# Three-phase transformer representation using FEMM, and a methodology for air gap calculation

**ABSTRACT** This work deals with the representation of three-phase and three-limb transformers in two dimensions using FEMM (finite element methods on magnetics). The transformer magnetization curve is obtained using the magnetization characteristic for the silicon steel used when assembling the transformer core, associated to the air gaps inherent to the assembling process. A method for the calculation of the air gaps reluctance is presented and it is based on the transformer magnetic circuit and on the maximum value in each winding for the magnetization current waveform, such current is obtained through the no-load test. The magnetic flux distribution in the values and the effects of inherent air gaps in the magnetization current is analyzed. Through the use of post-processing procedures, the transformer leakage reactance is calculated and compared with values obtained through the experimental method. Therefore, one of the objectives, is to explain a method to insert some of the transformer assembling imperfections in the computer model.

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**ABSTRACT:**Dryformer is an oil-free power transformer that its windings are made of high voltage cross-linked polyethylene (XLPE) cables. Leakage reactance calculation is a key step in transformer design process. Special structure of Dryformer makes conventional analytical methods useless for leakage reactance calculation in this type of transformer. In this paper, an analytical method is proposed for calculating leakage reactance of Dryformer based on its winding structure. The validity of proposed model is verified by means of 3D and 2D finite element models.01/2011; - W. Neves, F. J. A. Baltar, A. J. P. Rosentino, E. Saraiva, A. C. Delaiba, R. Guimaraes, M. Lynce, J. C. De Oliveira[Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents the mechanical effect on transformers when subjected to one of the power quality indicators, i.e., the electromagnetic transients. A common reason for internal faults in transformers windings is the weak insulation. This effect can be caused and accelerated by the electromechanical efforts produced by the inrush and short-circuit currents. Even though the transformer energizing is considered a normal operation, it produces high currents and this phenomenon significantly reduces the transformer life expectancy and may even lead to its instantaneous or timing destruction. Moreover, taking into account the same magnitude of current, the inrush currents can produce greater forces than those caused by short-circuit currents. Therefore, this papers aims to present and compare the electromechanical effects produced by the inrush and short-circuit currents in the transformers windings. To conduct this research, FLUX software in its 3D version, based on the finite element method (FEM) will be used. To highlight the overall model and the software performance, a laboratory 15 kVA transformer is utilized. It has been built with concentric double-layer windings and ferromagnetic core with three columns.01/2011;

Page 1

Proceedings of the 2008 International Conference on Electrical Machines

Three-Phase Transformer Representation Using

FEMM, and a Methodology for Air Gap Calculation

Elise Saraiva*, MSc; Marcelo L. R. Chaves, Dr.; José R. Camacho, PhD

School of Electrical Engineering, Universidade Federal de Uberlândia;

Av. João Naves de Ávila, 2121; Campos Santa Mônica; Uberlândia; Minas Gerais, Brazil

Tel: (34)-3239-4774, fax: (34)-3239-4704

*e-mail: elise.saraiva@yahoo.com.br

Abstract- This work deals with the representation of three-

phase and three-limb transformers in two dimensions using

FEMM (Finite Element Methods on Magnetics). The transformer

magnetization curve is obtained using the magnetization

characterisitic for the silicon steel used when assembling the

transfornmer core, associated to the air gaps inherent to the

assembling process. A method for the calculation of the air gaps

reluctance is presented and it is based on the transformer

magnetic circuit and on the maximum value in each winding for

the magnetization current waveform, such current is obtained

through the no-load test. The magnetic flux distribution in the

limb and yoke segments of the core are compared with design

values and the effects of inherent airgaps in the magnetization

current is analized. Through the use of post-processing

procedures, the transformer leakage reactance is calculated and

compared with values obtained through the experimental method.

Therefore, one of the objectives, is to explain a method to insert

some of the transformer assembling imperfections in the

computer model.

Paper ID 1032

978-1-4244-1736-0/08/$25.00 ©2008 IEEE

1

I.

INTRODUCTION

The behavioral study of electrical machines, generically, is

made through models based in the equivalent circuit [1].

Precision of results are reliant upon how the equivalent circuit

parameters are obtained. Most of the time, such parameters are

obtained through standard tests for each kind of machine, and

they take in consideration that the machine is symmetrical.

