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Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science

6.685 Electric Machinery

Class Notes 9: Synchronous Machine Simulation Models September 5, 2005c©2005 James L. Kirtley Jr.

1 Introduction

In this document we develop models useful for calculating the dynamic behavior of synchronousmachines. We start with a commonly accepted picture of the synchronous machine, assuming thatthe rotor can be fairly represented by three equivalent windings: one being the field and the othertwo, the d- and q- axis “damper” windings, representing the effects of rotor body, wedge chain,amortisseur and other current carrying paths.

While a synchronous machine is assumed here, the results are fairly directly applicable toinduction machines. Also, extension to situations in which the rotor representation must havemore than one extra equivalent winding per axis should be straightforward.

2 Phase Variable Model

To begin, assume that the synchronous machine can be properly represented by six equivalentwindings. Four of these, the three armature phase windings and the field winding, really arewindings. The other two, representing the effects of distributed currents on the rotor, are referredto as the “damper” windings. Fluxes are, in terms of currents:

[

λph

λR

]

=

[

Lph

M

MT LR

] [

Iph

IR

]

(1)

where phase and rotor fluxes (and, similarly, currents) are:

λph =

λa

λb

λc

(2)

λR =

λf

λkd

λkq

(3)

There are three inductance submatrices. The first of these describes armature winding induc-tances:

Lph

=

La Lab Lac

Lab Lb Lbc

Lac Lbc Lc

(4)

1

where, for a machine that may have some saliency:

La = La0 + L2 cos 2θ (5)

Lb = La0 + L2 cos 2(θ −2π

3) (6)

Lc = La0 + L2 cos 2(θ +2π

3) (7)

Lab = Lab0 + L2 cos 2(θ −π

3) (8)

Lbc = Lab0 + L2 cos 2θ (9)

Lac = Lab0 + L2 cos 2(θ +π

3) (10)

Note that, in this last set of expressions, we have assumed a particular form for the mutual in-ductances. This is seemingly restrictive, because it constrains the form of phase- to- phase mutualinductance variations with rotor position. The coefficient L2 is actually the same in all six of theselast expressions. As it turns out, this assumption does not really restrict the accuracy of the modelvery much. We will have more to say about this a bit later.

The rotor inductances are relatively simply stated:

LR

=

Lf Lfkd 0Lfkd Lkd 00 0 Lkq

(11)

And the stator- to- rotor mutual inductances are:

M =

M cos θ Lakd cos θ −Lakq sin θM cos(θ − 2π

3) Lakd cos(θ − 2π

3) −Lakq sin(θ − 2π

3)

M cos(θ + 2π3

) Lakd cos(θ + 2π3

) −Lakq sin(θ + 2π3

)

(12)

3 Park’s Equations

The first step in the development of a suitable model is to transform the armature winding variablesto a coordinate system in which the rotor is stationary. We identify equivalent armature windingsin the direct and quadrature axes. The direct axis armature winding is the equivalent of one ofthe phase windings, but aligned directly with the field. The quadrature winding is situated sothat its axis leads the field winding by 90 electrical degrees. The transformation used to mapthe armature currents, fluxes and so forth onto the direct and quadrature axes is the celebratedPark’s Transformation, named after Robert H. Park, an early investigator into transient behaviorin synchronous machines. The mapping takes the form:

ud

uq

u0

= udq = Tuph = T

ua

ub

uc

(13)

Where the transformation and its inverse are:

T =2

3

cos θ cos(θ − 2π3

) cos(θ + 2π3

)− sin θ − sin(θ − 2π

3) − sin(θ + 2π

3)

1

2

1

2

1

2

(14)

2

T−1 =

cos θ − sin θ 1cos(θ − 2π

3) − sin(θ − 2π

3) 1

cos(θ + 2π3

) − sin(θ + 2π3

) 1

(15)

This transformation maps balanced sets of phase currents into constant currents in the d-q frame.That is, if rotor angle is θ = ωt+ θ0, and phase currents are:

Ia = I cosωt

Ib = I cos(ωt−2π

3)

Ic = I cos(ωt+2π

3)

Then the transformed set of currents is:

Id = I cos θ0

Iq = −I sin θ0

Now, we apply this transformation to (1) to express fluxes and currents in the armature in the d-q

reference frame. To do this, extract the top line in (1):

λph = LphIph +MIR (16)

The transformed flux is obtained by premultiplying this whole expression by the transformationmatrix. Phase current may be obtained from d-q current by multiplying by the inverse of thetransformation matrix. Thus:

λdq = TLphT−1Idq + TMIR (17)

The same process carried out for the lower line of (1) yields:

λR = MTT−1Idq + LRIR (18)

Thus the fully transformed version of (1) is:

[

λdq

λR

]

=

[

Ldq

LC3

2LT

CL

R

] [

Idq

IR

]

(19)

If the conditions of (5) through (10) are satisfied, the inductance submatrices of (19) wind up beingof particularly simple form. (Please note that a substantial amount of algebra has been left outhere!)

