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• 3.1

Chapter 3 Modeling Of a Synchronous Machine

In Chapter 2 we have discussed about small-signal and transient stability of a

synchronous machine, connected to an infinite bus, represented by a classical model.

We have seen that there are several disadvantages of representing a synchronous

generator by a classical model. In this Chapter we will look at detailed modeling of a

synchronous machine.

3.1 Representation of Synchronous Machine Dynamics

While modeling a synchronous machine, different ways of representation,

conventions and notations are followed in the available literature. Hence, at the outset

the notations and conventions used for representing a synchronous machine should be

clear. In this course, IEEE standard (1110-1991) IEEE guide to synchronous

machine modeling has been followed for representing the synchronous machine.

(a)

• 3.2

(b)

Fig. 3.1: Synchronous machine (a) sectional view (b) Stator and rotor windings with

mmf along the respective axis.

The conventions and notations used along with their significance will be

explained in this Chapter. To model and mathematically represent a synchronous

machine first all the windings that need to be included in the model should be

identified. Consider sectional view of the synchronous machine shown in Fig. 3.1 (a).

The synchronous machine in Fig. 3.1 is a two pole salient machine. A general model

with n poles will be dealt latter in this Chapter.

The conductors anda a represent the sectional view of one turn of a-phase

stator winding. The dot in the conductor a represents current coming out of the

conductor and represents in to the conductor. By applying right hand thumb rule at

conductor anda a it can be observed that the mmf due to the conductors

• 3.3

anda a lie along axis marked A-axis. Similarly, the mmf due to ,b b and ,c c lie

along B and C axis, respectively. As an electrical circuit the stator can be represented

as three windings corresponding to three-phases, as shown in Fig. 3.1 (b). The three-

phase instantaneous ac voltages and currents in the stator windings are represented as

, ,a b cv v v and , ,a b ci i i . According to the generator convention, currents out of the

stator windings are considered as positive where as currents into the rotor windings

are considered as positive.

The rotor field is excited by a dc voltage represented as fdv with a field current

fdi . The mmf generated by the rotor field excitation lies normal to the pole surface,

along the direct axis or d-axis. The d-axis of the rotor is at angle m with respect to

the stator a a mmf axis that is A-axis. Angle m is in mechanical radians and in

case of two poles machine the electrical and mechanical angle are one and the same.

But in case of multiple poles the electrical angle is related to the mechanical angle

through the number of poles i.e. / 2s mP where P is the number of poles and s

is the rotor angle in electrical radians. In case of two poles machine the electrical and

mechanical rotor angle will be same as is the case for the synchronous machine shown

in Fig. 3.1. The analysis holds true for multiple pole machine as well but with the

additional condition that / 2s mP . For rest of the Chapters we will be expressing

the angle in terms of electrical radians unless specified other wise. The axis in

quadrature (leading or lagging by 90 ) with respect to the d-axis is called as

quadrature axis or q-axis. The q-axis can either be represented as leading d-axis or

lagging d-axis. Both the conventions are followed in the literature. However, here q-

axis is taken as leading d-axis according to the IEEE 1110-1991 standard.

Representing damper windings needs clarification. The damper windings are

copper bars placed usually in the slots of the pole face. The ends of the copper bars

are shorted forming a closed path for the currents to flow. The magnetic field

generated by these damper windings, due to currents circulating through these

windings, will be along the d-axis. However, the rotor core itself may act as closed

path for induced currents during non-synchronous operations. Hence, to properly

account for the action of the damper windings and damping effect of rotor core three

damper windings are considered. One damper winding represented as 1d , with a

• 3.4

voltage 1dv and current 1di , is considered whose mmf is along d-axis. Two damper

windings represented as 1 , 2q q , with a voltage 1 2,q qv v and current 1 2,q qi i are

considered whose mmf is along q-axis. In the d-axis and q-axis rotor windings the

current in to the winding is considered as positive.

For very accurate representation of synchronous machine, even more number

of damper windings may be considered along d and q axis. According to the number

of windings considered along each axis a model number is give as following 

Table 3.1: Classifications of synchronous machine model based on number of

windings in each axis

Number of windings in q-axis

0 1 2 3

Model

1.0

Model

1.1

Model

1.2

Model 2.1 Model

2.2

Number

of

windings

in

d-axis

1

2

3 Model 3.3

The first number in the model number given in Table 3.1 represents number of

windings in d axis and second number represents number of windings in q axis.

There should be at least one winding, field winding, in the d axis. Hence, the first

model 1.0 means that rotor is represented by one field winding, zero d-axis damper

winding and zero q-axis damper windings. From the view point of complexity, in the

representation of many windings along d axis and q axis, the maximum number

of winding that can be represented along any axis is fixed at 3. The model which is

shown in Fig. 3.1 (b) is 2.2 that is one field winding, one damper winding along d-

axis and two damper windings along q-axis. Model 2.2 is widely used in many

industry grade transient stability simulation softwares.

• 3.5

3.1.1 Stator and rotor winding voltage equations

Applying KVL at the stator windings the following equations can be written

aa s a

dv r idt

(3.1)

bb s b

dv r idt

(3.2)

cc s c

dv r idt

(3.3)

where, sr is the stator resistance and is assumed to be same in all the three

phases. The flux linkages in a, b, and c phases are represented as , ,a b c . The rate

of change of flux linkages in phase a, b and c lead to an induced emf (electro-motive

force) which is equal to the terminal phase voltage plus the drop in the stator

resistance (since we are using generator convention), as can be seen from equations

(3.1) to (3.3). Now applying KVL at the d and q axis rotor windings will give the

following expressions

fdfd fd fd

dv r i

dt

(3.4)

11 1 1

dd d d

dv r idt

(3.5)

11 1 1

qq q q

dv r i

dt

(3.6)

22 2 2

qq q q

dv r i

dt

(3.7)

Where, 1 1 2, , ,fd d q qr r r r and 1 1 2, , ,fd d q q are the rotor field, 1d, 1q and 2q winding

• 3.6

3.1.2 Stator and rotor windings flux linkage equations

The flux linkages of different windings can be expressed in terms of current

through the windings and inductance of the windings as:

1 1 21

1 1 21

1 1 22

abc abcss srrotor

fdafd a d a q a qa aa ab ac a

db ba bb bc b bfd b d b q b q

qc ca cb cc c cfd c d c q c q

qiL Li

iL L L LL L L i

iL L L i L L L L

iL L L i L L L L

i

(3.8)

abc ss abc sr rotorL i L i (3.9)

In equation (3.8) the diagonal elements of the matrix ssL represent the self

inductance of a, b, c windings and off-diagonal elements represent the mutual

inductance among a, b, c phases. The matrix srL represents the mutual inductance

between the stator and rotor windings. A similar expression for flux linkage of the

rotor windings can be written as

1

1 1 11 1 1 1

1 1 1 1 1 1 1 2

2 2 2 2 2 1 2 2

0 0

0 0

0 0

0 0abc

rotor rs rr

fdfd fd dfd fda fdb fdca

dfd d dd da db dcb

q qa qb qc q q q qc

q qa qb qc q q q qi

L L

L LL L Li L LL L Li

L L L L Li

L L L L L

1

1

2

rotor

fd

d

q

q

i

iiii

(3.10)

rotor rs abc rr rotorL i L i (3.11)

In the matrix rrL , the mutual inductance between the d-axis windings ( ,1fd

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