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Reduction of Stray Loss in Power Transformers Using HorizontalMagnetic Wall Shunts

Masood Moghaddami1, Student Member, IEEE, Arif I. Sarwat1, Senior Member, IEEEand Francisco de Leon2, Fellow, IEEE

1Department of Electrical and Computer Engineering, Florida International University, Miami, FL 33174 USA2Department of Electrical and Computer Engineering, Polytechnic Institute, New York University, Brooklyn, NY 11201 USA

The use of a horizontal arrangement of wall shunts is proposed in this paper as a cost-effective way to reduce stray losses inpower transformers. The paper compares the performance of horizontal wall shunts with the available alternative (vertical shunts).3D finite element analysis (FEA) is used for the calculation of stray losses in tank walls, and other structural parts. A novel hybridnumerical/analytical method is proposed for the calculation of stray losses inside the magnetic shunts. The proposed method is basedon double Fourier series expansions of the magnetic field distribution at the surface of the shunts which is determined using 3DFEA. A 200 MVA power transformer is investigated as a case study where the stray losses are calculated with and without verticaland horizontal shunts. A Parametric FEA is carried out to find the optimal placement of the horizontal shunts on the tank walls.Results show that the proposed horizontal magnetic shunts arrangement are as effective as conventional vertical shunts in reducingthe stray losses while reducing the weight of the shields, therefore providing a cost-effective method for magnetic shielding of thetransformer tank walls.

Index Terms—finite element method, magnetic shielding, power transformer, stray loss, wall shunts.

NOMENCLATURE

A Magnetic vector potential (V.s/m)H Magnetic field (A/m)B Magnetic flux density (Tesla)J Eddy-current density (A/m2)ω Frequency (rad/s)µ Magnetic permeability (H/m)ν Magnetic reluctivity (m/H)σ Electrical conductivity (S/m)P Power loss (W )

I. INTRODUCTION

STRAY load losses in transformer tanks and structural partscaused by flux escaping the core plays a key role in the

transformer overall performance and therefore reducing theselosses is of a great importance for manufacturers. Magneticwall shunts in power transformers are frequently used toreduce stray losses and eliminate possible hot spots in the tankand structural parts (see Fig. 1 (a)). These types of shieldscreate a low reluctance path for the stray flux and preventit from reaching the tank. Magnetic shunts are built fromlaminated steel packets to reduce losses due to the normalcomponent of the magnetic field to the shunt surface [1].

The design of magnetic shunts is always a great challengewhen designing large power transformers. Many studies on2D and 3D calculation of leakage magnetic field and eddycurrent losses in power transformers have been performed [1]–[16]. For calculation of stray fields in power transformers withmagnetic wall shunts, 3D modeling is needed to accurately

Manuscript received January 20, 2016; revised April 28, 2016; acceptedSeptember 9, 2016. Corresponding author: A. I. Sarwat (e-mail: asar-wat@fiu.edu).

This work was funded by the National Science Foundation under grantnumber CRISP-1541108.

compute the effect of different arrangements of magneticshunts on the stray losses.

Different structures of magnetic shunts in power transform-ers are studied in [16]–[25], using FEM analysis. Horizontalmagnetic shunts under the transformer yokes are studied in[20], [21]. Also, the effect of the lobe-type magnetic shieldingon stray losses of power transformer using 3D FEM analysisis studied. In [10], a numerical method based on networkapproach is presented and it is used in [21] to model theeffect of magnetic shunts on transformer stray flux. Manymethods are proposed for nonlinear laminated steel modelingand calculation of its stray loss [26]–[36]. The homogenizationmethod has been proposed in [26], [27] to model the nonlinearlaminated steel. In this method the solution needs to be ob-tained twice which doubles the calculation time. No study hasbeen performed to investigate the effectiveness of horizontalmagnetic wall shunts on transformers stray losses. This typeof magnetic shunt diverts the stray flux of the three phasesto itself and thus prevents the flux from entering the tank.Therefore this technique can effectively reduce stray losses inlarge power transformers.

