Abstract-- This paper aims at presenting investigation results

on the study of electromechanical forces in transformers due to

short circuit currents. A time domain transformer model using

magnetomotive forces and magnetic reluctances approach is

described. This software allows for simulating the transformer

transient and steady state behavior regarding electric, magnetic

and mechanical variables. To highlight the method potentiality

studies are carried out using a typical power transformer

operating under distinct conditions. Due to the lack of mechanical

stress reference values in the literature, a comparative analysis is

performed by comparing the time domain approach

computational results to similar ones derived from a finite element

program.

Index Terms-- Electromagnetic force, finite element method,

short-circuit currents, transformer modeling.

I. INTRODUCTION

URING normal lifetime, transformers are submitted to a

variety of electrical, mechanical and other stresses. One

of the most critical situations is related to the occurrence of

external short circuits. Under these circumstances, high

currents will circulate throughout the transformer windings and

high internal forces are produced. This mechanism can

strongly arise as a potential source for mechanically damaging

transformers as described in [1]. Techniques to mitigate the

impacts of these forces are also considered in this reference.

According to [2], solutions to overcome the high forces

occurring at the transformer windings during short circuit

conditions depend on the material selection and careful

structural design.

D

Focusing this subject, this paper is concentrated on the

development of a time domain transformer model using

magnetomotive forces and magnetic reluctances approach to

model transformers to evaluate the referred forces. This

method shows to be a powerful mean to achieve a

comprehensive view of the overall transformer magnetic and

electrical behavior under normal and disturbance conditions.

Using commercial 15kVA three-phase transformer

parameters, computational investigations are carried out to

A. C. de Azevedo, A. C. Delaiba and J. C. de Oliveira are with Federal

University of Uberlandia, Minas Gerais, Brazil (+55-34-3239-4701, E-mails:

anaclaudia.ene@gmail.com, delaiba@ufu.br, jcoliveira@ufu.br).

B. C. Carvalho is with Federal University of Mato Grosso, Mato Grosso,

Brazil (+55-65-3615-8732, E-mail: bcc@ufmt.br).

H. de S. Bronzeado is with the Companhia Hidro Elétrica do São

Francisco -CHESF, Brazil, (+55-81-3229-4105, E-mail:

hebron@chesf.gov.br).

illustrate the method application. Due to difficulties associated

to the lack of well established and accepted reference values,

the results are compared to corresponding analytical results

extracted from a traditional and well-accepted finite element

approach. The chosen package is a very well known program

known as Finite Element Method Magnetics (FEMM) which is

dedicated to the resolution of electromagnetic problems using

2D domain [3].

II. ELECTROMAGNETIC FORCES IN TRANSFORMERS

It is known that the electromagnetic forces occurring at the

transformer windings are generated by the interaction between

current density and leakage flux density. These forces can be

calculated using (1) [4].

f J X B→ → →

= (1)

In the above equation f

is the force density vector, J

is

the current density vector and B

is the leakage flux density

vector.

Fig. 1 is illustrates a typical leakage flux distribution within

the transformer windings. The principal flux density showed in

the figure is the axial component. Near to the winding ends, the

leakage flux bends towards to the core leg so as to make

shorter its return path. Therefore, at the top and bottom ends of

the windings it is possible to recognize the existence of both

flux axial and radial components.

Br

Ba

Ba

B

Fig. 1. Transformer flux distribution for concentric windings.

By taking into account a transformer with concentric

windings, the axial component of leakage flux density (Ba)

interacts with the current in the windings, producing a radial

force (Fr). This is a well established phenomenon and it is

responsible for the mutual repulsion between the inner and

Ana C. de AZEVEDO, MSc, Antônio C. DELAIBA, DSc, José C. de OLIVEIRA, PhD, Bismarck C.

CARVALHO, DsC and Herivelto de S. BRONZEADO, MSc, IEEE Member

Transformer mechanical stress caused by

external short-circuit: a time domain approach

1

outer windings. Similarly, the radial flux component (Br)

interacts with the corresponding current, producing an axial

force (Fa) which acts towards the compression or the expansion

of the windings [2].

