Calculation of copper losses in resistance spot welding transformer with space‐ and time‐dependent current density distribution, FEM and measurements

Calculation of copper lossesin resistance spot weldingtransformer with space-

and time-dependent currentdensity distribution,

FEM and measurementsJelena Popovic, Drago Dolinar, Gorazd Stumberger

and Beno KlopcicFaculty of Electrical Engineering and Computer Science,

University of Maribor, Maribor, Slovenia

Abstract

Purpose – So far the proposed analytical methods for calculation of copper losses are rathersimplified and do not include the time component in the basic partial differential equations, whichdescribe current density distribution. Moreover, when the physical parameters of the transformer(wire dimensions) are out of the certain range, the current density distribution approaches infinity. Thepurpose of this paper is to offer a generally applicable analytical method. The main goal of theproposed modification of the solution to the current density is improvement of the accuracy andstability of the analytical results.

Design/methodology/approach – This paper deals with the calculation of copper losses withvarious methods, which are based on a time-dependent electromagnetic field. Analytical method isbased on Maxwell equations and Helmholtz equation. Numerical calculation is performed with finiteelement method (FEM).

Findings – Analytical method is a very accurate and it gives results, which are very similar to theactual behaviour of the current density in the winding. However, the FEM analysis is easier tocomprehend, but yet very dependent on input parameters.

Research limitations/implications – The numerical analysis may not be accurate enough,because of the problems with the oscillation of the output welding current amplitude. To calculatecopper losses correctly, the output welding current must be equal in all test cases, especially during themeasurements.

Originality/value – When the physical properties exceed a certain range, the copper losses of theanalyzed welding transformer cannot be calculated with existing analytical methods. The newanalytical approach gives a far more realistic solution to the current density distribution and improvesthe accuracy and stability of the results.

Keywords Copper, Electric current, Finite element analysis, Welding, Transformers

Paper type Research paper

I. Introduction to the resistance spot welding systemResistance spot welding (RSW) system is one of the most widely used, inexpensiveand efficient material joining processes in the automotive industry. The basic structureof the discussed welding system is schematically shown in Figure 1. It consists of

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COMPEL: The International Journalfor Computation and Mathematics inElectrical and Electronic EngineeringVol. 30 No. 3, 2011pp. 996-1010q Emerald Group Publishing Limited0332-1649DOI 10.1108/03321641111110942

a semiconductor converter, single-phase transformer and full-wave rectifier mountedon the transformer’s output. The corresponding pulse width modulated (PWM)AC voltage u that supplies the transformer’s primary winding is generated by anh-bridge, with switching frequency of up to 5 kHz (Klopcic et al., 2008).

The major part of losses is diode losses and copper losses that appear in the RSWtransformer (TR) with integrated output diodes (D1, D2). The iron core losses (Klopcicet al., 2008; Podlogar et al., 2010) are much lower and can be neglected. This paper dealswith the determination of winding losses only. The nominal operating frequency ofthe analyzed RSW system is 1 kHz with an ambition to increase it up to 5 kHz.The transformer primary current i1 has a pulse form with amplitude up to 500 A.Primary winding (rectangular wire) consists of four coils with 14, 13, 14 and 14 turnsconnected in series, giving the total number of 55 turns. Secondary winding, which iscooled with water, consists of two single-turn coils, of which only one is active in eachhalf-period. The high magnitude of the DC welding current (from 10 kA up to 20 kA) isobtained by the rectifier (D1, D2) mounted on the secondary windings of the RSWtransformer. Nominal power of the RSW system at 80 per cent duty cycle is 110 kVA.Primary winding temperature is up to 1308C, whereas in spite of water cooling, thesecondary winding temperature increases up to 508C.

Determination of copper losses is possible with several analytical methods, finiteelement method (FEM) and measurements. An early work on this field suggestscalculation of copper losses by Maxwell equations (Dowell, 1966; Vandelac and Ziogas,1988; Robert et al., 1998). In more recent work, authors use Kelvin functions (Ferreira, 1992;Kazimierczuk, 2009). However, the problem is very complex and it is not sufficient toconsider the current density distribution over the given space (Sullivan, 1999). Besides, sofar proposed analytical methods are applicable only under certain geometrical limitations.Considering the time coordinate along with space coordinates it assures the stableand correct solution. The objective of this work is to carry out a detailed determinationof copper losses which are frequency dependent, due to the non-uniform currentdistribution caused by eddy currents. The AC resistance increases with frequency, whichmeans higher copper losses in transformer windings (Sippola and Sepponen, 2002;Dwight, 1922).

