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Published in IET Electric Power Applications

Received on 24th September 2012

Revised on 28th January 2013

Accepted on 31st January 2013

doi: 10.1049/iet-epa.2012.0293

ISSN 1751-8660

Dynamic modelling of the brushless doubly fed

machine

Paul C. Roberts1, Teng Long2, Richard A. McMahon3, Shiyi Shao4, Ehsan Abdi4,

Jan M. Maciejowski3

1

Sagentia Ltd., Cambridge, UK

2

GE Power Conversion, Rugby, UK

3

Department of Engineering, University of Cambridge, Cambridge, UK

4

Wind Technologies Ltd., Cambridge, UK

E-mail: teng.long@ge.com

Abstract: A coupled-circuit model for the brushless doubly fed machine (BDFM) has been developed. The transformation of the

model into the d–qaxis form, ultimately in a synchronous reference frame in which machine currents and voltages have constant

values in the steady state, has been carried out. A model-reduction technique is presented, which gives a concise representation of

the ‘nested-loop’rotor design using a single d–qpair. These models have been experimentally veriﬁed and give a convenient and

accurate way of calculating the dynamic behaviour of a BDFM. The ability to represent the BDFM with a single d–qpair

considerably simpliﬁes the design of suitable controllers.

Nomenclature

X

1

,X

2

,X

r

indicating a stator 1, 2 or rotor quantity X

p

1

,p

2

stator winding pole pairs

θ

r

,ω

r

rotor position, rad; angular velocity, rad/s

geffective air gap, m

l,dstack length, mean air gap diameter, m

α

c

,α

s

coil, slot pitch, rad

w

s

slot mouth opening, rad

Jmoment of inertia

T

e

,T

l

electrical, load torque

Rﬁeld of real numbers

1 Introduction

Contemporary interest in the brushless doubly fed machine

(BDFM) arises primarily from its potential as a generator in

wind turbines as an alternative to the widely used slip-ring

induction generator [1,2]. The BDFM allows the retention

of the proven and economical doubly fed arrangement

while the elimination of brushgear is predicted to lead to a

signiﬁcant improvement in reliability [1]. The BDFM can

also be used as drive [3,4].

Prototype BDFMs have been reported by several research

groups [5–9]. These machines comprise two windings on a

common stator core wound for different pole-pair numbers

(p

1

and p

2

) chosen so that there is no direct coupling

between them. A special rotor construction is used which

couples to both stator windings. In operation, one

winding is connected directly to the grid and the other is

supplied with variable voltage and frequency from a

converter.

Normally, the BDFM is run in a synchronous mode in

which the shaft speed has a ﬁxed relationship to the

frequencies supplied to the two windings [10]. However,

the BDFM is not stable in the open loop at all speeds [9],

so the ability to predict the dynamic behaviour of the

BDFM is important. In practice, a controller is necessary to

ensure stability and knowledge of machine dynamics is

needed to design these controllers.

The coupled-circuit model, ﬁrst applied to the BDFM by

Wallace et al.[

11,12] enables the dynamic performance to

be analysed. The coupled-circuit model is conceptually

straightforward but it has the disadvantage that it uses

position-dependent mutual inductance matrices.

Transformation into a d–qaxis form allows this position

dependence to be replaced by a speed dependence, as ﬁrst

shown for the BDFM by Li et al.[

13,14] and subsequently

by Boger et al.[

5]. The d–qaxis representation is

computationally simple and is well suited to controller

implementation. Several controllers based on a single d–q

pair for the rotor have been reported [7,15,16].

However, the nature of the most commonly used type of

rotor design, the nested-loop arrangement because of

Burbridge and Broadway [17], creates a difﬁculty. This

design has S=p

1

+p

2

evenly spaced sets of circuits, each

with Nindependent loops, as shown in Fig. 1. Kemp et al.

modiﬁed the model of Boger et al. to represent the rotor

using Nd–qpairs, where Nis the number of loops within

an individual nest [18]. Nevertheless, representing the rotor

with a single d–qpair is necessary for controller

implementation as well as being convenient for modelling [7].

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544 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

To address the problem, Boger et al.[5] summed the d–q

pairs for each loop for both the p

1

and p

2

couplings, and

then divided the sums by the number of loops, to give a

single d–qpair for the rotor, but did not test the validity of

this approach under dynamic conditions. There remains,

therefore a need for a veriﬁed procedure for deriving a

single d–qpair for nested-loop rotors with minimum loss of

accuracy. This paper describes an approach starting from a

coupled-circuit model [19]. Predictions from the full and

reduced models for the dynamic behaviour of a 4 pole/8

pole BDFM are compared with experimental data and also

to predictions using the method proposed by Boger et al.

2 Class of BDFMs to be analysed

In this paper we analyse BDFMs ﬁtting the following

deﬁnition:

1. The stator comprises two windings which do not couple

directly with pole pairs chosen such that |p

1

−p

2

|>1.

2. Rotors must have (or can be modelled as having)

S=p

1

+p

2

identical circuits spaced evenly around the

circumference. The circuits must be constructible from coil

groups of the form illustrated in Fig. 2.

3. The rotor may have multiple sets of circuits as described

above in (2); each set may be different from the other sets

and each set of circuits may be positioned at any angular

position around the rotor.

3 Coupled-circuit model

3.1 Electromagnetic equations

The coupled-circuit model was developed following the

procedure outlined by Wallace et al.[

12], which applies

standard coupled-circuit theory (see for example [20]) to

the BDFM. The model assumes that all the ﬂux leaving the

stator crosses the air gap and enters the rotor and that the

permeability of iron is inﬁnite, as in previous studies [6,

11]. It is worth noting that in a BDFM both the stator and

rotor have open slots.