In this paper, a horizontal wall shunt arrangement is pro-posed and its effectiveness is compared against conventionalvertical magnetic wall shunts in reducing the stray losses ofpower transformers. The investigation is carried out using 3Dfinite element analysis using COMSOL Multiphysics. Strayload losses on the tank walls and yoke beams are calculatedusing surface impedance boundary condition. Magnetic wallshunts are modeled with nonlinear anisotropic permeabilityand the corresponding losses are calculated using an analyticalmethod. The presented analytical method uses the magneticfield distribution calculated with 3D FEM, as boundary con-dition and Maxwell equations are solved inside the magneticshunts using double Fourier series expansion method. Fig.

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LV side mid

shunts

LV End side

shunts

LV TC side

shunts

End side

shunts

HV End side

shunts

HV mid

shunts

HV TC side

shunts

Core clamps

(yoke beams)

Laminated

Core

LV side top

shunts

LV side

Bottom shunts

HV side top

shunts

HV side

bottom shunts

End side

shunts

Core clamps

(yoke beams)

Laminated

Core

Core clamps

(yoke beams)

(a)

(b)

Fig. 1. Three-phase three-legged 200 MVA transformer; (a) conventionalvertical wall shunts; (b) proposed horizontal wall shunts.

1(b) depicts the 200 MVA sample transformer with the newproposed horizontal wall shunts.

II. 3D FINITE ELEMENT MODELING

The 3D finite element method for solving the eddy currentproblem uses the quasi-static magnetic vector potential PDEformulation given by:

∇× (v (∇×A)) = Je − jωσA (1)

Three phase full load current excitation is applied to wind-ings. Since the distance between the tank walls and the outerwindings are not equal in all directions, no symmetry boundarycondition is applied to the 3D model of the transformer tomaximize the accuracy of the analysis.

Fig. 2. Magnetization curve of the M-5 steel.

A. Impedance Boundary Condition

At the boundaries where the magnetic field penetrates onlya short distance into the boundary the Impedance BoundaryCondition (IBC) is used for approximating the magnetic fieldpenetration into the boundary. The IBC can be used to modela bounded domain as an unbounded region and is a validapproximation if the skin depth is small compared to the sizeof the conductor. The penetration depth δ is measured usingthe following equation:

δ =

√2

ωµσ(2)

The IBC boundary condition, which is a combination ofDirichlet and Neumann boundary conditions, proposes a rela-tionship between the value of magnetic vector potential A andits normal derivative at the boundary. This boundary conditioncan be written as the following equation:

∂A

∂n+ cA = 0 (3)

where c is a constant determined by the permeability andconductivity of the boundary material. Since in power trans-formers tank walls and yoke-beams are made of iron and thepenetration depth at 60 Hz for iron is less than 1 mm, in 3DFEM model of the transformer IBC is applied to the interiorboundaries of the tank walls and the exterior boundaries ofthe yoke-beams.

B. Magnetic Shunt Modeling

Magnetic shunts are constructed using laminated steel andtherefore should be modeled with nonlinear anisotropic perme-ability. The magnetization characteristic of M-5 steel (shownin Fig. 3) is used for modeling the nonlinearities of the shunts.The conductivity of the shunts is set to zero in the FEMmodel and then the eddy-current losses are calculated withthe analytical method described in Section III.

III. MAGNETIC SHUNT LOSS CALCULATION

The losses in the magnetic wall shunts can be calculatedusing a series expansion of eddy-current loss based on two-

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2L

W

d

x

y

z

Fig. 3. Laminated magnetic shunt with dimensions of W × 2L× d.

dimensional spatial harmonics of current density inside amagnetic wall shunt. This series can be expressed as follows:

P =∞∑m=1

∞∑n=1

Pmn (4)

where Pmn is the eddy-current loss due to spatial currentdensity harmonics. The current density harmonics in themagnetic shunt sheets can be calculated based on an analyticalapproach provided that the magnetic field distribution on themagnetic shunts surface is known. The analytical solution canbe obtained by solving the quasi-static field equations whichare expressed as follows [2]:

∇2H = jωµσH (5)∇.H = 0 (6)J = ∇×H (7)

Figure 3 shows a laminated magnetic shunt with dimensionsof W × 2L × d. It is assumed that spatial distribution of thenormal component of the magnetic field (Hz) on the surfacez = 0 could be written as a double Fourier series expansionas:

