Magnetic equivalent circuit of MF transformers: modeling

© 2018 Springer

Springer Nature, Electrical Engineering, Vol. 100, No. 4, pp. 2261-2275, December 2018

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameter Uncertainties

T. Guillod,F. Krismer,J. W. Kolar

This is a post-peer-review, pre-copyedit version of an article published in Electrical Engineering. The final authenticated version is available online at: https://doi.org/10.1007/s00202-018-0701-0

Electrical Engineering (2018) 100:2261–2275https://doi.org/10.1007/s00202-018-0701-0

ORIG INAL PAPER

Magnetic equivalent circuit of MF transformers: modeling andparameter uncertainties

Thomas Guillod1 · Florian Krismer1 · Johann W. Kolar1

Received: 9 November 2017 / Accepted: 17 May 2018 / Published online: 30 May 2018© Springer-Verlag GmbH Germany, part of Springer Nature 2018

AbstractMedium-frequency (MF) transformers are extensively used in power electronic converters. Accordingly, accurate modelsof such devices are required, especially for the magnetic equivalent circuit. Literature documents many different methodsto calculate the magnetizing and leakage inductances of transformers, where, however, few comparisons exist betweenthe methods. Furthermore, the impact of underlying hypotheses and parameter uncertainties is usually neglected. This paperanalyzes nine differentmodels, ranging from simple analytical expressions to 3D detailed numerical simulations. The accuracyof the different methods is assessed by means of Monte Carlo simulations and linearized statistical models. The experimentalresults, conducted with a 100 kHz/20 kW MF transformer employed in a 400VDC distribution system isolation, are inagreement with the simulations (below 14% inaccuracy for all the considered methods). It is concluded that, consideringtypical tolerances, analyticalmodels are accurate enough formost applications and that the tolerance analysis can be conductedwith linearized models.

Keywords Power electronics · Medium-frequency transformers · Equivalent circuit · Leakage · Magnetizing · Finite elementmethod · Measurement · Uncertainty · Tolerance · Monte Carlo simulation

1 Introduction

Many power electronic converters feature MF transformersfor voltage transformation, impedance matching, and gal-vanic isolation [1–4]. MF transformers are combining highefficiency, high power density, and fast dynamic. The analy-sis of such devices can be split into several categories:

– Magnetic: A magnetic equivalent circuit is usually usedto describe the magnetizing flux, the leakage flux, andthe voltage transfer ratio of a transformer [5–7].

– Electric:The electric field computation allows the extrac-tion of the parasitic capacitances and the insulationcoordination [4,8].

– Losses:Different methods exist for computing the wind-ing losses at MF (e.g., due to skin and proximityeffects) [6,9] and the core losses (e.g., due to hysteresisand eddy current losses) [6,10]. Additionally, the losses

B Thomas [email protected]

1 Power Electronic Systems Laboratory, ETH Zürich, Zürich,Switzerland

in the cooling system [1], the losses inside shielding ele-ments [4], and the dielectric losses [8] can be significantfor some designs.

– Thermal: A thermal model of the transformer and thecorresponding cooling system is used for the extractionof the temperature distribution [1,6].

These different categories are physically interconnected,and it appears that the extraction of the magnetic parametersis particularly important for determining the applied currents,the losses, parasitic resonances, etc.

Figure 1a illustrates the different magnetic fluxes, whichare defining the flux linkages of a transformer. The resultinginductance matrix, cf. Fig. 1b, can be expressed with equiv-alent circuits, which are analyzed in detail in “Appendix A.”Typically, the following magnetic parameters are used tocharacterize a transformer [6,11]:

– Voltage transfer ratio: The voltage transfer ratio is theratio between the primary and secondary voltages of atransformer. For a transformer with a high magnetic cou-pling factor, the voltage transfer ratio is almost equal tothe turns ratio and load independent (near the nominalload).

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2262 Electrical Engineering (2018) 100:2261–2275

(a)

|H| [

a.u.

]

|B| [

a.u.

]

Equivalent circuits

vs

is

vp

ip

vs

is

vp

ip

(b)

coreN

pi p

Npi p

Nsi s

Nsi s

Field patterns

Fig. 1 a Core-type transformer field patterns during rated operatingcondition: magnetic flux density inside the core (magnetizing flux) andmagnetic field in the winding window (leakage flux). b Inductancematrix with the corresponding (lossless) linear equivalent circuits

– Transformer magnetization: A magnetizing current isrequired for generating the flux inside the core of thetransformer (cf. Fig. 1a). The magnetizing current isrelated to the open-circuit inductances (self-inductances)of the transformer, which are measured from the primaryand secondary sides. The open-circuit inductances areimportant for determining the core losses and the satu-ration current [12]. The magnetizing current can also beused for the converter operation, typically for achievingzero-voltage switching (ZVS) [4,13].