The coupled-circuit model uses values for the mutual

inductance between two coils, requiring the calculation of

ﬂux densities and hence ﬂux linkages. Ampère’s law and

ﬂux conservation lead to an expression for the air gap

magnetic ﬂux density at angle θ(measured in a ﬁxed

reference frame), because of ‘unit current’in the kth coil

Bk(

u

)=Nk

m

0

g

2

p

−

a

ck

2

p

,

b

k,

u

,

b

k+

a

ck

−

a

ck

2

p

,otherwise

⎧

⎪

⎨

⎪

⎩

(1)

where the coil has N

k

turns, starts at angular position β

k

and

subtends an angle

a

ck, as illustrated in Fig. 2. The effective

air gap is g. This derivation can incorporate linear gradients

in the ﬂux density proﬁle across the slot if desired. The

mutual inductance between coil jand coil kis deﬁned as

the ﬂux linked by coil j,Ψ

j

, because of unit current in

coil k

Cj=Mjk ik=ld

2Nj

b

j+

a

cj

b

j

Bk(

u

)d

u

(2)

Equation (2) may be evaluated directly for coils with any

alignment relative to each other. Alternatively, using a

Fourier series representation, the mutual inductance

Fig. 1 Nested-loop rotor with three sets of loops in each of p

1

+p

2

nests

Fig. 2 Magnetic ﬂux and current density for a stator coil, all dimensions in radians except for g, the air gap width

www.ietdl.org

IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556 545

doi: 10.1049/iet-epa.2012.0293 &The Institution of Engineering and Technology 2013

between coil jand coil kcan be shown to be

Mjk =wd

2

2Nk

m

0

p

g

2Nj

p

n[N

sin n

a

ck/2

n

sin nwsk/2

nwsk/2

sin n

a

cj/2

n

sin nwsj/2

nwsj/2

p

cos

b

k+

a

ck

2+wsk

2−

b

j−

a

sj

2−wsj

2

(3)

where the subscript j/krefers to the jth/kth coil.

Having calculated the mutual inductance between coils the

effect of leakage ﬂux must be added. It is assumed that

leakage only modiﬁes the self-inductance of each coil, so

the total inductance is given by Mjj +Llj, where Lljis the

leakage inductance of the jth coil which can be estimated

by standard methods [21].

3.2 Circuit equations

For a particular relative position of rotor and stator, the circuit

equation is

˜

v=d

dt(˜

M˜

i)+˜

R˜

i(4)

where ˜

Mand ˜

Rare the mutual inductance and resistance

matrices for the (as yet unconnected) set of coils. The coil

resistances are calculated as usual [22]. The vectors ˜

vand ˜

i

are the voltages across and current through each coil.

The connection of the coils into machine winding phases is

now taken into account. If coils are connected in series then

the terminal voltage is simply the sum of the voltage across

individual coils, and the currents within each coil must be

equal. Therefore

v=Tc˜

v˜

i=TT

ci(5)

where v,iare vectors of terminal voltages across and currents

within each circuit, and Tcwill be a non-square matrix whose

elements are either 1, −1 or 0 to represent the appropriate

connection of coil voltages to form winding phases.

Therefore (4) can be transformed into a set of equations in

terms of the ‘circuit’currents and voltages, thus the circuit

resistance and inductance matrices are given by

R=Tc˜

RTT

cM=Tc˜

MTT

c(6)

Ris typically diagonal and Mcan be written in a form in

which portions are ‘circulant’(see [20, ch. 10]). The

circulant nature of Mwill be exploited in this paper.

3.3 Dynamic equations

The dynamic equations for the BDFM are

v=d

dt(Mi)+Ri Te=1

2iTdM

d

u

r

iJd

v

r

dt=Te−Tl(7)

where where Jis the moment of inertia of the machine and T

l

is the load torque. An expression for the electrical torque can

be derived from energy considerations [22]. Rewriting (7) as

d

dt

i

u

r

v

r

⎡

⎣⎤

⎦=

M−1−R−

v

r

dM

d

u

r

i+M−1v

v

r

1

2JiTdM

d

u

r

i−Tl

J

⎡

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎦

(8)

allows direct solutions using standard differential equation

solvers, noting that M, and hence dM/dθ

r

are position

dependent. Voltage and current vectors and mutual

inductance and resistance matrices may be partitioned into

stator 1, stator 2 and rotor portions. In the BDFM the rotor

voltage is always zero.

4 Rotor and synchronous reference frames

4.1 d–q axis model in the rotor reference frame

Although (8) can be used to determine the dynamic response

of a BDFM, a d–qrepresentation has advantages. At the same

time, it is implicit in the d–qtransformation that the stator–

rotor (and rotor–stator) mutual inductance terms are

well-approximated by the ﬁrst non-zero term of their

Fourier series representations given in (3). Under this

assumption, the rotor–stator coupling only takes place via

p

1

and p

2

ﬁelds.

Equation (8) is transformed with the aim of obtaining

time-invariant mutual inductance matrices. The following

transformation is used to align the reference frame with the

rotor angle

Tdq =Tdq0s1 00

0Tdq0s2 0

00Tdq0r

⎡

⎣⎤

⎦(9)

where Tdq0s1 ,Tdq0s2 and Tdq0rare the stator 1, stator 2 and

rotor transformation matrices, respectively.

In order to ensure that the model dynamics are preserved in

the transformation to d–qaxis, the transformation matrix, Tdq

must be invertible. This requires the matrices Tdq0s1 ,Tdq0s2

and Tdq0rto be invertible. The stator windings transform

according to [22], for stator x= 1 or 2 (see (10))

where p

x

is the pole-pair number of the winding.

For the rotor transformation, initially a BDFM rotor with a

single circuit per ‘nest’,thatis,Scircuits in total can be

considered. From [20, ch. 10], there exists a symmetric

Tdq0sx=

2

3

cos px

u

r

cos px

u

r−2

p

3px

cos px

u

r−4

p

3px

sin px

u

r

sin px

u

r−2

p

3px

sin px

u

r−4

p

3px

1

2

√1

2

√1

2

√

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(10)

www.ietdl.org

546 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

components transformation for an Sphase rotor which

diagonalises the rotor–rotor and rotor–stator inductance

matrices, leaving only two non-zero entries. Equation (11) (ﬁrst

suggested in [5]) can be seen to comprise the real and

imaginary parts of the ﬁrst non-constant row of a symmetrical

components transformation matrix, augmented a constant third

row. Crucially, (11) contains terms which are phase offset from

each other by 2π/(p

1

+p

2

) in electrical radians. Similarly, by

applying Lemma 1, it is clear that (11) will diagonalise a

circulant matrix, such as a mutual inductance matrix (see (11)).

Boger et al.[

5] also deﬁne a similar transformation for p

2

pole pairs and in all cases either the p

1

or p

2

transformation

can be used. The only difference being the negation of sin

terms in the p

2

case as

sin 2

p

p1

p1+p2

n

=sin 2

p

p1

p1+p2

n+2

p

p1+p2

p1+p2

n

for all integer n. Hence

sin 2

p

p1

p1+p2

n−2

p

p1+p2

p1+p2

n

=−sin 2

p

p2

p1+p2

n

(12)

The cosine terms behave similarly although they do not

change sign.