With the transformer operating under normal conditions, the

electromagnetic forces in the windings are small due to fact

that the currents and leakage magnetic fluxes are associated to

rated values. However, during external fault situations, the

currents and the fluxes are quite high. This produces new and

much higher electromagnetic axial a radial forces in the

windings.

As the leakage flux can be expressed as a function of the

current in the windings, the resultant force, as given in (1), will

be proportional to the squared current, independently of the

type of transformer windings arrangement [5].

Since the maximum forces are produced by three-phase

short circuits, it is usual to design transformers to withstand the

corresponding maximum peak using values derived from short-

circuited transformers connected to an infinite busbar [5]. In

this context, the equation to determine the highest short circuit

current level (Icc) is given in [6] and represented by expression

(2). The variable k is related to the asymmetry factor, Sn is the

transformer rated power in MVA, V is the rated voltage and Z

the transformer pu impedance.

62 10

3

ncc

k Si A

VZ= (2)

A. Radial Electromagnetic Forces for Concentric Windings

Template

The radial force components in a transformer with

concentric windings are calculated in accordance with the

procedures described in [6]. Fig. 2 taken from this reference

illustrates a transformer cross section with two concentric

windings. The resultant forces in the inner and outer windings

are also shown. In addition to the forces, it is also shown the

axial field density (Ba). This field is taken as constant all over

the extension comprising the outer and inner windings in their

internal area. Outside this region the flux is decreased until it

becomes null. Hence, the force, in per unit of length,

throughout the length of the coils, remains practically constant.

Ba

Distribution of

axial flux

Fr

Fr

h

Core

Fig. 2. Cross section of a transformer with concentric windings - axial field

density (Ba) and radial force (Fr).

By neglecting the flux at the winding ends, the

instantaneous ampere-turns for each winding (ni) is responsible

for producing the leakage field density (Ba) given by (3).

4

4 ( )

10a

niB T

π= (3)

This flux density will interact with the current and the

following mean radial force (Fr) will occur:

272 ( )

10mr

ni DF N

h

π −= (4)

In equation i is the current in transformer winding, n is the

number of turns, h is the length of the winding and Dm is the

winding mean diameter.

This force will act in such a way to produce a hoop stress

upon the outer windings and another effect towards the axis of

the limb belonging to the inner winding (compressive stress).

These forces are illustrated in Fig. 3.

Fig. 3. Forces producing hoop stress and compressive stress in concentric

windings.

Both, the mean hoop and compressive stress in the

concentric windings are calculated considering that the

winding has n turns with a cross section ac by (5) [6].

2

2

rmean

c

FN m

naσ = (5)

B. Axial Electromagnetic Forces for Concentric Windings

The analytical calculation of the radial leakage field, and

therefore, the axial force, is not as simple as the axial field

calculations [6]. Nevertheless, a well-accepted approach for

the above calculation is the Residual Ampere-turns Method.

This is based on the principle that any arrangement of

concentric windings can be split into two groups, each one

having balanced ampere-turns relationship. One produces the

axial field and the other the radial field [5].

Fig. 4 shows the radial field distribution and the axial forces

in an arrangement with asymmetrical windings. It can be noted

the height of the outer winding is shorter than that of the inner

winding. This asymmetry causes a great radial flux density in

the region where the imbalanced of ampere-turns occurs [7].

2

heff

Core

Br

Fa

BrFa

BrBr

Fa Fa

Radial

flux

Fig. 4. Radial field density and axial force [6].

To calculate the axial compression force by the mentioned

method, it is necessary to known the following parameters: heff

is the effective path length of the radial flux and depends on

the arrangement of taping; Br corresponds to the average radial

flux density; (1/2)a(ni) is the average value of ampere-turns,

being a the length of tap section expressed as a fraction of the

total length of the winding.

The average radial flux density is given by:

( )4

4

10 2r

eff

a niB T

h

π= (6)

To determine the axial force (Fa) for a transformer with tap

section in one end of the external winding, the following

equation is to be used:

( )2

7

2

10

ma

eff

a ni DF N

h

π π= (7)

By taking into account the physical arrangement given in

Fig. 4, the value of heff is 0.222h.