Ldq

=

Ld 0 00 Lq 00 0 L0

(20)

LC

=

M Lakd 00 0 Lakq

0 0 0

(21)

3

Note that (19) through (21) express three separate sets of apparently independent flux/currentrelationships. These may be re-cast into the following form:

λd

λkd

λf

=

Ld Lakd M3

2Lakd Lkd Lfkd3

2M Lfkd Lf

IdIkd

If

(22)

[

λq

λkq

]

=

[

Lq Lakq3

2Lakq Lkq

] [

IqIkq

]

(23)

λ0 = L0I0 (24)

Where the component inductances are:

Ld = La0 − Lab0 +3

2L2 (25)

Lq = La0 − Lab0 −3

2L2 (26)

L0 = La0 + 2Lab0 (27)

Note that the apparently restrictive assumptions embedded in (5) through (10) have resulted inthe very simple form of (21) through (24). In particular, we have three mutually independent setsof fluxes and currents. While we may be concerned about the restrictiveness of these expressions,note that the orthogonality between the d- and q- axes is not unreasonable. In fact, because theseaxes are orthogonal in space, it seems reasonable that they should not have mutual flux linkages.The principal consequence of these assumptions is the de-coupling of the zero-sequence componentof flux from the d- and q- axis components. We are not in a position at this time to determinethe reasonableness of this. However, it should be noted that departures from this form (that is,coupling between the “direct” and “zero” axes) must be through higher harmonic fields that willnot couple well to the armature, so that any such coupling will be weak.

Next, armature voltage is, ignoring resistance, given by:

V ph =d

dtλph =

d

dtT−1λdq (28)

and that the transformed armature voltage must be:

V dq = TV ph

= Td

dt(T−1λdq)

=d

dtλdq + (T

d

dtT−1)λdq (29)

A good deal of manupulation goes into reducing the second term of this, resulting in:

Td

dtT−1 =

0 −dθdt

0dθdt

0 00 0 0

(30)

4

This expresses the speed voltage that arises from a coordinate transformation. The two voltage/fluxrelationships that are affected are:

Vd =dλd

dt− ωλq (31)

Vq =dλq

dt+ ωλd (32)

where we have used

ω =dθ

dt(33)

4 Power and Torque

Instantaneous power is given by:P = VaIa + VbIb + VcIc (34)

Using the transformations given above, this can be shown to be:

P =3

2VdId +

3

2VqIq + 3V0I0 (35)

which, in turn, is:

P = ω3

2(λdIq − λqId) +

3

2(dλd

dtId +

dλq

dtIq) + 3

dλ0

dtI0 (36)

Then, noting that electrical speed ω and shaft speed Ω are related by ω = pΩ and that (36)describes electrical terminal power as the sum of shaft power and rate of change of stored energy,we may deduce that torque is given by:

T =3

2p(λdIq − λqId) (37)

5 Per-Unit Normalization

The next thing for us to do is to investigate the way in which electric machine system are nor-

malized, or put into what is called a per-unit system. The reason for this step is that, when thevoltage, current, power and impedance are referred to normal operating parameters, the behaviorcharacteristics of all types of machines become quite similar, giving us a better way of relatinghow a particular machine works to some reasonable standard. There are also numerical reasons fornormalizing performance parameters to some standard.

The first step in normalization is to establish a set of base quantities. We will be normalizingvoltage, current, flux, power, impedance and torque, so we will need base quantities for each ofthese. Note, however, that the base quantities are not independent. In fact, for the armature, weneed only specify three quantities: voltage (VB), current (IB) and frequency (ω0). Note that we donot normalize time nor frequency. Having done this for the armature circuits, we can derive eachof the other base quantities:

5

• Base Power

PB =3

2VBIB

• Base Impedance

ZB =VB

IB

• Base Flux

λB =VB

ω0

• Base Torque

TB =p

ω0

PB

Note that, for our purposes, base voltage and current are expressed as peak quantities. Base voltageis taken on a phase basis (line to neutral for a “wye” connected machine), and base current issimilarly taken on a phase basis, (line current for a “wye” connected machine).

