space harmonics in unified electrical-machine theory

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Space harmonics in unified electrical machine theory

Prof. J. L. Willems

Indexing terms: Machine

theory,

Differential equations, Harmonics

Abstract

The paper deals with the applicability of unified machine theory to electrical machines where space harmonics

can not be neglected. It is shown that there exists cases where a linear transformation can be determined to

transform the set of time-dependent linear differential equations for a machine at constant speed into

a

set

of linear time-invariant differential equa tions, even if space harmonics are taken into consideratio n. A

criterion for applicability is presented that involves the number of phases, the number of harmonics and the

order of the harmonics that have to be considered.

1

Introduction

A basic assumption necessary for the validity of unified

machine theory

1

is that each phase on stator and rotor

produces a sinusoidal space distribution of current density

and that, m oreover, the flux is a sinusoidal function of space.

An imp ortant consequence of this hypothesis is that the r oto r-

stator mutual inductances are sinusoidal functions of the

machine angle, and hence of time for machines running at

constant speed. This sinusoidal dependence is fundamental

for the validity of the linear transformation that reduces the

set of time-varying differential equations describing the

electrical machine into a set of time-invariant differential

equations.

In this paper, an attempt is made to generalise unified

machine theory to include the effect of space harmonics to

some extent. Therefore the linear-system approach to unified

machine analysis, as developed in an earlier paper,

2

is very

useful. Indeed, this approach leads to a straightforward

derivation of a linear transformation that reduces the set of

time-varying differential equations of the electrical machine

with space harmonics into a set of time-invariant ones. The

number of space harmonics that can be taken into considera-

tion depends on the number of phases of the nonsalient part

of the machine. It is also pointed out that the stationary

behaviour of a polyphase machine at the stator (or rotor)

terminals even holds for some configurations where space

harmonics are considered; this result is rather surprising,

since, in sinusoidal steady state, for example, the space

harmonics clearly produce harmonics in the rotor (or stator)

currents and voltages.

2 Mathematical model

Consider a machine without a commutator, with a

symmetrical m-phase stator and an «-phase rotor; it is not

assumed that the M-phase windings on the rotor are sym-

metric (in induction machines, the roles of stator and rotor

should be reversed). The rotor-rotor and the stator-stator

self and mutual inductances are independent of the rotor

position, since the uniform airgap corresponds to a smooth

magnetic structure. The mutual inductance between a rotor

phase and a stator phase varies with the rotor position. In

most applications, this dependence.is assumed to be sinu-

soidal ; however, in this paper, space harmonics are taken into

consideration with respect to the airgap flux. A mutual

inductance between a rotor winding and a stator phase is

hence a periodic function of the angular position 6 of the

roto r; the period is TT if the angle 6 is expressed in electrical

degrees (we only consider 2-pole machines in the sequel). In

most cases, this periodic function is odd; i.e.

M (0) = - M ( - 6)

so that it only contains odd harmonics. Let us first assume

Paper 6499 P,

first

received 5th April and in revised form 16th June

1971

Prof. Willems was previously with the Division of Engineering

&

Applied

Physics, Harvard University, Cambridge, Mass., USA , and is now with

the Engineering School, University of Gent, Gent, Belgium

1408

that

M

only contains a 3rd harmonic; the effect of further

harmonics is discussed later.

The differential equations of the machine can be written

by considering a network with time-dependent inductances

0)

In this equation, u and i are the column vectors of the voltage

and currents

« = [

u

s\

u

s2---

u

sm

u

r\

u

r2 • • • « , « ] '

v

= [h\is2 • • • W ' r l 'r 2 • • • '„,]'

having m + n components, with i

sk

and i

rk

the currents in the

fcih stator and rotor coil, u

sk

and u

rk

the voltages across the

terminals of these coils. The matrix R is the diagonal matrix

of the coil resistances

R =

2 )

with

l

m

the identity matrix of order

m , R

s

the resistance of any

stator coil and R

r

the diagonal matrix

/ ?

r

= d i a g (R

rl

,. .., R

rn

)

where R

rk

is the resistance of the Arth rotor coil. The m atrix

M(t)

is the inductance matrix

M

where

=

VM

SS

M , , l

3 )

=

M'

and

M,.

r

= M'

rr

are symmetrical matrices containing the self and mutual

inductance of stator coils and rotor coils, respectively, and

M

sr

= M'

rs

is the matrix of the mutual inductances between stator and

rotor coils.