Hz0(x, y) =

∞∑m=1

∞∑n=1

Hmn cos(mπxW

)sin(nπyL

)(8)

where Hmn are the double Fourier series coefficients, whichcan be calculated as follows:

Hmn =2

WL

∫ W

0

∫ L

−LHz0(x, y) cos

(mπxW

)sin(nπyL

)(9)

For calculation eddy-current loss of the magnetic shunt, firstthe loss corresponding to each spatial harmonic is calculatedand then the superposition is applied in order to calculate thetotal shunt loss. The spatial harmonic component of (8) couldbe written as:

Hz0(x, y) = Hmn cos(mπxW

)sin(nπyL

)(10)

This equation can be used as a boundary condition for (5),(6) and (7) in order to calculate magnetic field inside theshunt. By expanding (5) in the direction of z-axis the followingequation is derived:

∂2Hz

∂x2+∂2Hz

∂y2+∂2Hz

∂z2= jωµσHz (11)

According to (10) and (11), Hz can be written as follows:

Hz = Hmneβz cos

(mπxW

)sin(nπyL

)β2 = jωµσ +

(nπL

)2+(mπW

)2 (12)

where the permeability of µ is determined based on themagnetic flux density in each wall shunt, which is computedby the 3D FEA, as well as the magnetization characteristic ofthe M-5 steel, which is presented in Fig. 2.

Using (6) and (12), the following equation can be derived:

∂Hx

∂x+∂Hy

∂y+∂Hz

∂z= 0

∂Hx

∂x+∂Hy

∂y= −Hmnβe

βz cos(mπxW

)sin(nπyL

) (13)

Since the magnetic shunts are laminated, current cannot flowin the direction of z-axis, based on (7) it can be expressed asthe following equation:

Jz =∂Hy

∂x− ∂Hx

∂y= 0 (14)

It is assumed that the solution for Hx and Hy can be writtenas follows:

Hx = K1Hmneβz sin

(mπxW

)sin(nπyL

)Hy = K2Hmne

βz cos(mπxW

)cos(nπyL

) (15)

where K1 and K2 are unknown constants which can becalculated using (13) and (14) as follows:

K1 = − HmnL2mWβ

m2πL2 + n2πW 2

K2 =HmnLnW

2β

m2πL2 + n2πW 2

(16)

Thus, the final solution of Hx and Hy can be expressed asfollowing set of equations:

Hx = − HmnL2mWβ

m2πL2 + n2πW 2eβzHmne

βz sin(mπxW

)sin(nπyL

)Hy =

HmnLnW2β

m2πL2 + n2πW 2eβzHmne

βz cos(mπxW

)cos(nπyL

)(17)

Using solutions for magnetic field components (17), currentdensity distribution can be calculated using (7) and the fol-lowing equations can be deducted for the current densitycomponents Jx and Jy:

Jx =nHmn

[n2π2W 2 + L2

(m2π2 +W 2β2

)]Lπ (L2m2 + n2W 2)

×eβz cos(mπxW

)cos(nπyL

)

Jy =mHmn

[n2π2W 2 + L2

(m2π2 +W 2β2

)]Wπ (L2m2 + n2W 2)

×eβz sin(mπxW

)sin(nπyL

)(18)

Using (18), spatial distribution of eddy-current loss densitycan be expressed as follows:

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0 5 10 1510

-2

10-1

100

101

Convergence of eddy-current loss equation for magnetic shunt

Harmonic order

Erro

r (

%)

Fig. 4. The convergence of the eddy-current loss series expansion presentedin (4) for the case study transformer.

Pv =J.J∗

2σ=

H2mnL

2W 2µ2σω2

2π2 (L2m2 + n2W 2)2 e−2Real(β)

×[n2W 2 cos2

(mπxW

)cos2

(nπyL

)+m2L2 sin2

(mπxW

)sin2

(nπyL

)](19)

And the solution for calculation of loss due to normalcomponent of magnetic field at z = 0 surface can be derivedas follows:

Pmn =

∫ 0

−∞

∫ L

−L

∫ W/2

−W/2Pvdxdydz

=H2mnL

3W 3µ2σω2

8π2 (L2m2 + n2W 2)Real(β)(20)

Using (4) and (20), the total eddy-current loss of themagnetic shunt shown in Fig. 3 can be written as follows:

P =µ2σω2

8π2

∞∑m=1

∞∑n=1

L3W 3H2mn

(L2m2 + n2W 2)Real(β)(21)

Equation (21) presents the series expansion of the eddy-current losses of a magnetic shunt based on the Fourier seriesexpansion of magnetic field on its surface.

