Calculating Leakage Inductances of Split-Windings in Dry-Type Traction Transformers 17

Calculating Leakage Inductances ofSplit-Windings in Dry-Type Traction

Transformers

Davood Azizian1 , Mehdi Vakilian2 , Jawad Faiz3 , and Mehdi Bigdeli4 , Non-members

ABSTRACT

Calculating leakage inductances between windingsof multi-winding traction transformers is one of themost important schemes in design procedure. Inthis paper, a new analytical method and finite el-ement models have been introduced for electromag-netic modeling of traction transformers with four splitwindings. Using these methods, the leakage induc-tances between the split windings are calculated andthe methods have been validated with the help ofexperimental data gathered from a typical 4000kVAcast-resin dry-type transformer. Some reformationsare employed to improve the finite element resultsand effects of these reformations on accuracy of themodels are studied. Employing the validated models,the effects of middle yoke on the leakage inductancesof split-winding traction transformer are discussed inthis paper.

Keywords: Traction, Dry-Type Transformer, SplitWindings, Leakage Inductances, Analytic, Finite El-ement

1. INTRODUCTION

Along with the rapid development of subways andtraction networks, the converter systems have cometrue and widely used in power systems. The mainpart of these converter systems is its transformer,called as “traction transformer”. In order to protectagainst flame and explosion, nowadays traction trans-formers are manufactured as dry-type transformers.Fig.1 shows a converter system and its related cast-resin dry-type transformer. As shown in Fig. 1, thistype of transformer is usually manufactured from foursplit windings in which the high voltage windings are

Manuscript received on July 25, 2011 ; revised on October11, 2011.1 The author is with the Department of Electrical Engineer-

ing, Science and Research Branch, Islamic Azad University,Tehran, Iran, E-mail: d.azizian@srbiau.ac.ir2 The author is with the Sharif University of Technology,

Tehran, Iran, E-mail: vakilian@sharif.edu3 The author is with the Center of Excellence on Applied

Electromagnetic Systems, School of Electrical and ComputerEngineering, University College of Engineering, University ofTehran, Tehran, Iran, E-mail: jfaiz@ut.ac.ir4 The author is with the Department of Electrical Engineer-

ing, Science and Research Branch, Islamic Azad University,Tehran, Iran, E-mail: bigdeli aznu@yahoo.com

(A)

(B)

Fig.1: A) Schematic diagram of a 12-pulse con-verter in a traction system, B) traction transformerwith pancake windings

in parallel. As it is known, calculating leakage induc-tances of windings is an important factor in designprocedure of multi-winding traction transformers [1-4]. Leakage inductances of a concentric three-windingtransformer are discussed in [5]. Many different meth-ods have been presented for inductance calculationsin transformers with concentric and pancake wind-ings [6-10]. Authors introduce an analytic and asemi-analytic method for electromagnetic modelingof traction transformer in [11]. But there is a needfor a simple and fast method for leakage inductancecalculations in the split-winding traction transformerwith both concentric and pancake windings.

Finite element is an accurate method but when ashorter computation time is essential it is not pre-ferred and it is needed to introduce a fast and precisemethod. Thus in this paper, a new and simple an-alytic method is introduced for leakage inductancecalculations in the split-winding traction transformer

18 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.10, NO.1 February 2012

and using a typical 4000kVA dry-type traction trans-former, the results are validated with the help of finiteelement and experimental results. In the followingit is shown that the introduced analytic method is apowerful and accurate method that this is in additionof its small computation time. Afterwards, employ-ing the presented methods, the most effective param-eters on the leakage inductances of a traction trans-former are studied. Finally a refinement is proposedin the geometry of the split-winding traction trans-former and its effects on the leakage inductances arediscussed. It is shown that how the geometry can betailored to achieve the suitable leakage inductancesbetween split windings by inserting a middle yoke be-tween top and bottom windings.

2. ELECTROMAGNETIC MODELING INA SPLIT-WINDING TRANSFORMER

Fig.2 shows a scheme of one leg in a four-windingcast-resin dry-type transformer.

Fig.2: Schematic view of one leg in a four-windingdry-type Traction transformer

In order to measuring leakage inductance in a pairof windings in multi-winding transformer (Fig.2), itis convenient to use test circuit given in the Fig.3.

Fig.3: Circuit for measuring leakage inductance ofwinding i and j

If the nominal currents applied to the windings iand j (Ii for winding i, -Ij for winding j and zero for theother two windings), the magnetic vector potential isrelated to the leakage flux and can be calculated bysolving the Poisson’s equation as given in (1) [12].

∇2A− µε∂2A

∂t2= µ · J (1)

where,A: magnetic vector potential,J: current density,ε: permittivity, andµ: permeability.

Consequently, leakage inductance (seen from sidej) can be calculated, employing (2).

