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High Frequency Theory of Power Transformers
S. Luff and W. T. NorrisSchool of Engineering and Ap plied Science,
Aston University, Birmingham, B4 7ET, U.K..
I. A PRELIMINARY THEORY
A. Introduction
Approaches to analysing transformer windings at high frequencies began with
work using classical methods such as that of Rdenberg [1,2] on surge
propagation. More recently Wilcox et. al. [3-6], have used the Part ial Element
Equivalent Circuits [7-11] (PEEC) method, although without calling it this in their
work, to effect a reduction in the size of the problem. The primary stimulus for
such work is the problem of insulation breakdown under excessive stresses caused
by switching surges. These kinds of events can be approximately analysed through
a synthesis of frequency components using the Fourier method and these
frequencies often extend into very high values for short surge rise times. A more
modern manifestation of the same problem is in inverter sourced drives involving
coupling through transformers. In these cases the inverter waveform often
unavoidably contains many high harmonics which may induce harmful resonances
in the transformer coils.
In this paper the high frequency analysis is approached on the assumption that
sufficient computational power is available to apply a basic Multiconductor
Transmission Line (MTL) description of the problem [12-15]. The responses of
machine windings in high frequency ranges have been examined using MTL
approaches in previous work. Cornick et al., have investigated the problem of
surge propagation in transformer windings [16,17], as well the same problem in
motor windings [18,19], using straightforward presentations of MTL theory to
obtain close approximations to a limited amount of experimental data. Wright
et al. [20,21], have analysed the propagation of fast transient voltage surges
through an experimental arrangement of one phase of an induction motor coil
placed within a laminated iron core using an MTL approach and obtained fair ly
satisfactory modelling of the interturn transient voltages. Keerthipala andMcLaren [22] have performed a similar investigation using an analysis rather
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closer in spirit to that of the present paper.
Here the theory will be limited to situations in which the iron parts of the
transformer can be described electromagnetically by linear phenomenological
equations. Thus it can only apply to currents with sufficiently small low f requencycomponents. Within these limitations one can still take account of dielectric losses
and eddy current losses in the iron parts of the transformer. None of the previous
work cited using semi-analytical models has yet addressed the problem of
including non-linear material phenomena and the linear assumption enables the
use of frequency domain solutions and superposition to obtain the final solutions
to given problems. High frequency non-linear models at the detailed level
considered here would still present formidable computational and mathematical
problems.
B. Multiconductor Transmission Line Theory
A general multiconductor transmission line consists of an arbitrary array of
perfect conductors extending uniformly in one linear direction and embedded in a
non-conductive material medium of homogeneous and linear electric and magnetic
properties. Time dependent currents flowing in the conductors generate an
electromagnetic field. Fig. 1 illustrates the situation. The assumption is made that
this is the principle, or transverse electric-magnetic (TEM), solution to the full
Maxwell Equations describing the electromagnetic field. Other possible modes are
assumed to be cut off and in the usual manner for wave guides this imposes an
upper frequency limit on the validity of the theory such that a characteristic
wavelength of the field is of the order of the interconductor separations [23]. In
the case of a large power transformer of typical design this limit is of the order of
a few gigahertz. Within MTL theory transverse voltage differences can be defined
in the usual way between the surfaces of the conductors and a system of partial
differential equations relating these to the conductor currents can be derived
which is equivalent to the full Maxwell solution. At low frequencies the same
theory represents the usual lumped element circuit and in this sense any MTL
theory has validity right down to zero frequency modes.
We will use the same notations as Paul [15]. Given N+1 parallel conductors,
one is chosen as a zero of voltage potential (the ground, or reference potential)
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and given the index 0. The MTL equations describe the voltages and currents in
the remaining N conductors, which are functions of time t, defined relativistically,
and the third lengthwise coordinate of the conductors,z. The equations are
=
),(
),(
),(
),(
tz
tz
ttz
tz
z I
V
0C
L0
I
V(1)
where the 2N voltages and directed currents (assumed positive in the direction of
increasing z) are assembled into two column matrices
),(),(,),(),( tzItztzVtz jj == IV . (2)
C is a matrix of per-unit-length interconductor capacitances and L a matrix of per-
unit-length self and mutual inductances, computed from time independent two-
dimensional electrostatic and magnetic field solutions respectively. Paul [15]
gives a full description of the meaning of the elements of these matrices and
various methods for computing them. The geometric and material properties of the
multiconductor system determine these two-dimensional solutions and are
expressed through the resulting parameter matrices and do not depend on zor t.
The introduction of small resistive losses in the conductors and conduction
currents in the material medium around the conductors leads to the modified MTLequations
=
),(
),(
),(
),(
),(
),(
tz
tz
ttz
tz
tz
tz
z I
V
0C
L0
I
V
0G
R0
I
V, (3)
where R is a matrix of per-unit-length resistances and G a matrix of per-unit-
length inter-conductor conductances. Equation (3) is essentially a first order
perturbation theory whose predictions must be verified by comparison with
experiment. The entries of the parameter matrices in (2) and (3) depend on the
numbering system used for the conductors and much of the theory will require
convenient choices of this numbering.
Boundaries between materials of differing dielectric and magnetic properties
in the surrounding medium and effects due to eddy currents flowing within the
bodies of the conductors, no-longer assumed to be perfectly conducting, appear as
modifications of the fundamental matrices L and C in an obvious way, at least to a
first approximation. The effects of eddy currents within the laminated iron cores
and real conductors usually present in our problems may be approached by
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introducing an effective permeability relationship [24,25] and internal conductor
inductances [15,16,26]. Two useful relations that exist in the case of perfect
conductors embedded in a homogeneous medium with parameters , , , are:
N1LCCL == , (4)
N1LGGL == . (5)
Assuming a harmonic time dependence tie for all voltages and conductor
currents one obtains from (1) a system of coupled ordinary differential equations
for the complex phasor quantities now representing the voltages and currents,
=
=
)(
)(
)(
)(
)(
)(
z
z
z
zi
z
z
dz
d
I
VA
I
V
0C
L0
I
V . (6)
Equation (3) can be handled similarly to obtain a more general form of the
problem,
=
=
)(
)(
)(
)(
)(
)(
z
z
z
z
z
z
dz
d
I
VA
I
V
0Y
Z0
I
V, (7)
where the per-unit-length impedance matrix, Z , and admittance matrix, Y , are
defined by
+=
+=
CGY
LRZ
i
i. (8)
For engineering applications dc resistive conductor losses are often quite
negligible and dielectric losses in any insulating materials can be conveniently
handled in the frequency domain by introducing a complex permittivity
( )( ) tan1 i= in the usual way so that C becomes a complex matrix. It is useful
to introduce the voltage-current state vector
=
)(
)()(
z
z
z
I
VX (9)
so that (7) can be written
)()( zzdz
dXAX = . (10)
The solution of (10) expressing the 2N voltage and current phasors in terms of the
2N voltage and current phasors at the beginning of the line, 0=z , is well known:
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)0()exp()( XAX zz = . (11)
Other forms of solution are possible which generalize the classical formul of
transmission line theory, but in this paper only (11) is employed. Usually the
voltages and currents at some fixed distance along the MTL system from the
initial point are required, corresponding to a given geometrical dimension of
original structure. If this is L then
)0()()0()exp()( XXAX LLL == , (12)
where )(L is the chain parameter matrix for the system. The length parameter is
often omitted where it is implied from the discussion. )(L has various special
properties, one of which is the obvious relation
2N)()()()( 1== LLLL . (13)
Paul [15] gives other relations involving sub-blocks of )(L that depend on some
assumptions about the parameter matrices of the MTL system, but these have
limited use in numerical simulations involving lines composed of large numbers of
conductors.
C. Theory for a Single Transformer Coil
We firstly present a simple multiconductor transmission line model for a
single coil on a core leg of a transformer, which corresponds to the one used in the
most recent papers of Cornick et al. [16,17].