For the calculation of magnetic shunts losses based on thepresented analytical method, magnetic field distribution on theshunts surface is calculated with 3D finite element analysis(FEA) using COMSOL Multiphysics. The calculated magneticfield distribution is exported into MATLAB and the double-Fourier series coefficients of (8) are calculated and using (21)and thus the total loss of the magnetic shunt can be obtained.Fig. 4 shows the convergence of (21) for a sample verticalmagnetic shunt. This figure shows that by expanding thisequation up to the 15th component in both x and y axes,the convergence error would be less than 0.1%.

IV. RESULTS AND DISCUSSIONS

In this section the 200 MVA transformer shown in Figs.1(a) and 1(b) is considered as a case study to compute the

TABLE ICOMPARISON OF STRAY LOSSES IN DIFFERENT WALL SHUNT

ARRANGEMENTS

Parameter WithoutShunts

VerticalShunts

HorizontalShunts

Tank Loss (W ) 58993 5981 5458Yoke-Beam Loss (W ) 17893 5439 5368Shunt Loss (W ) − 1740 2236Total Loss (W ) 76886 13160 13062Total Loss Reduction (%) − 82.88 83.01Total Shunt Weight (kg) − 2671 2009

Fig. 5. The mesh of the case study transformer model without wall shuntsconsisting of 817,155 elements.

Fig. 6. Eddy current loss density (W/m2) distribution on transformers tankwithout any magnetic shunts.

stray losses of tank, yoke-beams, and magnetic shunts for threescenarios: transformer without magnetic shunts, transformerwith vertical magnetic shunts and transformer with horizontal

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magnetic shunts. In the following sections the results arepresented and comparisons are made.

A. Transformer without Magnetic Shunts

In this case, the transformer carrying full load is consideredand stray losses in tank walls and yoke-beams are calculated.The mesh of the 3D model consists of 817,155 elements andis shown in Fig. 5. The total stray loss is 76.89 kW. Theresults are presented in Table I. Fig. 6 shows the loss densitydistribution in W/m2 on tank walls. As it can be seen theloss density on the tank walls in the area between adjacentwindings is lower than other areas of the tank walls which isthe cause of leakage flux cancellation of adjacent windings. Onthe other hand, the maximum loss density occurs on the sidetank wall. Also, the loss density distribution have a completelydifferent pattern on main tank walls compared with side tankwall.

B. Transformer with Vertical Magnetic Shunts

In this case, vertical magnetic wall shunts are used to reducestray losses. In front of windings on each limb of the trans-former five laminated shunts are placed on the tank walls.Shunts are not placed on tap-changer side of the tank dueto its low stray loss. The mesh of the 3D model consistsof 2,182,302 elements. The calculation results are presentedin Table I. Fig. 7 shows the magnetic leakage flux lines.This figure, shows that flux lines are completely diverted bymagnetic shunts and as a result eddy current losses in the tankwalls and the yoke-beams are reduced 82.88% when comparedwith the transformer without wall shunts. The magnetic fluxdistribution in vertical magnetic shunts is shown in Fig. 8.This figure shows that flux is not distributed uniformly in theshunt packets and it is higher on the side shunt packets. Also,it can be seen that wall shunts in the center phase have a lowerflux density. Therefore, the thickness of these shunts can bereduced. Fig. 9 shows the induced current density streamlineson vertical shunt packet surfaces of one phase.