Lijl =

2×W ijm

I2j(2)

where, Wijm is the total energy stored in the magnetic

field of Fig.3, and it can be determined as shown in(3).

W ijm =

∫vol

A · Jdv (3)

To solve the partial differential equation (1) is veryintricate. Therefore, simplifying procedures shouldbe employed. In the following, two methods are in-troduced for solving equation (1) and calculating theleakage inductances of the multi-winding transformer.

2.1 Analytical Method

Consider a pair of windings as shown in Fig.4. Cal-culating the leakage inductance of these windings di-rectly, is very intricate. To simplifying the inductancecalculation, the leakage inductance of windings shownin Fig.4, can be subdivided to two components (theaxial and the radial) as presented in (4).

Lijl =

√L2lr + L2

lz (4)

where, Llr and Llz are the inductances related to leak-age flux in radial and axial direction.

Fig.4: Leakage flux density in axial and radial di-rections

To calculate Llz, the windings can be assumed asconcentric windings with equal heights (Fig.5).

Calculating Leakage Inductances of Split-Windings in Dry-Type Traction Transformers 19

Fig.5: Concentric windings for inductance calcula-tion due to the axial flux

It can be assumed that the leakage flux in Fig.5 hasonly one component in the axial direction. Thereforeusing the relation in [13], the leakage inductance ofthose windings can be determined as follows:

Llz=µ0πN

2

Hm/Krog−a

(B1Dm1

3+B2Dm2

3+BgDmg

)(5)

where, Krog-a is the Rogofsky factor that is relatedto flux fringing in the top and bottom of the windingsand it can be calculated as below [13]:

Krog−a =1− e−πHm/(B1+Bg+B2)

πHm/ (B1 +Bg +B2)(6)

Similarly, Llr can be calculated by assuming thewindings in the Fig.4 as two pancake windings withequal diameters as shown in Fig.6. It can be assumedthat the leakage flux in the Fig.6 has only one com-ponent in the radial direction and the leakage induc-tance of presented windings can be described as:

Llr =µ0πN

2Dm

Bm/Krog−r

(B1

3+

B2

3+Bg

)(7)

Where similarly, Krog−r can be calculated as

Krog−r =1− e−πBm/(H1+Hg+H2)

πBm/ (H1 +Hg +H2)(8)

Fig.6: Pancake windings with same diameter forinductance calculation due to the radial flux

2.2 Finite Element Method

Most convenient way to solve differential equationsgiven in the previous section is to use a numericalmethod. In this paper, finite element (FE) method[12] employed for electromagnetic modeling and in-ductance calculation in multi-winding traction trans-former (Fig.2). To simplify the model and to reducethe computation time of this modeling, it is suitableto model this three phase transformer as a singlephase transformer. In this paper 2D and 3D finiteelement models are used for inductance calculationof a single phase traction transformer. Fig.7 showssimplest 2D finite element model in which the core isassumed as an open leg.

Fig.7: Simple Axi-symmetric 2D FE Model

As the model shown in Fig.7 is not perfect for mod-eling the effect of core; the core model is modified asshown in the Fig.8(A).

Employing the axi-symmetric modeling, the effectof core in the 2D-F model is more than its effect in theactual core. To improve this modeling, it is better touse equivalent core model as shown in Fig.8(B) withgiven parameters in (9) and (10).

dyoke =r202r

(9)

drel.leg =r202Es

(10)

where,r0: actual radius of the yoker: radial distanceEs: width of the core window

Fig.9(A) shows the complete 3D finite elementmodel for mentioned traction transformer. To solvethe model shown in Fig.9(A), it would be very timeconsuming, however realizing the symmetry of geom-etry, it is sufficient to model only a quarter of this

20 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.10, NO.1 February 2012

(A)

(B)

Fig.8: Axi-symmetric 2D FE Model, B) full core:2D-F, C) improved core: 2D-I

transformer as it is shown in Fig.9(B). Using the finiteelement models, introduced in this section, leakage in-ductances of multi-winding traction transformer havebeen calculated in the next section.

(A) (B)

Fig.9: 3D FE models of the traction transformer-A) full model- B) quarter model

3. ANALYSIS OF INDUCTANCES IN MULTI-WINDING DRY-TYPE TRANSFORMR

Consider a typical 4000kVA dry-type tractiontransformer given in the Fig.10 and table 1 [14]. Trac-tion transformer explained in this table, had beenmodeled with the help of FEM and analytical meth-ods mentioned in the previous section and test pro-cedure given in the Fig.3 had been applied to all pairof the windings.