Reference [6] provides expressions to calculate

electromagnetic forces for other specific tap arrangements.

The axial force cause bending between radial spacers and in

this case the axial stress is given by (8) [7].

22

22

amean

F LN m

tbσ = (8)

where: Fax is distributed axial force - N/m

L is distance between stampings - m

b conductor axial dimension - m

t conductor radial dimension - m

III. TIME DOMAIN COMPUTATIONAL MODEL

Different approaches have been traditionally used to

computationally obtain time domain transformer models.

These methods are based on: electric equations (duality);

electric/magnetic equations and, through magnetomotive force

(mmf)/reluctance models [8]. Concerning this paper purposes,

the representation of three-phase core type transformer will be

carried throughout the mmf/reluctance model and a time

domain simulator. The choice of the mmf/reluctance method

for transformer modeling is particularly advantageous and

necessary, so it allows that the interactions between the

magnetic fluxes of different phases are taken in account, as

well as the distinct connections of the windings can easily be

accomplished. Using this software, a 100 MVA transformer

having four concentric windings per phase was implemented.

Table I shows the geometric, electric and magnetic

characteristics for the tested transformer.

TABLE I

CORE-TYPE TRANSFORMER CHARACTERISTICS: 15 KVA, THREE-PHASE

Leakage impedance 9.32%

Core loss 53.35kW

Frequency 60 Hz

VoltageOuter winding Inner winding

230 kV 138 V

Windings 2 4 1 3

Turns 756 64 394 98

Copper resistance 0.6883 Ω 0.0608 Ω 0.132 Ω 0.0438 Ω

External diameter 1290 m 1574 m 988 m 1460 m

Internal diameter 1112 m 1510 m 818 m 1410 m

Leg Yoke

Area 0.3775 m2 0.3775 m2

Medium magnetic

path length2.46 m 1.607 m

Magnetic flux density 1.61 tesla 1.614 tesla

Following the software procedures, Fig. 5 shows the three-

phase equivalent circuit.

mmF4

Ra

irR

Ra

irT

Rco

reR

Rco

reS

Rco

reT

RuRS RuST

RbRS RbST

Rle

ak

2

Rle

ak

2

Rle

ak

2

Ra

irS

mmF4 mmF4

mmF2 mmF2 mmF2

mmF1 mmF1 mmF1

mmF3 mmF3 mmF3

Rle

ak

1

Rle

ak

1

Rle

ak

1

Rle

ak

3

Rle

ak

3

Rle

ak

3

Fig. 5. Transformer equivalent electromagnetic model.

In Fig. 5: ℜcoreR, ℜcoreS and ℜcoreT are the non linear

reluctances for the left, central and right leg, respectively; ℜairR,

ℜairS and ℜairT are the linear reluctances of the air path flux

between the core and internal winding; ℜleak1, ℜleak2 and ℜleak3

represent the leakage linear reluctances associated to leakage

flux between inner and outer windings; ℜtRS, ℜtST, ℜbRS, and ℜ

bRS correspond to the non-linear reluctances associated with

flux in top and bottom yokes; mmf2 and mmf4 express the

magnetomotive forces associated to the outer windings

and;mmf1 and mmf3 are the magnetomotive forces produced

by the inner windings.

The described methodology requires the leakage fluxes for

calculating both the axial and radial electromagnetic forces. To

obtain these fluxes the corresponding magnetic area related to

the leakage effect. Fig. 6 illustrates the physical parameters

required to evaluate the necessary area (ALeak) [10].

Using the physical construction information illustrated in

Fig. 6, the required area between the outer and inner windings

can be calculated throughout (7).

3

X

X

X

X

X

X

X

X

X

X

Leakage

flux

Outer

windingInner

winding

d1 d0 d2

x

lm

y

H

i i

Fig. 6. Concentric windings physical parameters.

21 20

3 3Leak m

d dA l d m

= + +

(9)

Where: lm is the average length of the winding

circumference; d1 and d2 are the thickness of the outer and the

inner windings, respectively and; d0 is the space thickness

between windings.

IV. TIME DOMAIN AND FEMM RESULTS

The studies selected to be presented in this paper comprise

two situations concerning transformer operation: normal

functioning with rated loading and short circuit conditions.