– Transformer leakage: The leakage flux is the magneticflux which does not contribute to the magnetic couplingbetween thewindings (cf. Fig. 1a). For a transformerwithahighmagnetic coupling factor, the leakageflux is relatedto the short-circuit inductances,which aremeasured fromthe primary and secondary sides. The short-circuit induc-tances are important for determining the winding lossesand the short-circuit impedance of the transformer [5].Moreover, the short-circuit inductances can be used as aseries inductor for the power converter operation [14,15].

Therefore, the extraction of an accurate equivalent circuitis critical for the design and optimization of transformers.Inaccurate values can lead to increased losses [1,16] or toimproper converter operation [17]. The magnetic parametersof the transformer can also be used for diagnosis (productionand aging) [18]. This indicates that the selection of the mostsuitable (concerning accuracy, modeling cost, and computa-tional cost) computation method for the equivalent circuit isa challenging and important task.

Literature identifies different methods for the computa-tion of the equivalent circuit inductances: approximationsof the air gap fringing field [6,12,19,20], core reluctancecomputation [5,6], analytical computation of the leakagefield [5,6,21–23], semi-analytical computation of the leak-age field [5,7], and numerical simulations of the magneticfield distribution [5,21,24]. A detailed review of the existingmethods can be found in [25,26].

Table 1 MF transformer parameters

Parameter Value

Power P = 20.0 kW / S = 22.8 kVA

Excitation 400V (DC links) / 57A (RMS) / 100 kHz

Windings 6:6 shell-type / 2500 × 100µm HF litz wire

Insulation Mylar / 1 kV (RMS, CM) / 1 kV (RMS, DM)

Core 4 × E80/38/20 / ferrite / TDK N87 material

Air gaps 2 × 0.7mm (geometry) / 1.4mm (magnetic)

Terminations 2 × 220mm (parallel wires)

Volume/weight 80mm × 82mm × 154mm / 1.0 dm3/2.4 kg

Performance 99.65% @ 20 kW / 99.60% @ 10 kW

Nevertheless, deviations between the measured valuesand the expected values can arise from model inaccuracies,geometrical tolerances, material parameter tolerances, andmeasurement uncertainties. However, only few comparisonsbetween the methods can be found [25,27], where the impactof tolerances and uncertainties is ignored.

This paper analyzes the impact of tolerances on themagnetic equivalent circuit of MF transformers, which areemployed in higher power (several 1 kW to several 100kW)converter systems, for different computation methods. Sec-tion2defines the parameters of the considered100kHz/20kWMF transformer. Section 3 discusses nine different modelsfor computing the magnetic parameters: standard analyticalmethods (e.g., Rogowski and McLyman factors) [6,12,22],more elaborate semi-analytical methods (e.g., Schwarz–Christoffel mapping, mirroring method) [5,7,19], and finiteelement method (FEM) models (2D and 3Dmodels with dif-ferent levels of detail). Section 4 summarizes two approaches(Monte Carlo simulations and linearized statistical models)for assessing the impact of tolerances on the equivalent cir-cuit. Finally, Sect. 5 presents simulation and measurementresults for the considered MF transformer, which concludesthat simple analytical methods are very accurate and that lin-earized tolerance analyses are valid.

2 Considered transformer

Figure 2a depicts the considered 400V/100kHz/20kW DC–DC series-resonant LLC converter. This converter providesa galvanic isolation for a section of a 400V DC distribu-tion grid, e.g., for the next-generation datacenters [28,29].The converter is operated at the resonance frequency with50% duty cycle and acts as a “DC transformer,” i.e., advan-tageously provides voltage transformation nearly indepen-dently of the load conditions [14].

Figure 2b shows the considered MF transformer equiv-alent circuit and the corresponding short-circuit and open-

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Electrical Engineering (2018) 100:2261–2275 2263

400

V

(a)

154 mm

400

+400

Time [ s]

70

+70

0 5 10

v [V

]i [

A]

0

0

(c)

400

V

(b) (d)

vs

is

vp

ip

Np : Ns

Lp-M Ls-M

MLoc,p

Lp-M Ls-M

MLsc,p

Lp-M Ls-M

M Loc,s

Lp-M Ls-M

M Lsc,s

vpvs

ipis

ip+is

Fig. 2 a Considered 400V/100kHz/20kW DC–DC series-resonantLLC converter. bMF transformer equivalent circuit with the definitionsof the open-circuit and short-circuit inductances. c Voltage and current

waveforms for operation at the resonance frequency, ensuring a nearlyload independent voltage transfer ratio. d Constructed MF transformerprototype (cf. Table 1)

circuit inductances. The leakage inductance of the MF trans-former is part of the converter resonant tank [14,15], and themagnetizing inductance is used for achieving ZVS [13,30].Figure 2c shows the voltage and current waveforms appliedto the MF transformer. Since the leakage and magnetizinginductances of the MF transformer are actively used for theconverter operation, anymismatchbetween themeasured andsimulated values will lead to non-optimal operating condi-tions or will require an adaptation of the modulation scheme(e.g., dead times, switching frequency).

