A Practical Application of the Rabin´s Method for Inductance Calculation in Power Transformer Design

Paulo A. Pasquotto de Lima Departament of Transformer Engineering

WEG Energy Blumenau, Brazil

pasquotto@weg.net

Sérgio H.L. Cabral Departament of Elec. Engineering & Telecommunication

FURB- Fund. Universidade Regional de Blumenau Blumenau, Brazil scabral@furb.br

Abstract—The analysis of the performance of a power

transformer winding under the incidence of a voltage surge is one

of the most important concerns in the design of this equipment

that plays a fundamental role in power system. Basis of this

analysis lies in modeling power transformer winding with a

rather complex circuit composed by distributed inductances and

capacitances. However, since calculation of those capacitances is

somehow simpler than of inductances, some engineers may have

been led to believe in mistaken ideas like capacitances having

more importance than inductances. Thus, this work shows not

only how equally important inductances are as well as it presents

advantages in using a simplifying method for calculating winding

inductances. A comparison between experimental and calculation

data is presented.

Keywords-inductance; transformer; winding; voltage surge.

I. INTRODUCTION

Power transformer has a very important role in power systems by realizing providential changes of voltage levels by making viable the power system as a whole. Thus, since it is inherently submitted to high voltage stress as well as to incidence of several types of voltage surges, it is mandatory that every power transformer must withstand a minimum level of this solicitation, in accordance with its rated voltage. Therefore, power transformer designer must efficiently weigh the occurrence of critical voltage, limits of voltage withstand of dielectric material and costs. Today, this hard and complex exercise in industry invariably requires the use of computer simulation as a tool which no designer can waive. Even an experienced designer takes any decision about any design change only after checking results from a computer simulation based on a well-recommended modeling. For its turn, an electric circuit for modeling a transformer winding requires the correct calculation of four basic electric parameters: Inductance, capacitance, resistance and conductance. They may be lumped and/or distributed. Among them, inductance, including self and mutual, can be considered as the most complex and its complexity basically arises from evaluating magnetic field distribution within a volume or area. As a remarkable example, for the calculation of the inductance of

any simple and basic circuit a double integral is usually required and it often requires the use of elliptical functions [1]. On the other hand, the degree of complexity is quite lower for calculation of capacitance for that same simple circuit. Therefore, one of the aims of this work is to bring to the light a practical application of a specific and efficient method for simplifying calculation of the inductance of a power transformer winding that is not a simple circuit, but a rather complex one. This method is known as Rabin´s method [2,3] and an immediate result from its application is another important aim of this work that is to show that some ancient premises about the initial distribution of voltage surge along any power transformer winding under the incidence of a voltage surge are not correct. Especially in regard to the premise of being totally capacitive the ruling for the distribution of voltage surge along a winding at the very first instant of the incidence of the surge. With this idea, a practical case is taken for comparison between computer simulation and analytical results.

II. THE IMPORTANCE OF ANALYSIS OF TRANSFORMER

WINDING UNDER TRANSIENT STATE

The incidence of a lightning surge on any winding of a power transformers is such a frequent, harmful and thus important event that this theme has been studied by engineers ever since power transformer has been put in use [4,5]. The analysis of the behavior of a winding under the incidence of a surge is very important for industry engineers in view of better designing any power transformer. This kind of analysis requires significantly complex computational tools as well as knowledge of the very basics of the phenomenon at the same time that interest for this issue has been increasing by most of the engineers who deal with power transformer design. Some years ago, the personal experience was more intensively used for determining the insulation material and defining distances that should be applied to the design of a winding of power transformer. One of the most significant examples is in regard to the initial distribution of a voltage surge along a winding. In accordance with [5] this distribution is usually considered as being ruled by the distributed capacitances of a winding, only.

Therefore, any influence of inductances is systematically neglected in this case. In consequence, a wrong idea has become accepted for some designers of power transformer that is that from the revolution of the curve of very initial voltage surge distribution along a winding it would be possible to obtain the envelope of all the possible curves of voltage along winding and time [6]. Thus, if this would be true, there should be no need to spend time in somehow complex techniques for calculating inductances of a winding, but only capacitance should be preferentially calculated. That is why this initial voltage distribution is also called capacitive distribution.

Although the idea of the capacitive distribution was supposed as even being conservative, it is important to show that that it is not correct. In fact, right at moment of the very initial incidence of a voltage surge on a winding there is an immediate occurrence of resonant interaction among distributed inductances and capacitances that gives origin to significant differences between the real one and the capacitive voltage surge distribution along a winding. This fact shows how important is to take winding inductances into consideration even for the very initial instant, no matter how much the increase in complexity it may bring. Thus, with the aim of simplify inductance calculation of a winding this work shows the application of a method especially developed for simplifying calculation of inductances of this kind of circuit. This is known as Rabin´s method [2,3] and basis for showing its application is a practical air core winding with dimensions of a power transformer winding.