Fig.10: Three-Phase four-winding traction trans-former

Table 1: Conditions of test system under all type offaults

Winding HV LV1 LV2

Rated Voltage 4 MVA 2 MVA 2 MVARated Current 20000 750 750Line Current 0.1155 1.54 1.54Connection D y D

Layer Number 1 3 3

Axial Channel -3×10 1×12mm mm

Disk Number 11 1 1Radial Channel Between 9×12 mm

- -Disks 1×26 mm

Height 1230 mm1240 1240mm mm

Internal Diameter 654 mm 332 mm 929 mmExternal Diameter 799 mm 454 mm 982 mm

Es 1015Hf 1410Bs 320 mm

Fig.11 and Fig.12 show the leakage flux density forHV1-LV1 and LV1-LV2 tests.

Consequently, Fig.13 and Fig.14 show the relatedmagnetic energy for these leakage inductance tests.

Using the analytic method, leakage inductanceshave been calculated and the results are comparedto the FE and experimental results in table 2. It isseen that both 3D-FE and analytic methods are ac-curate, but as a shorter computation time is essentialin an automatic design procedure, the FE method isnot preferred and the presented analytic method is abetter choice.

Calculating Leakage Inductances of Split-Windings in Dry-Type Traction Transformers 21

Fig.11: Magnetic flux density due to LV1 -HV1 test

Fig.12: Magnetic Flux due to LV1 -LV2 test

Fig.13: Magnetic energy due to LV1-HV1 test

Fig.14: Magnetic energy due to LV1 -LV2 test

Table 2: Perunited leakage reactances of the typicaltraction transformer

As it can be seen from table 2, inductances be-tween common axis windings (i.e. LV1-LV2) are muchhigher than the leakage inductances of concentricwindings (i.e. LV1-HV1) of the same heights.

The inductances in the above table are calculatedand measured in the nominal currents. In many casesit can be assumed that these leakage inductances areconstant during the operation. But actually thesevalues will be varied according to their current. Thisphenomena that is shown in Fig.15 and Fig.16, is re-lated to nonlinearity of magnetic core. As it is clear,leakage inductance in a pair of concentric windingscan be assumed as a constant value, but the effect ofnonlinearity on leakage inductance of pancake wind-ings is considerable and it cannot be neglected. Inthe converter transformers it is important to keepthe leakage inductance of LV1-LV2 as high as it ispossible. In order to improve and increase leakage in-ductances between the axial windings in the tractiontransformer, it might be better to use a middle yoke.

Fig.17 and Fig.18 show the effect of middle yokeon the magnetic field and magnetic energy generatedin LV1-LV2 inductance test.

22 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.10, NO.1 February 2012

Fig.15: : Leakage Inductance of concentric wind-ings as a function of current

Fig.16: Leakage Inductance of pancake windings asa function of current

Table 3 compares the leakage inductances with andwithout middle yoke. As it is clear from table 3, theinductances of LV1-LV2, LV1-HV2 and LV2-HV1 in-crease rapidly with an increase in cross section ofmiddle yoke. But the middle yoke has a small andnegligible effect on inductances of LV1-HV1 and LV2-HV2.

Fig.17: Magnetic flux density due to LV1 -LV2 testin presence of middle yoke

Fig.18: Magnetic energy due to LV1 -LV2 test inpresence of middle yoke

Table 3: Effect of middle yoke on leakage induc-tances

4. CONCLUSIONS

Analytical and FEM based methods had been in-troduced in this paper for inductance calculation ofmulti-winding traction transformers. It had beenshown that FEM has better and most accurate re-sults. The accuracy of 3D model is much higher than2D models, but improved core 2D model (2D-I) hasalso a good result. Consequently, analytical methodgiven in this paper is a fast and simple method withenough accuracy for leakage inductance calculation in

Calculating Leakage Inductances of Split-Windings in Dry-Type Traction Transformers 23

multi-winding transformers and it can be used in anautomatic design procedure.

Leakage inductances of axial windings are aboutten times of leakage inductances in radial windingsand it had been shown that, adding a middle yokecan improve and increase this ratio. It is clear thatthe effect of this middle yoke on the leakage induc-tances of radial windings is not considerable in spiteof increase in leakage inductances of axial windings.

One can see that the nonlinearity has not consider-able effects on leakage inductances of concentric wind-ings, but this effect is not negligible when the leakageinductances between top and bottom windings areconsidered.

References

[1] D. Azizian, Design Schemes in TractionTransformer, Project Report, IRAN TRANS-FORMER RESEARCH INSTITUTE, 2007.

[2] D. D. Chase, A. N. Grain, “Split-Winding Trans-formers,” Electrical Eng., pp. 907-922, 1934.

[3] IEC Standard, Traction Transformers and In-ductors, IEC 310, Nov. 1991.

[4] D. Azizian, M. Vakilian, and M. Bigdeli, “Multi-winding traction transformer equivalent cir-cuit for short-circuit current determination,”20th Conference on Soft Magnetic Materials(SMM20), Greece, 2011.