Figure 2d depicts the constructed prototype, and the keyparameters are listed in Table 1. A large magnetizing currentis required for achieving complete soft switching [13,30].Accordingly, two air gaps are introduced into the magneticcircuit, which, due to the associated fringing field, compli-cates the computation of the equivalent circuit. The air gapsalso help the flux balancing between the paralleled core sets.Otherwise, this MF transformer features the most commondesign choices: E-shaped ferrite core, shell-type windings,and high-frequency (HF) litz wires [1–4]. However, the pre-sented methods can also be adapted to designs employingnanocrystalline cores, core-type windings, solid wires, foilconductors, etc.

Figure 3 shows the magnetic field and magnetic flux den-sity during short-circuit and open-circuit operations. Forshort-circuit operation, the energy is mostly stored in thewinding window and in the cable terminations. For open-circuit operation, the energy is stored in the air gaps and insidethe core. This indicates that the considered MF transformerfeatures a high magnetic coupling factor. In accordance,different methods can be used to extract the magnetic param-eters of the transformer.

3 Computationmethods

The extraction of the equivalent circuit depicted in Fig. 1bis described in this section for analytical, semi-analytical,

(a)

(d)(c)

(b)

Short-circuit

Short-circuit

Short-circuit

Open-circuit

0.0

1.0

|H|/

|Hm

ax| [

p.u.

]

0.0

1.0

|H|/

|Hm

ax| [

p.u.

]

0.0

1.0

|H|/

|Hm

ax| [

p.u.

]

0.0

1.0

|B|/

|Bm

ax| [

p.u.

]

side cut view

top cut view

front cut view

front cut view

Fig. 3 a Side view, b front view, and c top view of the magnetic fieldduring short-circuit operation. d Front view of themagnetic flux densityduring open-circuit operation. All the cut planes are intersecting at theMF transformer center point

and numerical methods. Additional information about theextraction of the equivalent circuit is given in “Appendices B,C, D.”

3.1 Analytical methods

Case: ip = 0 ∨ is = 0

First, the inductance is computed for the following oper-ation condition: ip = 0∨ is = 0. This computation is similarto the computation of inductors, which implies that the pre-sented methods can also be applied to inductors. Most of theanalytical methods are based on reluctance circuits:

– The core is modeled with a single reluctance, and thefringing field of the air gaps is neglected (single equiva-lent reluctance), cf. Fig. 4a [6].

– The core is modeled with a single reluctance, andthe McLyman factor (simple semi-empirical model) is

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(c)(b)(a) NiR c

ore,

lR g

ap,l

R cor

e,c

R gap

,c

R cor

e,r

R gap

,r

Ni

R gap

,l

R cor

eR g

ap,c

R gap

,r

NI

R gap

R cor

eNo fringing McLyman factor Schwarz-Christoffel mapping

Fig. 4 Reluctance circuits (shown for E-shaped gapped core). a Single-element core model without fringing field model [6], b single-elementcore model with a semi-empirical fringing field model (McLyman fac-tor) [12], and c detailed core model with a detailed fringing field model(Schwarz–Christoffel mapping) [5,19]. The red lines indicate the direc-tion of the magnetic flux density (streamlines)

used for describing the fringing field of the air gaps,cf. Fig. 4b [12].

– The core is described with a detailed reluctance model(limbs, yokes, and corners), and a 3D model of thefringing field is used (based on Schwarz–Christoffelmapping), cf. Fig. 4c [5,19]. However, the interactionsbetween the different air gaps and the placement of thewires are not considered.

The main hypothesis of these reluctance models is thata perfect magnetic coupling between the turns compos-ing a winding (same flux through all the turns) is given(cf. Fig. 3d). This implies that these computationmethods arelimited to transformers with high magnetic coupling factors.If required, the nonlinearities of the core material (flux den-sity and frequency dependences) can be considered for thereluctance computations, especially for designs with smallair gaps [12].

Case:+Npip = −Nsis

In a second step, the inductance is computed for the fol-lowing excitation: +Npip = − Nsis. With this operatingcondition, most of the energy is stored in the winding win-dow (cf. Fig. 3b) and the following computationmethods canbe used:

– A1Dapproximation of the leakagefield is computedwiththe Ampère’s circuital law, cf. Fig. 5a [6,7]. This modelis inaccurate for windings with a low fill factor, for largedistance between the windings and/or the core, and formodeling the leakage near the winding heads [7,16].

1D approx. Mirroring methodRogowski factor

(a) (b)

(c)(d)

Termination

Fig. 5 Leakage magnetic field distribution (shown for a 6:6 shell-typewindings). a 1D approximation (Ampère’s circuital law) [6,7], b 2Dapproximation of the field near the winding tips (Rogowski factor) [22],c 2D computation (mirroring method) [7,31], and d cable terminationfield (Biot–Savart law). The red lines indicate the direction of the mag-netic field (streamlines)

– The2Deffects occurringnear thewinding tips, cf. Fig. 5b,can be approximated with the Rogowki factor, which isa correction factor for the height of the windings [22].