II. Analytical approach for calculation of copper losses in transformerwindingsA. Basic Maxwell equations and time-dependent solution to the current densityThe simplest solutions of the field equations are those that depend only upon spacecoordinates. However, it is expected that the factors characterizing electromagneticfield depend also upon the time coordinate. Propagation of this elementary time-dependent

Figure 1.Shematic presentation

of the RSW system

uaUDC

PWMinvertor

TRRL

LL

D1

D2

i3

iL

i2i1

uub

uc

Copper lossesin resistancespot welding

997

space fields determine also in large part the propagation of the complex fields met with realproblems, such as the problem of current density distribution. The first step is todetermine the electromagnetic field completely. Any vector field is determined completelywhen its source and infinitesimal rotation of the vector field is defined. Two operators arecommonly used for this purpose, divergence and rotor operator. The Maxwell equationsfor field vectors (Stratton, 2007; Ida, 2004; Shadowitz, 1988) are given by following set ofexpressions (1):

7 £ Eþ m›H

›t¼ 0; 7 £H2 1

›E

›t2 sE ¼ 0;

7 ·E ¼ 0; 7 ·H ¼ 0;

ð1Þ

whereE is the electric field vector,H is the magnetic field strength vector,m represents thepermeability, 1 is the permittivity and s is the electrical conductivity, respectively. Withthe set of expressions (1), not only the electromagnetic field but also the relation betweenfield vectors is given. The relation between magnetic field and current density is given byequation (2):

7 £H2 1›E

›t2 sE ¼ 0; ð2Þ

where electric field E and current density J are related as equation (3):

J ¼ sE: ð3Þ

The electric field vector from equation (2) is replaced with expression (3). That gives thefollowing partial differential equation (4):

7 £H2 1s›J

›t2 J ¼ 0: ð4Þ

The following basic partial differential equation describes the behaviour of the currentdensity upon space and time coordinates equation (5):

72J2 ðm1v 2 þ jvmsÞ›J

›t¼ 0; ð5Þ

where v is the angular frequency and j is the imaginary unit. The expressionm1v 2 þ jvms is replaced with k 2 ¼ m1v 2 þ jvms. The parameter k depends uponelectric and magnetic characteristics of the region (Kalluri, 2010). In a conductive medium,k is always a complex quantity and it includes both a real and imaginary part, k ¼ a þ jb.Considering the 2D model, as it is shown in Figure 2(b), current density vector is in z-axisonly (Riley et al., 2006). The expression (5) can be re-written as equation (6):

›2J z

›z22 k 2 ›J z

›t¼ 0; ð6Þ

where Jz represents the current density in z-axis. In continuation of the paper, the currentdensity in z-axis is denoted as J.

The solution form is supposed on the basis of the mathematical properties of thecorresponding partial differential equation (6), which has been solved with theseparation of variables (Evans, 2010). So far existing solutions include a corresponding

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space coordinate only (Vandelac and Ziogas, 1988; Ferreira, 1992). The proposedsolution form to the given partial differential equation (6), which represents far morerealistic behaviour of the current density, is given by equation (7):

J ðx; tÞ ¼ J 1ðtÞebxcosðvt 2 axÞ þ J 2ðtÞe

2bxcosðvt þ axÞ; ð7Þ

where a and b represents the real and imaginary part of the parameter k, respectively.J1, J2 are corresponding constants obtained from boundary conditions and described asequation (8):

Figure 2.(a) Welding transformer

and (b) weldingtransformer cross-section

(a)

(b)

Layer begining, H(0)

Layer end, H(h)

Layer width, h

Core

H2O H2O

H2OH2O


999

J 1ðtÞ ¼

2k 2 ·H ðhÞþððk 3 ·H ð0Þ ·e2b ·h · cosðvtþahÞþ k 2 ·a ·H ð0Þ ·e2b ·h · sinðvtþahÞÞ

ðk · cosðvtÞþa · sinðvtÞÞÞ

k ·eb ·h · cosðvt2ahÞþa ·eb ·h · sinðvt2ahÞ2k ·e2b ·h · cosðvtþahÞ

2a ·e2b ·h · sinðvtþahÞ;

and:

J 2ðtÞ ¼ J 1ðtÞ þk 2 ·H ð0Þ

k · cosðvtÞ þ a · sinðvtÞ: ð8Þ

The corresponding constants depend upon the layer width h, magnetic field strength atthe beginning of the layer H(0) and magnetic field strength at the end of the layer H(h).Boundary conditions, as well as the current density distribution function, change withrespect to time. Magnetic field strength on the layer edges has been calculated with theexpression for magnetic field strength of a flat coil (Grover, 2004) as it can be concludedfrom Figure 3(a). Magnetic field strength on both edges represents a boundarycondition (Powers, 2010) at the particular point of time when the transient analysisstarts at t ¼ 0, and therefore, this estimation of the magnetic field is correct from themathematical point of view.

The layer is divided into finite number of points that produce finite number ofdifferential distances, as it is shown in Figure 3(b). Time is also discretized in regard toone period.

B. Calculation of copper lossesCopper losses per unit volume (W/m3) for the mth layer with width h and correspondingcurrent are calculated with the following definite integral equation (9):

Pmðh;TÞ ¼1

T

1

h

1

hs

Z T

0

Z h

0

Jmðx; tÞ · Jmðx; tÞ*dxdt; ð9Þ

where h represents the porosity factor, T is period and Jm(x,t) * is the complexconjugate of the function of current distribution Jm(x,t) in the mth layer. Porosity factoris calculated with expression (10):

h ¼Nbw

b; ð10Þ

where N is the number of turns, bw is the actual layer height, which includes copperonly and b is the total layer height with insulation included (Figure 3(b)). Porosityfactor of the primary windings is high, whereas the secondary wire has lower porosityfactor because it has a hole in it. Three transformers have been tested. Summarizedbasic physical transformer data are shown in Table I.

Electrothermal properties of the analyzed transformers are shown in Table II. Totalcopper losses for the analyzed welding transformer are given by equation (11):

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Test transformer Quantity Value

TR1 Primary wire area 6 £ 1.6 mmTR2 Primary wire area 3 £ 1.6 mmTR3 Primary wire area 2 £ 1.6 mmTR1, TR2, TR3 Primary turns 55TR1, TR2, TR3 Secondary wire area 12 £ 24 mmTR1, TR2, TR3 Hole size (in secondary wire) 5.5 £ 14.5 mmTR1, TR2, TR3 Secondary turns 2 £ 1TR1, TR2, TR3 Total layer height b 24 mmTR1, TR2, TR3 Porosity factor hprim 0.93TR1, TR2, TR3 Porosity factor hsec 0.71

Table I.Physical properties of the

test transformers

Figure 3.(a) The upper half of the

transformer and (b)vertical discretization of a

primary layer

Layer1prim

14I –55I 13I 14I 14Ihsec

In all consideredcases = 1.6 mm

Axi

s of

sym

men

try

y

xz

hprim

wire or layer width

Turn14

b b wTurn13Turn12Turn11Turn10Turn 9Turn 8Turn 7Turn 6Turn 5Turn 4Turn 3Turn 2Turn 1

Primary copper layerwith 14 turns

Note: Despite the same length in the figure, bw and b are not the same

hprim

1 ,000

Total = 1,000 points

Layer2sec

Layer3,4prim prim

Layer5sec

Layer6prim

(a)

(b)


1001

PCu ¼X5

m¼1

PCum¼

1

T

V prim14

hprimhprimsprim

Z T

0

Z hprim

0

J 1ðx; tÞ · J 1ðx; tÞ*dxdt

�

þV sec

hsechsecssec

Z T

0

Z hsec

0


þV prim13

hprimhprimsprim

Z T

0

Z hprim

0


þV prim14

hprimhprimsprim

Z T

0

Z hprim

0


þV prim14

hprimhprimsprim

Z T

0

Z hprim

0


�;

ð11Þ

where Vprim14 denotes the volume of the primary coils with 14 turns, Vprim13 presentsthe volume of a primary coil with 13 turns and Vsec is the volume of the secondary wire.

