 2 ⎟ +

2

3

2

3

2 cos cos sin θ π θ π θ π

3

2

3

⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥ Lskd r sin θ π

(5.2-9)

Lr

lkq mkq kq kq

kq kq lkq mkq

lfd mfd fdk

LL L

L LL

LL L =

+

+

+

1 1 12

12 2 2

0 0

0 0

0 0 d

0 0 L LL fdkd lkd mkd +

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

(5.2-10)

In (5.2-8) , L A > L B and L B is zero for a round rotor machine. Also in (5.2-8) and (5.2-

10) , the leakage inductances are denoted with l in the subscript. The subscripts skq 1,

skq 2, sfd , and skd in (5.2-9) denote mutual inductances between stator and rotor

windings.

The magnetizing inductances are defi ned as

L LL mq A B = − 3

2

( ) (5.2-11)

L LL md A B = +

3

2

( ) (5.2-12)

VOLTAGE EQUATIONS IN MACHINE VARIABLES 147

It can be shown that

L

N

N skq L kq

s

1 mq

1 2

3 = ⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜ ⎞

⎠

⎟ (5.2-13)

L

N

N skq L kq

s

2 mq

2 2

3 = ⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜ ⎞

⎠

⎟ (5.2-14)

L

N

N sfd L fd

s

= md

⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜ ⎞

⎠

⎟

2

3 (5.2-15)

L N

N skd L kd

s

= md

⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜ ⎞

⎠

⎟

2

3 (5.2-16)

L

N

N mkq L kq

s

1 mq

1

2 2

3 = ⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟ (5.2-17)

L

N

N mkq L kq

s

2 mq

2

2 2

3 = ⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟ (5.2-18)

L

N

N mfd L fd

s

= md

⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟

2 2

3 (5.2-19)

L N

N mkd L kd

s

= md

⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟

2 2

3 (5.2-20)

L

N

N

L

N

N

L

kq kq

kq

kq

mkq

kq

kq

mkq

1 2

2

1

1

1

2

2

= ⎛

⎝

⎜

⎞

⎠

⎟

= ⎛

⎝

⎜

⎞

⎠

⎟ (5.2-21)

L N

N

L

N

N

L

fdkd

kd

fd

mfd

fd

kd

mkd

= ⎛

⎝

⎜

⎞

⎠

⎟

= ⎛

⎝

⎜ ⎞

⎠

⎟ (5.2-22)

It is convenient to incorporate the following substitute variables, which refer the rotor

variables to the stator windings.

′ = ⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜ ⎞

⎠

⎟ i

N

N

i j

j

s

j

2

3 (5.2-23)

′ = ⎛

⎝

⎜

⎞

⎠

⎟ v

N

N j v s

j

j (5.2-24)

′ = ⎛

⎝

⎜

⎞

⎠

⎟ λ λ j

s

j

j

N

N (5.2-25)

148 SYNCHRONOUS MACHINES

where j may be kq 1, kq 2, fd , or kd .

The fl ux linkages may now be written as

l

l

abcs

qdr

s sr

sr T r

abcs

qdr ′

⎡

⎣

⎢ ⎤

⎦

⎥ =

′

′ ′

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ ′

⎡

⎣

⎢ ⎤

⎦

L L

L L

i

i 2

3

( ) ⎥ (5.2-26)

where Ls is defi ned by (5.2-8) and

′ = − ⎛

⎝

⎜ ⎞

⎠ Lsr ⎟

mq r mq r md r md r

mq r mq

LLLL

L L

cos cos sin sin

cos

θ θθθ

θ 2π

3

cos sin sin

cos

θ π θ π θ π r md r md r

mq

L L

L

− ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟ − ⎛

⎝

⎜ ⎞

⎠

⎟ 2

3

2

3

2

3

θ π θ π θ π

r mq r md r md r + LLL θ ⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠

⎟ +

2

3

2

3

2

3

cos sin sin 2

3

⎛ π

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

(5.2-27)

′ =

′ +

′ +

′ +

′

Lr

lkq mq mq

mq lkq mq

lfd md md

md lkd

LL L

L LL

LL L

L L

1

2

0 0

0 0

0 0

0 0 +

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥ Lmd

(5.2-28)

The voltage equations expressed in terms of machine variables referred to the stator

windings are

v

v

rL L

L rL

i abcs

qdr

s s sr

sr T r r

abc

p p

′ p p

⎡

⎣

⎢ ⎤

⎦

⎥ =

+ ′

′ ′ + ′

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ 2

3 ( )

s

qdr ′

⎡

⎣

⎢ ⎤

⎦

⎥ i (5.2-29)

In (5.2-28) and (5.2-29)

′ = ⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟ r N

N j r s

j

j

3

2

2

(5.2-30)

′ = ⎛

⎝

⎜ ⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟ L N

N lj L s

j

lj

3

2

2

(5.2-31)

where, again, j may be kq 1, kq 2, fd , or kd .