[5] F. de Leon and J. A. Martinez, “DualThree-Winding Transformers Equivalent CircuitMatching Leakage Measurements,” IEEE Trans-actions on Power Delivery, Vol. 24, No. 1, 2009.

[6] J. E. Hayek, and T. J. Sobczyk, “Analytic one-dimensional design method for railways tractiontransformers,” IEEE, 2003.

[7] F. D. Leon, and A. Semlyen, “Efficient calcula-tion of elementary parameters of transformers,”IEEE Trans. Power Del., vol. 7, no.1, pp. 376-382, Jan. 1992.

[8] ] J. Conway, “Inductance calculations for non-coaxial coils using Bessel functions,” IEEETrans. Magn., vol. 43, no.3, pp.1023-1034, Mar.2007.

[9] X. Margueron, J. P. Keradec, and D. Magot,“Analytical calculation of static leakage induc-tances of HF transformers using PEEC formu-las,” IEEE Trans. Industry App., vol. 43, no.4,pp. 884-892, Jul./Aug. 2007.

[10] K. J. Binns, The Analytical and Numerical Solu-tions of Electric and Magnetic Fields, John Wi-ley & Sons, New York, 1995, pp. 70-72.

[11] D. Azizian, M. Vakilian, J. Faiz, “Analyticand semi-analytic methods for electromagneticmodelling of multi-winding traction transform-ers,” 20th Conference on Soft Magnetic Materi-als (SMM20), Greece, 2011.

[12] S. J. Salon, Finite Element Analysis of ElectricalMachines, Kluwer Academic Publishers, 1995.

[13] S. V. Kulkarni and S. A. Khaparde, Trans-former Engineering Design and Practice, MarcelDekker, Inc., May 2004.

[14] Design Sheets for 4000kVA Transformer,ITRD025, Dry-Type Transformers TechnicalOffice, IRAN TRANSFO Co.

Davood Azizian received his B.Sc.and M.Sc. degrees in electrical en-gineering from the Zanjan University.He is now a Ph.D. student in IslamicAzad University, Science and ResearchBranch, Tehran, Iran. He has workin Iran Transformer Research Institute,R&D Department and Dry-Type Tech-nical Office of Iran Transfo Company,and Abhar branch of Islamic Azad Uni-versity. His fields of interests include

electromagnetics, thermal modeling and dry-type transform-ers. Mr. Azizian is now a lecturer and faculty staff of IslamicAzad University.

Mehdi Vakilian received his BSc inelectrical engineering (1978) and MSc(1986) in electric power engineeringfrom Sharif University of Technologyin Tehran and PhD in electric powerengineering from Rensselaer Polytech-nic Institute, Troy, NY, USA in 1993.He worked with Iran Generation andTransmission Company (Tavanir) (1981-1983), and then with Iranian Ministryof Energy (1984-1985). From 1986 he

joined the Faculty of Department of Electrical Engineering ofSharif University of Technology. From 2001 to 2003 he was As-sociate Professor and also Chairman of the Department. From2003 to 2004 and next from July 2008 to December 2008 hewas on leave of study at School of Electrical Engineering & Tel.of University of New South Wales, Sydney. He has been thedirector of the committee in charge of restructuring the Elec-trical Engineering Undergraduate Education in Sharif Univer-sity of Technology. From 2007 he is working as Professor inthis department, and from March 2010 he is also the Direc-tor of a research center for Management and control of PowerSystem. His research interests are; the transient modeling ofpower system equipments, design and modeling of power trans-formers, optimum design of high voltage equipments insulation,insulation monitoring, steady state analysis of power system,optimum design of distribution systems and distribution trans-formers.

Jawad Faiz received the Ph.D. degreein Electrical Engineering from the Uni-versity of Newcastle upon Tyne, Eng-land in 1988. He is now a Professor atSchool of Electrical and Computer Engi-neering, University College of Engineer-ing, University of Tehran. He has re-ceived a number of national and inter-national awards. His teaching and re-search interests are switched reluctanceand VR motors design, design and mod-

eling of electrical machines and drives, transformer modelingand design and fault diagnosis in electrical machinery.

24 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.10, NO.1 February 2012

Mehdi Bigdeli was born in 1981 inZanjan, Iran. He received the B.Sc. de-gree in electrical engineering from IranUniversity of Science and Technology(IUST) in 2004 and M.Sc. and Ph.D. de-gree from Faculty of Engineering of Zan-jan University and Islamic Azad Univer-sity, Sciences and Research Branch, in2006 and 2012 respectively. His researchinterests are in fault detection, transientmodeling and application of transform-

ers. Dr. Bigdeli is a faculty member and director of electri-cal engineering department of Islamic Azad University, ZanjanBranch.