Consider a single transformer coil positioned, for example, on the central leg
of a standard three-leg transformer core. We are not worried exactly how it is
wound for the present except for the observation that it will consist essentially of
individual turns distributed in a series of parallel planes. The whole coil is
imagined to be divided vertically down one side from the centre of the core leg
outwards. The external connections are assumed to lie in the same plane as the
division, the winding consisting of a series of full turns. The procedure is
indicated in Fig. 2 for a coil of four turns. Ignoring the small differences in the
relative circumferences of the turns this procedure gives the system of conductor
segments shown in the lower diagram, which can be taken to approximate a
multiconductor transmission line array if the original radius of the coil was very
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large compared with the depth of the winding, which is usually the case.
The model consists of the identification of currents and voltages between the
ends of the turn segments as indicated by dotted lines in the figure to recover a
circuit equivalent to the original coil. The positions of the terminals of thewinding in the array arise from the assumption about full turns. The length of each
conductor, L, is equal to the original circumference of each turn, the small
differences between them being neglected. At each end of the array there are four
voltages to ground and four currents flowing in the positive z direction, ( ))0(jV ,
)0(jI at 0=z and ( ))(LVj , ( )(LIj at Lz= . Application of MTL theory introduces
the electromagnetic characterization of the system and hence of the original coil.
The reference conductor ( 0)( =zV ) is taken to consist of the external transformer
casing and core taken as a whole and assumed grounded. Using phasor voltages
and currents proportional to tie , the output voltage and current are related to
those at the input of the coil by an overall chain parameter matrixO
through a
matrix equation
=
=
=
in
in
out
out
I
V
L
L
I
V
)0(X
)0(X
)(X
)(X
O
5
1O
8
4 . (14)
Identification of the segment ends provides six internal conditions of continuity of
voltage and current that were present in the original turns of the coil
)0()(
)0()(
)0()(
43
32
21
VLV
VLV
VLV
=
=
=
=
=
=
)0()(
)0()(
)0()(
43
32
21
ILI
ILI
ILI
, (15)
and the obvious external boundary conditions are
=
=
in
in
II
VV
)0(
)0(
1
1,
=
=
out
out
ILI
VLV
)(
)(
4
4. (16)
Two of the four quantities on the right of (16) are assumed to be known in order to
have a soluble problem: for our purposes inV andin
I . All these relations can be
written in terms of the voltage-current state vector for the array defined in (9).
Equations (16) become
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)0()(
)0()(
)0()(
43
32
21
XLX
XLX
XLX
=
=
=
=
=
=
)0()(
)0()(
)0()(
87
76
65
XLX
XLX
XLX
, (17)
while (16) become
=
=
in
in
IX
VX
)0(
)0(
5
1,
=
=
out
out
ILX
VLX
)(
)(
8
4. (18)
The indices are characteristic of the connection pattern of the original coil and all
these conditions on the system can be written as the single matrix equation
=
+
0
0
0
0
0
0
)(
0100
0010
0001
0000
0100
0010
0001
0000
)0(
1000
0100
0010
0001
1000
0100
0010
0001
in
in
I
V
LXX , (19)
or
SXJXI =+ )()0( L . (20)
which marks our first use of generalised connection matrices I and J. The standard
MTL solution given earlier introduces the electromagnetic characterization of the
system by further relating the state vectors at the beginning and end of the system
of conductor segments:
)0()()0()exp()( XXAX LLL == , (21)
or alternatively
)()()()()()exp()0( 1 LLLLLL XXXAX === .
From (20) and (21) )0(X can be eliminated to obtain
[ ] SXJI =+ )()(1 LL . (22)
The two columns ofO
can be obtained by solving for outVLX )( 4 = and
out
ILX )( 8 = in the two cases 0,1 ==inin
IV and 1,0 ==inin
IV . The solutions are
frequency dependent. That the desired linear relationship exists at all is clearly a
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result of (22). If internal voltages and currents of the coil are required then one
can use
)0()()0()exp()( XXAX zzz ==
to obtain them. For the strictly linear systems we are discussing the solution for an
arbitrary periodic source can be obtained by a Fourier synthesis of single
harmonic solutions. The solution is formal in the sense that we have not discussed
how to obtain the r equired parameter matrices and hence A, or how to compute the
exponential matrix (z); these are standard problems to be addressed by any ofthe present day numerical methods available.
One can immediately obtain the generalisation to any number, N, of turns in
the coil. One side of the winding is divided to obtain an approximate MTL array
of turn segments as before. If these are numbered by following the connectivity of
the original sequence of turns all that is required is to enlarge the matrices I, J,
and S, in the obvious way and solve for the appropriate state vector )(LX using
(22). The generalisations are:
[ ]
==
=
T
T
J0
0J
J1I
0
0
S )0(
)0(
,,
N
N
2N
1N
1N
in
in
I
V
. (23)
where2N
1 denotes the 2N-dimensional identity matrix and JN(0) the usual Jordan
block of size N.
D. Inclusion of a Secondary CoilA secondary coil can be introduced into the theory, a problem not considered
in the work of Cornick et. al. [16,17]. The winding is divided as before to obtain
an approximate MTL system, where we again assume full turns and that the
terminals of the coils lie in the plane of the axial division. The original
connections of the turns are then introduced as constraints. The only new feature
is the need to consider the numbering system for the analysis more carefully.
Consider a four turn primary coil with a two turn coil as secondary. The
construction of the MTL system is shown in Fig. 3 with the identification of the
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resulting conductor segments to regain the continuity of the original coils
indicated by dotted lines as before.
The exact details of the cross-section geometry are not important for
developing the overall form of the solution and one can assume that effects due tothe differing sizes of the primary and secondary conductor cross sections and their
relative positioning are automatically included through the corresponding MTL
parameter matrices. The primary turns are numbered as the first four conductors
and the secondary turns as the remaining two. A general characterization of the
two coil winding is in terms of an overall chain parameter matrix relating the two
output voltages and two output currents to those at the input terminals of the
device, as indicated in Fig. 3. For the four+two turn example this would be:
=
=
=
=
=
S
in
P
in
S
in
P
in
out
out
out
out
I
I
V
V
I
I
V
V
X
X
X
X
LX
LX
LX
LX
LI
LI
LV
LV
I
I
V
V
O
5
1
5
1
O
11
7
5
1
O
12
10
6
4
6
4
6
4
S
P
S
P
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)(
)(
)(
)(
)(
)(
)(
)(
, (24)
where the superscripts P and S refer now to primary and secondary coils. The
internal conditions of voltage and current continuity are
)0()(
)0()(
)0()(
43
32
21
VLV
VLV
VLV
=
=
=
=
=
=
)0()(
)0()(
)0()(
43
32
21
ILI
ILI
ILI
for the primary coil and
)0()( 65 VLV = )0()( 65 ILI =
for the secondary. The exterior boundary conditions are
P
1
P
4
)0(
)(
in
out
VV
VLV
=
=
=
=
P
4
P
4
)0(
)(
in
out
II
ILI
for the primary coil and
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S
5
S
6
)0(
)(
in
out
VV
VLV
=
=
=
=
S
5
S
6
)0(
)(
in
out
II
ILI
for the secondary. All these relations can be written in terms of the voltage-current
state vector )(zX for the array. The internal boundary conditions become
)0()(
)0()(
)0()(
43
32
21
XLX
XLX
XLX
=
=
=
=
=
=
)0()(
)0()(
)0()(
109
98
87
XLX
XLX
XLX
,
for the primary and
)0()( 65 XLX = )0()( 1211 XLX = ,
for the secondary. The exterior boundary conditions become
P
1
P
4
)0(
)(
in
out
VX
VLX
=
=
=
=
P
7
P
10
)0(
)(
in
out
IX
ILX
for the primary and
S
5
S
6
)0(
)(
in
out
VX
VLX
=
=
=
=
S
11
S
12
)0(
)(
in
out
IX
ILX
for the secondary. These conditions can be assembled into a single matrix equationof the same form as we used for a single winding
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=
+
0
0
0
0
0
0
0
0
)(
01
00
0100
0010
0001
0000
01
00
0100
0010
0001
0000
)0(
10
01
1000
0100
0010
000110
01
1000
0100
0010
0001
S
in
P
in
S
in
P
in
I
I
V
V
LX
X
, (25)
or, defining new matrices I, J, and S,
SXJXI =+ )()0( L . (26)
The solution is now formally the same as for the single winding
[ ] SXJI =+ )()(1 LL , (27)
where )(L is the chain parameter matrix for the MTL system we have
constructed. Solving (27) for 4 independent cases of the vectorS in a similar way
to the procedure for the single winding gives the elements ofO
at frequency .