Normalized, or per-unit quantities are derived by dividing the ordinary variable (with units) bythe corresponding base. For example, per-unit flux is:

ψ =λ

λB

=ω0λ

VB

(38)

In this derivation, per- unit quantities will usually be designated by lower case letters. Twonotable exceptions are flux, where we use the letter ψ, and torque, where we will still use the uppercase T and risk confusion.

Now, we note that there will be base quantities for voltage, current and frequency for each ofthe different coils represented in our model. While it is reasonable to expect that the frequency

base will be the same for all coils in a problem, the voltage and current bases may be different. Wemight write (22) as:

ψd

ψkd

ψf

=

ω0IdB

VdbLd

ω0IkB

VdbLakd

ω0IfB

VdbM

ω0IdB

Vkb

3

2Lakd

ω0IkB

VkbLkd

ω0IfB

VkdbLfkd

ω0IdB

Vfb

3

2M ω0IkB

VfbLfkd

ω0IfB

VfbLf

idikd

if

(39)

where i = I/IB denotes per-unit, or normalized current.Note that (39) may be written in simple form:

ψd

ψkd

ψf

=

xd xakd xad

xakd xkd xfkd

xad xfkd xf

idikd

if

(40)

It is important to note that (40) assumes reciprocity in the normalized system. To wit, the followingexpressions are implied:

xd = ω0

IdB

VdB

Ld (41)

6

xkd = ω0

IkB

VkB

Lkd (42)

xf = ω0

IfB

VfB

Lf (43)

xakd = ω0

IkB

VdB

Lakd

=3

2ω0

IdB

VkB

Lakd (44)

xad = ω0

IfB

VdB

M

=3

2ω0

IdB

VfB

M (45)

xfkd = ω0

IkB

Vfb

Lfkd

= ω0

IfB

Vkb

Lfkd (46)

These in turn imply:

3

2VdBIdB = VfBIfB (47)

3

2VdBIdB = VkBIkB (48)

VfBIfB = VkBIkB (49)

These expressions imply the same power base on all of the windings of the machine. This isso because the armature base quantities Vdb and Idb are stated as peak values, while the rotor basequantities are stated as DC values. Thus power base for the three- phase armature is 3

2times

the product of peak quantities, while the power base for the rotor is simply the product of thosequantities.

The quadrature axis, which may have fewer equivalent elements than the direct axis and whichmay have different numerical values, still yields a similar structure. Without going through thedetails, we can see that the per-unit flux/current relationship for the q- axis is:

[

ψq

ψkq

]

=

[

xq xakq

xakq xkq

] [

iqikq

]

(50)

The voltage equations, including speed voltage terms, (31) and (32), may be augmented toreflect armature resistance:

Vd =dλd

dt− ωλq +RaId (51)

Vq = ωλd +dλq

dt+RaIq (52)

The per-unit equivalents of these are:

vd =1

ω0

dψd

dt−

ω

ω0

ψq + raid (53)

7

vq =ω

ω0

ψd +1

ω0

dψq

dt+ raiq (54)

Where the per-unit armature resistance is just ra = Ra

ZB

Note that none of the other circuits in this model have speed voltage terms, so their voltageexpressions are exactly what we might expect:

vf =1

ω0

dψf

dt+ rf if (55)

vkd =1

ω0

dψkd

dt+ rkdikd (56)

vkq =1

ω0

dψkq

dt+ rkqikq (57)

v0 =1

ω0

dψ0

dt+ rai0 (58)

It should be noted that the damper winding circuits represent closed conducting paths on the rotor,so the two voltages vkd and vkq are always zero.

Per-unit torque is simply:Te = ψdiq − ψqid (59)

Often, we need to represent the dynamic behavior of the machine, including electromechanicaldynamics involving rotor inertia. If we note J as the rotational inertia constant of the machinesystem, the rotor dynamics are described by the two ordinary differential equations:

1

pJdω

dt= T e + Tm (60)

dδ

dt= ω − ω0 (61)

where T e and Tm represent electrical and mechanical torques in “ordinary” variables. The angle δrepresents rotor phase angle with respect to some synchronous reference.

It is customary to define an “inertia constant” which is not dimensionless but which neverthelessfits into the per-unit system of analysis. This is:

H ≡Rotational kinetic energy at rated speed

Base Power(62)

Or:

H =

1

2J(

ω0

p

)2

PB

=Jω0

2pTB

(63)

Then the per-unit equivalent to (60) is:

2H

ω0

dω

dt= Te + Tm (64)

where now we use Te and Tm to represent per-unit torques.