III THE RABIN´S METHOD

The Rabin´s method, or simply RM [2,3], is a method based on the evaluation of the distribution of the magnetic vector potential, A, within a given volume. It allows simplifying the evaluation of leakage of magnetic flux and respective inductance as well as forces that are quite applicable for transformer design. Since it was developed to be applied to windings its use gives a significant decrease in time for data processing if compared to the use of finite element method whereas it has a similar accuracy. Thus, all these features have taken this method to be increasingly applied for evaluating self and mutual inductances of power transformer windings, which are fundamental for analyses of transient state of these windings. Important information about this method is presented in the following sub sections.

A. General Description of Assumptions taken for applying

the MR

The very basis of the MR lies in the proposition that distribution of current density, J, along a winding axis may be represented by a Fourier series. Thus, this assumption yields an important restriction to its application that is that this method is limited to core type transformers, only [3]. Fortunately, this type is the most frequent for power transformer.

Some additional assumptions to be taken for the application of the MR are:

1) Evaluation of magnetic field is done for each of the limbs of the magnetic frame of the transformer, with no influence of any adjacent limb ;

2) Upper and lower horizontal branches of the magnetic frame as well as tank wall are considered as planes in which magnetic permeability is infinite;

3) Central column of the magnetic limb to which coils are laid around has also infinite magnetic permeability and its geometry is cylindrical and axis-symmetric, by including the tank wall;

4) Magnetic field is evaluated only within the window of the magnetic frame of the transformer;

5) Ruling equation for the evaluation of the magnetic field distribution within transformer window is based on application of set of Maxwell´s equation for magnetic vector potential, A , as in (1).

In accordance with assumption 3), above, a system of cylindircal coordinates is adopted and magnetic vector potential is considered as having only azimuthal component with dependence on radius, r, and on axial distance, z , as follows.

(1)

Where Jϕ is the current density, that has only azimuthal component. µo is the magnetic permeability of vacuum, since this value is the same for all the materials that fill the whole window space. Basically, oil, paper and metal.

B. Basics of the MR

Consider a core type power transformer of which the main part of its central limb is schematically presented in Fig. 1 through a cross section of the window of the magnetic frame.

Figure 1 – Cross section of the main part of the window of a core type transformer

ϕϕϕϕϕ µ J

z

A

r

A

r

A

rr

A⋅−=

∂

∂+−

∂

∂⋅+

∂

∂02

2

22

21

For the calculation of the magnetic vector potential that is distributed over the cross section of the part of the window, three regions are defined as:

1) Region I 1rrrC ≤≤ and jHz ≤≤0 :

It represents the inner layer of insulating material. Basically, insulating paper and oil channels.

2) Region II 21 rrr ≤≤ and jHz ≤≤0 :

It represents all the windings laid around the central limb. Basically, high and low voltage windings as well as a regulating winding, if applicable.

3) Region III ∞≤≤ rr 2 and jHz ≤≤0 :

It represents the outer layer of insulating material. Basically, a layer of insulating paper is laid on the surface of the winding whereas insulating oil fills the rest and thus most of the space of this region.

Therefore, for the regions I and III current density and thus right side of (1) is null. On the other hand, for the region II current density is not null but it presents several values that are distributed along z axis, as presented in Fig. 2. Thus, since values of current density within this region are discretized, in accordance with the very basis of the idea of Rabin the current density can be represented by a Fourier series with dependence on the axial component, z. And since Hj is the distance between horizontal yokes, the expression of the Fourier series will have 2Hj as the period [3].

Figure 2 – Distribution of current density along vertical axis, z.

Therefore, in accordance with Fig. 2 the expression for the current desity along z axis is

∑∞

=

⋅⋅⋅+= 1n j

nH

znJJzJ

πϕ cos)( 0

, (2)

for which

, (3)

and

. (4)

As a result of the application of (1) to each region, the

following equations are yielded. For the region I, the magnetic vector potential has the

following expression [3]:

, (5)

for which

, (6)

, (7)

, (8)

and

mrx = . (9)

I0, K0, I1 and K1 are modified Bessel´s functions of first and second kind with order zero and one, respectively.