The matrix

L

M ,

M

2

L

M

2

. .

M , . .

L. . .

4 )

is a constant matrix (with some negative offdiagonal entries).

It is at the same time symmetric and circulant;

2

i.e. the

rows can be obtained by circular permutuation. The (A:+l)th

row is obtained from the kth row by shifting all entries one

step to the right and by putting the last entry of the k th row

in the first column of the (k + l)t h row. All entries in M

s

, on

lines parallel to the m ain diagonal or the inverse diagonal are

equal.

Let a.

k

denote the angle between the axis of the kth rotor

coil and the first rotor coil. Using the assumption that the

PROC. IEE, Vol. 118, No. 10, OCTOBER 1971



stator-rotor mutual inductances consist of a first and a third

harmonic, the element on the k th row and pth column of the

matrix M

sr

is

M

Xk

co s i v —

(x

k

—

2{

P

- 1)77

m

+ M

3k

cos 3 i 6 - OL

U

-

2(

P

- 1)77

m

and hence

M

sr

= M

{

N

x

P

x

+ M

3

N

3

P

3

(5)

where M

{

an d M

3

are the constant n x n diagonal matrices,*

M

{

= diag ( M

n

, M

1 2

, . . . , M

Xn

)

M

3

= diag (M

3 1

, M

3 2

, . . . , M

3n

)

N

{

and N

3

a re t he cons tan t « x 2 ma tr ic e s

co s OL\ sin aj

cos a

2

sin a

2

.cosa

rt

s ina

n

.

3 =

cos 3a sin

3<x{

cos 3a

2

sin 3a

2

cos 3a

M

sin 3a.

and

Pi

and P

3

are the time-dependent 2 x ra matrices

27T

\ fa

2

(

m

- O^n

)

cos

{ 6

P . =

A

fa

27T

\ fa

2

(

m

- O^n '6 cos [6 ) . . .

co s

•{

6 — y

•a fa

l7T

\ • (a

2

(

m

— O^O

si n

V

sin ( 6 ) . . . s in -Id - — y

\ mJ V m J

~ „ / „ Z77\ „ C

n

2(m — 1)77^

cos30 c os 3 0 ) . . . c o s 3 ^ 0 - — — y

• ^n • ~ f n 277 \ . „ f

n

2 m — 1)771

s in 30 s in 3 ( 0 ) . . . s in 3 - {6 - — V

The rotor-inductance matrix is a symmetric constant n x n

matrix containing the constant self and mutual inductances

of the rotor coils. If the speed is constant (6 = cot), the

differential equations of the machine are linear but non-

stationary owing to the time-dependent coefficients. The aim

of unified machine theory is to introduce a linear transforma-

tion on the system variables to obtain a set of time-invariant

differential equations. However, the transformation matrix

that is used in classical unified machine theory does not work

here in most cases, because of the 3rd-harmonic terms in some

coefficients. In this paper, the possibility of obtaining a linear

transformation to achieve time invariance with space harm-

onics is discussed.