8

6 Equal Mutual’s Base

In normalizing the differential equations that make up our model, we have used a number of base

quantities. For example, in deriving (40), the per-unit flux- current relationship for the direct

axis, we used six base quantities: VB , IB, VfB , IfB , VkB and IkB. Imposing reciprocity on (40)results in two constraints on these six variables, expressed in (47) through (49). Presumably thetwo armature base quantities will be fixed by machine rating. That leaves two more “degrees offreedom” in selection of base quantities. Note that the selection of base quantities will affect thereactance matrix in (40).

While there are different schools of thought on just how to handle these degrees of freedom, acommonly used convention is to employ what is called the equal mutuals base system. The twodegrees of freedom are used to set the field and damper base impedances so that all three mutualinductances of (40) are equal:

xakd = xfkd = xad (65)

The direct- axis flux- current relationship becomes:

ψd

ψkd

ψf

=

xd xad xad

xad xkd xad

xad xad xf

idikd

if

(66)

7 Equivalent Circuit

∧ ∧ ∧∨ ∨

ra∩∩∩∩xal

⊃⊃⊃⊃xad

<<<

>>rkd

⊃⊃⊃⊃xkdl

∩∩∩∩xfl

∧ ∧ ∧∨ ∨

rf

vf

+

-

if

(ω0vd + ωψq)

+

-

-

id

ψd

+

-

Figure 1: D- Axis Equivalent Circuit

The flux- current relationship of (66) is represented by the equivalent circuit of Figure 1, if the“leakage” inductances are defined to be:

xal = xd − xad (67)

xkdl = xkd − xad (68)

xfl = xf − xad (69)

9

Many of the interesting features of the electrical dynamics of the synchronous machine may bediscerned from this circuit. While a complete explication of this thing is beyond the scope of thisnote, it is possible to make a few observations.

The apparent inductance measured from the terminals of this equivalent circuit (ignoring resis-tance ra) will, in the frequency domain, be of the form:

x(s) =ψd(s)

id(s)= xd

Pn(s)

Pd(s)(70)

Both the numerator and denominator polynomials in s will be second order. (You may convinceyourself of this by writing an expression for terminal impedance). Since this is a “diffusion” typecircuit, having only resistances and inductances, all poles and zeros must be on the negative realaxis of the “s-plane”. The per-unit inductance is, then:

x(s) = xd

(1 + T ′

ds)(1 + T ′′

d s)

(1 + T ′

dos)(1 + T ′′

dos)(71)

The two time constants T ′

d and T ′′

d are the reciprocals of the zeros of the impedance, which arethe poles of the admittance. These are called the short circuit time constants.

The other two time constants T ′

do and T ′′

do are the reciprocals of the poles of the impedance, andso are called the open circuit time constants.

We have cast this thing as if there are two sets of well- defined time constants. These are thetransient time constants T ′

d and T ′

do, and the subtransient time constants T ′′

d and T ′′

do. In manycases, these are indeed well separated, meaning that:

T ′

d ≫ T ′′

d (72)

T ′

do ≫ T ′′

do (73)

If this is true, then the reactance is described by the pole-zero diagram shown in Figure 2.Under this circumstance, the apparent terminal inductance has three distinct values, depending onfrequency. These are the synchronous inductance, the transient inductance, and the subtransient

inductance, given by:

x′d = xd

T ′

d

T ′

do

(74)

x′′d = x′dT ′′

d

T ′′

do

= xd

T ′

d

T ′

do

T ′′

d

T ′′

do

(75)

A Bode Plot of the terminal reactance is shown in Figure 3.If the time constants are spread widely apart, they are given, approximately, by:

T ′

do =xf

ω0rf(76)

T ′′

do =xkdl + xfl||xad

ω0rkd

(77)

10

c c× ×1

Td”

1

T ′

d

1

Tdo”

1

T ′

do

Figure 2: Pole-Zero Diagram For Terminal Inductance

@@@

@@@

1

T ′

do

1

T ′

d

1

Tdo”

1

Td”log ω

log |x(jω)|

Figure 3: Frequency Response of Terminal Inductance

Finally, note that the three reactances are found simply from the model:

xd = xal + xad (78)

x′d = xal + xad||xfl (79)

x′′d = xal + xad||xfl||xkdl (80)

8 Statement of Simulation Model

Now we can write down the simulation model. Actually, we will derive more than one of these, sincethe machine can be driven by either voltages or currents. Further, the expressions for permanentmagnet machines are a bit different. So the first model is one in which the terminals are allconstrained by voltage.

