Chapter 1

Introduction

The precise modeling of electromagnetic devices, involves magnetic materials, needs an accurate

representation its magnetic characteristics. The magnetic characteristics involve non-linearity and hysteretic

nature as well. The first effort to solve the magnetic field problems by finite element method were made in

the late 1960’s. The analysis of electrical machines by finite element method was first applied to

synchronous machines and dc machines in [1-2]. The operation of these machine types can be approximate

modeled by stationary fields They have been analyzed by using step by step methods to solve the time

dependence in [3] and by three dimensional finite element formulations in [4]. The filed computation with

nonlinear characteristics of an induction motor is done using an eddy current formulation and assuming

sinusoidal time variation. The time-stepping methods to calculate the time variation of magnetic fields in

the induction motors have been discussed in [5].

The core is the most difficult part of the transformer to model because of its highly nonlinear and multi-

valued (hysteretic) nature of magnetic characteristics and complex geometries. Several hysteresis models

have been proposed in the related literature. Among the models, the Preisach model and the Jiles-Atherton

(JA) model are the most widely used hysteresis models [6]. The JA model is particularly successful in field

computation, such as FEM, due to its easy implementation and relative simplicity with reasonable accuracy

[7-8]. It is relatively simple involving computation of only five parameters. However, it is very sensitive to

its parameter variations; determining the parameters with a reasonable accuracy is required [9]. A static

inverse JA model is presented in [10]. The report also describes FEM implementation of the using

differential reluctivity formulation and μ0 and M formulation. The report showed dynamic loss

incorporation with the differential reluctivity term but it is not straightforward with the μ0 and M

formulation.

1.1 Motivation and scope of work

Although the efficiency of a modern transformer may above 99%, the core loss cost still can be significant.

The core loss topic always attracted to the researchers. Despite of low values of core loss, there is a

continuous feeling of further improvement among the researchers. Even small improvement in the core loss

can make significant impacts on technical, economical and environmental issues. It requests the knowledge

of flux distribution in the core particularly on the core-joints, magnetization mechanism in the different

type of materials and performance of different materials in different type core structure. The literature

survey indicates no common agreement in the core design parameters. Further studies are still needed in

this area.

Another factor is the significance of core diagnosis that can make considerable impact on the total

owning cost of transformer with the prevention of long-term minor core faults. Existing core diagnosis

techniques are not much effective. There is still a need of efficient core diagnosis technique in today’s

environment. The precise core modeling is still be a challenge to researchers due to it complex

heterogeneous structure, anisotropic and multi-valued nonlinear hysteresis characteristics.

1.2 Outline of the thesis

In Chapter 2 , a general theory about finite element method and electromagnetic field theory. This

chapter describes the electromagnetic field theory, with potential function formulation, results in boundary

value problem

Chapter 3, describes the finite element formulation of electromagnetic fields.

Chapter 4, describes implematation of hysteresis by using various hysteresis models which have

been proposed in the literature lik Jiles-Atherton (JA) and the Preisach Model.

Chapter 5,describes the methods and ways for solving proposed problem

Chapter 6, in this chapter simulation and results are given for proposed

Chapter 7, in this chapter some conclusions are drawn from the analysis of proposed geometery of

core which and . Some ideas for future research on coupled field analysis of electromagnetic devices are

also given.

Chapter 2

Theory and application of finite element

method in electromagnetics

2.1 Theory of Finite Element Method

The finite element method has become a well established method in many fields of computer aided

engineering, such as, electromagnetic field computation, fluid dynamics and structural analysis. The finite

element method is basically, an efficient approximation method to solve the partial differential equations or

boundary-value problems, which are frequently occurred in different areas of engineering. There are three

main steps during the solution of partial differential equation (PDE) with the finite element method. First

the domain on which the PDE should be solved, descritized into finite elements. Depending on the

dimension of the problem, this can be triangles, squares, rectangles or tetrahedrons, cubes, or hexahedrons.