A particular result is available in the literature

3

concerning

the applicability of classical unified machine theory to

machines with space harmonics. Suppose that the rotor and

the stator have symmetrical 3-phase windings. Then the terms

in M

sr

involving the 3rd-harmonic terms assume the form

M

3

co s

3d

1

1

1

1

1

1

1

1

1

Introduce the change of variables

/„ =

T0)i

s

v

ns

= T(6)v

i =

where

i

s

, v

s

, i

r

and

v

r

are all the vectors of the currents and

voltages on stator and rotor, and i

ns

, v

ns

, i

nr

and v

nr

are the

transformed quantities. The transformation matrix is

1

cos

s in

1

1

cos

•

sin

*-T)

« » - T )

sin (0 - f )

fa

2 7 r

\

in f a — — J

It is then easy to show that the set of six time-varying

• A diagonal matrix is denoted by 'diag', a column vector by 'col'

PROC. IEE, Vol. 118, No. 10, OCTOBE R 1971

differential equations of the machine at constant speed is

transformed into a set of four time-invariant differential

equations and two time-varying

ones. The

latter two equations,

however, only involve the sums of the stator and rotor

currents and voltages. If the rotor and stator 3-phase windings

have an isolated neutral point, the sums of the stator currents

and of the roto r currents is zero; the same is then true for the

sums of the voltages, as can be seen from the equations. Thus

the only remaining differential equations are time-invariant.

It is thus concluded that, for this particular case, unified

machine theory is applicable.

This can be generalised to any machine having symmetrical

polyphase rotor and stator windings, with the same number

of phases m, where only the /nth harmonic appears in the

mutual inductances, and the neutral points of both stator

and rotor are isolated. This configuration is, however, very

particular. The purpose of this paper is to show that there

exist more general configurations where unified machine

theory can be applied, in the sense that a linear transforma-

tion can be found that reduces the set of time-varying

differential equatio ns of the machine to a set of time-in-

dependent differential equations. However, the transforma-

tion matrix will not be the same as the transformation matrix

used in classical unified machine theory, where no space

harmonics are considered.

Transformation matrix

Consider the linear nonstationary system

4(0 =

A(t)x(t) + B t)u{t)

y(t)

=

C(t)x(t)

(6)

with input u(t), output y(t) and state x(t). It was shown in

Reference 2 that an interesting result can be obtained if the

system matrix A(t) can be written as

A(t) =

exp

(-Ft)A

0

exp

( +

Ft)

. 7)

for some constant matrices

A

o

and

F.

Then the change of

variables

4

z(0 = exp (Ft)x(t) (8)

transforms the system equation (eqn. 6) into the set of equa-

tions

i(0 =

(A

o

+

F)x(t) +

exp

Ft)B t)u t)

|

^(0 = C(0

exp (-Ft)z(t) )

Since (/4

0

+ F) is a constant m atrix, this set of equations can

be solved using standard matrix exponential techniques or

Laplace-transform methods. A theorem has been proved in

Reference 2 that states conditions for eqn. 7 to hold:

Theorem 1: Suppose that A(t) is diagonalisable. Then A(t)

can be written in the form indicated by eqn. 7 if, and only if,

(a ) the eigenvalues of A(t) are constant

(b ) there exists a modal matrix 5( 0 of A(t) so that S(t)S(t)~

l

is a constant matrix.

The proof of this theorem and some important consequences

are discussed in Reference 2. Instead of the change of vari-

ables (eqn. 8), one can also use the linear transformation

*(0 = 5(0z(0

since this also yields a set of differential equations with time-

invariant matrix

A.

The mathematical model of the electrical machine at

constant speed is described by the differential equa tions with

time-dependent coefficients:

4 ( 0 =

-RM(t)

~

{

(10)

To apply the above considerations to this mathematical

model, the technique of Reference 2 is used. Therefore we try

to bring the inductance matrix M in the form of eqn. 7; since

1409



this matrix can be written in block form, and since M

rr

is

constant, we first write

M

ss

in the form of eqn. 7. The compu ta-

tion of the eigenvalues and eigenvectors of M

ss

is given in

Reference 2, and the real modal matrix

1/V2 si

1/V2 sin (0

1/V2 sin (

It is thus concluded that unified m achine theory still holds,

in the sense that the set of time-varying equations of a slip-

ring machine can be reduced to a set of time-invariant

differential equations (of a commutator machine), if the

S = J -

si n 0 —

m J

In

cos

(

cos (

COS0

e-

2

?