The extension to any numbers of primary and secondary turns,P
N andS
N , is
obtained by generalising I, J, and S in an obvious way:
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=
=
=
T
T
T
T
J
J
J
J
1
1
1
1
0
0
0
0
)0(
)0(
)0(
)0(
,
S
P
S
P
S
P
S
P
S
P
S
P
N
N
N
N
N
N
N
N
1N
S
1N
P
1N
S
1N
P
J
IS
in
in
in
in
I
I
V
V
. (28)
E. Retrograde Windings
The inclusion of a secondary coil introduces the question of the handedness of
the windings, which appears not to have been considered theoretically before.
There are two possibilities: the arrangement implicitly assumed by the diagrams in
which each winding has the same handedness, which is the usual case in
engineering applications, and the arrangement in which the secondary winding is
wound with the opposite sense to the primary.
To determine the necessary modifications to the present theory consider again
the four+two transformer winding of Section D, assuming as before that the
terminal leads of both windings have the same angular positions. A moment's
reflection shows that the effect of a retrograde secondary coil is to modify the
situation from that of Fig. 3 to that shown in Fig. 4. Therefore the external
boundary conditions for the secondary winding must be changed to
=
=
S
in
S
out
VLX
VX
)(
)0(
5
6
=
=S
in
S
out
ILX
IX
)(
)0(
11
12
whilst the internal conditions become
)()0( 65 LXX = )()0( 1211 LXX = .
Thus (25) must be changed to the following:
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=
+
0
0
0
0
0
0
0
0
)(
10
01
0100
0010
0001
0000
10
01
0100
0010
0001
0000
)0(
01
00
1000
0100
0010
000101
00
1000
0100
0010
0001
S
in
P
in
S
in
P
in
I
I
V
V
LX
X
. (29)
After modifying I and J accordingly the solution to the problem is still formally
the same as before,
[ ] SXJI =+ )()(1 LL .
At this point the methodology of solution becomes slightly different, since to
compute the entries ofO
one also requires the solution vector )0(X which
contains the output voltages and currents of the secondary winding. This can be
obtained with little extra effort from
)()()0( 1 LL XX = ,
because )(1 L has already been computed and only two entries of )0(X are
needed. The extension to the general case is obvious and the matrices of (29)
become
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=
=
=
S
P
S
P
S
P
S
P
S
P
S
P
N
N
N
N
N
N
N
N
1N
S
1N
P
1N
S
1N
P
)0(
)0(
)0(
)0(,
1
J
1
J
J
J
1
J
1
I
0
0
0
0
S
T
T
T
T
in
in
in
in
I
I
V
V
. (30)
F. Interleaved Windings
It is possible to apply the theory so far presented to windings obtained from
the original system by reconnecting the turns in a different order with only
minimal extra computational effort. Take a single coil such as was considered in
Section ID where, for example, the turns have been reconnected into an
interleaved type arrangement. One can assume the original coil was of the straight
forward disc wound variety and that this problem had been solved at the given
frequency. Within the limitations of the theory, the new interleaved winding is
equivalent to the same physical distribution of turns around the core leg, but with
a modified connection pattern [16,17].
Suppose, however, the theory is applied to the interleaved coil and the turn
segments of the MTL model are numbered in the sequential way used before,
following the electrical continuity turn by turn. The numbering of the turn
segments in the MTL system then differs from that used originally, but evidentlythe conditions of continuity of voltage and current will be given by the same
matrix equations in the resulting new numbering of the MTL conductor segments:
SXJXI =+ )(~
)0(~
L (31)
for the appropriate connection matrices I and J, and source vector S, given by
(23), where X~
is the state vector of the system in the new numbering scheme. The
terminal leads are assumed to move with their points of attachment under the
rearrangement. Thus the transformed external terminals may be at different turns
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compared to the original.
As an example, the original system might have appeared in cross section as
in the top part of Fig. 5, while a rearranged system might be numbered as in the
bottom part [27]; identical physically except for the renumbering of the conductorsby a permutation
: 11, 26, 32, 47, 53, 68, 74, etc.,
which is characteristic of the change in the winding pattern. Evidently we could
write down the MTL equations for the conductor system in either numbering
system just as well and in fact the matrices of parameters in the new numbering
system belonging to the interleaved winding are related to those in the original
numbering, belonging to the disc winding, by a permutation similarity
T~pMpM = , (32)
where M is a general parameter matrix such as C or L, and p is the orthogonal
row permutation matrix [28] obtained by permuting the rows of the N-dimensional
identity matrix according to :
=
OMMMMMMM
L
L
L
L
L
0000100
1000000
0000010
0100000
0000001
p ,N
TT 1pppp == . (33)
The transformation of A follows directly from equations (32),
T~PAPA = ,
=
p0
0pP . (34)
P is also a permutation matrix and we call the process of changing the winding
pattern in this way a turn permutation. The new chain parameter matrix for the X~
system can be obtained from that of the original arrangement by a permutation
similarity since
TTPPPAPA === )exp()~exp(~ LL . (35)
Since, from (31) and (36)
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)(~)(
~)0(
~)(
~)0(
~)
~exp()0(
~ 1LLLL XXXAX
=== (36)
the solution of the rearranged problem can be written
SXJI =+
)(
~
))(
~
(
1LL
.
However, by substitution of (35) one obtains
)(~
))(()(~
))(( 11 LLLL XPPJPIXJPPI TT +=+ .
It is easy to confirm that )()(~
LL XPX = so that this equation can be written as
SXJI =+ )()
~)(
~( 1 LL . (37)
which is what one would obtain by using the original numbering scheme for the
coil, but with I and J modified to take account of the rearranged winding pattern.
The desired transformations ofI and J are
PJJPII ==~
,~
. (38)
Evidently there is a choice of approaches in analysing the effect of a turn
permutat ion: either to treat I and J as invariants and determine the appropriate
chain parameter matrix~
for the rearranged system, or to obtain the transformedmatrices I and J while using the original chain parameter matrix . The choice isof little consequence here, but in the more involved model of Part II the approach
of transformation of the matrices I and J admits greater generalisation. One might
expect to be able to express the transformation in the mathematically more
satisfying form of a similarity, but the theory appears to dictate the type of
transformation in (38). The extension of the limited theory of this section to the
two coil example of Section D simply involves using the appropriate permutation
matrix that corresponds to the pattern of renumbering of the turn segments of both
coils.
II. AN EXTENDED THEORY
A. Introduction
The MTL transformer theory presented in Part I destroys a characteristic
topological property of the original arrangement, namely the looping of the turns
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17
around the core leg. This makes impossible the inclusion of the magnetic effects
of the iron core at power frequencies since appropriate parameter matrices could
not be determined and limits the theory to those sources containing only
frequencies so high that the core can be assumed to behave as a grounded shield; acounterpart to the steel casing outside the turns in this respect. At higher
frequencies the consideration of transformers with ferrite cores would also the out
of the question. There is also no way of describing the interactions of coils in a
multiphase transformer of the typical 3-leg design. We proceed to develop a
theory which allows the inclusion of these factors.
B. Theory of a Single Coil
Consider Fig. 6, the top part of which represents a schematic horizantal cross-
section of a single coil of four turns wound onto a square leg of a core. Assume
the turns are complete and leads are taken from one corner of the leg. As before
we do not concern ourselves with the exact spatial arrangement or cross sections
of the turns, merely assuming that they have a large diameter and each lies
essentially in a single transverse plane. The effects of spatial arrangement and
material properties appear through parameter matrices of capacitance, inductance,
and resistance in the MTL theory and these are assumed to be calculable using
standard methods.