The solution of PDE is approximated by piecewise continuous polynomials and the PDE hereby descritized

and split into finite number of algebraic equations. Thus, the aim is to determine the unknown coefficient of

these polynomials in such a way, that distance (which is defined by the norm in a suitable vector space)

from the exact solution becomes a minimum. Therefore, the finite element is essentially a variational

minimization technique [11].

Since the number of elements is finite, we have reduced the problem of finding a continuous solution to

our PDEs to calculating the finite number of coefficients of the polynomial. The solution of the Poisson’s

equation, which is required to calculate the magnetic vector potential, has to be solved for a given current

density distribution. One can write the Poisson’s equation in more general form as [11],

2u r f r (1)

In order to apply the finite element method, there are two methods are given in the corresponding literature

as the variational formulation and the Galerkin’s formulation. The variational formulation is based on the

energy minimization in the domain which is equivalent to the solution. The Galerkin’s methods leads to the

week formulation of the problem: we multiply Poisson equation by the test function u and integrate over

the solution domain

2( ). ( ) ( ) ( )u r v r dr f r v r dr

(2)

Integration by parts gives

u r v r dr u r v r dr f r v r dr

(3)

where, dr denotes the surface normal on the boundary. If appropriate boundary conditions define the value

of u (Dirichlet boundary condition) or of its derivative (Neumann boundary conditions) on the boundary, it

can be simplified (the homogeneous Neumann boundary conditions vanishes the term), and the Dirichlet

boundary conditions have to be apply in the equation given by,

( ) ( ) ( ) ( ) ( )u r v r dr gv r dr f r v r drn (4)

The exact solution ui can be approximated by a linear combination of trial functions,

0

( ) ( )n

h i i

i

u r u r

(5)

And we use a finite set of test functions φi

By inserting this expansion into the equation (5) and assume only Dirichlet boundary conditions, then it

becomes,

( ) ( ) ( ) ( )0

nu r v r dr f r v r dri i i i

i

(6)

A system of algebraic equations is obtained. It can be solved with any standard method for the solution of a

system of algebraic equations, such as Gauss method, the Cholesky decomposition or iterative scheme like

the conjugate gradient method.

Fig. 1. Problem domain and Boundary conditions

The value of the solution is explicitly defined on the boundary (or on a part of it). The value of solution can

be zero or non-zero on the boundary.

2.2 Application of FEM in Electromagnetic Field Computation

The theory of electromagnetics can be described by the Maxwell’s equations and constitutive equations.

The consistency of these equations (along with constitutive ones) is so high that very distinct phenomena

(like microwaves and permanent magnet fields) can be precisely described by these. The formulation and

the basic concepts of the electromagnetics are relatively simple but a realistic problem can be much

complex and difficult to solve. Due to complicated geometries, non-linearity, and hysteretic nature of

materials make it virtually impossible to find analytical solutions for such problems. Hence the numerical

methods have become widely used tools in electrical engineering nowadays.

The general, time dependent, Maxwell’s equations in differential form (also called point or local form) are

BE

t

(7)

DH J

t

(8)

D (9)

0B (10)

The differential form is more convenient in calculations using methods such as finite element method or

finite difference method, while integral form is more convenient in analytic calculation of fields and in

various integral methods of numerical calculations such as the method of moments and boundary element

methods.

The Potential functions are viewed as alternative representation of the electromagnetic field. These are the

simpler, more useful to describe the field properties rather than to use an abstract field variable like

magnetic flux density, magnetic field intensity etc. the magnetic vector potential function is defined based

on the solenoidal nature of the magnetic fields. The magnetic flux density B is solenoidal in nature (i.e.

0B ), it can be derived as the curl of another vector given by,

B A (12)

Here, A, is called the magnetic vector potential. The condition 0A is known as Coulomb gauge

condition. This relation is consistent with the field equations because as leads to continuity equation.