To make C

r1

invertible, it can be augmented according to

Tr1 =Cr1

C⊥

r1

(13)

where the rows of C⊥

r1 are orthonormal and span the orthogonal

complement to the row space of C

r1

. In the general case of a

rotor with Nloops for each of its Snests, the circuits can be

treated as Nsets of Sphase windings. Consequently, a

transformation comprising Ncopies of (11) augmented

appropriately makes the transformation invertible. The

transformation becomes, with Tdqr

[RNS×NS

Tdqr=Tr

T∅r

=

Cr1 00

0..

.

0

00Cr1

C⊥

r1 00

0..

.

0

00C⊥

r1

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(14)

Application of the transformations gives

idq0s1

idq0s2

idqr

i⊘r

⎡

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎦=Tdq

is1

is2

ir

⎡

⎢

⎣⎤

⎥

⎦

vdq0s1

vdq0s2

vdqr

v⊘r

⎡

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎦=Tdq

vs1

vs2

vr

⎡

⎢

⎣⎤

⎥

⎦

(15)

where idqr,vdqr

[R2Nand i⊘r,v⊘r

[RNS−2N.

By substituting these into (7) and applying the chain rule,

the full dynamic model transformed into the rotor reference

frame is obtained. In common with standard induction

machines, by using the properties of circulant matrices [20],

the transformed model comprises constant matrices, as

given in (16). It can be shown that the extra rotor current

states i⊘rarising from the transformation are independent of

all other currents in the machine and are ‘unobservable’

(their values cannot be determined from output

measurements), and also ‘uncontrollable’, since v⊘r=0

(since the rotor has no voltage source). They also have

stable dynamics (this can be seen if the transformed (7) are

written out in full, then the rotor–rotor matrix terms can be

shown to be negative deﬁnite), so these states can be

omitted and (7) written without loss of modelling accuracy as

vdq0=Rdq +

v

rQdq

idq0+Mdq

didq0

dt

Te=1

2iT

dq0Sdqidq0

with vdq0=

vdq0s1

vdq0s2

0

⎡

⎢

⎣⎤

⎥

⎦, and idq0=

idq0s1

idq0s2

idqr

⎡

⎢

⎣⎤

⎥

⎦

(16)

Q

dq

arises from application of the chain rule, and S

dq

contains

elements from Q

dq

and arises for the same reason. The form

of S

dq

is similar to that of S

sync

, as in (30).

The equivalent circuit of the full dq model using the rotor

reference frame, for the example machine described in

Appendix 1 is shown in Fig. 3, where

c

ds1 =Ldqs1 iqs1 +

Mdqsr11 iqr1 +Mdqsr11 iqr2 +Mdqsr11 iqr3 and

c

ds2 =Ldqs2 iqs2 +

Mdqsr21 iqr1 +Mdqsr21 iqr2 +Mdqsr21 iqr3 . The ﬁgure shows the

d-axis equivalent circuit. The q-axis equvialent circuit is

similar, however Qvqs1 =−p1

v

r

c

qs1 and Qvqs2 =−p2

v

r

c

qs2

with similar expressions for

c

qs1 and

c

qs2 . There is no

Fig. 3 Full dq-axis equivalent circuit of a three loop per nest

BDFM in the rotor reference frame (only d-axis shown)

Cr1 =

2

p1+p2

cos(0)cos 2

p

p1

p1+p2

cos 2

p

2p1

p1+p2

··· cos 2

p

p1+p2−1

p1

p1+p2

sin(0)sin 2

p

p1

p1+p2

sin 2

p

2p1

p1+p2

··· sin 2

p

p1+p2−1

p1

p1+p2

⎡

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎦

(11)

www.ietdl.org

IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556 547

doi: 10.1049/iet-epa.2012.0293 &The Institution of Engineering and Technology 2013

mutual coupling between the d- and q-axis of the equivalent

circuit.

4.2 d–qaxis model in a synchronous reference

frame

The d–qaxis model derived in the previous section can be

transformed into any arbitrary reference frame, but a frame

which is in synchronism with currents in the two stators has

the convenience of making these currents constant in value.

Zhou and Spée [23] proposed such a frame, but their

derivation is only for a current-fed BDFM with Srotor

circuits, each having a single loop [23].

The transformation matrix required is a rotation matrix (the

zero-sequence in the stators need not be transformed). Thus

for a reference frame rotation of ψ(which may be time

varying), the transformation matrix is

Tdq0rot (

c

)=

cos(

c

) sin(

c

)0

−sin(

c

) cos(

c

)0

001

⎡

⎣⎤

⎦

=Tdqrot (

c

)0

01

(17)

When the BDFM is operating in synchronous mode, the shaft

angular velocity is related to the supply angular frequencies

by [9]

v

r=

v

1+

v

2

p1+p2

(18)

and the angular frequency of the currents in both stator

windings and the rotor, in the rotor reference frame, is

given by

v

s=p1

v

r−

v

1=−p2

v

r+

v

2(19)

From (19), note that currents in the rotor reference frame

rotate in the opposite sense to those in stator 2, but in the

same sense to those in stator 1. For this reason, when

transforming the BDFM d–qaxis model into another

reference frame, the stator 2 transformation must be in the

opposite sense to those applied to stator 1 and the rotor.

Furthermore, (19) shows that rotating the reference frame

by t

0

v

sdtleads to constant currents in this new reference

frame.

Therefore the following rotations are applied

c

s1 =

f

s1 +t

0

v

sdtstator 1 (20)

c

s2 =

f

s2 −t

0

v

sdtstator 2 (21)

c

r=t

0

v

sdtrotor (22)

where φ

s1

and φ

s2

are constant angular offsets which may be

freely chosen. Choosing values of

f

s1 =

f

11and

f

s2 =−

f

21aligns stators 1 and 2 to the rotor coils, the

effect of which is to diagonalise some, or all, of the stator–

rotor mutual inductance terms. An invertible state

transformation matrix may therefore be made with multiple

copies of (17) with the choices of φas described above,

concatenated into a full transformation matrix.