Assume the coil is divided as indicated by two diagonal planes d and d'
containing the z-axis to obtain four arrays of turn segments rather than the one we
had before. However, those conductor segments on opposite sides of the core leg
are to be considered as the elements of single MTL systems, as they could be if
their small curvatures are ignored. In each system this allows the approximate
inclusion of the core by modifications of the electromagnetic properties of the
intervening space and hence of the appropriate parameter matrices, possibly with
some allowance for a slow variation of these with position along the side of the
core leg. Once again the transformer casing is to be taken as the grounded
reference conductor.
This construction gives two MTL systems, X and Y, as il lustrated in the lower
part of Fig. 6, assumed to have a common length, L, although this is not essential
to the theory. The origin of each MTL system and the directions of increasing
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18
distance are indicated. The nth turn is divided into successive segments numbered
n, n, n+N, and n+N, since this gives certain convenient patterns in the matrices
that occur in the general solution. The voltage and current phasors for each of the
arrays X and Y are assembled into voltage-current state vectors for each array in
the usual way. We seek an overall chain parameter matrixO
for the coil at a
given frequency such that:
=
=
=
in
in
out
out
I
V
-I
V
)0(X
)0(X
)0(Y
)0(Y
O
9
1O
16
8 . (39)
Remembering the definition of the state vector and that positive currents are in the
direction of increasing distance along an MTL array, identification of the voltages
and currents to recover the original coil circuit gives the following internal
conditions of continuity of voltage and current, written in terms of the relevant
voltage-current state vectors:
)0()(
)0()(
)0()(
)0()(
44
33
22
11
YLX
YLX
YLX
YLX
=
=
=
=
,
=
=
=
=
)0()(
)0()(
)0()(
)0()(
1212
1111
1010
99
YLX
YLX
YLX
YLX
at corner A,
)()(
)()(
)()(
)()(
84
73
62
51
LXLY
LXLY
LXLY
LXLY
=
=
=
=
,
=
=
=
=
)()(
)()(
)()(
)()(
1612
1511
1410
139
LXLY
LXLY
LXLY
LXLY
at corner B,
)()0(
)()0(
)()0(
)()0(
88
77
66
55
LYX
LYX
LYX
LYX
=
=
=
=
,
=
=
=
=
)()0(
)()0(
)()0(
)()0(
1616
1515
1414
1313
LYX
LYX
LYX
LYX
at corner C,
and
=
=
=
)0()0(
)0()0(
)0()0(
47
36
25
XY
XY
XY
at corner D.
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19
The input and output voltage and current phasors, of which only two are assumed
to be unknown quantities, enter into the external boundary conditions:
=
=
in
in
IX
VX
)0(
)0(
9
1
,
=
=
out
out
ILY
VLY
)(
)(
16
8
.
All these conditions are expressed by the two matrix equations:
)(
)0()(
L
L
Y
1
1
Y1
1
X
4
4
4
4
+
=
(40)
and
)(
)0()0(
)0(
)0(
7
7
L
I
V
in
in
Y
1
1
YJ
J
X
0
0
4
4
T
4
T
4
+
=+
, (41)
where X and Y are the standard state vectors for the two systems. We write (40)
and (41) as
)()0()( 424
1 LL YtYtX += (42)
and
)()0()0( 444
3
4LYtYtXs +=+ . (43)
To proceed to the solution requires further relations between the four unknown
state vectors in (42) and (43) provided by MTL theory. In the previous model
these were given by the chain parameter matrix )(L in the form of a simple expo-
nential. In the present case one has to consider the possible mutual coupling of the
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20
systems X and Y through the fields they generate in the core leg. Fortunately this
is negligible. Paul [15] derives the MTL equations as modified by the presence of
an incident electromagnetic field and in our case the forcing functions that appear
vanish when we take the external casing of the transformer as the referenceconductor in our systems. All the coupling is already included in the analysis
through the conditions of continuity in (42) and (43). Thus the usual MTL chain
parameter matrices can be computed for the two systems X and Y as if they were
isolated from each other in space and this gives the required extra relations:
==
==
)0()0()()(
)0()0()()(
YYY
XXX
LL
LL, (44)
where we abbreviate without fear of confusion. From (42), (43), and (44), by
successive substitution,
( ) ( )4434444344
4
1
4
2
4
1
4
2
)0()0()0()(
)0()()0()0()0()(
sYtYtsYtYt
XXYtYtYtYt
++=++=
==+=+
L
LL
and expanding and collecting terms gives equations for the determination of )0(Y :
444
43
42
41 )0( sYtttt =+ . (45)
The two columns ofO
in (39) can be obtained by solving (45) for two indepen-
dent cases of the input voltage and current phasors to obtain the output voltage
and current phasors, which are elements in )0(Y . When the fields are excluded
from the core leg at high frequencies (45) simply represents in a more obscure
form essentially the solution we derived in Part I.
The extension to a winding of N turns with the equivalent MTL systemsnumbered in the generalisation of Fig. 6 is clear. We only need to display the
general forms of the matrices, since the subsequent expression of the solution in
matrix algebra is unchanged:
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21
=
=
N
NN
2
N
N
N
1
1
1t
1
1
t
=
=
N
NN
4
N
N
N
3(0)
(0)
1
1t
J
J
tT
T
. (46)
=
12N
1N2N
0
0s
in
in
I
V
Despite the utility of the solution represented by (45) for numerical evaluation
further theoretical development requires an alternative formulation in which the
solution appears in the form we obtained in Part I. In fact (42) and (43) can be
written, in the general case of N turns, as the single matrix equation
=
+
N4
N
N
24N
N
4N4
N
1N4
N
34N
)(
)(
)0(
)0(
0
s
Y
X
t1
t0
Y
X
t0
t1
L
L. (47)
Introducing the state spinor components
=
)0(
)0()0(
Y
X ,
=
)(
)()(
L
LL
Y
X ,
and spinor source
=
N4
N
N
0
sS (48)
(47) can be written
NNN )()0( SJI =+ L , (49)
with
=
=
N
24N
N
4N4N
N
1N4
N
34NN ,t1
t0J
t0
t1I . (50)
Equations (44) can be combined into
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22
=
)0(
)0(
)(
)(
Y
X
0
0
Y
X
L
L,
or using the state spinor components,
)0()( =L ,
= 0
0. (51)
From (49) and (51) it is apparent that the problem for the state spinor components
is formally identical to that obtained in Part I. The equations determining the
solution are
NNN )0()( SJI =+ , (52)if one solves for )0( . The relation of to the equations describing the currentsand voltages of the double system represented by is similar to that of the chain
parameter matrices of each individual MTL system to their state vectors. In fact
XAX
X=
dx
dand YA
Y
Y=
dy
d(53)
and hence the governing equations for a state spinor component are
)()()(
Y
Xuu
du
ud AA0
0A=
= , (54)
introducing a dummy variable u for eitherx ory. The required solution to this
system of differential equations is
)0()0()0(
)exp(
)exp()0()exp()(
Y
X
=
=
==
0
0
A0
0AA
L
LLL . (55)
The simple relation of to the individual chain parameter matrices of the X andY systems derives from the absence of any incident field coupling between the
two systems in equations (53) and the resulting quasi-diagonal structure of A in
(54). In general one might wish to examine more involved analyses which admit
such a coupling, for instance in approximations for non-linear magnetisation of
the core or for the curvature of the turn segments.
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23
C. Theory of Two Coils
Consider the case of two coils wound on a single core leg. Divide each one
into turn segments by the diagonal axial planes dand d' and assume coil 1 has N1
turns and coil 2 has N2, the turns are complete, and the terminal leads of both coilsare at corner A of the core leg. The situation is indicated schematically in Fig. 7
for the case N1=4, N2=2. As before, identification of the segments on opposite
sides of the leg creates two MTL systems X and Y in the x and y directions,
respectively. Systems X and Y both include turn segments from each of the coils
and the electromagnetic coupling between them will appear in the MTL solutions
because of the appropriate cross coupling terms in the parameter matrices. We use
a double indexing for the voltages and currents in each of the complete X- and Y-
systems, ( ) ( jj IV , , where denotes the number of the coil, 2,1= , and the
indexing of the segments belonging to coil corresponds to the numbering
scheme we used in the previous section for a single coil.