In the case of the magnetic vector potential

1s

AA J V

t t

(13)

Using the relations B=µH and D= E and vector identity which is given as,

2A A A (14)

The Equation (2) can be rewritten as in generalize form as,

2

2

2

V AA A J

t t

(15)

In the case of low frequency (static and quasi-static cases ), last term can be neglected and if there is no

electric scalar potential is present, then the resultant equation will be

2A A J (16)

Now, using the Coulomb gauge, the resultant equation will be partial differential equation in terms of

magnetic vector Potential

2 A J (17)

It can be solved with knowledge of magnetic vector potential on the boundary of the solution domain [16].

Chapter 3

FEM formulation for electromagnetic field

computation

3.1 Magneto-static Analysis

In the magneto static analysis, the quantities are not time dependent. This analysis is generally required for

the problems of steady flow of dc electric current, permanent magnets etc. [11].The governing equation for

the magneto-static case can be derived from (15) and it can be given as,

2 2

2 2

1 1s

d A d AJ

dx dy (18)

By using the finite element formulation as discussed in section (II-A), it will result in the algebraic equation

of the form

S A J (19)

The linear magneto-static analysis has been done for a bar plate problem. The finite element mesh and flux

distribution is shown in Fig.2 and Fig.3 respectively.

Fig 2.Finite Element Mesh in Problem Domain

Fig 3. Flux Lines

3.2 Time-Harmonic analysis

In the quasi-static case (with eddy currents are considered), the governing equation can be given as,

1SV

t t

A AA J (20)

Or it can also be written as,

21S

t

AA J (21)

In eq. (21), the first term on RHS indicates the source current density and the second term indicates the

eddy current density. The above equation can be solved with transient formulation as well as time harmonic

case (in case of linear analysis).

The FEM formulation for time-harmonic analysis is similar to magneto static analysis as described above.

Due to presence of eddy currents in time-harmonic analysis, there will be an additional term in the resultant

FEM equation for time-harmonic analysis as given in

r i r i r iS A jA j T A jA J jJ (22)

In comparison to magneto-static analysis, all the quantities in time-harmonic analysis are in complex form.

The linear magneto-static analysis has been done for the geometry shown in Fig.4. The finite element mesh

and flux distribution at different frequencies is shown in Fig.5,6,7, and 8.

Fig.4.Geometry for Time-Harmonic Analysis

Fig.5 Meshing of Time-Harmonic Geometry

Fig.6.Flux Distribution at 0 Hz

Fig.7.Flux Distribution at 50 Hz

Fig.8.Flux Distribution at 300 Hz

3.3 Non-Linear Electromagnetic FEM Analysis

The magnetic materials are characterized by nonlinear and hysteretic magnetic characteristics. This section

discussed the nonlinear characteristics without hysteresis. These materials are characterized by a nonlinear

B-H curve and the permeability is field dependent (B or H). The governing equation for a non-linear

magnetostatic case can be written as,

21A J

B (23)

The above equation can be solved using iterative methods such as fixed-point and Newton-Raphson

methods. The rate of convergence of Newton-Raphson method is higher than the fixed-point method but it

needs appropriate initial values. The Newton-Raphson method is described below.

3.3.1 The Newton-Raphson Method

It is based on the Taylor series method. The series is truncated after its first term for simplicity. For a

column vector P, which is a function of a vector X, the truncated Taylor series is, for X close to Xm

m m

PP X X

X

(24)

where, P

X

is the Jacobian of P at Xm.. For example, P and X have three components for three variables

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 2 3

P P P

x x x

P P PP

X x x x

P P P

x x x

If Xm is not too far from the solution of P, we find that Xm+1 in the relation as,

1 0m m m

m

PP X X

X

(25)

Or,

1m m

m

PX P

X

(26)

and the updation can be done in unknown variables as,

1 1m m mX X X (27)

It will continue until the convergence is reached.

