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I E E E

4th International Conference ELMECO

Naczw, PolandSeptember 2003

COMPUTATIONAL SOLUTIONS OF STEADY AND

TRANSIENT STATES IN TRANSFORMERS USING FEM

Dariusz CZERWISKI, Ryszard GOLEMAN, Leszek JAROSZYSKI

Institute of Electrical Engineering and Electrotechnologies,Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland

e-mail: leszekj@weber.pol.lublin.pl

Abstract

Electromagnetic Field Theory is often perceived as one of the most difficult

subjects of electrical engineering. Difficulties arise from a very complex mathematical

background. Necessity of some simplifications and idealisations of analysed objects

takes place. The investigation of real live electromagnetic systems, where

phenomenological interpretation of obtained results is the most important, requires

some modern numerical methods placed in user-friendly operational environment. The

Finite Element Method (FEM) gives very versatile and powerful solution of that

problem. However, despite its advantages, its usage is relatively expensive and difficult

especially for slightly experienced users. Progress in the software development enabled

a significant simplification of the user interface in packages and made FEM veryattractive for every PC user.

Keywords: electromagnetic field, transformers.

1. INTRODUCTION

Electromagnetic Field Theory is often perceived as one of the most difficult subjects

of electrical engineering. Difficulties arise from a very complex mathematical background.

Necessity of some simplifications and idealisations of analysed objects takes place. The

investigation of real live electromagnetic systems, where phenomenological interpretation

of obtained results is the most important, requires some modern numerical methods placed

in user-friendly operational environment. The Finite Element Method (FEM) gives very

versatile and powerful solution of that problem. However, despite its advantages, its usageis relatively expensive and difficult especially for slightly experienced users [1, 2]. Progress

in the software development enabled a significant simplification of the user interface in

packages and made FEM very attractive for every PC user.

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2. PROBLEM DESCRIPTION

Problems of the single phase transformer with ferromagnetic core, set of parallel

current conducting strips and three phase transformer with five legs ferromagnetic core

were solved as a samples of FEM computer aided solutions. Numerical models of these

problems have been built. Firs and second one were solved using QuickField software [3]

and the last one using FLUX2D package [4].

3. MAGNETOSTATIC FIELD ANALYSIS

The problem of a single-phase transformer has been the example of computer aided

computations of magnetic field (Fig. 1). The transformer consists of low voltage and high

voltage windings wounded on a ferromagnetic core. The core is made of a transformer plate

(ET-6). Computations assume the plane geometry and a simplified mathematical model thatassumes the isotropy of the core, the uniform distribution of current density in a cross-

section of each winding. Moreover, power losses in the core and insulation are neglected.

Fig. 1. Right upper corner of a single-phase transformer.

Current densities in the low voltage winding and in the high voltage winding are

Jn = 1.5106 A/m2 and Jw = 0.99810

6 A/m2, respectively. The magnetic field generated in

the transformer can be expressed by means of the non-linear Poissons equation in a currentarea

zzz Jy

A

yx

A

x )

1()

1(

(1)

and the Laplaces equation in a non-current area

0)1

()1

( y

A

yx

A

x

zz

(2)

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where: Az - a vector magnetic potential component normal to the model cross-section,

Jz - current density, - medium magnetic permeability.

The mathematical model assumes Neumanns boundary conditions along the axes OXand OY because of the symmetry

0n

Az

(3)

and the Dirichlets condition Az = 0 on the other edges where magnetic fields areneglectable.

Fig. 2. Magnetic flux distribution and field

lines in a single phase transformer.

Fig. 3. Magnetic density distribution and

field lines in a single phase transformer.

This vector magnetic potential distribution enables computations of magnetic field

intensity (Fig. 2) and magnetic density (Fig. 3) in the whole area. Magnetic density values

reach 1.63 T in straight line segments but in the corner is much smaller because of the

bigger cross-section of a magnetic core.

The influence of a transformer tank on the field distribution in the transformer has

been analyzed. The results indicate that if the transformer is placed in the ferromagnetic

tank the outflow of the magnetic leakage is different. Some part of this flux is closed with

the tank wall, which reflects in the comparison between the distributions of Hx component.

4. THREE PHASE TRANSFORMER WITH FIVE LIMB CORE

The numerical model of the power transformer of 250 MVA was assumed. This model

is based on real transformer build for industrial purposes. Calculations were made for

unsteady and steady states with fixed time step equal 0.001 s. Transient magnetic problem

was coupled with circuit analysis. High and low voltage windings of the transformer areconnected in star and stabilizing winding is connected in delta Fig. 4.

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Fig. 4. Connection diagram of the windings of the transformer.

