Space Harmonics in Unified Electrical-machine Theory
-
Upload
dhirajbharat20 -
Category
Documents
-
view
219 -
download
0
Transcript of Space Harmonics in Unified Electrical-machine Theory
8/19/2019 Space Harmonics in Unified Electrical-machine Theory
http://slidepdf.com/reader/full/space-harmonics-in-unified-electrical-machine-theory 1/5
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
8/19/2019 Space Harmonics in Unified Electrical-machine Theory
http://slidepdf.com/reader/full/space-harmonics-in-unified-electrical-machine-theory 2/5
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
8/19/2019 Space Harmonics in Unified Electrical-machine Theory
http://slidepdf.com/reader/full/space-harmonics-in-unified-electrical-machine-theory 3/5
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
8/19/2019 Space Harmonics in Unified Electrical-machine Theory
http://slidepdf.com/reader/full/space-harmonics-in-unified-electrical-machine-theory 4/5
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
8/19/2019 Space Harmonics in Unified Electrical-machine Theory
http://slidepdf.com/reader/full/space-harmonics-in-unified-electrical-machine-theory 5/5
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