3.3.2 Non-linear Magnetostatic Analysis with Magnetic Vector Potential

Formulation

The resultant FEM matrix system of equations for a non-linear case is given as [11],

SJ A R (28)

The right-hand side vector R is called a residual vector, can be obtained as

R SS A Q (29)

where, A is the unknown magnetic vector potential and Q is the source vector. The column matrix P is the

matrix product SSA where X has been replaced by A . This matrix product Pi(k) is obtained by assembling

the terms below as,

3

1 2i l k l k l

i

P k q q r r AD

(30)

The general term of the Jacobian P

X

matrix can be determined as [11],

2 3

21

1

2 2i n k n k k l k l l

ln n

BP k q q r r q q r r A

A D D AB

(31)

We need to calculate2

n

B

A

, which can be determined as,

3

1

i i i i

i

A p q x r y A

(32)

Now magnetic flux density B can find using the relation B A as,

A AB i j

y x

(33)

2 3

21

2l n l n l

ln

Bq q r r A

A D

(34)

Now the global Jacobian matrix SJ can be obtained by assembling the terms given below,

3 3

2 21 1

4, , , ,i l l

l l

J n k S k n S n l A S k l AD B

(35)

where,

,2

k l k lS n k q q r rD

(36)

The non-linear magneto-static analysis has been done for a bar plate problem. The flux distribution with

MATLAB code is shown in Fig.9. The MATLAB results are validated with commercial FEMM software.

Fig.9.Non-linear analysis (Flux Density)

Fig.10.FEMM analysis (Flux Density)

Chapter 4

Implementation of Hysteresis in FEM

modeling

The unique feature of the magnetic materials is magnetic hysteresis. Various hysteresis models have been

proposed in the literature. The Jiles-Atherton (JA) and the Preisach model are most frequently used

hysteresis models in electromagnetic field problems [6]. These models can be coupled with the Maxwell’s

equations in order to obtain accurate solutions for electromagnetic field problems. The Jiles-Atherton

Hysteresis model

The Jiles-Atherton model is based on energy balance equation. In the JA model, the magnetization M is

decomposed into its two components, viz. reversible Mrev and irreversible component Mirr. The model can

be represented by the following first- order differential equation [9],

0

( ) ( )

( ) ( ) ( )

an irr an

an irr

M H M H dMdM dMc

dH k M H M H dH dH

(37)

where, M is the total magnetization, H is the applied magnetic field, Man is the anhysteretic

magnetization, 0 is the magnetic permeability of free space, and is the directional parameter defined as,

1 for 0 and ( ( ) ( )) >0

1 for 0 and ( ( ) ( )) <0

an

an

dH dt M H M H

dH dt M H M H

(38)

The anhysteretic magnetization Man is defined as,

coth e

an e s

e

H aM H M

a H

(39)

where, He is the effective field and can be written as,

eH H M (40)

The five physical parameters of the JA model and their physical interpretations are given as,

Ms (A/m) : Saturation magnetization

a (A/m) : Form-factor or shape factor

k (A/m) : Domain wall pinning constant (irreversible magnetization component)

c (dimensionless): Domain wall bowing parameter (reversible magnetization component)

α (dimensionless): Mean field parameter. (inter-domain coupling)

The values of these parameters can be determined by parameter identification process using iterative or

optimization techniques. The model is relatively simpler one to implement in numerical techniques of field

computation. We will describe here implementation of JA model in finite element method for

electromagnetic field problem. In the original JA model, we can obtain magnetic induction B by giving

magnetic field H as an input to the model. However, when using a magnetic vector potential formulation B

is obtained directly from the computed magnetic vector potential at each time step. To overcome this

problem, an inverse JA model proposed in [10] allowing the magnetic induction B as an input and gives the

magnetic field H as an output.

Chapter 5

Methodology

5.1 Original JA hysteresis model

In the original JA model, the magnetization M is decomposed into its two components, viz. reversible Mrev

and irreversible component Mirr. The model can be represented by the following first- order differential

equation [51],

0

( ) ( )

( ) ( ) ( )

an irr

an irr

an

M H M H

k M H M HdM

dH dM dMc

dH dH

(41)

where, M is the total magnetization, H is the applied magnetic field, Man is the anhysteretic

magnetization, 0 is the magnetic permeability of free space, and is the directional parameter has the

value +1 if dH/dt >0 and -1 if dH/dt < 0.