– The 2D field distribution can be computed with themirroring method (method of images) where the effectof the magnetic core is replaced by current images,cf. Fig. 5c [7,31,32]. The equivalent inductance can beextracted from the energy stored in the field [32], or,more conveniently, directly from the inductance matrixbetween the wires [33,34]. The boundary conditions nearthe winding heads and the finite core permeability can beconsidered [7]. However, this method is only accurate ifthe leakage field can be approximated in 2D (cf. Fig. 3b,c).

– For a transformer with a reduced leakage inductance,the energy stored inside the cable terminations cannot beneglected [35]. For calculating the corresponding induc-tance, the inductance of two parallel wires (Biot–Savartlaw) can be used, cf. Fig. 5d [36]. Nevertheless, the fielddistribution near the cable terminations is intrinsically a3D problem, which cannot be perfectly modeled analyt-ically (cf. Fig. 3a).

If required, the aforementioned methods can be extendedto other transformer geometries (e.g., core-type windings,three-phase windings, interleaved windings) [23] or fortaking the frequency dependencies of the short-circuit induc-tances into account (especially for designwith solidwires) [21].

Circuit extraction

The equivalent circuit can be extracted from the obtainedvalues (ip = 0 ∨ is = 0 and + Npip = − Nsis) such

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that the energy stored in the equivalent circuit matches withthe computed cases. This procedure is explained in detailin “Appendix B.” It should be noted that the analyticalmethods are inaccurate for transformers with low couplingfactors (k < 0.95), where the aforementioned hypothesis(e.g., reluctance circuit, energy distribution) are not validanymore (cf. “Appendix D”).

3.2 FEM simulations

Circuit extraction

The equivalent circuit of a transformer can also be obtainedfrom numerical simulations, where FEM is typically selec-ted [2–4,37]. The exact geometry, the nonlinearities of thecore material, and the frequency dependences of the cur-rent distribution (e.g., skin and proximity effects) can berepresented. For extracting the equivalent circuit, which fea-tures three degrees of freedom, different simulations arerequired. In this paper, the extraction of the equivalent circuitis done with the following numerically stable procedure. Theenergy inside the system is extracted for the following cases:ip �= 0∧ is = 0 (energy in the primary self-inductance, Lp),ip = 0 ∧ is �= 0 (energy in the secondary self-inductance,Ls), and+Npip = − Nsis (approximately, for high magneticcoupling factors, the energy in the leakage magnetic field).Then, the equivalent circuit is determined such that the storedenergymatcheswith the simulations (cf. “AppendixC”). Thismethod is also applicable to transformers with low magneticcoupling factors.

2Dmodeling

Figure 6a depicts a 2D FEMmodel with a simplified windingstructure, and Fig. 6b considers the actual placement of thewires. For both cases, a planar and an axisymmetricmodel areused for modeling the core window and the winding heads,respectively (E-shaped core, shell-type windings). Differentsymmetry axes are used for reducing the size of the model.

3Dmodeling

Figure 7a shows a simple 3D model with simplified core andwinding structures (single bodies). Figure 7b considers theexact shape of the cores (core sets) and the different wires.These two models neglect the pitch of the windings and thecable terminations, such that different symmetry planes exist.Finally, Fig. 7c represents a full 3D model of the transformerand the cable terminations, which does not feature any sym-metry plane.

(b)

(a)

2D FEM simple

2D FEM complexeven sym.

rot.

sym

.

winding heads

odd

sym

.

even sym.

core window

even sym.od

d sy

m.

core window

even sym.

rot.

sym

.

winding heads

Fig. 6 a 2DFEMmodel with a simplifiedwinding geometry and bwitha detailed winding geometry. For both cases, a different model is usedfor the core window and the winding heads. The symmetry axes arealso indicated

4 Statistical analysis

The aforementioned computation methods for the short-circuit (Lsc,p and Lsc,s) and open-circuit (Loc,p and Loc,s)inductances (cf. Fig. 2b and Sect. 3) are subject to theuncertainties of the used geometrical and material parame-ters. Therefore, the question arises if accurate calculationsare actually necessary. However, the measurement of theuncertainties requires the construction of many identicaltransformers and therefore cannot be conducted during thedesign of a transformer. Accordingly, in the following, thetolerance analysis is conductedwith the computationmodels.

4.1 Considered parameters

The different uncertain parameters are named xi , the toler-ances ± δi , and the nominal values xi,0. These parametersrepresent the uncertainties linked with the geometrical tol-erances and the material parameter tolerances. For thestatistical analysis, the following expressions are considered:

L = FL (x1, x2, . . .) , (1)

L0 = FL(x1,0, x2,0, . . .