The state vector of the X- system is numbered in the obvious (sequential) way
by the equations
( )( )( )( )
( )( )( )( )
=
=
=
=
)(
)(
)(
)(
)(
)(
)(
)(
)(,
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
L
L
L
L
LI
LI
LV
LV
L
I
I
V
V
X
X
X
X
j
j
j
j
X
X
X
X
j
j
j
j
I
I
V
V
X
I
I
V
V
X , (56)
where the voltages and currents belong to the X- MTL segments; analogous
definitions serve for the Y system. There will again be negligible coupling
between the X- and Y- systems as constructed, and writing
=
=
= )(
)()))()
))(LL
L
L
0,,
0
00
Y
X
Y
X
,
for source frequency we have
)0()0()0()exp(
)exp()0()exp()(
Y
X =
=
==
0
0
A0
0AA
L
LLL . (57)
We can also define state spinor components for each individual coil of the kind
used in the previous section,
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( )( )( )
( )
( )( )( )
( )
=
=
=
=
=
=
)(
)(
)(
)(
)(
)(
)(
)(
)(
)()(,
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0()0(
L
L
L
L
LI
LV
LI
LV
L
LL
I
V
I
V
Y
Y
X
X
Y
j
Y
j
X
j
X
j
Y
Y
X
X
Y
j
Y
j
X
j
X
j
I
V
I
V
Y
X
I
V
I
V
Y
X.(58)
Because of the choice of numbering and the definitions of equations (58) the
results of the previous section can be used to write down all the conditions of
continuity for the two coils as
=+
=+
2
2
2
2
2
1
1
1
1
1
)()0(
)()0(
SJI
SJI
L
L
, (59)
where
1I
,
1
J ,
1
S , etc., are the connection matrices and spinor sources ofappropriate dimensions obtained from (48) and (50); we abbreviate the notation
somewhat. Equations (59) can be written as
=
=
2
1
22
11
2
2
1
1
22
11
)(
)0(
)(
)0(
S
S
JI00
00JI
JI00
00JI
L
L. (60)
If the voltage and current components of are expanded into those of the separateMTL systems we get
=
=
)(
)()0(
)0(
)(
)(
)0(
)0(
)(
)0(
)(
)0(
2
2
2
2
1
1
1
1
2
2
1
1
L
L
L
L
L
L
Y
X
Y
X
Y
X
Y
X
. (61)
From (58)
=
)0(
)0(
)0(
)0(
)0(
)0(
2
2
1
1
2
1
X
X
X
X
I
V
I
V
X
X
and comparing this with (56) gives
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)0()0()0(
)0(
2
1
N2
N2
N2
N2
2
1
2
2
1
1
Xqq
qX
1
1
1
1
X
X=
=
=
, (62)
where q is an appropriate permutation matrix. Relations analogous to those in (62)
exist forX(L), Y(0), and Y(L), and we can write
QY
X
Y
X
q
q
q
q
Y
Y
X
X
Y
Y
X
X
=
=
)(
)(
)0(
)0(
)(
)(
)(
)(
)0(
)0(
)0(
)0(
2
1
2
1
2
1
2
1
L
L
L
L
L
L, (63)
where Q is also a permutation matrix. Evidently
=
=
)(
)(
)(
)(
)0()0(
)0(
)0(
)(
)(
)(
)(
)0()0(
)0(
)0(
)(
)(
)0(
)0(
)()(
)0(
)0(
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
4N
4N
4N
4N
4N
4N
4N
4N
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
L
L
L
L
L
L
L
L
L
L
L
L
Y
Y
X
X
Y
Y
X
X
P
Y
Y
X
X
Y
Y
X
X
10000000
00100000
00001000
00000010
01000000
00010000
00000100
00000001
Y
X
Y
X
Y
X
Y
X
,(64)
where P is another permutation matrix and combining this with (61) and (63)
gives the relationship RPQ == . It is a simple exercise in partitionedmultiplication to find that
=
2
2
2
2
1
1
1
1
q00q
q0
0q
q0
0q
q0
0q
R . (65)
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Substitution into (60) gives the conditions of continuity for the double winding
[ ] SJIJI =+= )()0( L , (66)
where
[ ]
=
=
2
1
22
11
,S
SSR
JI00
00JIJI . (67)
Equation (66) has the same form as the continuity conditions for the single coil
example of the previous section, to which it reduces as a special case if the
matrices are interpreted correctly when 0N2= . Substituting from (57) into (66)
the solution to the double winding problem is represented by equations of the
same form as before
SJI =+ )0()( . (68)With the developments of the next section in mind we also write (68) in the form
[ ] S0
01JI =
)0(
)0(
. (69)
This same expression could also have been obtained for the solution in the single
coil theory of the previous section.
D. Mult iple Coi ls
The treatment above can be extended to any number of coils in the winding.
Let us suppose there are K coils of the kind we considered earlier numbered
K,,2,1 KK , wound onto a single core leg, with K21 N,,N,N KK turns in each,
respectively. As before, each coil is divided into turn segments by the diagonal
planes dand d' and the special case is assumed wherein the terminals of every coil
are at Corner A. The resulting segments in the x- and y- directions are then
assembled into MTL systems X and Y, respectively. The voltages and currents of
the conductor segments belonging to each individual coil are numbered in the
same way as before and a double indexing used for the voltages and currents of
the entire X- and Y- systems, ( ) ( jj IV , , where denotes the index of the coil and
j the indexing of the segments belonging to coil , according to the standard
single coil scheme.
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State vectors for the X- and Y-systems are defined by the obvious extension
of (56) and we can then follow through an argument in the general case quite
analogous to that given above for two coils. We only give the results. The required
generalizations are
( )( )
( )( )( )
( )
( )( )
( )( )( )
( )
=
=
=
=
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(
)(,
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
)0(
XK
X
2
X
1
X
K
X
2
X
1
XK
X
2
X
1
X
K
X
2
X
1
XK
X
2
X
1
X
K
X
2
X
1
XK
X
2
X
1
X
K
X
2
X
1
L
L
L
L
L
L
LI
LI
LI
LV
LV
LV
L
I
I
I
V
V
V
j
j
j
j
j
j
j
j
j
j
j
j
I
I
I
V
V
V
X
I
I
I
V
V
V
X
M
M
M
M
M
M
M
M
, (70)
with Y(0) and Y(L) given analogously. The conditions of continuity have the same
form as before
[ ] SJI = , (71)
where now
[ ]
=
=
K
2
1
KK
22
11
,
S
S
S
SR
JI
JI
JI
JIMOO
, (72)
with
=
K4
24
14
q1
q1
q1
R M , (73)
where
=
+
+
K11-21
K11-21
2Nr
N2r2NrN2r2NrN2r2NrN22NrN22NrN2
2NrN2r2NrN2r2NrN2r2NrN22NrN2
2NrN2
001
00
0
00000000
00000000
001000q
LL
LL
r
. (74)
The electromagnetic characterization of the coils is given by the appropriate MTL
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28
solution, (57), and from (57) and (71) the full solution to the problem for given
monochromatic inputs S again appears in the form
[ ]S
0
01JI =
)0(
)0(
.
By finding solutions for the required number of linearly independent sources we
can construct an overall chain parameter matrix for the multiple coil winding at
the given frequency just as we did for the single coil case.
Although we have assumed the K coils are wound onto a single core leg this is
unnecessary since the matrix formulation of the theory applies equally well if they
are wound on different legs of a given core. The new situation is reflected in the
changes brought about in the relevant parameter matrices belonging to the two
MTL systems that are formed, which depend on the relative positions of the turn
segments and their magnetic and electric interactions. The problem is the
computational one of determining the various capacitances and inductances. The
situation for three coils each wound onto one leg of a three leg core is shown in
Fig. 8, which indicates the divisions of the coils into segments by suitable
diagonals and the assignment of the resulting segments to two MTL systems, X
and Y. Other positions for the various coil terminals can be introduced using the
theory developed in the Part III.