Derivative of Mirr with respect to He can be given as,

irr an irr

e

dM M M

dH k

(42)

The relation between anhysteretic and reversible magnetization and total magnetization are given as,

rev an irrM c M M (43)

rev irrM M M (44)

The anhysteretic magnetization Man is defined as,

coth e

an e s

e

H aM H M

a H

(45)

where, He is the effective field and can be written as,

eH H M (46)

and, the constitutive relation is given as,

0B H M (47)

where, a, α, c, k, and Ms are the five parameters which can be determined from the measured hysteresis

curve using a parameter identification procedure as discussed in [50].

The hysteresis equation (41) is derived from an energy balance equation. The supplied magnetic energy

appears either as a change in magnetization in the form of magneto-static energy or be dissipated due to

irreversible change of magnetization in the form of hysteresis loss. The magnetization can be represented

by anhysteretic magnetization, as given by (45), when the hysteresis loss is zero.

Additional losses which generally occur in a conducting magnetic material are the classical eddy current

loss and anomalous loss [52]. The classical eddy current loss per unit volume is proportional to the square

rate of change of magnetic induction and to square of the thickness of material. This component of the loss

assumes fine enough sample to neglect the skin-effect. The classical eddy current instantaneous power loss

per unit volume is given as,

22 B

2

ECdW d d

dt dt

(48)

where, d and ρ represent thickness (m) and resistivity (Ω-m) respectively. B is the magnetic induction. β

is a geometrical parameter (β=6 for lamination, β=16 for cylinder)

Anomalous loss has been treated on the basis of a statistical approach to the loss phenomenology in [52].

The anomalous loss results from domain wall motion during the change in the domain wall configuration in

magnetization process. It is generally expressed as,

1 3

2 20 BA

GdwHdW d

dt dt

(49)

where, G is a dimensionless coefficient of eddy current damping (0.1356), w indicates the width of

laminations (m) and H0 characterizes the statistical distribution of the internal domain wall field and takes

into account the grain size.

5.2 Inverse Dynamic JA Model

The energy balance equation in the presence of classical eddy current and anomalous loss can be written as

[46],

0 0

2

0

3 2

1

an e e

irre e

e

a

M H dH M H dH

dM dBk c dH k dt

dH dt

dBk dt

dt

(50)

Equation (50) can be converted into a manageable form by assuming B=μ0M, which is generally valid in

case of soft magnetic materials [46], and divided by μ0. It leads to,

1 2

1

an e e

irre e e

e e

a ee

M H dH M H dH

dM dB dMk c dH k dH

dH dt dH

dB dMk dH

dt dH

(51)

Differentiating (51) with respect to He and using the relations given in (43-44) it becomes,

anan d

e e e

dMdM dMM M k k c P t

dH dH dH (52)

The dynamic loss term Pd (t) can be expressed as, 1 2

d e a

dB dBP t k k

dt dt

(53)

where, ke and ka are dynamic loss parameters, which can be defined as,

2

1

20

2e

a

dk

GdwHk

(54)

Equation (52) can be rewritten as,

and an

e e

dMdMk P t M M k c

dH dH

(55)

From the effective flux density, we have Be = μ0He and then we have,

0e

e

dB

dH (56)

Using equation (56), equation (55) can be expressed as,

0an

d ane e

dMdMk P t M M k c

dB dH

(57)

And the term, e

dM

dB can also be written as,

e e

dM dM dB

dB dB dB (58)

Using the relationships, given in (46) and (47), flux density can be expressed as,

0 0eB B M M (59)

Differentiating (59) with respect to Be and substituting in (58), gives,

01 1e

dM dBdM

dB dM dB

(60)

Now, using the relation (60), equation (57) can be written as,

0

( )

1 ( )

M an an e

M an an e d

M M k c dM dHdM

dB M M k c dM dH k P t

(61)

5.3 Implementation of the JA hysteresis model in FEM computations

The JA model can be implemented in FEM calculations using some standard iterative methods such as

fixed point method and Newton-Raphson method. Another alternative is time stepping method using

inverse JA hysteresis model which is based on differential reluctivity approach.