m/

m)

sin

sin (ft

sin (ft

A

4TT\

m)

8TT\

co s f

COS

(

cos f t

' 4T T \

is obtained, w here

j8,, j8

2

, . . .

are arbitrary. This modal matrix

satisfies the requirements of Theorem 1. Moreover, M can

be written in the same form if the stator-rotor inductance

matrix can be expressed as

M

sr

= WS'

11)

where

W

is a constant matrix. This is only possible if the

number m of stator phases exceeds seven; there are then two

columns in S where the difference between the arguments of

the successive elements is 3.(27r/m). By taking the corres-

ponding elements

j8,-

equal to 30, we obtain the eigenvectors

sin 30

sin

30 - l-

sin 3(0 —

A

and

cos 3d

cos 3(6 — 27r/m)

cos 30 - ,-n\m)

Using these eigenvectors in the modal matrix

S,

it is easily

checked from eqn. 5 that eqn. 11 holds. This yields

M =

SAS' SJV1

= VDV

• • 12)

where A = diag (A

o

, A

b

A

2

,.. . )

is the matrix of the eigenvalues of M

ss

(Reference 2),

W=M

{

N

{

Q

X

+ M

3

/V

3

<2

3

\ [~0 0 0 0 0 0 1 0 . . .

|_0 0 0 0 0 1 0 0 . .

where / is the

n

x « identity matrix, and

D

~ \_w

Hence

RM(t)~

l

= V(t)CV(t)'

where C = RD~

X

.

13)

This shows that RM(t)~

l

is of the form in eqn. 7, so that the

transformation suggested above can be used to transform the

machine equations (eqn. 10) into a set of time-invariant

differential equations. The transformed variables are obtained

by means of the linear transformation

This leads to the set of time-invariant equations

_. . _ dL

(14)

0 5)

where F = — V' V is constant. Since the transformation is

orthogonal, the power is easily computed:

P =

i'

n

Ri

n

+

i'

n

D—

+

i'

n

FDi

n

(16)

where each of the three terms can be physically interpreted.

2

1410

number of phases of the symmetric stator (or rotor) is

sufficiently high. However, the linear transformation (eqn. 14)

is not the same as the transformation used in classical unified

machine theory.

2

4 Discussions and gene ralisations

In the previous Section, it was shown tha t th e ideas of

unified machine theory can also be applied where the rotor-

stator mutual inductances are nonsinusoidal functions of the

rotor angle, but contain a third harmonic provided that the

number of phases on the polyphase stator is at least seven.

For this case, a transformation on the voltage and current

vectors has been displayed, which transforms the set of time-

dependent linear differential equations of the electrical

machine at constan t speed into a set of time-invarian t dif-

ferential equations. The derivation of this transformation

matrix is a straightforward application of the linear-system-

theory approach to unified machine theory.

2

It is easy to extend the ideas of the previous section to cases

where the mutual inductances between stator and rotor coils

contain more harmonics. If the highest harmonic is k, a

transformation matrix achieving time invariance can be

constructed along the lines set forth in Section 3, if the number

of phases on the symmetric polyphase stator (or rotor) is at

least 2k + 1. This same property also holds if even harmonics

occur, which is excluded, however, if the inductance is an odd

function of the rotor angle.

Suppose that the number of stator phases is such that the

above condition is satisfied and the transformation matrix

can be constructed. If the stator phases are connected to

identical impedances, the relationship between the trans-

formed stator voltages and currents is also time-invariant.

2

Since the rotor voltages and currents are invariant under th e

transformation, it is clear that the machine is a time-invariant

input-output system, seen from the rotor terminals. This

result is rather surprising, since it shows that, in sinusoidal

steady state, the rotor voltages produced by sinusoidal cu rrent

sources at the rotor terminals do not contain time harmonics,

although the stator voltages and currents do contain

harmonics owing to the space harmonics. This property

would be much more difficult to prove using standard

techniques for machine analysis. In most cases where the

stationarity property is important (e.g. induction machines),

the roles of rotor and stator are inversed.