The state variables are the two stator fluxes ψd, ψq, two “damper” fluxes ψkd, ψkq, field flux ψf ,and rotor speed ω and torque angle δ. The most straightforward way of stating the model employscurrents as auxiliary variables, and these are:

idikd

if

=

xd xad xad

xad xkd xad

xad xad xf

−1

ψd

ψkd

ψf

(81)

[

iqikq

]

=

[

xq xaq

xaq xkq

]

−1 [

ψq

ψkq

]

(82)

11

Then the state equations are:

dψd

dt= ω0vd + ωψq − ω0raid (83)

dψq

dt= ω0vq − ωψd − ω0raiq (84)

dψkd

dt= −ω0rkdikd (85)

dψkq

dt= −ω0rkqikq (86)

dψf

dt= ω0vf − ω0rf if (87)

dω

dt=

ω0

2H(Te + Tm) (88)

dδ

dt= ω − ω0 (89)

and, of course,Te = ψdiq − ψqid

8.1 Statement of Parameters:

Note that often data for a machine may be given in terms of the reactances xd, x′

d, x′′

d, T′

do andT ′′

do, rather than the elements of the equivalent circuit model. Note that there are four inductancesin the equivalent circuit so we have to assume one. There is no loss in generality in doing so.Usually one assumes a value for the stator leakage inductance, and if this is done the translation isstraightforward:

xad = xd − xal

xfl =xad(x

′

d − xal)

xad − x′d + xal

xkdl =1

1

x′′

d−xal

− 1

xad− 1

xfl

rf =xfl + xad

ω0T ′

do

rkd =xkdl + xad||xfl

ω0T ′′

do

8.2 Linearized Model

Often it becomes desirable to carry out a linearized analysis of machine operation to, for example,examine the damping of the swing mode at a particular operating point. What is done, then,is to assume a steady state operating point and examine the dynamics for deviations from thatoperating point that are “small”. The definition of “small” is really “small enough” that everythingimportant appears in the first-order term of a Taylor series about the steady operating point.

Note that the expressions in the machine model are, for the most part, linear. There are,however, a few cases in which products of state variables cause us to do the expansion of the

12

Taylor series. Assuming a steady state operating point [ψd0 ψkd0 ψf0 ψq0 ψkq0 ω0 δ0], the first-order (small-signal) variations are described by the following set of equations. First, since theflux-current relationship is linear:

id1

ikd1

if1

=

xd xad xad

xad xkd xad

xad xad xf

−1

ψd1

ψkd1

ψf1

(90)

[

iq1ikq1

]

=

[

xq xaq

xaq xkq

]

−1 [

ψq1

ψkq1

]

(91)

Terminal voltage will be, for operation against a voltage source:

Vd = V sin δ Vq = V cos δ

Then the differential equations governing the first-order variations are:

dψd1

dt= ω0V cos δ0δ1 + ω0ψq1 + ω1ψq0 − ω0raid1 (92)

dψq1

dt= −ω0V sin δ0δ1 − ω0ψd1 − ω1ψd0 − ω0raiq1 (93)

dψkd1

dt= −ω0rkdikd1 (94)

dψkq1

dt= −ω0rkqikq1 (95)

dψf1

dt= −ω0rf if1 (96)

dω1

dt=

ω0

2H(Te1 + Tm1) (97)

dδ1dt

= ω1 (98)

Te = ψd0iq1 + ψd1iq0 − ψq0id1 − ψq1id0

8.3 Reduced Order Model for Electromechanical Transients

In many situations the two armature variables contribute little to the dynamic response of themachine. Typically the armature resistance is small enough that there is very little voltage dropacross it and transients in the difference between armature flux and the flux that would exist inthe “steady state” decay rapidly (or are not even excited). Further, the relatively short armaturetime constant makes for very short time steps. For this reason it is often convenient, particularlywhen studying the relatively slow electromechanical transients, to omit the first two differentialequations and set:

ψd = vq = V cos δ (99)

ψq = −vd = −V sin δ (100)

The set of differential equations changes only a little when this approximation is made. Note,however, that it can be simulated with far fewer “cycles” if the armature time constant is short.

13

9 Current Driven Model: Connection to a System

The simulation expressions developed so far are useful in a variety of circumstances. They are,however, difficult to tie to network simulation programs because they use terminal voltage as aninput. Generally, it is more convenient to use current as the input to the machine simulation andaccept voltage as the output. Further, it is difficult to handle unbalanced situations with this setof equations.