III. FEM calculation of copper lossesNumerical model is based on FEM (Bastos and Sadowski, 2003). Pre-processing, transientmagnetic analysis and post-processing are carried out with programme package Maxwell(Programme tool Maxwell, 2008). Figure 4 shows the 2D finite element (FE) model of thewelding transformer. The transient solver in Maxwell supports the coil terminals andwinding definitions. It is possible to define coil terminals in models, which is necessary forthe connection with an external circuit. Instantaneous power p is calculated by using


TR1 Resistance of primary winding 45.58 mVTR2 Resistance of primary winding 94.86 mVTR3 Resistance of primary winding 146.32 mVTR1, TR2, TR3 Operating temperature of primary winding 1308CTR1, TR2, TR3 Operating temperature of secondary winding 508C

Table II.Winding resistances andtemperatures of the testtransformers

Figure 4.Welding transformerMaxwell 2D FE model

Axis of symmentry

Primary wirecross-section

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Helmholtz equation (Vandelac and Ziogas, 1988) whereas copper losses PCu are calculatedwith FEM by equation (12):

PCu ¼1

T

Z T

0

pðtÞdt; ð12Þ

where p is the sum of instantaneous power losses in primary and secondary windings andT is a corresponding period. Transient analysis is more accurate for the 2D model, becausethe number of FEs (50,000) is proportionally high in comparison to the 3D model (500,000).However, the 3D model, although closer to the actual physical model, is left off.

Owing to the skin concentration of current, AC inductances and resistances are not equalto their DC equivalents (Ferreira, 1992; Reatti and Kazimierczuk, 2002). The well-knownfact is that impact of the skin effect on inductances and resistances becomes more obviouswith the increasing frequency (Ida and Bastos, 1997). In non-conducting regions, magneticfield is computed from the magnetic scalar potential. In conductive regions, magnetic vectorpotential is used for field calculation. The eddy current field solver calculates the eddycurrents by solving the field equation (13):

7 £1

mð7 £AÞ ¼ ðsþ jv1Þð2jvA2 7FÞ; ð13Þ

whereA is magnetic vector potential andF is electric scalar potential, respectively. Currentdensity is calculated from magnetic vector potential.

The parameters are set up as follows: number of simulation steps is 500 andadaptive mesh control is included. The FE transformer model is placed in a three timeslarger surrounding region. Sufficiently large outer region is very important for the FEMcorrect and accurate calculation. A calculation performed on the quad-core PC lasts forapproximately 4 h.

IV. Experimental measurements of copper lossesMeasurements have been carried out on three test models TR1, TR2 and TR3 (Table I).Figure 5 shows the experimental set up. A controller system dSPACE 1103 has beenapplied for data acquisition, pulse-width modulation and control of the inverter.The current sensors and Rogowsky coil have been used for measurements of theprimary and welding current, while differential probes have been used formeasurement of voltages. Measurements have been carried out for frequency rangefrom 1 up to 5 kHz with the step of 0.5 kHz.

The goal has been to keep the same welding current during all measurements. Onlyin that case, the copper losses could have been comparable for different wire dimensionsand different frequencies. Copper losses have been calculated with followingequation (14):

pCu ¼ i1 · u2 i2L ·R2 pcore 2 udiode · iL; ð14Þ

where pCu are the total instantaneous copper losses, i1 is measured primary current, u isthe measured primary voltage, iL is the measured output welding current, R is theohmic resistance of the load, pcore are core losses and udiode is diode voltage drop.The welding current iL depends on the amplitude of the input PWM voltage.Unfortunately, due to the equipment limitations, the amplitude could not have been


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increased at higher frequencies. Therefore, the welding current amplitude has beenset up with the corresponding width of the primary voltage. In comparison to thecopper losses, core losses are relatively low (core losses are 4 per cent of copperlosses). However, core losses have been measured as well with an open circuit test(Podlogar et al., 2010).

Owing to the high welding current, the rectifier power diodes at the output have hadpower dissipation between 10 and 12 kW (Backlund and Toker, 2004). The voltage dropudiode has varied between 1 and 1.2 V, depending on the welding current and operatingfrequency. Copper losses, which had included the power dissipation of the diodes, havebeen calculated by equation (15):

PCu ¼1

T

Z T

0

pCuðtÞdt; ð15Þ

where PCu are copper losses and pCu are the instantaneous copper losses.Since that skin depth for copper equals around 2.1 mm (Garg, 2008) at the operating

frequency 1 kHz, the most inconvenient results had been expected for the tested modelTR3, which has also been confirmed.