Consider a symmetric polyphase stator (or rotor) with an

even number of phases. Suppose that the k + 1 . . . 2£th

phases are taken away, but that the 1st, 2nd, . . . £th phases

now have the voltages v

l

—

v

k+l

and currents //

— i^+i',

noth-

ing has been changed as far as the airgap field, roto r cu rrents

and voltages, or mechanical torque are concerned. A sym-

metrical stator with k identical coils distributed along half

the boundary and with an angle 2TT/2A: between each of them

will be called here a symmetrical 2A;/2-phase stator or a

stator with semi2fc phases.

Using the above argument, or applying the technique of

Section 3 directly to a machine with a 2/:/2-phase stat or, it is

readily seen that unified machine theory can be applied to

such machines provided that k> a, where a is the order of

the highest harmonic in the rotor-stator mutual inductances,

and provided that all harmonics are of odd order. Stationary

of machines with a semieven number of phases can also be

PROC. IEE, Vol. 118, No. 10, OCTOB ER 1971



discussed in a similar way. For example, consider an induction

machine with four phases on the rotor with their axis at 45°

angles. The machine is stationary

at

the stator ports even

if

the mutual inductances contain

a

3rd harmonic. The discus-

sion

at

the end

of

Section 2 shows that this result is true for

induction machines with symmetrical 3-phase rotors

and

stators w ith isolated neutral points. The case considered here

yields

a

stationary behaviour, even

if

the neutral points are

not isolated, and even if the stator is not symmetrical.

The condition obtained

in the

previous section gives

a

sufficient condition

so

that

all

harmonics

up to a

certain

order can be taken into account, and so that unified machine

analysis can still be applied. However, the following dis-

cussion shows that unified machine theory is also applicable

in other cases. Suppose that a machine satisfying the condition

that the kth harmonic in the mutual inductances can be taken

into account by using the following column in the trans-

formation matrix

k(m -

1)2TT

col cos (kcot), . . ., co s

•{

kcot —

,

r .

„

, . / ,

k(m-

1)2TT

col sin

(kcot), . . .,

sin •< kcot

If now the mutual inductances do not contain the kth har-

monic, but contain one harmonic of order pm

+

k for some

integer p, the technique can also be applied. Indeed, use the

following columns in the transformation matrix:

col cos

{ k +

pm)cot},.

..,

(k + pm)cot —

2-n(m-\)k

m

2iT(m-\)k

col sin

{(k + pm)cot}, . .

., sin <

(k +pm)cot >

where the equali ty

co s {(k + pm(cot —

l27T/m)}

= co s i (k + pm)cot —

sin { (k + pm)(cot — l2Trjm)} = sin

<

(k + pm)cot

kl27r\

has been used repeatedly. The same property

is

true

for an

harmonic of order pm-k for some ineger p.

The above column vectors are eigenvectors

of

the stator-

inductance matrix, as can easily be concluded from eqn. 11.

The thus obtained modal matrix still satisfies SS '= constant.

If both rotor and stator are symmetric and have polyphase

structure with an unequal number of phases, unified machine

theory (without considering space harmonics) can be applied

by transforming either

the

stator

or the

rotor variables.

Where space harmonics are considered, however,

it

is better

to transform the variables

on

the m ember with the higher

number of phases, since this makes it possible to take

a

larger

number of space harmonics into consideration. For squirrel-

cage induction machines,

the

rotor has

a

high number

of

phases,

so

that

the

transformation

of

the rotor variables

enables one to take

a

large num ber

of

space harm onics into

consideration.