C. Transformer with Horizontal Magnetic Shunts

In this case, horizontal and vertical magnetic wall shuntsare used to reduce stray losses. In front of three phases of thetransformer, six horizontal laminated shunts are placed on thetank walls (three shunts at top and three shunts at bottom).The mesh of the 3D model consists of 1,853,022 elements.The calculation results are presented in Table I. Fig. 10 showsthe magnetic leakage flux lines. This figure, shows that fluxlines are completely diverted by the horizontal magnetic shuntsand as a result the total eddy current losses in the tank wallsand the yoke-beams are reduced by 83.01% compared withthe transformer without wall shunts. Based on Table I it canbe seen that by using horizontal shunts the total weight of theshunts is reduced by 25% when compared with vertical shuntswhile stray losses are almost the same. Thus using horizontalshunt walls would be very cost-effective. The magnetic fluxdistribution in horizontal and vertical magnetic shunts is shownin Fig. 11. Comparing Fig. 8 and Fig. 11 it can be concluded

Fig. 7. Magnetic leakage flux lines diverted by vertical magnetic shunts.

Fig. 8. Magnetic flux distribution (Tesla) in vertical magnetic shunts.

that flux density in the horizontal arrangements of shunts arehigher than vertical arrangements and as a result the lossdensity in horizontal shunts will be higher. Also, it can be seenin Table I that although the total shunt weight with horizontalshunts is 25% lower compared with vertical shunts but theshunt losses is 28.5% higher.

Also, parametric FEA were carried out to find the optimalposition of the horizontal magnetic shunts for maximumperformance. In Fig. 12, the losses versus the gap betweenyokebeams and horizontal magnetic shunts are presented. Thisfigure shows that with 150 mm overlap of the horizontal shuntsand yokebeams, the transformer will have the minimum straylosses.

V. CONCLUSIONS

A new method for magnetic shielding of the tank wallsin large power transformers has been proposed in this paper.The presented method is based on a horizontal arrangementof magnetic shunts on tank walls. It is shown that horizontal

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Fig. 9. Induced current density streamlines on vertical shunt packets surfaces.

Fig. 10. Magnetic leakage flux lines diverted by horizontal magnetic shunts.

shunts are as effective as vertical shunts in reducing stray loadlosses. However, horizontal shunts weigh 25% less than ver-tical shunts. Hence, the proposed horizontal magnetic shuntsarrangement is very cost effective. The proposed combinedFEM and analytical method provides an effective and accu-rate method for calculation of induced current densities andcorresponding eddy-current losses in magnetic wall shunts.It is shown that the flux is not uniformly distributed in theshunt packets; it is higher in the side packets. Also, wall shuntpackets in front of the center limb have lower flux densities.Consequently the thickness of these shunts can be reduced.Furthermore, parametric FEA can be used to find the optimalplacement of the magnetic shunts on the tank walls.

REFERENCES

[1] S. V. Kulkarni and S. Khaparde, Transformer engineering: design andpractice. CRC Press, 2004, vol. 25.

Fig. 11. Magnetic flux distribution (Tesla) in horizontal and vertical magneticshunts.

-400 -300 -200 -100 0 100 200 300 4002

3

4

5

6

7

8

9

10

Lo

ss (

kW

)

Vertical displacement of horizontal shunts (mm)

Tank Loss

Yokebeam Loss

Magnetic Shunt Loss

(a)

(b)

-400 -300 -200 -100 0 100 200 300 40013

14

15

16

17

18

Lo

ss (

kW

)

Vertical displacement of horizontal shunts (mm)

Total Loss

Fig. 12. Parametric FEA results on vertical position of horizontal shunts: (a)The losses in tank walls, yokebeams and magnetic shunts, (b) Total losses.

[2] R. M. Del Vecchio, B. Poulin, P. T. Feghali, D. M. Shah, and R. Ahuja,Transformer design principles: with applications to core-form powertransformers. CRC press, 2010.

[3] K. Karsai, D. Kerenyi, and L. Kiss, “Large power transformers,” 1987.[4] Z. Zhu, D. Xie, G. Wang, Y. Zhang, and X. Yan, “Computation of 3-

d magnetic leakage field and stray losses in large power transformer,”IEEE Trans. Magn., vol. 48, no. 2, pp. 739–742, Feb 2012.

[5] J. Smajic, G. D. Pino, C. Stemmler, W. Mnig, and M. Carlen, “Numericalstudy of the core saturation influence on the winding losses of tractiontransformers,” IEEE Trans. Magn., vol. 51, no. 3, pp. 1–4, March 2015.