), (2)

where L is the computed inductance, L0 the nominal value,and FL the function describing the inductance computation

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(a)(a

(b)

(c)

3D FEM simple

3D FEM intermediate

3D FEM complex

even

odd

odd

odd

odd

odd

odd

even

Fig. 7 a 3D FEM model with a simplified winding and core geometry,bwith a more detailed winding and core geometry, and cwith the actualtransformer geometry (with the cable terminations). The consideredsymmetry planes are also indicated

Table 2 Uncertain Parameters (input)

Parameter

nx Number of uncertain parameters

xi Uncertain value of the parameter

xi,0 Nominal value of the parameter

± δi Tolerance around xi,0

Worst-case analysis

fi,wc Probability density function (uniform distribution)

MCi,wc Monte Carlo samples

Normal distribution analysis

fi,nd Probability density function (normal distribution)

σi,nd Standard deviation of fi,nd

pi,nd Confidence interval for σi,nd

MCi,nd Monte Carlo samples

method (cf. Sect. 3). Tables 2 and 3 summarize the vari-ables used to describe the uncertain input parameters and theinductances, respectively.

Table 3 Uncertain Parameters (output)

Inductance

L Uncertain value of the inductance

L0 Nominal value of the inductance

FL Function describing the computation method

CL Computational cost of FL

Lmeas Nominal value of the measured inductance

{L}meas Measurement tolerance interval

Worst-case analysis

± δL,wc Tolerance around L0 (linearized analysis)

MCL,wc Monte Carlo samples

{L}lin,wc Tolerance interval (linearized analysis)

{L}MC,wc Tolerance interval (Monte Carlo simulations)

Normal distribution analysis

fL,nd Probability density function (linearized analysis)

σL,nd Standard deviation of fL,nd (linearized analysis)

pL,nd Confidence interval for σL,nd (linearized analysis)

± δL,nd Tolerance around L0 (linearized analysis)

MCi,nd Monte Carlo samples

{L}lin,nd Tolerance interval (linearized analysis)

{L}MC,nd Tolerance interval (Monte Carlo simulations)

In the following, a simple exemplary function FL isconsidered for explaining the tolerance stacking analysis.Afterward, in Sect. 5, the defined analysis framework isthen used for analyzing the tolerances for the open-circuitand short-circuit inductances of the considered transformer(cf. Sect. 2), using the aforementioned computation methods(cf. Sect. 3).

4.2 Worst-case analysis

For a worst-case analysis, the variables xi are assumed tobe uniformly distributed in the tolerance intervals and inde-pendent of each other. The probability density distributionfunction, fi,wc, can be set to

fi,wc (xi ) = funiform(xi , xi,0 − δi , xi,0 + δi

), (3)

where funiform describes the probability density distributionfunction of the continuous uniformdistribution.With the helpof FL (cf. (1)), the probability distribution of L can be com-puted and the extrema can be extracted, cf. Fig. 8a.

For the magnetic equivalent circuit, many variables arepresent, FL is nonlinear, and FL , for some computationmeth-ods, cannot be expressed analytically. For these reasons, theprobability density distribution function is obtained bymeansof Monte Carlo simulations [38]. In the course of the MonteCarlo simulations, a large number of random parameter com-binations is generated with respect to the input probability

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(a)

Normal distributionUniform distributionp(

x 1 )

p(x 2

)p(

L)

Input 1: x1

Input 2: x2

Output: L Output: L

Input 2: x2

Input 1: x1

p(x 1

)p(

x 2 )

p(L

)

(b)

x1,0

MC1,wcf1,wc

x2,0

MC2,wcf2,wc

x1,0

MC1,nd

f1,nd

x2,0

MC2,nd f2,nd

L0

MCL,wc MCL,nd

fL,nd

2 1 2 1

2 2 2 2

{L}lin,wc {L}MC,wc {L}MC,nd{L}lin,nd L0

Fig. 8 a Worst-case analysis with uniform distributions. b Statis-tical analysis with normal distributions. Monte Carlo simulationsand linearized computations are compared. The following exemplaryparameters are used: nx = 2, x1 ± δ1 = 0 ± 1.0, x2 ± δ2 = 0 ± 1.5,FL = x1 + ((x2/2) − 0.2)2. The confidence intervals (pi,nd and pL,nd)are set to 95%. The number of Monte Carlo samples is 104

distributions (cf. (3)) and the corresponding inductance val-ues are computed. Then, it is straightforward to extract theinterval, {L}MC,wc, where the inductance value is fluctuating.