Results obtained from the time-domain model are firstly given

and these are later compared to the Finite Element Method

Magnetics (FEMM) approach.

Case A. Time-Domain Results

The results achieved with normal operating conditions are

given in Figs. 7 and 8. Due to the fact there are a large number

of variables to be visualized; only those at the outer winding

are illustrated.

Fig. 7(a) shows the supply three-phase voltages and Fig.

7(b) the corresponding transformer magnetic flux density. The

maximum flux density value found is of 1.6 T and this is in

agreement with expected values.

0.2 0.22 0.24 0.26 0.28 0.3-200

-150

-100

-50

0

50

100

150

200x 10

3 vTvSvR

time (s)

vo

ltage

(V)

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time (s)

mag

net

ic i

nduct

ion (T

)

BTBSBR

(a) (b)

Fig.7. Transformer normal operation: (a) Supplied voltage; (b) Magnetic field

density.

Fig. 8(a) gives the load currents and Fig. 8(b) highlights the

leakage magnetic flux in the three spaces between the four

windings. The maximum value is low and it has reached a peak

value of about 194 mT.

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

-600

-400

-200

0

200

400

600

time (s)

curr

ent (A

)

iTiSiR

RMS: 416,06

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

time (s)

leak

age

mag

neti

c f

lux d

ensi

ty (

T)

Bleak_12 =195 mT

Bleak_23 =31 mTBleak_34 =20 mT

(a) (b)

Fig.8. Transformer normal operation: (a) Current waveform (b) Leakage flux

density.

Once the short circuit is established a new set of results are

obtained. Fig. 9(a) provides the currents waveform and shows

that the phenomenon occurs at 200 ms. The related leakage

flux is given in Fig. 9(b). It can be noticed that the maximum

value for this variable is 3.5 T. Again, this is within standard

values during short-circuit conditions.

0.15 0.2 0.25 0.3 -15000

-10000

-5000

0

5000

10000

time (s)

curr

ent

(A)

0.35

iR(peak)= -11061 A

iS (peak)=

8.103.3 A

i T(peak)= 8.311.2 A

0.15 0.2 0.25 0.3

time (s)

0.35-2

-1

0

1

2

3

4

leak

age

mag

net

ic f

lux d

ensi

ty (T

)

Bleak_12 =3,5 T

Bleak_34 =360 mT

Bleak_23 =920 mT

(a) (b)

Fig. 9. Short circuit condition: (a) currents waveform; (b) leakage flux density.

By knowing the time domain current, the radial force can be

founded throughout expression (4). The final result associated

to the highest current phase, for both the normal and short-

circuit conditions are given in Figs. 10 and 11, respectively for

the external and internal winding.

0.1 0.11 0.12 0.13 0.14 0.150

5000

10000

15000

20000

25000

30000

35000

40000

time (s)

rad

ial

forc

e (N

)

36397 N

0.1 0.11 0.12 0.13 0.14 0.15 0

5000

10000

15000

20000

25000

time (s)

rad

ial

forc

e (N

)

20344 N

(a) (b)

Fig.10. Radial force at normal operation. (a) external winding - 2; (b) internal

winding - 1.

0.15 0.2 0.25 0.30

2

4

6

8

10

12

14

time (s)

radia

l fo

rce

(N)

(x106)

12,3 M N

0.35 0.15 0.2 0.25 0.30

1

2

3

4

5

6

7

8

time (s)

rad

ial

forc

e (N

)

0.35

(x106)

7,0 M N

(a) (b)

4

Fig.11. Radial force in short-circuit: (a) external winding; (b) internal

winding.

A comparison between the results clearly shows that the

radial force occurring at the windings, during short circuit

situation, is increased severely with regard to the normal value.

The mechanical stresses associated to the radial force are

shown in Fig. 12 for the phase that present highest value.

0.2 0.22 0.24 0.26 0.28 0.30

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

time (s)

stre

ss (

N/c

m2)

Inner winding (1):

5927.0 N/cm2

Outer winding (2):

9172.7 N/cm2

Fig.12. Mechanical stress during short-circuit: phase R

The stress mechanical was showed only for short-circuit

condition, since for normal condition it had a small value.