The leakage reactance of a transformer and the core

magnetization curve are essential for the model, mainly when

the study is based on the magnetic flux distribution of the

whole machine. In such cases, the effects of winding and core

symmetry must be taken in consideration [2]. Therefore the

computation of leakage reactance and the determination of

magnetization curve through various analytical and numerical

methods don’t have the required precision, especially when it

is a three-limb transformer core [3]. In this kind of transformer

the magnitude of the magnetization current in each of the

phases are not the same and generically the radial and axial

dimensions of high and low voltage windings are different [4].

Added to this fact, it is difficult to make tests for the

determination of the transformer magnetization curve in its

higher saturation levels, as such the ones that happen during

the transformer energization process.

In this paper is presented a technique for the representation

of three-phase and three-limb transformers with the aid of

FEMM [5], free-source software that is widely used by the

academic community to simulate electromagnetic phenomena

in a two-dimension environment.

The computation methodology is the finite elements method

and it is structured in three stages: pre-processing, processing

and post-processing [6]. The main characteristic of this

technique here presented is the use of a silicon steel

magnetization curve used in the core, associated to small air

gaps, that comes from the assembling joints in the silicon steel

sheets when mounting the core. The idea consists into the

establishment of the possibility of modeling the existence of

such air gaps due to the assembling and to distribute the effect

along the transformer core. From such constraints, to build a

magnetic circuit formed by magnetomotive sources and

reluctances of silicon steel parts in the core, and the air gaps.

Reluctances of silicon steel parts are obtained from the

expected flux, magnetic characteristics of the silicon steel

sheets and magnetic core geometrical dimensions. Reluctances

associated to air gaps are related to the design peculiarities and

core assembling aspects for each transformer. In such a way,

those characteristics must be computed after the transformer

assembling, taking as a basis the maximum absolute value in

the magnetization current waveform in each winding, obtained

from the no-load test. Depending upon the nature of the

problem, the air gap reluctances can be associated in order to

make the computation simpler and to reduce the computational

effort.

The FEMM program does the transformer representation in

two dimensions only, so it has some limitations for the

representation of coils and limbs in cylindrical shapes along

the third dimension (iron core depth), some adaptation is

necessary in the geometric dimensions for the results to be in

agreement with laboratory experiments.

II. TRANSFORMER CHARACTERISTICS

In this work will be used a transformer specially built for

laboratory tests, with characteristics typical of a distribution

transformer. This transformer has similar primary and

secondary windings, being both low voltage windings, to make

easier laboratory measurements. On the other hand, the

assembling characteristics of such coils follow the typical

standards for the low and high voltage windings of a

distribution transformer. Table I presents other significant

transformer characteristics.

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Proceedings of the 2008 International Conference on Electrical Machines

2

TABLE I

TRANSFORMER CHARACTERISTICS

Transformer power

Voltage

Connection type

Copper wire dimensions

Number of turns

15 kVA

Wind. ext. & int. 220 V

Wind. ext. & int. wye

Wind. ext. & int. 3.5x4.5 mm

Wind. ext. & int. 66

limb

yoke

limb

yoke

limb

yoke

limb

yoke

3.47 %

0.0049996 m2

0.0052826 m2

0.0047496 m2

0.0050185 m2

0.080 m

0.066 m

1.55 Tesla

1.44 Tesla

Aparent cross section area

Net cross section area

Width

Magnetic flux density

Impedance in percent

Figure 1 shows the upper view of the transformer core and

windings, Figure 2 shows de front view, and Figure 3 shows

the B-H curve for the silicon steel sheet used in the assembling

of the core (provided by ACESITA). Such figures show in

detail some core and winding sizes, those dimensions will be

of extreme importance for the transformer modeling using

FEMM.

outer winding

80

13

9.5

inner winding

air space between windings

Ø85

106

132

151

Ø

core

87

Ø

Ø

Ø

winding and the core

air space between the inner

9.5

80.04

Fig. 1. An internal upper view of the transformer (dimensions in millimeters).

66

260

66

80 8380

163

8380

Fig. 2. Transformer core front view (dimensions in millimeters).

In Figures 1 and 2 can be observed that the limbs and the

windings have the cylindrical shape, and this format along the

third dimension can not be accurately represented in FEMM.

Hence, some geometrical changes are necessary to keep the

same area for the core and windings.