V. ResultsThe results have been obtained analytically, numerically and experimentally. All threeapproaches carry a certain accuracy and confirmation of how the welding transformerwould behave under the different circumstances. The deviation of the results, caused bynumerical error or measurement error, is approximately 20 per cent, which refers to therelative error of the copper losses calculation.

The first set of the results shows calculated and measured primary voltage, primarycurrent and welding current (Figures 6-8).

Analytical analysis presumes the realistic input quantities (not ideal behaviour). Incases of different frequencies only the time frame is different, because the periodchanges. Note that in Figure 7, the primary current has not reached its final value,because of the transient phenomenon caused by an ohmic-inductive load.

Current density distribution determined with the analytical analysis is shown inFigure 9. The current density is shown for one certain moment t ¼ 0.14 ms along all

Figure 5.Experimental setup

Inverter

Transformer

Load

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Figure 7.Primary current i1

for f ¼ 1 kHz

Measured primary current

Primary current calculated with FEM

0

–200

–100

0

100

200

0.5 1

t (ms)

i 1 (

A)

1.5 2

Figure 6.Primary voltage u

for f ¼ 1 kHz

1,000 Measured primary voltage

Primary voltage calculated with FEM

500

0

–500

u (V

)

t (ms)

–1,0000 0.5 1 1.5 2

Figure 8.Welding current iL

for f ¼ 1 kHz

14

12

10

8

6i L(k

A)

4

2

00 2

t (ms)

Welding current calculated with FEMMeasured welding current

4


1005

layers, as they are placed physically inside the welding transformer. The information ofthe precise time t ¼ 0.14 ms is very important, because the current has just started toincrease (Figure 7, the blue line) and the amplitude has not been reached yet. Just for thecomparison, in the case of DC input current of the same amplitude (140 A), theapproximate value of the average current density in one primary coil would be between15 £ 106 A/m2 and 43 £ 106 A/m2 due to the different wire cross-section. Maximumcurrent density of the secondary wire would be approximately 45 £ 106 A/m2.

In this case, the situation is rather complicated, because the current densitydistribution is different for each moment (Vitelli, 2004; Ferkal et al., 1996). As it is shownin Figure 9, the proximity effect has the largest impact on the first primary layer, evenat the beginning of the time period (t ¼ 0.14 ms) as the current density increases uprapidly (Becker, 1982). That is because the secondary wire (with high current in kA atfrequency of 1 kHz) is placed on its right side. The current density of the secondarylayer is almost equal, because the primary layers obviously do not affect it. The thirdlayer is, however, not affected much; it keeps its original current density distribution(but scaled) because the influences balance among each other. The fourth layer isaffected by secondary and primary layer on its left side. The fifth primary layer isphysically far away from other layers since there is one more non-active secondarylayer between the fourth and the fifth primary layer (Figure 3(a)). The current densityamplitude increases for factor 10 when the input current reaches its maximum (e.g.t ¼ 0.64 ms) in comparison to the case of DC input current. Note that the scale varies,because the values are not in the same range.

The second set of the results refers to the comparison of copper losses calculationwith different methods. The meaning of the term “layer” is already explained beforewith Figure 3. Primary winding is constructed with much thinner and longerrectangular wire, which increases DC resistance, especially at higher temperatures.Owing to the lowest DC resistance, initial prediction of lower copper losses leanstowards tested welding transformer TR1.

Comparison of the results for transformers TR1, TR2 and TR3 is shown in Figure 10.Analytical analysis matches very well to the experimental results, which is the

Figure 9.2D presentation of currentdensity distribution intransformer layers forTR2

Cur

rent

den

sity

dis

trib

utio

n (A

/m2 )

0Layer 1

1.77 0 0 0Layer 2

Note: The result is shown for the certain moment t = 0.14 ms and f = 1 kHz when only the firstsecondary winding is active

Layer 3 Layer 4 Layer 5 Layer 6

2 1.8 6 2 3

Not active

× 107 × 106 × 105 × 105 × 104COMPEL30,3

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consequence of the accurately calculated both time- and space-dependent current density.FEM analysis results strongly depend on initial parameters and mesh. Therefore, theagreement among other results is the best in the case of the TR1 (Figure 10(a)). The reasonis more evenly distribute FEs, since the primary wire cross-section represents relativelylarge area in comparison to the other areas in the 2D FE model regions. Thin wire meanssmaller cross section, and, consequently, non-uniformly distributed FEs of different sizes,despite the tools, which perform the mesh refinement.