Case B. FEMM Results

To verify the previous results accuracy to obtain

transformers internal forces, the solution would be a

comparison with experimental results, however, no reference

publication concerning the tested transformer or others were

found. In this way, the analysis was carried out throughout two

software performance. One being the mentioned time domain

program and the other the well accepted finite element based

FEMM. Due to the fact the latter is a commercial program no

further discussions are made in relation to its potentiality and

use. This can be found in [3].

Fig. 13 provides the magnetic flux density distribution

during normal transformer operation for a particular instant.

Fig. 13. Magnetic flux density with normal transformer conditions.

Following standard procedures, the rated load current peak

values of 355A in the HV e 590 A in the LV were injected in

the windings. As it should be expected, with normal operating

conditions, the magnetic flux stays predominantly in the

ferromagnetic material.

On the other hand, the effect of applying an external short-

circuit, represented by an injection of 6510 A in the HV and

11061 in the LV windings can be seen in Fig. 14. The flux

density pattern can be easily seen. The leakage flux has been

largely increased due to the short-circuit conditions. The short

circuit current level was determined in accordance with (2)

using an asymmetry factor of 1.6. To provide means for a

comparative analysis, the chosen instant to represent the

magnetic situation of Fig. 14 is the same as with normal

conditions.

Fig.14. Magnetic flux density with short circuit conditions.

So as to have a better understanding about the leakage flux

distribution along the winding extension, Fig. 15 the flux

behavior in relation to the physical turns dimension. By

comparing the normal to the short-circuit situations, it can be

concluded that the leakage flux of about 200 mT went up to

about 3.6 T.

0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

1 1600height winding (mm)

leak

age

flu

x d

ensi

ty (

T)

normal condition short-ciricuit

800

Fig.15. Leakage flux density against winding height: normal and short circuit

conditions.

As the required information to feed equation (4) is known,

the forces can be readily calculated. These are given in the

following section.

Table II synthesizes the main results obtained from the

Time Domain and the FEMM approach.

TABLE II

RESULTS SUMMARY WITH NORMAL AND SHORT-CIRCUIT CONDITIONS

Normal Operation

Time domain FEMM

Leakage magnetic flux density [T] 195x10-3 200x10-3

Magnetic flux density [T] 1.6 1.6

Radial force [N]

Outer winding (2) 36397 36890

Inner winding (1) 20344 20580

Short Circuit Condition

Leakage magnetic flux density [T] 3.5 3.6

Magnetic flux density [T] - 1.32

Radial force [N]

Outer winding (2) 12300x103 13470x103

Inner winding (1) 7000x103 7540x103

5

The results derived from the two methods are in good

agreement. Therefore, by taking the FEMM calculations as

reference value, the time domain software performance has

shown to be appropriated for this paper purposes, i.e, to

calculate internal forces and further estimation of equipment

mechanical stress due to short-circuit conditions.

Mechanical stress was estimated only using the time domain.

Nevertheless, if the forces calculated by two numerical

methods are in agreement, hence the mechanical stresses also

they are. The maximum value founded for this variable was

9173 N/cm2. This is closer of hoop stress is closer of the

maximum hoop stress permissible in the copper which is 9806

N/cm2.

V. CONCLUSIONS

Using mmf and reluctance time domain transformer

modeling strategy, this paper has proposed a way to calculate

mechanical stresses occurring in the windings with short-

circuit conditions. Due to the absence of real site values, the

results were compared to as well established software based on

finite elements. The calculated values have demonstrated that

the transformer performance with normal and short-circuit

conditions are in close agreement. Therefore, as for the present

comparison, the time domain approach can be assumed as a

good procedure to achieve transformer internal forces and

subsequently stress evaluation. However, it must be stressed

the investigations were carried out for radial forces and

stresses. The paper did not took into account the axial force

and nor the stresses caused for them. However, this matter will

be investigated in the next paper.

VI. REFERENCES

[1] Kulkarni, S. V., Khaparde, S. A., “Transformer Engineering - Design

and Practice”, Marcel Dekker, Inc, New York, 2004.