Trot =diag Tdq0rot

c

s1

,Tdq0rot

c

s2

,Tdqrot

c

r

,...,

Tdqrot

c

r

(23)

Substituting (20)–(22) into (23) gives the complete

synchronous reference frame transformation

Tsync =diag Tdq0rot t

0

v

sdt

Tdq0rot

f

s1

T−1

dq0rot t

0

v

sdt

Tdq0rot

f

s2

Tdqrot t

0

v

sdt

,...,Tdqrot t

0

v

sdt

(24)

The transformed voltages and current are given by

idq0s

s1

idq0s

s2

idqs

r

⎡

⎢

⎣⎤

⎥

⎦=Tsync

idq0s1

idq0s2

idqr

⎡

⎢

⎣⎤

⎥

⎦

vdq0s

s1

vdq0s

s2

vdqs

r

⎡

⎢

⎣⎤

⎥

⎦=Tsync

vdq0s1

vdq0s2

vdqr

⎡

⎢

⎣⎤

⎥

⎦(25)

Substituting (25) into (16), and rearranging gives the full

dynamic equations in state-space form (see (26))

d

dt

is

dq0s1

is

dq0s2

is

dqr

u

r

v

r

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎦

=

M−1

sync −Rsync −Qsync

is

dq0s1

is

dq0s2

is

dqr

⎡

⎢

⎣⎤

⎥

⎦+

vs

dq0s1

vs

dq0s2

0

⎡

⎣⎤

⎦

⎛

⎜

⎝⎞

⎟

⎠

v

r

1

2J

is

dq0s1

is

dq0s2

is

dqr

⎡

⎢

⎣⎤

⎥

⎦

T

Ssync

is

dq0s1

is

dq0s2

is

dqr

⎡

⎢

⎣⎤

⎥

⎦−Tl

J

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(26)

www.ietdl.org

548 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

and

RsyncWTsync RdqT−1

sync (27)

Qsync

v

1,

v

r

W

v

rTsyncQdq T−1

sync +TsyncMdq

d

dtT−1

sync (28)

MsyncWTsync MdqT−1

sync (29)

SsyncWTsync

00Qdqsr1

00Qdqsr2

QT

dqsr1 QT

dqsr2 0

⎡

⎢

⎣⎤

⎥

⎦T−1

sync (30)

As a result of the transformation to the synchronous reference

frame, R

sync

,S

sync

and M

sync

become constant-valued, and

Q

sync

is linearly dependent on ω

1

and ω

r

only.

As R

dq

is diagonal and the resistance values appear in pairs

it is easy to show that

Rsync =TsyncRdqT−1

sync =Rdq

because each resistance pair may be represented at the identity

multiplied by a scalar which therefore commutes with the

transformation matrix.

Similarly the stator–stator portions of M

dq

must remain

unchanged under the transformation as they too have the

same structure. From consideration of the rotor–rotor terms,

as given in (50), it can be seen that the rotor–rotor terms

remain unchanged, as all the rotor transformation matrices

have been chosen to be the same. The stator–rotor portions

of M

dq

which comprise a scaled rotation matrix of angle

f

11for stator 1 and

f

21for stator 2 will be diagonalised,

and the remaining terms will be rotated by −

f

11or −

f

21,

respectively. Hence M

sync

consists of constant terms only.

As Q

dq

is closely related to M

dq

similar comments apply, in

particular it may be shown that in all cases [19, p. 161]

Qsync

v

r,

v

1

=X1TsyncMdq T−1

sync

+−p1

v

r+

v

1

TsyncMdq T−1

syncX2(31)

where (see equation at the bottom of the page)

Therefore as M

sync

is constant, Q

sync

(ω

r

,ω

1

) is linearly

dependent on ω

1

and ω

r

, and furthermore it may be veriﬁed

that in the steady state in the synchronous mode (that is

when (18) holds for all time), the currents and voltages in

the rotor and stator are all constant-valued.

5 Model order reduction for ‘nested-loop’

design rotors

The nested-loop-type rotor leads to a d–qmodel with one

rotor d–qpair per loop within each nest (i.e. Nrotor d–q

pairs), so the task is to reduce this to a single pair with

minimum loss of accuracy. The ﬁrst step is to diagonalise

the rotor mutual inductance matrix, M

r

, by means of an

orthogonal transformation –this will always be possible as

M

r

is symmetric. Furthermore, the state order will be chosen

such as to order the eigenvalues in decreasing order from

the top left. This can be achieved with the following state

transformation

is

˜

ir

=I0

0T

is

ir

(32)

where Tis a matrix of the eigenvectors of M

r

ordered

appropriately.

With M

r

diagonal, the BDFM equations are partitioned into

three parts: stator, two rotor states, remainder of the rotor

X1=

p1

v

r

010

−100

000

⎡

⎢

⎣⎤

⎥

⎦

p2

v

r

010

−100

000

⎡

⎢

⎣⎤

⎥

⎦000

000

000

⎡

⎢

⎣⎤

⎥

⎦..

.

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

X2=

010

−100

000

⎡

⎢

⎣⎤

⎥

⎦0−10

100

000

⎡

⎢

⎣⎤

⎥

⎦010

−100

000

⎡

⎢

⎣⎤

⎥

⎦..

.

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

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IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556 549

doi: 10.1049/iet-epa.2012.0293 &The Institution of Engineering and Technology 2013

states, as follows

vs

0

0

⎡

⎢

⎣⎤

⎥

⎦=

Ms˜

Msr1

˜

Msr2

˜

MT

sr1

˜

Mr10

˜

MT

sr20˜

Mr2

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

d

dt

is

˜

ir1

˜

ir2

⎡

⎢

⎣⎤

⎥

⎦

+

Rs00

0˜

Rr1

˜

Rr12

0˜

RT

r12

˜

Rr2

⎡

⎢

⎣⎤

⎥

⎦+

v

r

Qs˜

Qsr1

˜

Qsr2

00 0

00 0

⎡

⎢

⎣⎤

⎥

⎦

⎛

⎜

⎝⎞

⎟

⎠

×

is

˜

ir1

˜

ir2

⎡

⎢

⎣⎤

⎥

⎦(33)

It should be noted that the off-diagonal resistances will, in

general, be non-zero. In all nested-loop rotors that have

been simulated, the terms ˜

Msr2,˜

Rr12 , and ˜

Qsr2are small,

and hence truncating the system by removing ˜

ir2is a good

approximation, since this means these extra rotor current

states have very little inﬂuence on the machine dynamics.

Physically, a small value of ˜

Msr2, (and hence ˜

Qsr2)

corresponds to having a large transformer turns ratio

between the stator and that part of the rotor state. This

means that smaller voltages are induced, and if the

resistances are similar, then the current will be smaller. The

effect of diagonalisation (and ordering) of M

r

has been to

prioritise the effect of the stator coupling on the various

parts of the rotor. Hence by keeping only the states

corresponding to the most signiﬁcant M

r

eigenvalues, the

system can be reduced with minimum error.

The reduction algorithm can be summarised as:

1. Compute T, a matrix of eigenvectors of M

r

ordered so that

the corresponding eigenvalues decrease from left to right.