The analyzed welding transformers are not suitable for frequency higher than 1 kHz,since copper losses quickly increase with the frequency. Table III shows copper losses,which are estimated on the basis of the results of all three methods.

As it is clear from Table III, tested transformer TR1 has shown the bestperformance.

VI. ConclusionAnalysis of the methods for calculation of copper losses in the RSW transformer hasshown considerable difference among numerical and experimental results. Moreover,the difference or even illogical result has occurred when comparing numerical withanalytical results. That is why the modification and improvement of the so far developedanalytical methods is important. In such way, the research offers a generally applicable

Figure 10.Copper losses with

increasing frequency

17TR 1: rectangular wire 6 mm × 1.6 mm TR 1: rectangular wire 3 mm × 1.6 mm

TR 1: rectangular wire 2 mm × 1.6 mm

15

13

11

9

7

51 1.5 2 2.5 3 3.5 4 4.5 5 1 1.5 2 2.5 3 3.5 4 4.5 5

1 1.5 2 2.5 3f (kHz)

Notes: (a) For TR1; (b) for TR2; (c) for TR3

f (kHz)

(a) (b)

(c)

PC

u (k

w)

PC

u (k

w)

PC

u (k

w)

f (kHz)

3.5 4 4.5 5

17

15

13

11

9

7

5

17

15

13

11

9

7

5

Analytical

Experiment

FEMAnalytical

Experiment

FEM

Analytical

Experiment

FEM


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analytical method. The improved analytical method considers a time-dependent currentdensity distribution as well as the dynamic of the analyzed problem. Excellentagreement among measured and analytically calculated copper losses throughout awide range of the operating frequency fully confirms the correctness of the proposedanalytical method.

VII. Future researchMinimizing the size of the transformer and higher operating frequency are the maingoals for the future research. The new winding design has already been proposed.The results are very promising pointing out the possibility of increasing the frequencyat least to 5 kHz without major changes in the RSW system.

References

Backlund, B. and Toker, O. (2004), Designing Large Rectifiers with High Power Diodes, ABBSwitzerland Ltd Semiconductors, Lenzburg, Product information.

Bastos, J.P.A. and Sadowski, N. (2003), Electromagnetic Modelling by Finite Element Methods,Marcel Dekker, New York, NY.

Becker, R. (1982), Electromagnetic Field and Interactions, Dover, Mineola, NY.

Dowell, P.L. (1966), “Effects of eddy currents in transformer windings”, Proc. Ins. Elec. Eng., pt.B, Vol. 113 No. 8, pp. 1387-94.

Dwight, H.B. (1922), “Skin effect and proximity effect in tubular conductors”, TransactionsA.I.E.E., Vol. 41, pp. 189-918.

Evans, L.C. (2010), “Partial differential equations, graduate studies in mathematics”, AmericanMathematical Society, Vol. 19.

Ferkal, K., Poloujadoff, M. and Dorison, E. (1996), “Proximity effect and eddy current losses ininsulated cables”, IEEE Tran. on Power Delivery, Vol. 11 No. 3, pp. 1171-8.

Ferreira, J.A. (1992), “Analytical computation of AC resistance of round and rectangular Litz wirewindings”, IEEE Proceedings-B, Vol. 139 No. 1, pp. 21-5.

Garg, R. (2008), Analytical and Computational Methods in Electromagnetics, Artech House,Norwood, MA.

Grover, F.W. (2004), Inductance Calculations, Dover, New York, NY.

Ida, N. (2004), Engineering Electromagnetics, 2nd ed., Springer, New York, NY.

Ida, N. and Bastos, J.P.A. (1997), Electromagnetics and Calculation of Fields, 2nd ed., Springer,New York, NY.


TR1 Copper losses 6 kWPrimary current peak value 120 AWelding current 11 kA



Table III.Copper losses

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Kalluri, D.K. (2010), Electromagnetics of Time Varying Complex Media, 2nd ed., Taylor &Francis, London.

Kazimierczuk, M.K. (2009), High-Frequency Magnetic Components, Wiley, New York, NY.

Klopcic, B., Dolinar, D. and Stumberger, G. (2008), “Advanced control of a resistance spotwelding system”, IEEE Tran. on Power Electronics, Vol. 23 No. 1, pp. 144-52.