Vector potential in our model is given by equation (4)

zzzz A

dt

dJ

y

A

yx

A

x

)

1()

1( (4)

which is coupled with equations of an electrical transformer circuit.

0

CBA

CHVBHVCCBBCB

CHVAHVCCAACA

iii

dt

d

dt

diRiRee

dt

d

dt

diRiRee

(5)

0

)()(

)()(

1111

1111

cba

cc

bbcccLVbbbLV

cLVbLV

cc

aacccLVaaaLV

cLVaLV

iii

dt

diL

dt

diLiRRiRR

dt

d

dt

d

dt

diL

dt

diLiRRiRR

dt

d

dt

d

(6)

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022222

2222

222

2222

2222

2222

dt

diLiR

dt

diLiR

dt

diLiR

iRiRiRdt

d

dt

d

dt

d

dt

diLiRiR

dt

d

dt

diLiRiR

dt

d

dt

diLiRiR

dt

d

cccc

bbbb

aaaa

cCWcCWbCWbCWaCWaCWcCWbCWaCW

cccccCWcCW

cCW

bbbbbCWbCW

bCW

aaaaaCWaCW

aCW

(7)

where: - magnetic flux linkage with winding dependent on subscript, a, b, c - low

and middle voltage phases, A, B, C - high voltage phases, LV - low voltage windings,

HV - high voltage windings, CW - stabilizing windings, R - resistance of windings,

L - inductance of windings, Ra1, Rb1, Rc1, Ra2, Rb2, Rc2 - loads resistance of the LV windings

and CW windings, La1, Lb1, Lc1, La2, Lb2, Lc2 - loads inductance of the LV windings and CW

windings.

Magnetic core of the transformer is made of ET6 laminations. Ratio of the yoke to

main limb surfaces in our model is equal 0.58. Surfaces of the extreme yokes and limbs are

reduced to 0.43 of the corresponding main limb.

Rated power of the transformer is 250 MVA and for stabilizing winding is 50 MVA.

Delta voltage of the high and low voltage windings is equal 400 kV and respectively

123 kV. Phase to phase voltage of the stabilizing winding is equal 31.5 kV.

Fig. 5. Currents in the transformer

primary windings.

Fig. 6. Time characteristic of the flux density in

the middle limb of the transformer.

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a)

b)

Fig. 7. Comparison of the flux density distribution in the core of the transformer at different

time steps: a) t=0.1925s, b) t=0.1955s.

Obtained results enable us to tell, that unsteady state lasts until time is equal 0.3 s.

Supplying voltage of the transformer is sinusoidal with constant amplitude. Currents in theprimary windings of the transformer are also sinusoidal however after the switching in

current waveforms appear the transient terms (Fig. 5). Waveforms and distribution of the

magnetic flux density in the middle limb and transformer are shown at the Fig. 6 and Fig. 7.

From Fig. 8 and Fig. 9, we can see that waveforms of the flux density in the external limbs

and voltage on the inductance of the low voltage load are distorted and they consist the

transient terms.

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Fig. 8. Flux density in the external limb

of the transformer.

Fig. 9. Voltage on the inductance

of the low voltage load.

5. CONCLUSION

The new generation of FEM software with an integrated, user-friendly programming

environment greatly facilities Finite Element Analysis of electromagnetic systems for

tutorial and professional applications. We have to care about the geometry and shapes of

finite elements because of accuracy of the FEM.

Computational model of the transformer can be used in solution of additional winding

losses.

Model of parallel strips with ferromagnetic plate enables to obtain distribution of the

magnetic field and value of the eddy current losses.

The presented numerical model of the five limb transformer permits to determine

multidimensional calculations at the both transient and steady state. It is possible to change

the dimensions of our model e.g.: limbs, yokes and windings during calculations.

Required values of the flux density in the limbs and yokes can be obtained due tomodification of the ratio of the limbs to yokes cross-sections.

Stabilizing winding connected in delta provides to the transformer odd harmonics of

the order of 3n and permits pure waveform of the magnetic flux density in the main yokes.

Waveforms of the flux density in the external limbs and yokes are distorted.

REFERENCES

[1] S. Bolkowski, M. Stabrowski, J. Skoczylas, J. Sroka, J. Sikora, S. Wincenciak,

Komputerowe metody analizy pola elektromagnetycznego, Warszawa, WNT,1993.

[2] D. S. Burnett, Finite Element Analysis, From Concepts to Applications, Addison-

Wesley Publishing Company, 1988.

[3] Students' QuickField - Finite Element Analysis System, V. 3.4 - User's Guide, Tera

Analysis, 1995.

[4] FLUX2D V. 7.20, Users Guide, CEDRAT, August 1996.