[2] IEEE Guide for Failure Investigation, Documentation and Analysis for

Power Transformers and Shunt Reactors, IEEE Standard C57.125, 1991.

[3] Meeker, D., (September 2006) “Finite Element Method Magnetics -

User's Manual Version 3.4” [online]. Available: http://femm.berlios.de.

[4] Yun-Qiu, T., Jing-Qiu, Q., Zi-Hong, X., “Numerical Calculation of

Short Circuit Electromagnetic Forces on the Transformer Winding”,

IEEE Transaction on Magnetic vol. 26, No.2, March, 1990.

[5] Heathcote, J. Martin, “J&P Transformer Book”, 12th ed., Oxford,

Elsevier Science Ltd, 1998.

[6] Waters, M., “The Short-Circuit Strength of Power Transformers”,

McDonald & Co. Ltd, London, 1966.

[7] The Short Circuit Performance of Power Transformers, Brochure CIGRE

WG 12.19, 2002.

[8] Apolônio, R., Oliveira, J. C., Bronzeado, H. S., Vasconcellos, A. B.,

“The Use of Saber Simulator for Three-Phase Non-Linear Magnetic

devices Simulations: Steady-State Analysis”, The 7th Brazilian Power

Electronic Conference, Fortaleza, Ceará, Brazil, September, 2003, pp

524-529.

[9] Yacamini, R., Bronzeado, H. S., “Transformer Inrush Calculations using

a Coupled Electromagnetic Model”, IEE Proc. Sci Meas. Technol. Vol.

141, no. 6, November, 1994, pp 491-498.

[10] Slemon, G. R., “Magnetoelectric Devices: Transducers, Transformers

and Machines”, 5th ed, John Wiley and Sons, 1966.

VII. BIOGRAPHIES

Ana Claudia de Azevedo was born in São Gonçalo

do Abaeté, Brazil, in 1972. She received her BSc

from the Federal University of Mato Grosso, Brazil

in 1999, and her MSc degree from the Federal

University of Uberlandia, Brazil, in 2002. She is

currently working towards her PhD degree. Her

research interest areas are: Electromechanical Stress

in Power Transformers.

Antônio Carlos Delaiba was born in Botucatu,

Brazil, in 1954. He graduated from Federal

University of Barretos, Brazil in 1979. He received

his MSc degrees from of University of São Paulo,

Brazil in 1987 and his PhD degree from the Federal

University of Uberlandia, Brazil in 1997. He is

currently a professor and a researcher at the Faculty

of Electrical Engineering, Federal University of

Uberlândia. His research interest areas are Power

Quality, Transmission, and Distribution of Energy.

José Carlos de Oliveira was born in Itajuba, MG,

Brazil. He received his BSc and MSc degrees from

the Federal University of Itajubá–Brazil, and his

PhD degree from the University of Manchester -

Institute of Science Technology – Manchester-UK.

He is currently a professor and a researcher at the

Faculty of Electrical Engineering, Federal

University of Uberlandia - Brazil. He has taught

and published in a variety of subjects related to

Electrical Power Systems and Power Quality.

Bismarck Castillo Carvalho was born in Roboré,

Santa Cruz, Bolívia, in 1957. He received his BSc

from the Federal University of Mato Grosso, Brazil

in 1981 and his MSc and PhD degrees from the

Federal University of Uberlandia, Brazil. His

research interest areas are: Electrical Power System,

Power Quality and Renewable Energy concerned to

Wind Power.

Herivelto de Souza Bronzeado (M’ 97) was born

in Remígio, Paraiba, Brazil, on April 2, 1953. He

was graduated by the Universidade Federal da

Paraíba, Brazil, in July 1975, and since then he

works for the Companhia Hidro Elétrica do São

Francisco – CHESF, being responsible for Power

Quality and R&D Projects. He received his MSc

degree in Power System Engineering, in 1993, from

the University of Aberdeen, Scotland (UK). He is

also the chairperson of the Study Committee C4 of

Cigré-Brasil and the IEEE Joint Chapter

PES/IAS/PELS, Northeast 1, Section Bahia, Brazil.

His research interests include transformer modeling

for transient studies and Power Quality problems.

6