Podlogar, V., Klopcic, B., Stumberger, G. and Dolinar, D. (2010), “Magnetic core model of amidfrequency resistance spot welding transformer”, IEEE Tran. on Magnetics, Vol. 46No. 2, pp. 602-5.

Powers, D.L. (2010), Boundary Value Problems and Partial Differential Equations, 6th ed.,Elsevier, London.

Programme tool Maxwell (2008), Programme tool Maxwell 2D and Maxwell Circuit Editor,Ansoft, Canonsburg, PA.

Reatti, A. and Kazimierczuk, M.K. (2002), “Comparison of various methods for calculating theAC resistance of inductors”, IEEE Tran. on Magnetics, Vol. 38 No. 3, pp. 1512-18.

Riley, K.F., Hobson, M.P. and Bence, S.J. (2006), Mathematical Methods for Physics andEngineering, 3rd ed., Cambridge University Press, New York, NY.

Robert, F., Mathys, P. and Schauwers, J.P. (1998), “Ohmic losses calculation in SMPStransformers: numerical study of Dowell’s approach accuracy”, IEEE Tran. on Magnetics,Vol. 34 No. 4, pp. 1255-7.

Shadowitz, A. (1988), The Electromagnetic Field, Dover, Mineola, NY.

Sippola, M. and Sepponen, R.E. (2002), “Accurate prediction of high-frequencypower-transformer losses and temperature rise”, IEEE Tran. on Power Electronics,Vol. 17 No. 5, pp. 8351-847.

Stratton, J.A. (2007), Electromagnetic Theory, Wiley, Hoboken, NY.

Sullivan, C. (1999), “Optimal choice for number of strands in a Litz-wire transformer winding”,IEEE Tran. on Power Electronics, Vol. 14 No. 2, pp. 283-91.

Vandelac, J. and Ziogas, P.D. (1988), “A novel approach for minimizing high-frequency copperlosses”, IEEE Tran. on Power Electronics, Vol. 3 No. 3, pp. 266-77.

Vitelli, M. (2004), “Numerical evaluation of 2D proximity effect conductor losses”, IEEE Tran.on Power Delivery, Vol. 19 No. 3, pp. 1291-8.

About the authorsJelena Popovic received the BSc degree in Electrical Engineering in 2007 from Faculty of ElectricalEngineering and Computer Science, University of Maribor, where she is currently a TeachingAssistant and Junior Researcher. She is enrolled in a PhD programme. Her PhD thesis includeselectromagnetic effects in devices based on analytical equations and numerical analysis. The mainresearch interest is, however, application of solutions to the actual problems in industry.Jelena Popovic is the corresponding author and can be contacted at: [email protected]

Drago Dolinar (M’82) received the BSEE, MSc and PhD degrees in Electrical Engineering fromthe University of Maribor, Maribor, Slovenia, in 1978, 1980 and 1985, respectively. Since 1981, hehas been with the Faculty of Electrical Engineering and Computer Science of the University ofMaribor where he is currently a Professor. His current research interests include modelling andcontrol of electrical machines. Drago Dolinar is a member of the Institution of Engineering andTechnology (IET), London, UK, ICS, the Conseil International des Grands Reseaux Electriques(CIGRE) and the Slovenian Simulation Society (SloSim).

Gorazd Stumberger (M’92) was born in 1964 in Ptuj, Slovenia. He received the BSEE, MSc andPhD degrees in Electrical Engineering from the University of Maribor, Maribor, Slovenia, in 1989,


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1992 and 1996, respectively. In 1989, he joined the Faculty of Electrical Engineering andComputer Science, University of Maribor, where he is currently a Professor. His current researchinterests include design, modelling, analysis and control of electrical machines. Stumberger is amember of the Conseil International des Grands Reseaux Electriques (CIGRE) and the ICS.

Beno Klopcic received the BSc degree in Electrical Engineering from the University inLjubljana, Slovenia, in 1985 and the MSc and PhD degrees in Electrical Engineering from theUniversity of Maribor, Maribor, Slovenia, in 2005 and 2007, respectively. He is currently aResearcher with Indramat elektromotorji, Rexroth Bosch Group, Skofja Loka, Slovenia. Hisresearch interests are mainly concerned with the design and analysis of electrical machines anddesign and analysis of power electronic converters.

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