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[6] B. Bai, Z. Chen, and D. Chen, “Dc bias elimination and integratedmagnetic technology in power transformer,” IEEE Trans. Magn., vol. 51,no. 11, pp. 1–4, Nov 2015.

[7] L. Li, S. Niu, S. L. Ho, W. N. Fu, and Y. Li, “A novel approach toinvestigate the hot-spot temperature rise in power transformers,” IEEETrans. Magn., vol. 51, no. 3, pp. 1–4, March 2015.

[8] O. Br, G. Koczka, G. Leber, K. Preis, and B. Wagner, “Finite elementanalysis of three-phase three-limb power transformers under dc bias,”IEEE Trans. Magn., vol. 50, no. 2, pp. 565–568, Feb 2014.

[9] C. Liao, J. Ruan, C. Liu, W. Wen, and Z. Du, “3-d coupledelectromagnetic-fluid-thermal analysis of oil-immersed triangular woundcore transformer,” IEEE Trans. Magn., vol. 50, no. 11, pp. 1–4, Nov2014.

[10] D. Albertz and G. Henneberger, “Calculation of 3d eddy current fieldsusing both electric and magnetic vector potential in conducting regions,”IEEE Trans. Magn., vol. 34, no. 5, pp. 2644–2647, Sep 1998.

[11] J. Olivares, R. Escarela-Perez, S. Kulkarni, F. de Leon, andM. Venegas-Vega, “2d finite-element determination of tank walllosses in pad-mounted transformers,” Electric Power Systems Research,vol. 71, no. 2, pp. 179 – 185, 2004. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0378779604000410

[12] C. Carpenter, “A network approach to the numerical solution of eddy-current problems,” IEEE Trans. Magn., vol. 11, no. 5, pp. 1517–1522,Sep 1975.

[13] G. Paoli, O. Biro, and G. Buchgraber, “Complex representation innonlinear time harmonic eddy current problems,” IEEE Trans. Magn.,vol. 34, no. 5, pp. 2625–2628, Sep 1998.

[14] M. Moghaddami, A. Moghadasi, and A. I. Sarwat, “An algorithm forfast calculation of short circuit forces in high current busbars of electricarc furnace transformers based on method of images,” Electric PowerSystems Research, vol. 136, pp. 173 – 180, 2016. [Online]. Available:http://www.sciencedirect.com/science/article/pii/S0378779616300037

[15] A. Basak and H. Kendall, “Leakage flux in the steel tank of a 2.5-kva single phase transformer,” IEEE Trans. Magn., vol. 23, no. 5, pp.3831–3835, Sep 1987.

[16] L. Li, W. N. Fu, S. L. Ho, S. Niu, and Y. Li, “Numerical analysis andoptimization of lobe-type magnetic shielding in a 334 mva single-phaseauto-transformer,” IEEE Trans. Magn., vol. 50, no. 11, pp. 1–4, Nov2014.

[17] C. Hernandez, M. A. Arjona, and J. P. Sturgess, “Optimal placementof a wall-tank magnetic shunt in a transformer using fe models and astochastic-deterministic approach,” in 2006 12th Biennial IEEE Confer-ence on Electromagnetic Field Computation, 2006, pp. 468–468.

[18] M. A. Tsili, A. G. Kladas, P. S. Georgilakis, A. T. Souflaris, andD. G. Paparigas, “Geometry optimization of magnetic shunts in powertransformers based on a particular hybrid finite-element boundary-element model and sensitivity analysis,” IEEE Trans. Magn., vol. 41,no. 5, pp. 1776–1779, May 2005.

[19] C. Yongbin, Y. Junyou, Y. Hainian, and T. Renyuan, “Study on eddycurrent losses and shielding measures in large power transformers,” IEEETrans. Magn., vol. 30, no. 5, pp. 3068–3071, Sep 1994.

[20] M. Djurovic and J. Monson, “3-dimensional computation of the effectof the horizontal magnetic shunt on transformer leakage fields,” IEEETrans. Magn., vol. 13, no. 5, pp. 1137–1139, Sep 1977.