The inclusion of multiple coils on different core legs means that the Y-system
of coil segments is not an MTL system in the usual sense of the definition and the
application of the general theory requires some comment in these cases. If the Y-
system conductor segments were isolated in a dielectric medium, the interactions
between the segments from different coils would be negligible. Then the usual
parameter matrices fo r the Y system would be constructed and contain no cross-
coupling between such segments; the standard MTL solution,
)0()()0()exp()( YY YYAY LLL ==
would in actual fact simply corresponds to a set of disjoint systems, one on each
core leg, and the previous analysis would go through unchanged. However, the
transformer core couples all these Y segments magnetically and cross coupling
inductances should be included in the parameter matrix L to account for this. One
has to consider whether this is permissible and the theoretical position is not clear.
It seems that if one computed approximate delayed partial inductances (equivalent
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29
to a phase factor in the frequency domain) in the same way one applies a PEEC
analysis [8-11], the MTL theory should then extend to the new situation: if it does,
then the analysis goes through again. Perhaps one can hope that the results of
numerical simulation and measurement will establish whether such a procedure isvalid.
III. WINDING SYMMETRIES
A. Introduction
It is possible to extend the theory of Part II to winding arrangements obtained
from the original system by applying certain symmetry transformations, or
winding symmetries. These include the retrograde windings and rearrangements
obtained from turn permutations considered previously in Part I.
B. Symmetries of One Coil
We begin by considering the single N turn coil of Section IIB subjected to aturn permutation, where for example the turns are reconnected into an interleaved
type arrangement. A new interleaved winding is equivalent to a modified
connection pattern of the conductor segments of both the X and Y systems in the
theory of Part II. Suppose, however, that the two MTL systems of the theory are
numbered in the sequential way used before by following the electrical continuity
turn by turn. The numbering of the turn segments in each system then differs from
that used originally, but the conditions of continuity of voltage and current will be
given by the same matr ix equations found before:
[ ] SJI =~
, (75)
or
SJI =+ )(~
)0(~
L , (76)
as long as the voltage-current state vectors X~
and Y~
are assembled in the
standard way. The terminal leads are assumed to move with their points of
attachment under the transformation. Thus the transformed external terminals may
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be at different turns as compared to the original ar rangement.
The original X system could, for example, have appeared in cross section as
in the top of Fig. 9 while a possible rearranged system might be numbered as in
the bottom diagram; identical physically except for the renumbering of theconductors by a permutation
: 11, 26, 32, 47, 53, 68, 74, etc.,
which is characteristic of the change in the winding pattern. As in Section IF the
matrices of parameters in the new numbering system belonging to the interleaved
winding are related to those in the original numbering, belonging to the disc
winding, by a permutation similarity:
TpMpM =
~
, (77)
where M is a general parameter matrix such as C or L and p is the orthogonal row
permutat ion matrix obtained by permuting the rows of the 2N-dimensional identity
matrix according to :
=
OMMMMMMM
LL
L
L
1000000
0000010
0100000
0000001
p , N1ppppTT
== . (78)
The transformation ofX
A follows from equations (77)
TPAPA
XX
~= ,
=
p0
0pP (79)
just as in the earl ier discussion. The new chain parameter matrix for the X~
system
can be obtained from that of the original arrangement by a permutation similarity
since
TTPPPAPA === )exp()~exp(~ XX LL . (80)
The Y system of conductor segments is renumbered in the same way as the
X system because the turns are rearranged as intact units, therefore
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TPPAA
YY
~= , (81)
TTPPPAPA === )exp()~exp(~ YY LL . (82)
The MTL solution for the coil expressed within the transformed numbering is
)0(~~)0(
~~
~
)0(~
)~
exp(
)~
exp()0(
~)
~exp()(
~
Y
X =
=
==
0
0
A0
0AA
L
LLL (83)
and from (80) and (82)
T
T
T
PP0
0PP
0
0
=
=
=
~
~
~ ,
=
P0
0P . (84)
Using (76) and (83) the solution of the rearranged problem is
SJI =+ )0(~
)~( .However, by substitution of (84) we obtain
SJIJITT
=+=+ )0(~
)()0(~
)( .
It is easy to confirm that )0()0(~ = so that this equation can be written as
SJI =+ )0()~~
( .which is what we would obtain by using the original numbering scheme for the
coil, but with I and J modified to take account of the rearranged winding pattern.
The desired transformations ofI and J are
JJII ==~
,
~
.
All this parallels the discussion of Section IF. However, the approach of
transformation of the matrices I and J can be developed to include many other
possible rearrangements of the coil . We begin by expressing the transformed
solution for a winding permutation in a modified form.
Corresponding to the winding permutation represented by define a 16N-dimensional orthogonal permutation matrix by
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=
=
=
p
p
p
p
p
p
p
p
P
P
P
P
0
0
)( . (85)
The solution of the transformed problem can then be written
[ ] S0
01JI =
)0(
)0()(
, (86)
which includes the un-transformed problem of the Part II as a special case.
( 1)(e ). The continuity conditions are evidently
[ ] SJI = )( . (87)We now show that for a larger group of winding symmetries )(g exist for each
symmetry g such that the solution for the transformed coil is always expressed by
a matrix equation like (86) and the matrices T)(g constitute a faithful
representation of the winding symmetry group as (orthogonal) matrices. For the
special case of a turn permutation )( is quasi-diagonal and (86) is equivalent to
the picture in which the I and J are invariant and the appropriate chain parameter
matrix is obtained by a similarity transformation from the original.
For simplicity assume that the coil has a circular (or possibly square) cross
section. Consider the symmetries represented by reflexions in planes parallel and
perpendicular to the core leg laminations, the y-z and x-z planes respectively,
together with rotations through multiples of 900
around the core axis. These
transform the conductor segments of the X and Y systems into themselves and
form a group, G, isomorphic to C4v . This group includes reflections in both of the
planes containing the core diagonals and the zaxis, d and d', and has order 8. By
applying these symmetries the coil is changed in its handedness and the terminals,
which are assumed to follow their points of attachment to the respective turn
segments, are repositioned.
The general cases can be obtained by considering the example of a four turn
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coil. Consider firstly a reflection of such a coil across the y-z plane,yz
, as
depicted in Fig. 10. Suppose we divide the turns in the transformed coil into the
usual segments and distribute them between the two MTL systems X~
and Y~
.