An alternative to this set would be to employ the phase currents as state variables. Effectively,this replaces ψd, ψq and ψ0 with ia, ib, and ic. The resulting model will, as we will show, interfacenicely with network simulations.

To start, note that we could write an expression for terminal flux, on the d- axis:

ψd = x′′did + ψf

xad||xkdl

xad||xkdl + xfl

+ ψkd

xad||xfl

xad||xfl + xkdl

(101)

and here, of course,x′′d = xal + xad||xkdl||xfl

This leads us to define a “flux behind subtransient reactance”:

ψ′′

d =xadxkdlψf + xadxflψkd

xadxkdl + xadxfl + xkdlxfl

(102)

So thatψd = ψ′′

d + x′′did

On the quadrature axis the situation is essentially the same, but one step easier if there is onlyone quadrature axis rotor winding:

ψq = x′′q iq + ψkq

xaq

xaq + xkql

(103)

wherex′′q = xal + xaq||xkql

Very often these fluxes are referred to as “voltage behind subtransient reactance, with ψ′′

d = e′′qand ψ′′

q = −e′′d. Then:

ψd = x′′did + e′′q (104)

ψq = x′′q iq − e′′d (105)

Now, if id and iq are determined, it is a bit easier to find the other currents required in thesimulation. Note we can write:

[

ψkd

ψf

]

=

[

xkd xad

xad xf

] [

ikd

if

]

+

[

xad

xad

]

id (106)

and this inverts easily:

[

ikd

if

]

=

[

xkd xad

xad xf

]

−1 ([

ψkd

ψf

]

−

[

xad

xad

]

id

)

(107)

14

The quadrature axis rotor current is simply:

ikq =1

xkq

ψkq −xaq

xkq

iq (108)

The torque equation is the same, but since it is usually convenient to assemble the fluxes behindsubtransient reactance, it is possible to use:

Te = e′′q iq + e′′did + (x′′d − x′′q)idiq (109)

Now it is necessary to consider terminal voltage. This is most conveniently cast in matrixnotation. The vector of phase voltages is:

vph =

va

vb

vc

(110)

Then, with similar notation for phase flux, terminal voltage is, ignoring armature resistance:

vph =1

ω0

dψph

dt

=1

ω0

d

dt

T−1ψdq

(111)

Note that we may define the transformed vector of fluxes to be:

ψdq

= x′′idq + e′′ (112)

where the matrix of reactances shows orthogonality:

x′′ =

x′′d 0 00 x′′q 0

0 0 x0

(113)

and the vector of internal fluxes is:

e′′ =

e′′q−e′′d

0

(114)

Now, of course, idq = Tiph, so that we may re-cast (111) as:

vph =1

ω0

d

dt

T−1x′′Tiph + T−1e′′

(115)

Now it is necessary to make one assumption and one definition. The assumption, which isonly moderately restrictive, is that subtransient saliency may be ignored. That is, we assumethat x′′d = x′′q . The definition separates the “zero sequence” impedance into phase and neutralcomponents:

15

x0 = x′′d + 3xg (116)

Note that according to this definition the reactance xg accounts for any impedance in the neutralof the synchronous machine as well as mutual coupling between phases.

Then, the impedance matrix becomes:

x′′ =

x′′d 0 00 x′′d 00 0 x′′d

+

0 0 00 0 00 0 3xg

(117)

In compact notation, this is:

x′′ = x′′dI + xg

(118)

where I is the identity matrix.Now the vector of phase voltages is:

vph =1

ω0

d

dt

x′′diph + T−1xgT iph + T−1e′′

(119)

Note that in (119), we have already factored out the multiplication by the identity matrix. Thenext step is to carry out the matrix multiplication in the third term of (119). This operation turnsout to produce a remarkably simple result:

T−1xgT = xg

1 1 11 1 11 1 1

(120)

The impact of this is that each of the three phase voltages has the same term, and that isrelated to the time derivative of the sum of the three currents, multiplied by xg.

