high frequency theory of power transformers

7/28/2019 High Frequency Theory of Power Transformers

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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


18/44

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.


19/44

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


20/44

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:


21/44

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


22/44

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.


23/44

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,


24/44

24

( )( )( )

( )

( )( )( )

( )

=

=

=

=

=

=

)(

)(

)(

)(

)(

)(

)(

)(

)(

)()(,

)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


25/44

25

)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)


26/44

26

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.


27/44

27

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


28/44

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


29/44

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


30/44

30

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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31

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


32/44

32

=

=

=

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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33

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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35

=

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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36

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


37/44

37

=

==

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


38/44

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

,


39/44

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:


40/44

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


41/44

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


42/44

42

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

[1] R. Rdenberg, "Electromagnetic Waves in Transformer Coils Treated by Maxwell's Equations", Journal

of Applied Physics, Vol. 12, March 1941, pp. 219-229.

[2] Reinhold Rdenberg, Electrical Shock Waves in Power Systems, Harvard University Press, Cambridge,

1968.

[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,

pp. 121-128.

[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.

Develop., September 1972, pp. 470-481.

[8] A. E. Ruehli, "Equivalent Circuit Models for Three-Dimensional Multiconductor Systems", IEEE

Transactions on Microwave Theory and Techniques, Vol. 22, No. 3, March 1974, pp. 216-221.

[9] A. E. Ruehli, "Survey of Computer-Aided Electrical Analysis of Integrated Circuit Interconnections",

I.B.M. J. Res. Develop., Vol. 23, No. 6, November 1979, pp. 626-639.

[10] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, "Resistive and Inductive Skin Effect in

Rectangular Conductors", I.B.M. J. Res. Develop., Vol. 23, No. 6, November 1979, pp. 652-660.

[11] P. A. Brennan, N. Raver, and A. E. Ruehli, "Three-Dimensional Inductance Computations with Partial

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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43

[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

International Conference on Electrical Machines and Drives, IEE Conference Publication 468,

September 1999.

[25] P. J. Tavner and R. J. Jackson, "Coupling of discharge currents between conductors of electrical

machines owing to laminated steel core", IEE Proceedings, Vol. 135, Pt. B, No. 6, November 1988,

pp. 295-307.

[26] R. L. Stoll, "Approximate formula for the eddy-current loss induced in a long conductor of rectangular

cross-section by a transverse magnetic field", IEE Proceedings, Vol. 116, No. 6, June 1969, pp. 1003-

1008.

[27] A. C. Hall, "The Impulse Strength of Transformer Windings", I.E.E., Students Quarterly Journal, Vol.

33, No. 131, March 1963.

[28] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

[29] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.

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

7/28/2019 Hi

high frequency theory of power transformers

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