Magnetization Curve

0

5

10

15

20

25

0,11101001000

Magnetic Field - H [Oe]

Magnetic Induction - B [KG]

Grade : E-005

Thickness: 0,30 mm

Frequency: 60 Hz

Fig. 3. Magnetization characteristic for the silicon steel sheets E005 provided

by ACESITA.

In Table II can be seen the peak values for the

magnetization currents in each of the three phases, obtained

through the no-load test, with the wye, without earth,

connection.

TABLE II

PEAK VALUE FOR THE MAGNETIZATION CURRENT

IA [A] IB [A]

2.34 1.54

IC [A]

2.34

III. TRANSFORMER MODELING USING FEMM

A. Magnetic core considerations

The FEMM program does electromagnetic simulations in

two dimensions only, and is not suitable for the correct

representation of the transformer limbs (three teeth cross) and

windings (cylinder) in its original shape, due to the fact that all

the geometric representation is made using horizontal and

vertical dimensions for a given depth. In this case it is used as a

depth the height of the pack of magnetic sheets from which the

core is made, it is equivalent to 80.04 mm as shown in Figure

1.

Therefore, the limb magnetic section will be represented in

the rectangular form with equivalent area. So, the width used

for the limb will be 62.46 mm. This modification is made in

such a way not to change the average length of the magnetic

circuit.

Making such adjustments and inserting in FEMM all the

necessary dimensions shown in Figure 2, as well as the

magnetization curve for the iron core, taking in consideration

the lamination fill factor as 0.95, a simulation is made taking in

consideration the no-load transformer with normal excitation

(at nominal voltage), at the instant where the current in phase

A is at its peak value. In this simulation, currents at phases B

and C were adjusted in such a way that the magnetic flux

distribution imposed by the applied voltage is obtained. In this

case the values used for the current in the simulation are

presented in Table III.

Page 3

Proceedings of the 2008 International Conference on Electrical Machines

3

TABLE III

CURRENT IN PHASES A, B AND C, USED IN THE SIMULATION

IA [A] IB [A]

2.340 1.122

IC [A]

1.218

The simulation result is shown in Figure 4, which presents

the core magnetic flux distribution in the transformer. It can be

verified that the values of magnetic flux density in the limb and

yoke relatively to phase A are, respectively, 1.73 T and 1.64 T.

Fig. 4. Results obtained for a 62.46 mm limb width.

It can be observed that the values are not the expected, 1.55

T for the limb and 1.44 T for the yoke. Such discrepancy can

be justified by the existence of small air gaps that take place

during the core assembling. Such air gaps, in spite of being

small, are significative in the composition of the necessary

magnetomotive force for the magnetic flux, and they were

responsible for the no-load current imposition.

B. Air gap insertion

For the insertion of air gaps in the model, it is necessary to

know the size of them. However, this is data is not given by the

manufacturer and is impossible to be measured. In this way,

the option was to compute such values through calculations

properly based, having as a basis the magnetization current

peak value.

To carry out such computation it will be considered that the

transformer is connected to a three-phase symmetrical voltage

source, thus, the flux distribution relative to the instant in

which the magnetic flux in the central limb reaches its peak

value can be seen in Figure 5. The condition of symmetry for

the voltage source imposes that when the flux assumes its peak

value in one limb (central), the flux in the other two limbs will

match the half of the value and in the opposite direction.

The air gaps that may be arising from the assembling of

sheets can be seen in Figure 5, which are represented by its

respective magnetic reluctances (ℜ1, ℜ2, ℜ3, ℜ4, ℜ5, ℜ6 e ℜ7).

In this figure is observed the air gaps of superior and inferior

yokes in the same side of the magnetic core, having as a

reference the central limb, are always in series with the air gap

of the external limb at the same side. This allows their

association in a unique air gap. In this way, with the purpose of

computation, it can be considered the magnetic core composed

by three air gaps: the first relative to the left external limb and

superior and inferior yokes at the left side denoted by (ℜe), the

second related to the central limb (ℜc), and the third related to

the right external limb and superior and inferior yokes at the

right side (ℜd).

Fig. 5. Magnetic flux distribution for the no-load transformer, with the flux

peak value at the central limb.

The air gaps can be all with different sizes since that their

dimensions rely on specific details of the steel sheets cutting

process and their assembling in the core composition. Such air

gaps have direct influence in the transformer magnetization

current in each phase; this is the reason why they are

determined based in the values of such magnitudes.