2. Partition Tinto T1T2

"#

, where T

1

is two columns wide.

3. Apply the resulting state ‘transformation’

˜

i=I0

0TT

1

i

which reduces the state order.

Thus application of the above algorithm to (33) leads to

vs

0

=Ms˜

Msr1

˜

MT

sr1

˜

Mr1

d

dt

is

˜

ir1

+Rs0

0˜

Rr1

+

v

r

Qs˜

Qsr1

00

$%

is

˜

ir1

(34)

This can be rearranged and reincorporated into the d–q(16),

effectively giving a single-loop (N= 1) representation. In this

case the equivalent circuit is a simpliﬁed form of Fig. 3,

where there is only a single rotor circuit.

Fig. 4 Responses to a step reduction in speed from 550 to 450 r/min

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550 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

6 Model verification

6.1 Experimental arrangement

The models were veriﬁed by tests on a frame size 180 4 pole/8

pole BDFM with a nested-loop rotor comprising six nests of

three loops (S=6,N= 3). Details of this machine are given

in Appendix 1. Tests were carried out in an open-loop stable

region of operation to avoid artefacts from a controller. The

BDFM could be driven as a generator, via a torque

transducer, by a DC drive operated in torque control mode.

The output of the torque transducer, which has a bandwidth

of 5 kHz, was used as the load torque for modelling

purposes, thereby avoiding the need to include the

characteristics of the DC drive in the models. The total

moment of inertia of the BDFM’s rotor and coupling to the

torque transducer is 0.13 kgm

2

.

Stator 1 (4-pole) was supplied through a variable

transformer with 240 V

rms

at 50 Hz and stator 2 (8-pole)

was supplied from an inverter. The actual voltages of all

three-phases applied to both stators were recorded for use

as inputs to the models.

6.2 Comparison of models

In the ﬁrst test, the machine was run initially at 550 r/min,

with zero generating torque; stator 2 was supplied with

30 V

rms

at 5 Hz. The frequency was then changed from

5to −5 Hz causing the rotor to decelerate to 450 r/min,

resulting in transients in torque and machine currents.

Taking the measured stator voltages and load torque as

inputs, the models were used to calculate the transients

in electromagnetic torque, speed and stator currents. A

comparison between predictions from the coupled-circuit

model, the full and reduced d–qmodels, and the

experimental data are shown in Fig. 4. During this test

the change of frequency was achieved by interchanging

the sequence of two of the phases connected to stator 2,

but the phase shown in Fig. 4is the one which was

unaltered. The speed transient predicted by the

coupled-circuit model matches the experimental result

whereas the predictions from the d–qmodels, although

close, show some small oscillations. The same pattern is

evident in the predictions of electromagnetic torque. The

stator currents derived from the coupled-circuit model

again match the experimentally measured value but

those obtained from the d–qmodels do not have such

good agreement. The 4-pole winding is shown on an

expanded time-scale in this, and subsequent ﬁgures, for

clarity.

In the second test, the machine was run at 550 r/min with a

generating torque of 45 Nm which was rapidly reduced to

zero at t= 0.8. Stator 2 was supplied with 30 V

rms

at 5 Hz.

Fig. 5 Responses to a step reduction in torque from 45 Nm to zero in generating mode

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Once again, the measured stator voltages and load torque

were used as inputs to the models. The calculated transients

in electromagnetic torque are shown, and the calculated

transients in speed and stator currents are compared with

measured values in Fig. 5. All the models give predictions

for the electromagnetic torque which are close, and the

predictions for the speed transient accord with the

experimental result. The greater ripple in torque and speed

observed in this test arise as the DC machine is energised.

At full load all the models give values for the stator

currents that are close to the measured current but the

agreement is not so good for the 8-pole stator current at a

zero load torque.

The coupled-circuit model gives good predictions of

machine performance under the conditions tested. The full

d–qmodel, which only considers stator–rotor coupling by

the principal ﬁelds, also gives generally good predictions.

However, the degree of damping is underestimated,

reﬂecting that fact that coupling via space harmonics

cannot be completely represented by additional

inductances, and the values for stator currents at low load

torques show discrepancies, perhaps because of saturation.

The loss of accuracy resulting from reducing the full d–q

model to one d–qpair is not signiﬁcant. In practice, the

reduced d–qmodel has been successfully used as the

basis for the control of BDFMs with ratings up to 250

kW [24].

6.3 Comparison with summed loop representation

In a previous study, Boger et al.[

5] reduced a model for a

nested-loop rotor with three loops to a single d–qpair by

taking the average of the d–qpairs for the three loops. The

predictions from the summed loop method and the

coupled-circuit model developed in this paper to the same

step change in speed as used earlier are compared in Fig. 4,

which also includes experimental data. The predictions from

the sum of loops method are signiﬁcantly in error. Recent

work [25] shows that coupling between the loops via space

harmonics of the principal ﬁelds manifests itself as a

reduction in rotor leakage inductance. Neglecting this, as in

the approach described in [5], leads to a signiﬁcant error in

rotor inductance, and to a degree in rotor resistance, and

hence to erroneous predictions of machine dynamics (Fig. 6).

7 Conclusions

The coupled-circuit model has been shown to be an effective

way of modelling the dynamic behaviour of the BDFM with

good accuracy yet low computational cost. The

coupled-circuit model conveniently leads to a d–q

representation of the BDFM, but the rotor normally will

have multiple d–qpairs. A model-reduction technique has

been developed which enables a single d–qpair to be

derived. This reduced model gives good predictions of

Fig. 6 Comparison of mode- reduction techniques for a step change of speed as in Fig. 4

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552 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

machine performance. The ultimate accuracy of any of the

models depends on the precision with which machine

parameters can be calculated and the model does not

include saturation of the iron circuit.

The reduced model can be used to generate a single d–q

pair for a rotor for use in implementing a controller. In

addition, in a synchronous reference frame, the model can

be linearised about an equilibrium point, allowing standard

eigenvalue-based stability analysis and advanced control

system design methods to be used.