The third and final term in (119) describes voltages induced by rotor fluxes. It can be writtenas:

1

ω0

d

dt

T−1e′′

=1

ω0

d

dt

T−1

e′′ +1

ω0

T−1de′′

dt(121)

Now, the time derivative of the inverse transform is:

1

ω0

d

dtT−1 =

ω

ω0

− sin(θ) − cos(θ) 0− sin(θ − 2π

3) − cos(θ − 2π

3) 0

− sin(θ + 2π3

) − cos(θ + 2π3

) 0

(122)

Now the three phase voltages can be extracted from all of this matrix algebra:

va =x′′dω0

diadt

+xg

ω0

d

dt(ia + ib + ic) + e′′a (123)

vb =x′′dω0

dibdt

+xg

ω0

d

dt(ia + ib + ic) + e′′b (124)

vc =x′′dω0

dicdt

+xg

ω0

d

dt(ia + ib + ic) + e′′c (125)

16

Where the internal voltages are:

e′′a = −ω

ω0

(e′′q sin(θ) − e′′d cos(θ))

+1

ω0

cos(θ)de′′qdt

+1

ω0

sin(θ)de′′ddt

(126)

e′′b = −ω

ω0

(e′′q sin(θ −2π

3) − e′′d cos(θ −

2π

3))

+1

ω0

cos(θ −2π

3)de′′qdt

+1

ω0

sin(θ −2π

3)de′′ddt

(127)

e′′c = −ω

ω0

(e′′q sin(θ +2π

3) − e′′d cos(θ +

2π

3))

+1

ω0

cos(θ +2π

3)de′′qdt

+1

ω0

sin(θ +2π

3)de′′ddt

(128)

This set of expressions describes the equivalent circuit shown in Figure 4.

∩∩∩∩x′′d

∩∩∩∩x′′d

∩∩∩∩x′′d

+ −

e′′c+ −

e′′b+ −

e′′a

∩∩∩∩xg

-ia

-

ib

-ic

vc

vb

va

Figure 4: Equivalent Network Model

10 Restatement Of The Model

The synchronous machine model which uses the three phase currents as state variables may nowbe stated in the form of a set of differential and algebraic equations:

dψkd


dψkq


dψf

dt= −ω0rf if (131)

dδ

dt= ω − ω0 (132)

17

dω

dt=

ω0

2H

(

Tm + e′′q iq + e′′did)

(133)

where:[

ikd

if

]

=

[

xkd xad

xad xf

]

−1 ([

ψkd

ψf

]

−

[

xad

xad

]

id

)

and

ikq =1

xkq

ψkq −xaq

xkq

iq

(It is assumed here that the difference between subtransient reactances is small enough to beneglected.)

The network interface equations are, from the network to the machine:

id = ia cos(θ) + ib cos(θ −2π

3) + ic cos(θ +

2π

3) (134)

iq = −ia sin(θ) − ib sin(θ −2π

3) − ic sin(θ +

2π

3) (135)

and, in the reverse direction, from the machine to the network:

e′′a = −ω

ω0


+1

ω0

cos(θ)de′′qdt

+1

ω0

sin(θ)de′′ddt

(136)

e′′b = −ω

ω0


3) − e′′d cos(θ −

2π

3))

+1

ω0

cos(θ −2π

3)de′′qdt

+1

ω0

sin(θ −2π

3)de′′ddt

(137)

e′′c = −ω

ω0


3) − e′′d cos(θ +

2π

3))

+1

ω0

cos(θ +2π

3)de′′qdt

+1

ω0

sin(θ +2π

3)de′′ddt

(138)

And, of course,

θ = ω0t+ δ (139)

e′′q = ψ′′

d (140)

e′′d = −ψ′′

q (141)

ψ′′

d =xadxkdlψf + xadxflψkd

xadxkdl + xadxfl + xkdlxfl

(142)

ψ′′

q =xaq

xaq + xkql

ψkq (143)

18

11 Network Constraints

This model may be embedded in a number of networks. Different configurations will result indifferent constraints on currents. Consider, for example, the situation in which all of the terminalvoltages are constrained, but perhaps by unbalanced (not entirely positive sequence) sources. Inthat case, the differential equations for the three phase currents would be:

x′′dω0

diadt

= (va − e′′a)x′′d + 2xg

x′′d + 3xg

−[

(vb − e′′b ) + (vc − e′′c )] xg

x′′d + 3xg

(144)

x′′dω0

dibdt

= (vb − e′′b )x′′d + 2xg

x′′d + 3xg

−[

(va − e′′a) + (vc − e′′c )] xg

x′′d + 3xg

(145)

x′′dω0

dicdt

= (vc − e′′c )x′′d + 2xg

x′′d + 3xg

−[

(vb − e′′b ) + (va − e′′a)] xg

x′′d + 3xg

(146)

12 Example: Line-Line Fault

We are not, however, constrained to situations defined in this way. This model is suitable forembedding into network analysis routines. It is also possible to handle many different situationsdirectly. Consider, for example, the unbalanced fault represented by the network shown in Figure5. This shows a line-line fault situation, with one phase still connected to the network.