The non-linearity, for the magnetic circuit, associated to the

hysteresis cycle define the distorted waveform for the

magnetization current, although, can be shown that the peak

value for the magnetization current match the peak value for

the magnetic flux in the same winding. However, the flux

distribution shown in Figure 5 requires the peak value for the

magnetization current at the central limb winding, but at the

windings in the other two limbs, the condition of half of the

flux peak doesn’t imply in half of the peak value for the

magnetization current. Other important issue to be taken into

consideration is that the hysteresis cycle is not being taken in

consideration by FEMM, consequently, the measured current

waveform can not be the same used in the simulation. Hence,

only the magnetization current peak values measured at the

windings are used.

Different values of magnetic flux lead to different values of

core reluctance. These values can be easily computed through

the use the magnetic material B-H curve.

Considering IA, IB, e IC, the transformer magnetization

current peak values in the windings respectively of left

external, center and right external limbs, a system of equations

can be written for the magnetomotive forces considering the

instants in which the magnetic flux has its peak value in each

limb.

1) Condition 1: Maximum flux at the central limb.

Figure 6 pictures the transformer equivalent magnetic circuit

at the instant which the flux reaches its peak value at the

central limb (phase B). This situation has been depicted in

Page 4

Proceedings of the 2008 International Conference on Electrical Machines

4

Figure 5 and the flux in the central limb is symmetrically split

to the external limbs imposed by the voltage source.

2

e

N i

x

1B

N I

x

c

Ø

Ø

2

N i

x

2

d

Ø

L2

L1

L2

Y2 Y2

Y2Y2

Fig. 6. Transformer equivalent magnetic circuit for the no-load condition, at

the instant of maximum flux at the central limb.

Where:

φ

N

ℜL1 e ℜ Y1

– Magnetic flux [Wb];

– Winding number of turns;

– Limb and yoke reluctances for the flux peak

value (φ ) [H-1];

– Limb and yoke reluctances for the half of the

flux peak value (φ /2) [H-1];

– Transformer magnetization current peak

value in the central limb winding [A];

– Instantaneous values for phases A and C

currents, at the instant which the current at

phase B is a maximum [A];

– Right, left and center limb air gap

reluctances, respectively [H-1];

According Figure 6, and with the analysis of the closed

circuit formed by the center and left limbs, it can be obtained

the following equation:

ℜL2 e ℜY2

IB

i1 e i2

ℜd, ℜe ,ℜc

B1 L1 L2

Y2ec

NI Ni

+

2

+ ℜ ×

2

φ

φ

φ

φφ

×+× = ℜ× + ℜ×

+ ℜ×+ ℜ ×

(1)

With the knowledge that the current at phase B (IB) is

equivalent to the algebraic sum of currents in the other two

phases (i1 e i2), condition imposed by the inexistence of a

neutral connection, the following relationship is them true:

iI -i

=

With the analysis of the closed circuit formed by the right

and center limbs in Figure 6, it is obtained equation (3), in

which the value of i2 is replaced by equation (2).

2B1

(2)

B1 L1L2 Y2

cd

2NI Ni

+ ℜ

2

2

φ

φφ

φ

φ

××−×= ℜ× + ℜ××+

+ ℜ ×+ ℜ ×

(3)

Adding equations (1) and (3) the following equation is

obtained:

B L1L2 Y2e

cd

3 N I

×

2 2

φ

+ ×ℜ × +ℜ ×

2

2

+ ×ℜ × +ℜ ×

2

φ

φφ

φ

φ

×= ×ℜ × +ℜ ×+

(4)

2) Condition 2: Maximum flux at the left external limb.

With the same reasoning of Condition 1, only that now is

considered that the flux reaches its maximum in the left

external limb (phase A), them equation (5) is obtained.

3 NI2

φ

××= ×ℜ × +ℜ × + ×ℜ ×

AL1L2 Y1

φ

Y2ecd

4

2

φ

+ℜ ×

22

φφ

φ

φ

+ℜ × + ×ℜ ×+ℜ ×

(5)

3) Condition 3: Maximum flux at the right external limb.

Using the same procedure applied in Conditions 1 and 2,

only with the fact that the flux reaches its peak value in the

right external limb (phase C), this in this case equation (6) is

obtained.