The governing equation for diffusion equation;

21S

t

AA J (62)

If we neglect the eddy current term t

Ain above equation, we get;

21S

A J (63)

In case of ferromagnetic materials µ will be a function of B or H and above problem can be treated as

nonlinear as well as hysteretic in nature. The problem with hysteresis characteristics can be solved with

differential reluctivity.

The generalize relationship between vector B and H can be expressed as [11];

dd dΗ B (64)

The differential reluctivity can be expressed for 2-D case as,

2 2

x x y y

d

x y

H B H B

B B

(65)

The final equation to be solved is

( ( )) ( ) ( ( )) ( )d dt t t t t t A J A H (66)

Equation (66) is solved iteratively incorporation with the JA hysteresis equation (37). The third term can be

considered as a secondary source.

At each time step, B(t+Δt) is obtained with the computed A(t+Δt) for each mesh element. The inverse JA

model is used to obtain Hx(t+Δt) and Hy(t+Δt) for the components of magnetic induction Bx(t+Δt) and

By(t+Δt) respectively. ΔHx and ΔHy are evaluated as from the known Hx (t) and Hy (t) from previous

time step. vd is calculated using eq. (65). The elemental matrices are recalculated with the new vd and

again solved (66). This procedure is repeated until the convergence is achieved.

Chapter 6

Simulation and Results

Fig.11-15 are the output which got for designed core of transformer

Fig.11.Computed hysteresis loop at a point in Yoke

Fig.12.Computed hysteresis loop at a point

Fig.13.Current Waveform

Fig14.Magnetic Field at a point

Fig.15.Flux Density at a point

Fig.16(a).Flux at 10 Hz

Fig.16(b).Flux at 50 Hz

Fig.16(c).Flux at 100 Hz

The finite element mesh and flux density at different frequencies is shown in Fig.17(a),(b),(c).

Fig.17(a).Flux density at 10 Hz

Fig.17(b).Flux density at 50 Hz

Fig.17(c).Flux density at 100 Hz

Chapter 7

Conclusion

The report discussed the FEM formulation for the static, time-harmonic and non-linear magnetostatic

analysis. The above formulation has been applied in the 2-D field analyses. The all formulation has been

implemented with MATLAB programming with the validation of results by commercial FEM softwares.

The nonlinear magnetostatic problem is the first step towards hysteresis implementation in FEM. The

theoretical work of hysteresis is completed in stage BTP project stage-1 and implementation of hysteresis

in FEM is completed in this stage. The report describes FEM implementation of the JA model. Using the

model, one can incorporate the dynamic losses in the field analyses in a direct way. Both the differential

reulctivity and M and μ0 formulations can be used in the dynamic cases.

This work presents a detailed literature review on the transformer core design, diagnosis and modeling. The

reported work shows a need of further work on the core design to improve the performance of transformer.

The minor core faults in core may lead to increased core-loss, which in turn increase the transformer

capitalizing cost significantly. The possibility of using the Jiles-Atherton hysteresis model in the core

diagnosis, as discussed in this report, may be extended to determine the excess stress level, interlamination

short-circuits and deformation in transformer core. The JA hysteresis model may be helpful to determining

the building factor of transformer core and separation of core loss in hysteresis, eddy current and

anomalous losses as well.

This report presented the anisotropic modeling of core with the assumption of B and H is in same

direction. The assumption is not realistic in Grain-oriented materials. These materials show some finite

angle between B and H due to M. The discussed anisotropic model may be extended to consider the angle

between B and H. In two dimensional FEM core modeling, the interlaminar flux component has been

ignored in literature. However, experimental studies showed its significant impact on the core loss and local

flux distributions. The core modeling may be extended to consider these effects.

Future work

Till now we worked on single phase,three limed transformer core,the most popular joint used at the

intersection of the center limb and the yokes in the 40-90 degree T-joint.It is simple and economical to

manfacture because the laminations are cut out either 40 degree or 90 degrees to the rolling direction and

with careful cutting procedure the amount of waste material is very small.In future we are going to work on

three phase,three limed transformer core and another type of joint configuration referred to as the Y-45

degree T-joint.

Chapter 8

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