These are numbered, as was the original coil, by following the turns along the path
of electrical continuity between the new terminal leads. In other words, it is the
numbering scheme obtained by application of the symmetryyz
. The necessary
conditions of continuity of voltage and current of the transformed coil can be
simply written in this numbering as
SJI =+ )(~
)0(~
L , (88)
or
[ ] [ ] SJIJI ==
~
)(~
)0(~
L
, (89)
using the same matrices I and J as for the original coil treated within the original
numbering scheme. Define the matrices
=
N
N
N
N
10
01
10
01
and
=
01
10
01
10
N
N
N
N
. (90)
Then the following relations link the voltage and current vectors in the two
systems of numbering for the conductor segments:
==
==
)()(~)0()0(~
)0()(~
)()0(~
LL
LL
YYYY
XXXX
. (91)
The transformations with arise from the opposite senses of current flow in some
parts of the transformed systems as compared with the originals. The
transformations involving the matrix are not quite so obvious, but depend on the
symmetry mapping any complete turn into itself and the numbering system for the
segments. Consideration of a few simple examples will convince the reader of the
relationships that ensue; the result of applyingyz
to a representative Y-system is
indicated in Fig. 11. Equations (91) can be combined to give the relations of the
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34
state spinor components in the two numbering schemes:
)()0(
)(
)(
)0(
)0(
)0(
)(
)0(~
)0(~
)0(~
L
L
LL
+
=
+
=
=
=
00
0
0
00
Y
X
00
0
Y
X
0
00
Y
X
Y
X
)()0(
)(
)(
)0(
)0(
)(
)0(
)(~
)(~
)(~
L
L
L
LL
LL
+
=
+
=
=
=
0
00
00
0
Y
X
0
00
Y
X
00
0
Y
X
Y
X
(92)
Thus the state spinor transforms as )(~yz
= , where )(yz
is the orthogonal
(and symmetric) 8N-dimensional matrix
=
000
000
000
000
)(yz
. (93)
Substituting directly into (89) gives the continuity conditions for the transformed
coil expressed within the original referencing system:
[ ] SJI = )( yz . (94)
Next consider a reflection of the coil across the x-zplane,xz
. We obtain two
transformed MTL systems X~
and Y~
and proceed to number these sequentially in
their new arrangement as before, following the electrical continuity. The situation
here is indicated in Fig. 12, and instead of (91) we obtain the relations
==
==
)0()(~
)()0(~
)()(~
)0()0(~
YYYY
XXXX
LL
LL. (95)
This gives )(~xz
= , where )(xz
is the orthogonal (and symmetric) 8N-
dimensional matrix
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=
000
000
000
000
)(xz
. (96)
The continuity conditions for the transformed coil in terms of the original
referencing system are then, from (89),
[ ] SJI =)(xz
. (97)
The final case that needs detailed consideration is a 900 rotation of the coil about
the core leg axis. The transformed systems that result on applying the sequential
numbering of the segments are indicated in Fig. 13 and we have the relations
==
==
)0()(~
)()0(~
)()(~
)0()0(~
XYXY
YXYX
LL
LL. (98)
Since and commute the order of their product in equations (98) is irrelevant.
From these )C(~ 4= , where )C( 4 in this case is the orthogonal (but notsymmetric) 8N-dimensional matrix
=
000
1000
000
0010
)C( 4 . (99)
The continuity conditions in terms of the original referencing system are now of
course
[ ] SJI =)C( 4 . (100)
The three symmetries 4yzxz Cand,, generate G whose order is 8, as we
mentioned before. The full group of coil symmetries G is obtained by adjoining
turn permutations of the type considered at the start of this section. The
symmetries in G evidently commute with any turn permutation and G is a normal
subgroup ofG . To obtain the order ofG we need to consider the number of turn
permutat ions admitted by the coil . Both X and Y systems contain 2N conductor
segments, but not all permutations of these correspond to turn permutations. Turn
segments on one side of the core must map only to those on the same side; the
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permutat ion on the other side is then the mirror image of this. Thus there are only
N! admissible turn permutations, G is isomorphic toNSG , and has order 8(N!).
One also notes that of the admissible turn permutations many are impossible for
technical reasons and the subset of those remaining will not necessarily constitute
a sub-group ofG .
The matrices T)(g constitute a representation of G as (orthogonal) permuta-
tion matrices, where if G2gg , we have the correspondence
( ) TTT )()()( 12 gggggg = 1oo (101)
under the usual law of matrix multiplication. As an example consider the
symmetry that results from two successive rotations through angle 900. From (101)
we have
=
==
000
000
000
000
000
1000
000
0010
000
1000
000
0010
)C()C()C( 442 (102)
and becauseyzxz
=2
C we also have
=
==
000
000
000
000
000
000
000
000
000
000
000
000
)()()C( xzyz2 . (103)
Since and commute (102) and (103) are the same, as they should be, and one
verifies that they represent the correct relations in the equation )C(~ 2= thatresult from rotating the original coil through 180
0
. As another example we
consider reflection in the diagonal plane d. One has yzd = 4C , so that
=
==
000
000
000
000
000
1000
000
0010
000
000
000
000
)C()()( 4yzd . (104)
On the other hand the symmetry resulting from reflection in the other diagonal d'
is given by4xz
C=d'
, so that
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=
==
000
000
000
000
000
000
000
000
000
1000
000
0010
)()C()( xz4d . (105)
One can verify that (104) and (105) are indeed the correct -matrices. In each case
the solution to the problem transformed by symmetry Gg is obtained from
equations of the form obtained at the end of the last section:
[ ] S0
01JI =
)0(
)0()(
g . (106)
is the appropriate chain parameter matrix which is computed as before fromthose of the X and Y systems. The group of symmetries also includes reflection of
the coil through a medial plane if this is a symmetry, since this is equivalent to a
certain turn permutation.
C. Winding Symmetries for Two Coils
So far we have looked at the symmetries of a single coil and it is obvious that
the effects of these transformations on the actual electromagnetic characteristics
of the coil, in other words the actual solution to the single harmonic problem
obtained by solving (106), must be quite limited. Therefore we now consider the
symmetries of a winding containing several coils, and begin with the case of two
coils wound onto a single core leg, considered already in Section IID.
We can apply reflexions in the x-z, y-z, d and d' planes, together with
multiples of rotations of 900 about the zaxis, separately to either coil to obtain
various differing arrangements and the group of symmetries thus obtained will be
isomorphic to GG . The full group of symmetries GG is found by adjoining to
this group the winding re-arrangements represented by turn permutations. These
commute with the symmetries of GG and by similar considerations to those
used previously one sees that GG is isomorphic to21
NN + SGG , has order
( )( )!NN6421
+ , and has GG as a normal subgroup. GG is also a subgroup,
containing precisely those symmetries that preserve the associations of the coilsegments with their original coils; it has order ( )( )!N!N64
21. GG is a normal
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38
subgroup of GG , but GG is not a normal subgroup of GG . As before many
turn permutations are technically impossible to achieve and the subset of those
remaining will not necessarily constitute a sub-group of either GG or GG .
To represent the effects of the double coil symmetries on the electromagnetic
solution in the same way as for a single winding in the previous section, we return
to equation (60) of Section IIC:
=
=
=
2
1
22
11
2
1
22
11
2
2
1
1
22
11
)(
)0(
)(
)0(
S
S
JI00
00JI
JI00
00JI
JI00
00JI
L
L.(107)
Suppose a symmetry ( ) GG hg, is applied to the double coil. If the numbering
systems of the primary and secondary parts of the MTL systems are also
transformed as we did in the derivations of the previous section, then using the
obvious extension of the notation used there the continuity conditions of the
transformed winding can simply be written down as
SS
S
JI00
00JI
JI00
00JI=
=
=
2
1
22
11
22
11
~
~
~
2
1. (108)
Using the representation of G already derived we have the relationships
1g=~ and 2h=~ so that
))
))
21
21
=
=
=
h
g
h
g
0
0
0
0
~
~
~ . (109)
But from Section IIC R= , so)
)))
21
R0
0
0
0
=
=
h
g
h
g~
~
. (110)
Substitution into (108) gives the conditions of continuity of the transformed
winding expressed within the or iginal numbering scheme:
SR0
0
RRJI00
00JI T=
)
)h
g22
11
,
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39
or equivalently, from (67),
[ ] [ ] ( ) SJIR0
0RJI
T ==
))
h)(g,h
g. (111)
The correspondence defined by
( ) ( ) R0
0R
T
T
TT
=
)(
)()(
h
ghg,hg,
(112)
gives a representation of GG as matrices of dimension )NN(16 21 + . If
( ) ( ) GG 2211
and h,gh,g , from (101) and (112)
( ) ( ) ( ) R0
0R
T
=
)()(
)()(,,,
21
21
hh
gghhgghghg
12121122
. (113)
To obtain the -matrix that represents a general turn permutation we need only
repeat the arguments used earlier, since they can be carried over unchanged to the
present case. In th is way we find that for a turn permutation , )( is again the
quasi-diagonal permutation matrix
=
=
=
p
p
p
p
p
p
p
p
P
P
P
P
0
0
)( , (114)
where p is the appropriate 2 ( )21
NN + dimensional permutation matrix. If is in
fact a turn permutation that preserves the identities of the primary and secondary
coils then it has the form ( ) GG 21
, for certain permutations of the two coils
1 and
2 , respectively. Therefore it must be the case that )( in (114) and
( ))( 21 , given by (112) coincide:
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40
( ))()()(
)( 212
1
,=
=
=
R00
R
p
p
p
p
p
p
p
p
T
.