∩∩∩∩x′′d

∩∩∩∩x′′d

∩∩∩∩x′′d

+ −

e′′c+ −

e′′b+ −

e′′a

∩∩∩∩xg

-ia

-

ib

va

∧ ∧ ∧∨ ∨

ra

∧ ∧ ∧∨ ∨

ra

∧ ∧ ∧∨ ∨

ra

Figure 5: Line-Line Fault Network Model

In this situation, we have only two currents to worry about, and their differential equationswould be:

dibdt

=ω0

2x′′d(e′′c − e′′b − 2raib) (147)

diadt

=ω0

x′′d + xg

(va − e′′a − raia) (148)

and, of course, ic = −ib.Note that here we have included the effects of armature resistance, ignored in the previous

section but obviously important if the results are to be believed.

19

13 Permanent Magnet Machines

Permanent Magnet machines are one state variable simpler than their wound-field counterparts.They may be accurately viewed as having constant field current. Assuming that we can define theinternal (field) flux as:

ψ0 = xadif0 (149)

13.1 Model: Voltage Driven Machine

We have a reasonably simple expression for the rotor currents, in the case of a voltage drivenmachine:

[

idikd

]

=

[

xd xad

xad xkd

]

−1 [

ψd − ψ0

ψkd − ψ0

]

(150)

[

iqikq

]

=

[

xq xaq

xaq xkq

]

−1 [

ψq

ψkq

]

(151)

The simulation model then has six states:

dψd


dψq


dψkd


dψkq


dω

dt=

ω0

2H(ψdiq − ψqid + Tm) (156)

dδ

dt= ω − ω0 (157)

13.2 Curent-Driven Machine Model

In the case of a current-driven machine, rotor currents required in the simulation are:

ikd =1

xkd

(ψkd − xadid − ψ0) (158)

ikq =1

xkq

(ψkq − xaqiq) (159)

Here, the “flux behind subtransient reactance” is, on the direct axis:

ψ′′

d =xkdlψ0 + xadψkd

xad + xkdl

(160)

and the subtransient reactance is:x′′d = xal + xad||xkdl (161)

20

On the quadrature axis,

ψ′′

q =xadψkq

xad + xkql

(162)

andx′′q = xal + xaq||xkql (163)

In this case there are only four state equations:

dψkd


dψkq


dω

dt=

ω0

2H

(

e′′q iq + e′′did + Tm

)

(166)

dδ

dt= ω − ω0 (167)

The interconnections to and from the network are the same as in the case of a wound-fieldmachine: in the “forward” direction, from network to machine:


3) + ic cos(θ +

2π

3) (168)


3) − ic sin(θ +

2π

3) (169)

and, in the reverse direction, from the machine to the network:

e′′a = −ω

ω0


+1

ω0

cos(θ)de′′qdt

+1

ω0

sin(θ)de′′ddt

(170)

e′′b = −ω

ω0


3) − e′′d cos(θ −

2π

3))

+1

ω0

cos(θ −2π

3)de′′qdt

+1

ω0

sin(θ −2π

3)de′′ddt

(171)

e′′c = −ω

ω0


3) − e′′d cos(θ +

2π

3))

+1

ω0

cos(θ +2π

3)de′′qdt

+1

ω0

sin(θ +2π

3)de′′ddt

(172)

13.3 PM Machines with no damper

PM machines without much rotor conductivity may often behave as if they have no damper windingat all. In this case the model simplifies even further. Armature currents are:

id =1

xd

(ψd − ψ0) (173)

iq =1

xq

ψq (174)

21

The state equations are:

dψd


dψq


dω

dt=

ω0

2H(ψdiq − ψqid + Tm) (177)

dδ

dt= ω − ω0 (178)

13.4 Current Driven PM Machines with no damper

In the case of no damper the machine becomes quite simple. There is no “internal flux” on thequadrature axis. Further, there are no time derivatives of the internal flux on the d- axis. The onlymachine state equations are mechanical:

dω

dt=

ω0

2H(ψ0iq + Tm) (179)

dδ

dt= ω − ω0 (180)

The “forward” network interface is as before:


3) + ic cos(θ +

2π

3) (181)


3) − ic sin(θ +

2π

3) (182)

and, in the reverse direction, from the machine to the network, things are a bit simpler than before:

e′′a = −ω

ω0

ψ0 sin(θ) (183)

e′′b = −ω

ω0

ψ0 sin(θ −2π

3) (184)

e′′c = −ω

ω0

ψ0 sin(θ +2π

3) (185)

(186)

22

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