The final conclusion

is

that

a

linear transformation exists

to transform the set of time-dependent differential equations

describing

a

machine with uniform airgap and without com-

mutator into

a set of

time-invariant differential equatio ns

(for constant machine speed) in the following cases:

(a)

If

one member (rotor

or

stator)

of

the machine has

a

symmetrical m -phase winding; the space harmonics can then

be taken into consideration

if,

at m ost, one belongs to any of

the following sets:

(i) the set

of

harmonics

of

rank p

{

m

+

1 or p

{

m — 1 for

some integer p

{

(ii) the set

of

harmonics

of

order p

2

m

+ 2

or p

2

m

+ 2

for

some integer p

2

(iii) the set of harmonics of order pm ± —-— (for m odd)

or pm ± (— — J (for m even) for some integer p.

(b )

If both rotor and stator have a symmetrical polyphase

winding, it is sufficient that the condition above is true for one

of them.

In some particular cases, it is also possible to take harm-

onics

of

order pm (and pm/2,

if

m

is

even) into account;

therefore one should check if homopolar currents and voltages

can occur.

If

the number

of

phases m

of

the symmetrical

polyphase winding

is

odd, and

if

only odd harmonics

are

present, the above rule implies that

all

harmonics

of

order

smaller than m

can be

taken into consideration. T his

is

interesting

for

squirrel-cage induction moto rs where

the

number of phases on the rotor equals the number of bars.

It

is of

course clear tha t only where the machine speed

is

constant will the transformed set of equations be linear and

time-invariant.

If

the speed

is

not constant, the transformed

set of equations only depends on the rotor speed, but not on

the angular position

of the

roto r; this

is

interesting

for

numerical computation.

A number of papers are available in the electrical-engi-

neering literature that deal with the effect of space harmonics

on the analysis and the performance

of

electrical machines.

Naser

5

applies the unified-machine-theory ideas to machines

with space harmonics, but his results and method are only

valid if the condition o btained in the present paper h olds; the

analysis presented in a recent repo rt by Bausch and Weis

6

only

yields the complete solution

of

the electrical-machine equa-

tions

if

this same condition is true, which clearly restricts the

number and the orders

of

the space harmonics. Barton and

Dunfield

7

'

8

have obtained

as

sufficient condition

for the

applicability of unified machine theory that all harmonics be

of odd order and less than the number

of

rotor phases;

as

indicated above, this is

a

particular case of the m ore general

sufficiency condition obtained here. An additional feature of

the present paper with respect to earlier studies is that uni-

fied-machine-theory techniques are introduced

by

means

of

linear-system-theory methods; this yields

an

a priori deriva-

tion

of

the linear transformation;

2

most earlier papers use

2-axis theory, and the usefulness of the proposed linear trans-

formations

is

usually only checked a posteriori.

The im-

portance

of

space harmonics

is

shown

by

Dunfield

and

Barton,

9

and, in

particular

for

reluctance machines,

by

Lawrenson et a/.

10

5 Example

Consider

a

synchronous machine with symmetrical

8/2-phase stator and two damping coils and

a

d.c. field coil

on the rotor. The stator is connected to a symmetrical poly-

phase current source with phase currents I\/(2) cos cot — 8),

V ( 2 ) cos {cot -

(rr/4)

- 8}, V (2 ) cos {cot - (TT/2) - 8)},

/\/(2) cos {cot —

(3TT/4)

— 8}, and the sinusoidal steady state

is considered. The rotor-stator mutual inductances are

as-

sumed to contain

a

1st and

a

3rd harmonic. The transforma-

tion to be used for this example

is

in = Ti i = T

«„ = Tu u= T'u

n

where

T= 7

sin co t

co s

cot

sin 3cot

co s

3cot

sin

co s

sin 3

co s 3

(cot

(cot

(cot

(cot

4/

~

4 7

47

47

sin

co s

sin 3

cos 3

(cot

(cot

(cot

(cot

2 /

- f

7T\

27

sin

co s

sin 3

co s 3

(cot

(cot

(cot

(cot

3TT\

~ TJ

~ ~4~)

3TT\

~ TJ

PROC. 1EE, Vol. 118, No. 10, OCTOBER 1971 1411



The transformed stator currents are 2 / sin 8, 21 cos 8, 0 and

0.