8 References

1 Arabian-Hoseynabadi, H., Tavner, P.J., Oraee, H.: ‘Reliability

comparison of direct-drive and geared-drive wind turbine concepts’,

Wind Energy, 2010, 13, (1), pp. 62–73

2 Logan, T., Warrington, J., Shao, S., McMahon, R.A.: ‘Practical

deployment of the brushless doubly-fed machine in a medium scale

wind turbine’. Eighth Int. Conf. Power Electronics and Drive Systems,

Taiwan, November 2009, pp. 470–475

3 Li, R., Spee, R., Wallace, A.K., Alexander, G.C.: ‘Synchronous drive

performance of brushless doubly-fed motors’,IEEE Trans. Ind. Appl.,

1994, 30, (4), pp. 963–970

4 Shao, S., Abdi, E., McMahon, R.: ‘Operation of brushless doubly-fed

machine for drive applications’. Fourth IET Int. Conf. Power

Electronics, Machines and Drives, York, UK, April 2008, pp. 340–344

5 Boger, M., Wallace, A., Spee, R., Li, R.: ‘General pole number model of

the brushless doubly-fed machine’,IEEE Trans. Ind. Appl., 1995, 31,

(5), pp. 1022–1028

6 Williamson, S., Ferreira, A.C.: ‘Generalised theory of the brushless

doubly-fed machine. Part 2: model veriﬁcation and performance’,IEE

Proc. Electr. Power Appl. 1997, 144, pp. 123–129

7 Poza, J., Oyarbide, E., Sarasola, I., Rodriguez, M.: ‘Vector control

design and experimental evaluation for the brushless doubly fed

machine’,IEE Proc. Electr. Power Appl., 2009, 3, (4), pp. 247–256

8 Runcos, F., Carlson, R., Sadowski, N., Kuo-Peng, P., Voltolini, H.:

‘Performance and vibration analysis of a 75 kw brushless double-fed

induction generator prototype’. 41st IEEE Industry Application

Society Annual Meeting, Florida, October 2006, pp. 2395–2402

9 McMahon, R.A., Roberts, P.C., Wang, X., Tavner, P.J.: ‘Performance of

BDFM as generator and motor’,IEE Proc. Electr. Power Appl., 2006,

153, (2), pp. 289–299

10 Williamson, S., Ferreira, A.C., Wallace, A.K.: ‘Generalised theory of the

brushless doubly-fed machine. Part 1: analysis’,IEE Proc. Electr. Power

Appl., 1997, 144, pp. 111–122

11 Spee, R., Wallace, A.K., Lauw, H.K.: ‘Performance simulation of

brushless doubly-fed adjustable speed drives’. Industry Application

Society Annual Meeting, October 1989, vol. 1, pp. 738–743

12 Wallace, A.K., Spee, R., Lauw, H.K.: ‘Dynamic modelling of brushless

doubly-fed machines’. Conf. Record of the IEEE Industry Applications

Society Annual Meeting, October 1989, vol. 1, pp. 329–334

13 Li, R., Wallace, A., Spee, R.: ‘Dynamic simulation of brushless doubly-

fed machines’,IEEE Trans. Energy Convers., 1991, 6, (3), pp. 445–452

14 Li, R., Wallace, A.K., Spee, R.: ‘Two axis model development of

cage-rotor brushless doubly-fed machines’,IEEE Trans. Energy

Convers., 1991, 6, (3), pp. 453–460

15 Zhou, D., Spee, R., Alexander, G.C.: ‘Experimental evaluation of a rotor

ﬂux oriented control algorithm for brushless doubly-fed machines’,

IEEE Trans. Power Electron., 1997, 12, (1), pp. 72–78

16 Shao, S., Abdi, E., Barati, F., McMahon, R.A.: ‘Stator-ﬂux-oriented

vector control for brushless doubly-fed induction generator’,IEEE

Trans. Ind. Electron., 2009, 56, (10), pp. 4220–4228

17 Broadway, A.R.W., Burbridge, L.: ‘Self-cascaded machine: a low-speed

motor or high frequency brushless alternator’,IEE Proc., 1970, 117,

pp. 1277–1290

18 Kemp, A., Boger, M., Wiedenbrug, E., Wallace, A.K.: ‘Investigation

of rotor-current distributions in brushless doubly-fed machines’. 31st

IEEE Industry Applications Conf. (IAS96), October 1996, vol. 1,

pp. 638–643

19 Roberts, P.C.: ‘A study of brushless doubly-fed (induction) machines’.

PhD dissertation, University of Cambridge, 2005

20 White, D., Woodson, H.: ‘Electromechanical energy conversion’(John

Wiley & Sons, New York, 1959)

21 Draper, A.: ‘Electrical machines’(Longmans, 1981, 2nd edn.)

22 Krause, P.C., Wasynczuk, O., Sudhoff, S.D.: ‘Analysis of electric

machinery and drive systems’(IEEE Press Wiley, New York, 2002,

2nd edn.)

23 Zhou, D., Spee, R.: ‘Field oriented control development for

brushless doubly-fed machines’. IEEE Industry Application Society

Annual Meeting (IAS1996), Taipai, Taiwan, October 1996, vol. 1,

pp. 304–310

24 McMahon, R.A., Abdi, E., Malliband, P.D., Shao, S., Mathekga, M.E.,

Tavner, P.J.: ‘Design and testing of a 250 kw medium-speed brushless

dﬁg’. Sixth IET Int. Conf. Power Electronics, Machines and Drives

(PEMD 2012), March 2012, pp. 1–6

25 McMahon, R., Tavner, P., Abdi, E., Malliband, P., Barker, D.:

‘Characterising rotors for brushless doubly-fed machines (bdfm)’. XIX

Int. Conf. Electrical Machines (ICEM2010), Rome, September 2010,

pp. 1–6

26 Davis, P.J.: ‘Circulant matrices’(John Wiley & Sons, 1979)

27 Strang, G.: ‘Introduction to applied mathematics’(Wellesley-Cambridge

Press, 1986)

28 Zhou, K., Doyle, J.C., Glover, K.: ‘Robust and optimal control’

(Prentice-Hall, 1996)

9 Appendix 1

9.1 Parameters of the prototype machine

General physical dimensions of the BDFM are given in

Table 1. The stator mutual inductance (excluding leakage),

M

ss

follows (see (35))

Table 1 BDFM design specifications

machine stack length w= 189.9 mm

air gap diameter d= 174.5 mm

effective air gap width g= 0635 mm

stator

number of slots n

s

=48

winding layers startor 1: double

startor 2: double

number of pole pairs p

1

=2

p

2

=2

number of turns per-phase per-pole pair Nph1=80

Nph2=80

number of slots that the winding is

short-pitched by

mp1=2

mp2=1

wire cross-sectional area As1=1.13 mm2

As2=1.13 mm2

slot mouth w0s=3.2×10−3m

rotor

number of slots n

r

=36

bar cross-sectional area A

r

=85mm

2

slot mouth w0r=2mm

The rotor is a ‘nested-loop’design consisting of six ‘nests’of

three concentric loops of pitch 5/36, 3/36 and 1/36 of the rotor

circumference. Each nest offset by 1/6 of the circumference, for

the details see [17].