3 N I2

φ

φ

φ

+ℜ × +ℜ × +ℜ × + ×ℜ ×

CL1L2Y1

Y2ecd

4

φ

2

22

φφ

φ

×× = ×ℜ × +ℜ × + ×ℜ ×

(6)

The equation system as a function of the magnetization

currents in each phase, fluxes and reluctances of limbs and

yokes and for the air gaps in the core is them computed

through the use of equations (4), (5) and (6). Solving the three

equation system it is possible to acquire the values of air gap

reluctances, as shown in equations (7), (8) and (9):

()

eABC

5 III

39

162

+×ℜ + ×ℜ

(

)

L1L2

Y1Y2

N

×

1

63

φ

ℜ =××−−−××ℜ+ ×ℜ+

(7)

()(

)

cABCL1L2

Y1Y2

N

×

1

9

I5 I

+ ×

I63

3

88

φ

ℜ =× −−−××ℜ+ ×ℜ+

+ ×ℜ − ×ℜ

(8)

()(

)

dABCL1L2

Y1Y2

N

×

1

9

II5 I

+ ×

63

3

162

φ

×ℜ + ×ℜ

ℜ = × −−−××ℜ+ ×ℜ+

+

(9)

4) Routine for the determination of magnetic material

reluctance in the condition of flux peak value.

The magnetic flux and the magnetic induction can be

computed respectively by equations (10) and (11).

V

×

44. 4fN

×

=

φ

(10)

m

B

A

φ

=

(11)

Where:

V

φ – Magnetic flux (peak) [Wb];

N – Turns number;

F

– Frequency [Hz];

B

– Magnetic induction [T];

Am – Transversal cross section area for the ferromagnetic

core [m2].

– Winding voltage rms value [V]

Page 5

Proceedings of the 2008 International Conference on Electrical Machines

5

Once obtained the value of B, the magnetic field H can

determined trough the material B against H curve. Thus, the

permeability can be computed through equation (12):

B

H

where:

µ

– Absolute magnetic permeability [H/m];

H – Magnetic field [A/m].

Equation (13) is used to estimate limb (ℜL) and yoke (ℜY)

reluctances.

l

A

µ

×

Where:

ℜ

– ℜL1, ℜL2, ℜY1 or ℜY2 reluctances [H-1];

l – Average length of limb or yoke [m];

A – Transversal section for the limb or yoke [m2].

With the substitution of ℜL1, ℜL2, ℜY1 and ℜY2 in equations

(7), (8) and (9), it is possible to obtain reluctances ℜe, ℜc and

ℜd.

The air gap lengths can be estimated through expression (13)

using the air permeability as µo = 4π10-7H/m.

TABLE IV

VALUES OBTAINED FOR THE CALCULATION OF AIR GAP THICKNESS

Variable

φ

µL (Bmáx)

µL (Bmáx/2)

µY (Bmáx)

µY (Bmáx/2)

ℜL1

ℜL2

ℜY1

ℜY2

C. Air gap calculation

The computation methodology was presented in the previous

section, so, in this section will be shown only the values for

each of the parameters used to estimate the thickness of each

air gap. Table IV shows such figures.

Once determined the reactance of limbs and yokes for the

two flux conditions, it will be possible to compute reluctances

ℜe, ℜc and ℜd through equations (7), (8) and (9), obtaining in

this case:

=ℜ

µ =

(12)

ℜ =

(13)

Rating

0.007223 Wb

0.0303 H/m

0.0388 H/m

0.0366 H/m

0.0385 H/m

1,806.65 H-1

1,410.86 H-1

887.43 H-1

843.63 H-1

-1

e

H 20,294.19

-1

c

H 7,547.36

=ℜ

-1

d

H 2,0294.19

=ℜ

Using equation (13), and also the figures already known, the

values of air gap thickness are obtained, as shown below.

ll

de

. 0

==

lc

045 . 0

=

As expected, the air gaps are very thin, almost insignificant,

when compared with the length of limbs and yokes, due to this,

they are generally neglected. However, reluctances of such air

mm 12

mm

gaps can be larger than the reluctances of the magnetic parts;

thus, the magnetomotive force required to establish the

magnetic flux in the air gap is larger.

When implementing such air gaps in the transformer model

in FEMM, Figure 7 is obtained after running a simulation. In

this figure it is possible to note a reduction in the values of

magnetic flux density. The flux density at the limb and yokes

dropped respectively to 1.54 T and 1.46 T. Thus, it is a truth

that the air gaps are important for the establishment of the real

transformer core magnetization characteristic.