However, Monte Carlo simulations require the computa-tion of a large number of designs. Alternatively, the functionFL can be linearized near the nominal values of the differentparameters, which leads to [39]

δL,wc =n∑

i=1

∣∣∣∣∣∂FL

∂xi

∣∣∣∣xi,0

δi

∣∣∣∣∣(4)

{L}lin,wc = [L0 − δL,wc, L0 + δL,wc], (5)

where δL,wc is the obtained tolerance and {L}lin,wc the cor-responding interval. Beside the reduced computational cost,the linearization allows the analysis of the sensitivity of theinductance value with respect to the different parameters. Ifthe linearization is valid, the ranges {L}MC,wc and {L}lin,wcshould match together. In Fig. 8a, some differences can beobserved due to the strong nonlinearities of the consideredexemplary function FL .

4.3 Normal distribution analysis

The aforementioned worst-case analysis is often too conser-vative. For this reason, normal (Gaussian) distributions areoften used for tolerance stacking analysis [39]. The proba-bility density distribution function, fi,nd, and the standarddeviation, σi,nd, can be set to

fi,nd (xi ) = fnormal(xi , xi,0, σi,nd

), (6)

σi,nd = δi√2erf−1

(pi,nd

) , (7)

where fnormal is the probability density distribution functionof the normal distribution and erf−1 the inverse Gauss errorfunction [40]. The confidence intervals, pi,nd, are describingthe percentage of the data lying within the accepted toler-ances±δi .With these definitions, the probability distributionof L can be computed, cf. Fig. 8b.

Again, the probability density distribution function canbe obtained by means of Monte Carlo simulations, where thenonlinearities of FL are considered [38]. The interval, wherepL,nd percent of the values are located (confidence interval),is depicted as {L}MC,nd.

Alternatively, the linearization of FL can be conducted. Alinear combination of normal distributions results in a normaldistribution, which leads to

fL,nd (L) = fnormal(L, L0, σL,nd

), (8)

σL,nd =√√√√

n∑

i=1

(∂FL

∂xi

∣∣∣∣xi,0

σi,nd

)2

, (9)

where fL,nd (L) is the probability density distribution func-tion and σL,nd the standard deviation. With the confidenceinterval, pL,nd, the tolerance on the inductance value can beexpressed as

δL,nd = σL,nd√2erf−1 (

pL,nd), (10)

{L}lin,nd = [L0 − δL,nd, L0 + δL,nd], (11)

where δL,nd is the obtained tolerance and {L}lin,nd the cor-responding interval. In Fig. 8b, the mismatch between theprobability density distribution functionobtainedwithMonteCarlo simulations and fL,nd (L) results from the nonlineari-ties of the chosen exemplary function FL .

5 Results

The MF transformer depicted in Sect. 2 is considered. Firstthemagnetic parameters of theMF transformer aremeasured.In a second step, the aforementioned computation methods(cf. Sect. 3) and tolerance stacking analysis (cf. Sect. 4) areapplied. Finally, the measurement and computation resultsare compared.

5.1 Measurements

The open-circuit (Loc,p and Loc,s) and short-circuit (Lsc,p

and Lsc,s) inductances (cf. Fig. 2b) are extracted from small-

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103 104 105 106 107

-10

10

30

50

70

104 105 106 107 108

-20

0

20

40

60

10 30 50 70

40.0

45.0

50.0

55.0

60.0

20 80 140 200

1.45

1.50

1.55

1.60

1.65

1.70

103 104 106

55.0

56.0

57.0

58.0

58.0

103 106 107

1.45

1.50

1.55

1.60

1.65

1.70Z s

c [dB

]L s

c [µH

]

Z oc [

dB]

L oc [

µH]

L oc [

µH]

L sc [

µH]

Open-circuit / Zoc( f ) Short-circuit / Zsc( f )

Open-circuit / Loc( f ) Short-circuit / Lsc( f )

Open-circuit / Loc( ibias ) Short-circuit / Lsc( ibias )

Frequency [Hz] (b)Frequency [Hz] (a)

Frequency [Hz] Frequency [Hz] (d)(c)

(f)(e)]A[tnerruC Current [A]

105 104 105

Zoc,p

Zoc,s

Zsc,p

Zsc,s

Loc,p

Loc,s

Lsc,p

Lsc,s

Loc,p

Loc,s Lsc,p

Lsc,s

OP OP

OP OP

OPOP

Fig. 9 a Measured open-circuit and b short-circuit impedances for thesetup shown in Fig. 2d [41]. c Open-circuit and d short-circuit induc-tances for different frequencies [41]. e Open-circuit and f short-circuitsmall-signal differential inductances for different bias currents [42].The primary and secondary windings are the outer and inner windings(with respect to the central limb), respectively. Themeasurement uncer-tainties (shaded area) and the operating points (“OP,” cf. Fig. 2c) areindicated

signal impedance measurements [43]. Figure 9a, b depictsthe measured impedances, where the resonance frequenciesof the MF transformer can be seen. Figure 9c, d shows theextracted nominal inductances (Lmeas) with the correspond-ing measurement uncertainties ({L}meas). The inductancesare extracted with a resistive–inductive series equivalent cir-cuit.