Mss =10−3×

237.4−108.6−108.6000

−108.6 237.4−108.6000

−108.6−108.6 237.4000

000244.8−109.9−109.9

000−109.9 244.8−109.9

000−109.9−109.9 244.8

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(35)

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IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556 553

doi: 10.1049/iet-epa.2012.0293 &The Institution of Engineering and Technology 2013

The rotor mutual inductance matrix (excluding leakage) is

(see (36))

The leakage inductance of each stator phase and rotor loop

are given in Table 2. The ﬁrst non-zero harmonic terms of the

stator–rotor mutual inductances are given in Table 3. The

stator per-phase and rotor loop resistances are given in

Table 4.

10 Appendix 2

10.1 Resistance and mutual inductance matrices

The class of BDFMs analysed, as deﬁned in Section 2, leads

to the following structure for resistance and mutual

inductance matrices. The mutual inductance structure is a

Table 2 Leakage inductance

Stator 1 Stator 2 Inner loop Middle loop Outer loop

3.8 mH 9.0 mH 1.69 μH 1.76 μH 1.83 μH

Table 3 First non-zero terms of the stator–rotor mutual

inductances

Inner loop Middle loop Outer loop

stator 1 169 μH 487 μH 745 μH

stator 2 167 μH 425 μH 483 μH

Mrr =10−8×

535 −16 −16 −16 −16 −16 528 −48 −48

−16 535 −16 −16 −16 −16 −48 528 −48

−16 −16 535 −16 −16 −16 −48 −48 528

−16 −16 −16 535 −16 −16 −48 −48 −48

−16 −16 −16 −16 535 −16 −48 −48 −48

−16 −16 −16 −16 −16 535 −48 −48 −48

528 −48 −48 −48 −48 −48 1558 −144 −144

−48 528 −48 −48 −48 −48 −144 1558 −144

−48 −48 528 −48 −48 −48 −144 −144 1558

−48 −48 −48 528 −48 −48 −144 −144 −144

−48 −48 −48 −48 528 −48 −144 −144 −144

−48 −48 −48 −48 −48 528 −144 −144 −144

496 −80 −80 −80 −80 −80 1487 −240 −240

−80 496 −80 −80 −80 −80 −240 1487 −240

−80 −80 496 −80 −80 −80 −240 −240 1487

−80 −80 −80 496 −80 −80 −240 −240 −240

−80 −80 −80 −80 496 −80 −240 −240 −240

−80 −80 −80 −80 −80 496 −240 −240 −240

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣−48 −48 −48 496 −80 −80 −80 −80 −80

−48 −48 −48 −80 496 −80 −80 −80 −80

−48 −48 −48 −80 −80 496 −80 −80 −80

528 −48 −48 −80 −80 −80 496 −80 −80

−48 528 −48 −80 −80 −80 −80 496 −80

−48 −48 528 −80 −80 −80 −80 −80 496

−144 −144 −144 1487 −240 −240 −240 −240 −240

−144 −144 −144 −240 1487 −240 −240 −240 −240

−144 −144 −144 −240 −240 1487 −240 −240 −240

1558 −144 −144 −240 −240 −240 1487 −240 −240

−144 1558 −144 −240 −240 −240 −240 1487 −240

−144 −144 1558 −240 −240 −240 −240 −240 1487

−240 −240 −240 2454 −400 −400 −400 −400 −400

−240 −240 −240 −400 2454 −400 −400 −400 −400

−240 −240 −240 −400 −400 2454 −400 −400 −400

1487 −240 −240 −400 −400 −400 2454 −400 −400

−240 1487 −240 −400 −400 −400 −400 2454 −400

−240 −240 1487 −400 −400 −400 −400 −400 2454

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(36)

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554 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293

direct consequence of having p

1

+p

2

identical circuits

uniformly spaces around the rotor circumference. The

resistance matrix, Ris typically diagonal, although if the

machine has a squirrel-cage rotor it will be symmetric

circulant. See [19] for further details.

R=diag Rs1,Rs2 ,Rr

(37)

Rs1 =diag R1,R1,R1

,Rs2 =diag R2,R2,R2

(38)

Rr=diag diag Rr1,Rr1 ,...,Rr1

,...,diag RrN,RrN,...,RrN

(39)

M=

Ms1 0Msr1

0Ms2 Msr2

MT

sr1 MT

sr2 Mr

⎡

⎣⎤

⎦(40)

Ms1 =

L1M1M1

M1L1M1

M1M1L1

⎡

⎢

⎣⎤

⎥

⎦,Ms2 =

L2M2M2

M2L2M2

M2M2L2

⎡

⎢

⎣⎤

⎥

⎦

(41)

Mr=

Mr1Mr12 ···

MT

r12 Mr2

..

.

..

...

...

.

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

(42)

Msr1

Msr2

=Msr11 Msr12 ··· Msr1N

Msr21 Msr22 ··· Msr2N

(43)

where MrN[RS×Sand is symmetric circulant and

MrN[RS×Sis (non-symmetric) circulant. See [26]or[27,

ch. 4] for details on circulant matrices. A typical M

sr1

and

M

sr2

element is given by (see (44))

where Q

1

=p

1

(2k−1), Q

2

=p

2

(2k−1), k= {1, 2, 3, …},

S=p

1

+p

2

,p1,p2[Nare the number stator windings 1

and 2 pole pairs, M1(n), M2(n)[Rare the coefﬁcients for

the nth absolute harmonic of mutual inductance, and

b

1,

b

2[Rare offset angles between the ﬁrst rotor coil and

the ﬁrst phase of stator windings 1 and 2, respectively.

When transformed into d–qaxes the following results are

obtained, where x= 1 for stators 1 and 2 for stator 2

quantities

Qdq0sx=

0pxLx−Mx

0

−pxLx−Mx

00

000

⎡

⎣⎤

⎦(45)

Mdq0sx=

Lx−Mx00

0Lx−Mx0

00Lx+2Mx

⎡

⎣⎤

⎦(46)

(see (47) and (48))

where Qdq0s1 ,Qdq0s2 ,Mdq0s1 ,Mdq0s2

[R3×3,Qdq0s1r ,Qdq0s2r ,

Mdq0s1r ,Mdq0s2r

[R3×2Nand are all constant (i.e.

independent of t,θ

r

,ω

r

etc.). Algebraic representations of

M1srdq1 ,M2srdq1 ,

f

11,

f

21etc. in terms of their coupled

circuit may be derived using Lemma 1. In the case of a

‘nested-loop’rotor

f

11=

f

1Nand

f

21=

f

2Nas loops

within a nest are concentric.