The -matrices representing the full group of double coil symmetries in GG can
be obta ined by composing those given by (112) and (114). The equations
determining the solution for the problem after transformation by any symmetry
GG g are then
[ ] S0
01JI =
)0(
)0()(
g . (115)
In general it will probably not be possible to reduce the dimensions of the matrix
inversion required to solve (115) using simple algebraic manipulation as we were
able to do in the first steps of the discussion of the single coil problem, when we
derived equation (45) of Section II. The earlier result depended on a special form
of the matrices which will not always be present after a winding symmetry has
been applied.
D. Winding Symmetries for Multiple Coils
It is possible to extend the treatment of Section IID to the representation of
the symmetries of any number of coils in a winding. Suppose that there are K
coils. The symmetries ofG and G can be applied to any one of them individually
and the group of symmetries of the whole winding is at least
( )timesKGGG L and contains ( )timesKGGG L as a normal subgroup,
among others. The representation of such symmetries as -matrices follows as a
natural extension of (112) using the appropriate matrix Rgiven by the formul in
Part II. To these we can add all the turn permutations represented by (114). It is
not possible to go any further than this in generality because the technically
allowable permutations depend on how the individual coils are arranged on the
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41
core legs, for obvious reasons. In fact this consideration also applies to the case of
2 coils: if these were wound onto two separate legs the allowable symmetries
would exclude all turn permutations that mix up the turns from the two coils.
Symmetries arising from possible whole-scale transpositions of individual coilsare represented by certain turn permutations, but these are only equivalent to
renumbering the coils in the model; they can be treated in either way.
Finally we remark on the generality of the idea of a winding symmetry. So far
the winding symmetries have been treated as actual rotations and reflections.
However, there is nothing in the theory to limit its application in this way. The
symmetries discussed can be extended in an obvious way by interpreting them as
changes in the connection pattern and sense of positive current flow in the system
of conductor segments that makes up a coil in the model, regardless of whether the
segments would actually map into each other physically under the given
symmetry. Interpreted in this way these pseudo-symmetries can be applied to any
coil, whatever its cross section.
IV. CONCLUSION
In this paper the theory of the multiconductor transmission line description of
the electromagnetic response of a power transformer at high frequencies has been
extended considerably and the resulting equations presented in a matrix form
suitable for implementation by any standard software program that handles matrix
algebra. The connection between the MTL method and certain geometrical
transformations of a winding has been elucidated and expressed in a covariant
form.
V. MATHEMATICAL APPENDIX
A. The Kronecker Product
Suppose )C(M][ , nmija =A and )C(M][ , qpijb =B are two complex matrices.
The Kronecker product ofA and B, )C(, nqmpMBA , is defined in block form as
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the matrix
=
BBB
BBB
BBB
BA
mnmm
n
n
aaa
aaa
aaa
LMOMM
L
L
21
22221
11211
. (116)
The product occurs as a convenient notation in many applications of mathematics
and the description of matrix equations is a typical one [29]. It has a number of
obvious linear properties, while for our applications it is important to remember
that in general ABBA . Howevermnmnnm
11111 == , =nm
10
mnmn001 = , and CBACBA = )()( .
VI. ACKNOWLEDGEMENTS
The Authors are grateful to the United Kingdom Engineering, Physics, and
Science Research Council (EPSRC) for funding through Grant GR/L08656.
VII. REFERENCES
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of Applied Physics, Vol. 12, March 1941, pp. 219-229.
[2] Reinhold Rdenberg, Electrical Shock Waves in Power Systems, Harvard University Press, Cambridge,
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[3] D. J. Wilcox, M. Conlon, D. J. Leonard and T. P. McHale, "Time-domain modelling of power
transformers using modal analysis", IEEE Proc.- Electr. Power. Appl., Vol. 144, No.2, 1997, pp. 77-84.
[4] D. J. Wilcox, "Theory of transformer modelling using modal analysis", IEE Proc., C, 138, (2), 1991,
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[5] D. J. Wilcox and T. P. McHale, "Modified theory of modal analysis for the modelling of multiwindingtransformers", IEE Proc., C, 139, (6), 1992, pp. 505-512.
[6] D. J. Wilcox, W. G. Hurley, T. P. McHale, and M. Conlon, "Application of modified modal theory in the
modelling of practical transformers", IEE Proc., C, 139, (6), 1992, pp. 513-520.
[7] A. E. Ruehli, "Inductance Calculations in a Complex Integrated Circuit Environment", I.B.M. J. Res.
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Element Equivalent Circuits", I.B.M. J. Res. Develop., Vol. 23, No. 6, November 1979, pp. 661-668.[12] S. A. Levin, "Electromagnetic Waves Guided by Parallel Wires", Trans. A.I.E.E., 46, 1927, pp. 983-989.
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[13] L. A. Pipes, "Steady-State Analysis of Multiconductor Transmission Lines", J. App. Phys., 12, 1942,
pp 782-799.
[14] W. T. Weeks, "Multiconductor Transmission Line Theory in the TEM Approximation", I.B.M. J. Res.
Develop., Nov. 1972, pp. 604-611.
[15] Clayton R. Paul, Analysis of Multiconductor Transmission Lines, New York, Wiley, 1994.
[16] K. Cornick, B.Filliat, C.Kieny, and W. Mller, "Distribution of Very Fast Transient Overvoltages inTransformer Windings", Cigr, 1992 Session, 12-204.
[17] J. L. Guardado, V. Carrillo, and K. J. Cornick, "Calculation of Interturn Voltages in Machine Windings
During Switching Transients Measured on Terminals", IEEE Transactions on Energy Conversion,
Vol.10, No.1, 1995, pp. 87-94.
[18] J. L. Guardado, K. J. Cornick, V. Venegas, J. L. Naredo, and E. Melgoza, "A Three-Phase Model for
Surge Distribution Studies in Electrical Machines", IEEE Transactions on Energy Conversion, 1997,
Vol.12, No.1, pp. 24-31
[19] J. L. Guardado and K. J. Cornick, "The Effect of Coil Parameters on the Distribution of Steep-Fronted
Surges in Machine Windings", IEEE Transactions on Energy Conversion, 7, 1992, pp. 552-559.
[20] M. T. Wright, S. J. Yang, K. McLeay, "General theory of fast-fronted interturn voltage distribution in
motor stator windings", IEE Proceedings, 130, 1983, pp. 245-256.
[21] M. T. Wright, S. J. Yang, K. McLeay, "The influence of coil and surge parameters on transient interturn
voltage distribution in stator windings", IEE Proceedings, 130, 1983, pp. 257-263.
[22] W. W. L. Keerthipala and P. G. McLaren, "Modelling the Effects of Laminations on Steep Fronted
Surge Propagation in Large ac Motor Coils", IEEE Transactions on Industry Applications, Vol. 27,
No. 4, July/August 1991, pp. 640-644.
[23] E. U. Condon, "Principals of Microwave Radio", Rev. Mod. Phys., 14, 1942, pp. 341-389.
[24] S. D. Garvey, W. T. Norris, S. Luff, and R. Regan, "Prediction of High Frequency Characteristics of the
Windings of Large Electrical Machines: A Lumped-Parameter Reluctance-Network Analysis", Ninth
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[25] P. J. Tavner and R. J. Jackson, "Coupling of discharge currents between conductors of electrical
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VIII. BIOGRAPHIES
Stefan Luff is a graduate of the University of Oxford and obtained his
Ph. D., at the University of Surrey studying the properties of high performance
synthetic lubricants using nuclear magnetic resonance spectroscopy. At present he
is a Research Fellow at Aston University, Birmingham, England, specializing in
the problem of high frequency electrical phenomena in large machine windings.
William Tobius Norris is a graduate of the University of Cambridge,
England, and obtained his DSc from the Massachusetts Institute of Technology in
the United States of America. He worked for ten years on speculative,
environmental, and trouble-shooting aspects of electricity generation and
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