For their turn, expressions of magnetic vector potential for respective remaining regions, I and III, are :

, (10)

and

, (11)

for which L1(x) is the Struve´s modified function with order one,

, (12)

, (13)

and

. (14)

( )∫ ⋅⋅=1

0

1

x

n dttKtE

( )∫ ⋅⋅=2

1

1

x

x

n dttKtC

( )( ) n

c

cn C

xK

xID ⋅=

0

0

jH

nm

π=

( )

( ) ( )( )zm

xLxKF

xIE

m

Jr

r

rrrJA

n n

n

nII ⋅⋅

⋅−⋅+

+⋅

⋅⋅+

−

⋅−

⋅⋅⋅= ∑

∞

=

cos

2362 1 11

1

20

2312

00 πµµ

( ) ( )[ ] ( )zmxKGm

Jr

r

rrJA

n

nnIII ⋅⋅⋅⋅⋅+⋅

⋅

−⋅⋅= ∑

∞

=

cos6 1

120

31

3200 µ

µ

( )( )

( ) ( )∫∫ ⋅⋅−⋅⋅⋅=22

1 0

110

0

xx

xc

cn dttItdttKt

xK

xIF

∫ ⋅⋅=jH

j

dzzJH

J0

0 )(1

ϕ

∫

⋅⋅⋅⋅=

jH

jj

n dzH

znJ

HJ

0

cos2 π

( ) ( ) ( )[ ] ( )zmxKDxICm

Jr

rrJA

n

nnnI ⋅⋅⋅+⋅⋅⋅+⋅

−⋅⋅= ∑

∞

=

cos2 1

11201200 µ

µ

axial distance, z

( )( )

( ) ( )∫∫ ⋅⋅+⋅⋅⋅=2

1

2

1

110

0

x

x

x

xc

cn dttItdttKt

xK

xIG

Thus, for the evaluation of inductance of a transformer winding it should be considered as divided into several sections. In this way, expression for the inductance is :

, (15)

for which Ap is the magnetic vector potential of the region

where lies the section p whereas Ip and Iq are currents for the respective sections p and q and Vq is the cylindrical volume related to section q. On the other hand, Jq is the current density that is considered as constant along section q and thus given by

, (16) for which Nq is the number of turns of the winding for the respective section q whereas riq and r0q are the internal radii that are applied for the evaluation of magnetic vector potential in section q.

For calculation of self inductance of each section p and q are considered as being equal to each other.

Therefore, for sections of different windings (15) and (16) gives origin to

,(17)

whereas for sections of a same winding (15) and (16) give origin to

, (18)

C. Application of the MR - Example

For showing the application of the MR for inductance calculation and its consequently positive results, part of a high voltage winding of a power transformer was taken as being the basis for a test in which the influence of the inductance has been shown as very important. This winding is a half of a typical 138 kV-15 kV (Y-∆) -31.25 MV with 42 interweaved disks divided into 21 pairs and it was submitted to a lightning impulse voltage test. Additional data of the winding as well as of the test setup apparatus are presented in [7].

Thus, for submitting this winding to a lightning impulse test some adaptation was made on it. For example, a grounded aluminum foil was internally laid on the winding for playing the role of the transformer core with its zero potential, only. On the other hand, another grounded aluminum foil was laid externally as a usual shielding of a power transformer. Fig.

2.a shows the winding before the use of the external shielding foil and part of the test setup, whereas Fig. 2.b shows the winding within the shielding.

Figure 2.a – Test setup apparatus and air core winding before the use of the external shielding foil

Figure 2.b – Test setup apparatus and air core winding after the use of the external shielding foil

Application of lightning impulse has the aim of evaluating the very initial distribution of impulsive voltage along the winding length. Thus a low voltage recurrent 1.2/50µs – 430 V impulse generator was connected to the winding with a grounded end and voltage was measured with several voltage probes connected along it. Fig. 3 shows the schematic diagram of the test setup as well some geometric data of the winding and aluminum foils.

qp

qpq

pqII

dVAJM

⋅

⋅⋅=∫

( ) ( )qqi

qq

qrrzz

INJ

012 −⋅−

⋅=

( ) ( ) ( )

⋅⋅⋅

⋅⋅+−⋅−⋅

⋅

⋅

⋅⋅= ∫∑

∞

=

oq

iq

mr

mrn

qnpn

pniqqipp

qp

qp

j

pq dttKtm

JJGrrrr

JJ

II

HM 1

14

,,,0

330

,0,00

3

µπ

( ) ( ) ( )

( ) ( ) ( )

⋅−⋅+⋅

⋅+

+

−−−⋅−−⋅

⋅

⋅⋅

⋅⋅=

∫∫∫∑∞

=

oq

iq

oq

iq

oq

iq

mr

mr

mr

mr

pn

mr

mr

pn

n

qnpn

iqq

iqqipiqpp

qp

qp

j

pq

dttLtdttKtFdttItEm

JJ

rrrrrrrr

JJ

II

HM

11,1,1

4

,,

440

0333

00,0,0

0

2

23

π

µπ

Figure 3 – Schematic diagram of the experimental apparatus for voltage application to the winding. Some distances, in mm, are also presented

Thus, for a comparison to the experimental case,

capacitances and inductances were calculated for modeling the circuit of the practical winding and a computer simulation of the application of a lightning impulsive voltage was performed.