Table 4 Stator windings per-phase and rotor loops resistances

Stator 1 Stator 2 Inner loop Middle loop Outer loop

2.08 Ω3.55 Ω104 μΩ 119 μΩ 134 μΩ

Msr11

Msr21

=

&

n[Q1

M11(n) cos

u

r−

b

1

n

···

&

n[Q1

M11(n) cos

u

r−2

p

3p1−

b

1

n

···

&

n[Q1

M11(n) cos

u

r−4

p

3p1−

b

1

n

···

&

n[Q2

M22(n) cos

u

r−

b

2

n

···

&

n[Q2

M22(n) cos

u

r−2

p

3p2−

b

2

n

···

&

n[Q2

M22(n) cos

u

r−4

p

3p2−

b

2

n

···

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

&

n[Q1

M11(n) cos

u

r−(S−1)2

p

S−

b

1

n

&

n[Q1

M11(n) cos

u

r−(S−1)2

p

S−2

p

3p1−

b

1

n

&

n[Q1

M11(n) cos

u

r−(S−1)2

p

S−4

p

3p1−

b

1

n

&

n[Q2

M22(n) cos

u

r−(S−1)2

p

S−

b

2

n

&

n[Q2

M22(n) cos

u

r−(S−1)2

p

S−2

p

3p2−

b

2

n

&

n[Q2

M22(n) cos

u

r−(S−1)2

p

S−4

p

3p2−

b

2

n

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(44)

Mdq0sxr=Mxsrdq1

cos

f

x1

−sin

f

x1

sin

f

x1

cos

f

x1

00

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

... MxsrdqN

cos

f

xN

−sin

f

xN

sin

f

xN

cos

f

xN

00

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

(47)

Qdq0sxr=Mxsrdq1px

sin

f

x1

cos

f

x1

−cos

f

x1

sin

f

x1

⎡

⎣⎤

⎦... MxsrdqN px

sin

f

xN

cos

f

xN

−cos

f

xN

sin

f

xN

⎡

⎣⎤

⎦

⎡

⎣⎤

⎦(48)

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IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556 555

doi: 10.1049/iet-epa.2012.0293 &The Institution of Engineering and Technology 2013

The dq transformed resistance matrices will be diagonal. In

the case that R

r

is diagonal then

Rdqr=diag diag Rr1,Rr1

,...,diag RrN,RrN

(49)

where Rdqr

[R2N×2N. When R

r

is not diagonal then Rdqrmay

be derived using Lemma 1 (see (50))

where Mdqr

[R2N×2Nand all elements therein are constants.

11 Appendix 3

11.1 Circulant matrices result

Lemma 1: Given a circulant matrix (not necessarily

symmetric), C[Rn×n,n[N≥3, then

Q

Q⊥

CQ

Q⊥

T

=|

l

|cos (/

l

) sin (/

l

)

−sin (/

l

) cos (/

l

)

00

0

l

00

00S

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

where

l

,

l

0[R1and S[R(n−3)×(n−3) and

Q=

2

n

cos(

f

) cos 2

p

p

n+

f

··· cos 2

p

(n−1)p

n+

f

sin(

f

) sin 2

p

p

n+

f

··· sin 2

p

(n−1)p

n+

f

1

2

√1

2

√··· 1

2

√

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

where p[N,p=nt,t[N, and Q

⊥

an orthogonal matrix

whose rows span the orthogonal complement of the

subspace spanned by the rows of Q. Furthermore

l

=&n−1

i=0ciwiand

l

0=&n−1

i=0ci, where c

i

is the element on

the ith column of row 1 of C, and w= exp(j2π/n). Finally

if Cis symmetric then: cos(/

l

)=+1 and sin(/

l

)=0.

Proof: The proof follows the same steps whether or not Cis

symmetric, therefore without loss of generality we assume C

is symmetric. It may be shown that each column of Q

T

is an

eigenvector with corresponding eigenvalues λ,λ,λ

0

[18,

Lemma A.14] [26]. Therefore we may write

CQT=QT

l

00

0

l

0

00

l

0

⎡

⎢

⎣⎤

⎥

⎦

⇒QCQT=QQT

l

00

0

l

0

00

l

0

⎡

⎢

⎣⎤

⎥

⎦

it is easy to show that QQ

T

=Ihence

⇒QCQT=

l

00

0

l

0

00

l

0

⎡

⎣⎤

⎦

We now consider the entire transformation. First note that an

orthogonal Q

⊥

exists as QQ

T

=I[28, p.19]. Therefore we

may write

Q

Q⊥

CQ

Q⊥

T

=QCQTQCQ⊥T

Q⊥CQTQ⊥CQ⊥T

The term Q

⊥

CQ

T

is zero because

CQT=QT

l

00

0

l

0

00

l

0

⎡

⎣⎤

⎦

and by deﬁnition Q

⊥

Q

T

= 0. For similar reasons QCQ

⊥T

is

also zero (C

T

is also circulant and hence diagonalised by

Q

T

). It may be veriﬁed that QCQ

T

is diagonal [19, Lemma

A.14] [26]. Therefore

Q

Q⊥

CQ

Q⊥

T

=

l

00

0

l

0

00

l

0

⎡

⎣⎤

⎦0

0Q⊥CQ⊥T

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

Lemma 2: Given a symmetric circulant matrix,

C[Rn×n,n[N≥3, that is, a circulant matrix such that

C=C

T

, then

QC d

dtQT

=d

f

dt

0

l

0

−

l

00

000

⎡

⎣⎤

⎦

where λand Qare deﬁned as in Lemma 1.

Proof: The proof follows a similar approach to Lemma 1, and

can be found in full in [19, Lemma A.15].

Mdqr=

Ldqr1 0

0Ldqr1

Mdqr12aMdqr12b

−Mdqr12bMdqr12a

···

Mdqr12a−Mdqr12b

Mdqr12bMdqr12a

..

..

.

.

.

.

.

··· LdqrN0

0LdqrN

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(50)

www.ietdl.org

556 IET Electr. Power Appl., 2013, Vol. 7, Iss. 7, pp. 544–556

&The Institution of Engineering and Technology 2013 doi: 10.1049/iet-epa.2012.0293