Fig. 7. Transformer simulation in FEMM with the insertion of air gaps.

Beyond the consideration for the current peak at phase A,

simulations were made considering instants in which currents

in phases B and C have their respective peak values. In all the

simulated cases it was observed that the magnetic flux density

for the limbs and yokes, for the maximum flux condition,

where very close of expected values, 1.55 T and 1.44 T for

limbs and yokes respectively.

D. Calculation of leakage reactance

Leakage reactance can be estimated using the same magnetic

structure already incorporated at the FEMM program [1, 7].

The recommendation is to use the total energy stored in the

volume of air represented by the core window around the coils,

when it is applied the same magnetomotive force (mmf) in

both windings (primary and secondary). In this situation the

magnetization is negligible. Such energy can be obtained by

the post-processing phase, considering the volume defined by

the window area and the core depth [8]. In this case, the

following adjustment should be made: divide the calculated

energy by the core depth and multiply by the length of the

average core window circunference. The leakage inductance

can be calculated through the use of equation (14).

2 w

L

I

Where:

L – Leakage inductance [H];

w – Total energy stored in the air volume [J];

I – Winding current [A].

2

×

=

(14)

Page 6

Proceedings of the 2008 International Conference on Electrical Machines

6

In the case under investigation, it was applied the rated

current in each phase individually, in both windings. Table V

shows the values for the leakage reactance found for each of

the phases. Must be highlighted that the value obtained for the

short-circuit test (load test) is 3.47%, and the average value

obtained in the simulation with FEMM is 3.5345%, which

gives us a percent difference of 0.065%.

TABLE V

PERCENT LEAKAGE REACTANCE OBTAINED THROUGH THE SIMULATION

USING FEMM

XA [%] XB [%]

3.5341 3.5347

XC [%]

3.5348

IV. CONCLUSIONS

This work highlights the use of air gaps arising from the

assembling procedures for the magnetic steel sheets in the

composition of the transformer core. Such air gaps with the

companion of the silicon steel sheet magnetization curve given

by the manufacturer makes dispensable the setting up of the

transformer magnetization, which some times is unworkable.

The air gaps reluctances and, consequently, their lengths,

are estimated as a function of magnetic material reluctances for

the limb and yokes and magnetization current peak values,

which must be obtained from the no-load current waveform.

The core magnetic flux distribution in its various instants

can be determined by imposing the current instantaneous

values in the windings, which can be obtained through

measurements or time domain simulations.

It is possible to verify through expressions (4), (5) and (6)

that if the mentioned air gaps did not exist the magnetization

current peak value will be approximately 15% of the measured

value. The meaning is: the air gaps, although small, are

responsible for the larger part of the magnetization current.

Once configured the transformer core and the average

diameter of coils in the FEMM program, many variables can

be found in the post-processing stage, such as the leakage

reactance, mechanical stress due to magnetic generated forces

in the windings and others. It is clear that for the leakage

reactance a very close value to that measured in the short-

circuit test was obtained.

ACKNOWLEDGMENTS

We would like to thank the monetary assistance from the

Brazilian Ministry of Education by way of its grant agency

CAPES – Coordenação de Aperfeiçoamento de Pessoal de

Nível Superior, and the institutional support of Universidade

Federal de Uberlândia, School of Electrical Engineering

through its Electrical Systems Dynamics Group and Rural

Electricity and Alternative Energy Sources Laboratory.

REFERENCES

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International Conference on Publication Electrical Machines and

Systems, 2005. (ICEMS 2005). pgs: 1725- 1730 - Vol. 3.

[2] WATERS, M. - “The Short-Circuit Strength of Power Transformers”

5. ed. [S.l.]: Macdonald & Co., London, 1966.

[3] FUCHS, E.F.; YOU, Y. - “Measurement of λ-i Characteristics of

Asymmetric Three-Phase Transformers and Their Applications” IEEE

Transactions on Power Delivery, Vol 17, Nº 4, October 2002.

[4] HEATHCOTE, J. M. - “J and P Transformer Book” 12. ed. [S.l.]:

Elsevier Science Ltd, Oxford, 1998.

[5] AZEVEDO, A.C.; REZENDE, I.; DELAIBA, A.C.; OLIVEIRA, J.C.;

CARVALHO, B.C.; BRONZEADO, H.C. - “Investigation of

Transformer Electromagnetic Forces Caused by External Faults Using

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