[21] M. Djurovic and J. E. Monson, “Stray losses in the step of a transformeryoke with a horizontal magnetic shunt,” IEEE Transactions on PowerApparatus and Systems, vol. PAS-101, no. 8, pp. 2995–3000, Aug 1982.

[22] A. D. Pasquale, G. Antonini, and A. Orlandi, “Shielding effectiveness fora three-phase transformer at various harmonic frequencies,” IET Science,Measurement Technology, vol. 3, no. 2, pp. 175–183, March 2009.

[23] Z. Song, Y. Wang, S. Mou, Z. Wu, Y. Zhu, B. Xiang, and C. Zhou, “Tanklosses and magnetic shunts in a three phase power transformer,” in Elec-trical Machines and Systems (ICEMS), 2011 International Conferenceon, Aug 2011, pp. 1–4.

[24] F. Dughiero and M. Forzan, “Transient magnetic fem analysis forthe prediction of electrodynamic forces in transformers with magneticshunts,” in Magnetics Conference, 2002. INTERMAG Europe 2002.Digest of Technical Papers. 2002 IEEE International, April 2002, pp.EV5–.

[25] Z. Song, Y. Wang, S. Mou, Z. Wu, Y. Zhu, B. Xiang, and C. Zhou, “Theedge effects of magnetic shunts for a transformer tank,” in ElectricalMachines and Systems (ICEMS), 2011 International Conference on, Aug2011, pp. 1–4.

[26] H. Kaimori, A. Kameari, and K. Fujiwara, “Fem computation ofmagnetic field and iron loss in laminated iron core using homogenizationmethod,” IEEE Trans. Magn., vol. 43, no. 4, pp. 1405–1408, April 2007.

[27] L. Krahenbuhl, P. Dular, T. Zeidan, and F. Buret, “Homogenizationof lamination stacks in linear magnetodynamics,” IEEE Trans. Magn.,vol. 40, no. 2, pp. 912–915, March 2004.

[28] I. Sebestyen, S. Gyimothy, J. Pavo, and O. Biro, “Calculation of losses inlaminated ferromagnetic materials,” IEEE Trans. Magn., vol. 40, no. 2,pp. 924–927, March 2004.

[29] J. Gyselinck, L. Vandevelde, J. Melkebeek, P. Dular, F. Henrotte, andW. Legros, “Calculation of eddy currents and associated losses inelectrical steel laminations,” IEEE Trans. Magn., vol. 35, no. 3, pp.1191–1194, May 1999.

[30] A. G. Jack and B. C. Mecrow, “Calculation of three-dimensional elec-tromagnetic fields involving laminar eddy currents,” IEE Proceedings A- Physical Science, Measurement and Instrumentation, Management andEducation - Reviews, vol. 134, no. 8, pp. 663–671, September 1987.

[31] K. Preis, O. Biro, and I. Ticar, “Fem analysis of eddy current losses innonlinear laminated iron cores,” IEEE Trans. Magn., vol. 41, no. 5, pp.1412–1415, May 2005.

[32] R. F. Hemmings and G. D. Wale, “Heating in transformer cores dueto radial leakage flux. part 1: Experimental models and test results,”Electrical Engineers, Proceedings of the Institution of, vol. 124, no. 11,pp. 1064–1072, November 1977.

[33] C. J. Carpenter, K. O. Sharples, and M. Djurovi, “Heating in transformercores due to radial leakage flux. part 2: Computed results,” ElectricalEngineers, Proceedings of the Institution of, vol. 124, no. 12, pp. 1181–1186, December 1977.

[34] C. J. Carpenter, “Theory of flux penetration into laminated iron andassociated losses,” Electrical Engineers, Proceedings of the Institutionof, vol. 124, no. 7, pp. 659–664, July 1977.

[35] C. J. Carpenter and M. Djurovi, “Three-dimensional numerical solutionof eddy currents in thin plates,” Electrical Engineers, Proceedings ofthe Institution of, vol. 122, no. 6, pp. 681–688, June 1975.

[36] V. C. Silva, G. Meunier, and A. Foggia, “A 3-d finite-element compu-tation of eddy currents and losses in laminated iron cores allowing forelectric and magnetic anisotropy,” IEEE Trans. Magn., vol. 31, no. 3,pp. 2139–2141, May 1995.