These measurements are taken with an Agilent 4924Aprecision impedance analyzer [41]. The uncertainties arecomposed of the measurement device tolerances and anabsolute error. The absolute error, which is estimated frommeasurements, is set to 20 nH for short-circuit measurements(interconnection inductances) and to 100 nH for open-circuitmeasurements (interconnection inductances and slight vari-ation of the length of the air gaps in case the core halvesare pressed together). These measurements have been suc-

cessfully reproduced (less than 1% deviation) with an Omi-cron Bode 100 network analyzer [44,45].

Figure 9c, d indicates that the frequency dependences ofthe inductances are negligible due to the used ferrite cores andHF litz wires [6,21]. The slight increase of the open-circuitinductances near 1MHz is due to the capacitive currents(resonance frequency of 3.2MHz) [4]. The slight decreaseof the short-circuit inductances near 10MHz is explainedby high-frequency effects (induced currents in the wind-ings and permeability change of the core) [9,21]. However,near the operating frequency (100 kHz), the impact of thesehigh-frequency effects is negligible (constant open-circuitand short-circuit inductances below 1MHz).

However, the MF transformer is not operated with small-signal excitations. For this reason, the MF transformer hasalso been measured with a ed-k DPG10/1000A power choketester [42]. This device applies a current ramp to the MFtransformer and measures the induced voltage. This enablesthe extraction of the small-signal differential inductance(L = ∂Ψ /∂i) for different bias currents, cf. Fig. 9e, f. Theuncertainties are composed of the measurement device toler-ances and an absolute error. The absolute error is set to 50 nHand 120 nH, respectively, for short-circuit (interconnectioninductances) and open-circuit measurements (interconnec-tion inductances and variation of the length of the air gaps).

Figure 9e, f shows that the open-circuit inductances arealso linear below 40A (magnetizing current), which meansthat theMF transformer is operated well below the saturation(cf. Fig. 2c). As expected, the short-circuit inductances areindependent of the current.Due to the highmagnetic couplingfactor of the MF transformer (k = 0.986), the inductancesmeasured from the primary and secondary sides are verysimilar (Np = Ns).

For these reasons, the small-signal inductances (ibias =0A) at the switching frequency ( f = 100 kHz) are extractedfor the comparison with the computations. Moreover, onlythe averages of the primary-side (Loc,p and Lsc,p) andsecondary-side inductances (Loc,p and Lsc,p) are further con-sidered (called Loc and Lsc, cf. Fig. 2b). The obtained nomi-nal values (Lmeas) and measurement uncertainties ({L}meas)are used for comparing the measurement results with the dif-ferent computation methods.

5.2 Computations

The computation methods described in Sect. 3 are combined,as shown in Table 4, for extracting the equivalent circuit ofthe MF transformer. For the analytical models, the compu-tational cost, CL , is measured in floating point operations(FLOPs) [46]. For FEMmodels, the computational cost,CL ,is expressed in degrees of freedom (DOFs) [37], given thatthree operating points are solved for extracting the equivalentcircuit (cf. Sect. 3.2).

123


Table 4 Computation methods

Name Model CL nx

Analytic simple Figures 4a/5a 50 FLOPs 12

Analytic intermediate Figures 4b/5b 150 FLOPs 14

Analytic complex Figures 4c/5c 200 kFLOPs 16

Analytic termination Figures 4c/5c/5d 200 kFLOPs 20

2D FEM simple Figures 6a/5d 8 kDOFs 19

2D FEM complex Figures 6b/5d 12 kDOFs 20

3D FEM simple Figures 7a/5d 0.9MDOFs 17

3D FEM intermediate Figures 7b/5d 1.4MDOFs 17

3D FEM complex Figure 7c 7.2MDOFs 16

For the statistical analysis, the methods described inSect. 4 are used. The confidence intervals for the parameters(pi,nd) and for the inductances (pL,nd) are set to 95%. Thenumber of uncertain parameters, nx , is indicated in Table 4.The uncertain parameters can be classified into different cat-egories:

– The geometrical and material tolerances are mostlyextracted from the datasheets (e.g., core, coil former, andlitz wire). The uncertainties linked to the constructionof the MF transformer (e.g., air gaps, insulation dis-tances) are estimated. The main tolerances are the airgaps length (0.7 mm ± 0.05 mm), the distance betweenthe twowindings (0.75 mm±0.3 mm), thewire diameter(7.2 mm±0.2 mm), the core permeability (2200±500),and the core cross section (1600 mm2 ± 40 mm2).

– Some geometrical parameter uncertainties are also linkedto the geometrical approximations included in the mod-eling. The fact that a 3D geometry is modeled in 1D/2D(e.g., winding heads and cable terminations) introducesadditional uncertainties since some geometrical parame-ters in the 1D/2D models do not exist in the 3D models.

– The simplifications used in the different computationmethods (e.g., 1D/2D modeling, no fringing field model,no cable terminationsmodel) are intrinsic inaccuracies ofthe methods and therefore are not considered as uncer-tainties.