For the calculation of inductances through the MR method a program named IND [8] was used, which generated the inductance matrix presented in Fig. 4. For the sake of simplicity and clearness the upper diagonal terms of the matrix were omitted since this matrix is diagonally symmetric. Dimension of the matrix is 21 x 21 since 21 is the number of pairs of interweaved disks.

Figure 4 – Inductance matrix of the practical winding, in mH.

In accordance with Fig. 1, since the winding taken for this test has neither tank or magnetic core, some additional premises were taken in regard to input data to IND program that uses the MR for inductance calculation. Basically, measures were taken for adapting the real geometry of the practical winding to that of Fig 1. Thus, as an example of measure, the height of the window of the magnetic frame was set to 10000 mm, that is significantly high enough for

eliminating the influence of both upper and lower horizontal yokes.

For their turn, capacitances of this same winding were calculated by specific means[8] and then both capacitance and inductance matrixes were taken as input data to a digital program TRANEM [8] for the simulation of the application of lightning impulsive voltage.

At last, Fig. 4 shows the distribution of voltage surge to the ground along the winding to the ground for the very initial instant (t = 0 +) for the experimental case, for the same case simulated by digital program TRANEM based on MR method and for the theoretical capacitive distribution.

Figure 4 – Comparison of results for three different cases : experimental, computer simulation (TRANEM / MR) and theoretical capacitive

From Fig. 4 it can be seen that capacitive distribution

really fails in predicting the very initial voltage distribution along winding since it is quite different from the experimental and thus the real one. Basically, values predicted by the capacitive distribution are lower than real values, which may cause failure in designing winding if based on this distribution. By extension, a technique of revolution of this curve for obtaining an envelope of all possible voltage curves for the following instants is definitively mistaken. On the other hand, values of distribution obtained from digital program TRANEM, that is based on MR shows a good concordance with the real distribution. This important result shows how important is to consider all inductances of a winding even for the very initial voltage surge distribution. Therefore, simplifying characteristics of the proposed method for inductance calculation are clearly welcome.

III. CONCLUSION

Inductances of any winding have been shown as being equally fundamental as its capacitances for determining the shape of the very initial voltage surge distribution along the winding length. Therefore, the misconceptions in taking only capacitive distribution as well as its extension to a technique for enveloping all possible values of transient voltage have no agreement with reality.

A possible reason for these mistaken ideas may lie in the fact that former studies about this issue used to take a theoretical step voltage as the voltage surge incident to a winding. On the other hand, real voltage surges have a double

exponential aspect that does not allow the sudden change of value as a step voltage does.

On the other hand, the suggestion of the use of a somehow old but not well-known method for inductance calculation , that is the Rabin’s method may contribute significantly for simplifying the analysis of performance of a winding under transient state.

REFERENCES

[1] F. W. Grover, Inductance Calculations: Working Formulas and Tables,

1946 & 1973. Dover Phoenix Edition 2004.

[2] L.Rabins, "Transformer Reactance Calculations with Digital Computers", AIEE Trans., Vol. 75 Pt. I, July 1956, pp. 261–267.

[3] R. M. Del Vecchio , B. Poulin , P. T. Feghali , D. M. Shah and R. Ahuja, Transformer Design Principles With Applications to Core-Form Power Transformers, Second Edition. CRC Press 2010. pp. 327-346.

[4] S.H.L Cabral, T.I.A.H Mustafa, H.A.D Almaguer, T. Santos and C.V. Nascimento, "The role of the distribution transformer in the transference of voltage surges," Transmission and Distribution Conference and

Exposition: Latin America (T&D-LA), 2010 IEEE/PES , vol., no., pp.727-731, 8-10 Nov. 2010.

[5] S.H.L. Cabral and A. Raizer, "Single-phase insulator transformers," Potentials, IEEE , vol.21, no.5, pp. 35- 38, Dec 2002/Jan 2003.

[6] A.N. Greenwood, Electrical Transients In Power Systems. 2. Ed. New York : John Wiley & Sons, 1991, pp. 327-346.

[7] S.H.L. Cabral and S.L. Bertoli, “Application of the method of residues in comparison to TLM method in a practical case. In: COMPUMAG 2009 - 17 th Conference on the computation of electromagnectic fields, p. 857-859.

[8] P. A. P. de Lima., “Estudo Análise da Distribuição de transitórios de Tensão em Enrolamentos de Transformadores Imersos em Óleo”. M.Sc. Dissertation. Universidade Regional de Blumenau, Brazil. In Portuguese, Nov. 2011. Available in https://docs.google.com/open?id=0B0OIGFb3BvMTTXRQUTFQT1BTdjIwWVFXWTdjeEtWdw

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