The damping-coil currents

are

zero because steady state is

considered.

The

machine equations, with U

ni

denoting

the

transformed stator voltages,

are

U

nl

= 2^, /s in 8 - 2w A

2

/cos 8 + ^{2)M

f

I

f

U

nl

= 2R

s

Icos 8 — 2coA,/sin

5

E

f

=R

f

I

f

f

I

f

where My-and M

3

yare the amplitudes

of

the fundamental and

3rd harmonic

in the

mutual inductance between

the

field coil

and

a

stator coil; Rf, /y-and £^ are the field resistance, cu rrent,

and voltage, respectively.

The

above solutions immediately

show that

the

stator voltages

are not

sinusoidal polyphase

quantities,

but

contain

a

3rd

harmonic; this

is

due

to the

presence

of

the space harmonic. I t would

be

much harder

to

obtain this solution by means

of

standard analysis techniques.

This

is

even more true

for

the solution

of

problems involving

transient machine behaviour, since

the

reduction

of

a

set of

time-varying differential equations

to

a

set

of time-invariant

differential equations considerably simplifies

the

solution of

the problem.

6 Conclusions

In this paper,

it

has

been shown that,

in

some cases

where space harmonics are taken into consideration,

a

linear

transformation can be set up to transform the nonstationary

equations describing an electrical machine to stationary

equations. This generalises unified machine theory

to

deal

with some cases where space harmonics

are not

negligible.

The paper shows that an interesting relationsh ip exists

between unified machine theory

and

linear system theory,

which is mainly used

for

the study

of

linear con trol systems.

7 Acknowledgments

The author gratefully acknowledges discussions with

Prof. R.

W.

Brockett at Harva rd University; this research

was partially supported

by

NASA Grant

NGR

22-007-172.

8 References

1 WHITE, D. c , and WOODSON, H. H. : Electromechanical energy

conversion' (Wiley, 1959)

2 WILLEMS, J . L. : A system theory approach to unified electrica l

machine theory', Internal. J. Control, 1971 (to be published)

3

JONES,

c . v.:

The

unified theory of electrical mach ines' (Butter-

worths, 1967)

4

WILLEMS, J. L . :

A

new

derivation to

the

transformation matrices

in generalized machine theory', Internal. J. Elec.

Eng.

Educ,

1971

(to be

published)

5 NASER, s. A. : 'Electromechanical energy conversion in nm-winding

double cylindrical structures

in

the presence

of

space harm onics' ,

IEEE Trans., 1968, PAS-87, pp. 1094-1106

6

BAUSCH,

H., and WEIS, M.

:

Hauptachsentransformation der

Induktionsmaschine mit Kafig Laufer' (to be published)

7 BARTON, T. H., and DUNFIELD, J. c.: 'Polyphase to two-axis trans-

formations for real windings', ibid., 1968,

PAS-87,

pp. 1342-1346

8 DUNFIELD, J. c , and BARTON,

T.

H. : 'Axis transformations for prac-

tical primitive machines',

ibid.,

1968,

PAS-87, pp . 1346-1354

9 DUNFIELD, J. c , and BARTON,

T .

H. : 'Effect of m.m.f. and permeance

harmonics in electrical machines, with special reference to syn-

chronous machines' , Proc.

IEE,

1967, 114 (10), pp. 1443-1450

10 LAWRENSON, P . J. MATHUR, R . M. a nd MURTHY VAMARAJU, S. R.:

' Importance of winding and permeance harmonics in the prediction

of reluctance-motor performance', ibid., 1969,

116,

(5), pp. 781-787

1412

PROC. IEE, Vol. 118, No. 10, OCTOBER 1971

space harmonics in unified electrical-machine theory

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