The computational cost of the FEM models is too highfor Monte Carlo simulations. On the other hand, the simpleanalytical models (“Analytic simple,” “Analytic intermedi-ate,” and “Analytic complex”) neglectmany parameters (e.g.,placement of the wires, cable terminations). Therefore, thefull statistical analysis is performed with the model “Ana-lytic termination,” where 2 × 103 Monte Carlo samples areconsidered.

Figure 10 shows the obtained tolerances for the induc-tances, which are not negligible. It appears that the linearized

52.0 55.5 59.0 62.5 66.00.0

0.1

0.2

1.3 1.4 1.5 1.6 1.7 1.80.0

4.0

8.0

(a)

p(L s

c ) [

1/µH

]

Lsc (b)]Hµ[

Loc [µH]

p (L o

c ) [

1/µH

]

Open-circuit inductance / distribution

Short-circuit inductance / distribution

L0

MCL,nd

L0

MCL,nd

{L}MC,wc

{L}lin,wc

{L}lin,nd

{L}MC,nd

fL,nd

fL,nd

{L}MC,wc

{L}lin,wc

{L}lin,nd

{L}MC,nd

Fig. 10 a Tolerance analysis for the open-circuit and b short-circuitinductances.Worst-case andnormal distribution analyzes are performedbymeans ofMonte Carlo simulations and linearization (cf. Sect. 4). Themodel “Analytic termination” is considered (cf. Table 4)

tiucric-trohStiucric-nepO

(b)(a)

L ,w

c = 6

.87

µH

L ,w

c = 0

.22

µH

air gap length: 49%

model approx.: 13%

core size: 24%

core permeability: 14%

pri./sec. insulation: 36%

model approx.: 25%

termination: 7%

turn insulation: 13%

winding length: 9%

window size: 3%litz wire diameter: 7%

termination: 0%

Fig. 11 a Sensitivity analysis for the open-circuit and b short-circuit inductances. The model “Analytic termination” is considered(cf. Table 4) and a linearized worst-case analysis is applied (cf. (4))

statistical analysis is valid (compared to Monte Carlo sim-ulations). The linearized analysis allows the extraction ofthe sensitivities of the different parameters (cf. (4)), whichis shown in Fig. 11. The main uncertainties are originatingfrom the air gaps length, the size of the core, the permeabilityof the core, the insulation distances, and the simplification ofthe 3D geometry (depicted as “model approx.”).

Therefore, the linearized statistical analysis is appliedto all the computation methods described in Table 4. Thisgreatly reduces the computational cost since only 2nx + 1equivalent circuit computations are required. The obtainedvalues (L0, {L}lin,wc, and {L}lin,nd) are used for comparingthe computations and the measurements.

123


Analyt

ic sim

ple

Analyt

ic ter

minatio

n

2D FEM sim

ple

2D FEM co

mplex

Agilen

t 429

4A

3D FEM sim

ple

3D FEM in

termed

iate

3D FEM co

mplex

Analyt

ic int

ermed

iate

Omicron

Bode 1

00

ed-k

DPG10/10

00A

20

10

0

+10

+20

Open-circuit / comparison / Loc,ref = 56.090 µHD

evia

tion

[%]

L oc [

µH]

0

30

60

Analyt

ic co

mplex

refe

renc

e

{L}lin,nd

{L}lin,wc

{L}meas

L0

Lmeas

Fig. 12 Comparison between measurements (cf. Sect. 5.1) and com-putations (cf. Sect. 5.2) for the open-circuit inductance (Loc). Thereference value is the measurement obtained with an Agilent 4294Aprecision impedance analyzer [41]

5.3 Comparison: measurements and computations

Figures 12 and 13 show the comparison betweenmeasure-ments and computations for the open-circuit and short-circuitinductances, respectively. The most accurate measurements,i.e., the values obtained with the Agilent 4294A precisionimpedance analyzer [41], are selected as reference values forthe extraction of the relative errors. The deviations betweenthe nominal values and the reference value are smaller than14% for all the methods.

For the open-circuit inductance, the method “Analyticsimple” (cf. Table 4) underestimates the inductance, whichis due to the neglected fringing field of the air gaps. For allthe other methods, the tolerances ({L}lin,wc and {L}lin,nd)match with the measurement uncertainties. The methodsbased on Schwarz–Christoffel mapping (“Analytic complex”and “Analytic termination”) are in good agreement with the“3D FEM complex” method (1% error).

For the short-circuit inductance, the methods whichneglect the cable terminations (“Analytic simple,” “Ana-lytic intermediate,” and “Analytic complex”) underestimatethe inductance. For all the other methods, the tolerances({L}lin,wc and {L}lin,nd) match with the measurement uncer-tainties. The method “Analytic termination” is also in goodagreement with the “3D FEM complex” method (5% error).

Therefore, it can be concluded that the complex computa-tion methods (2D and 3D FEM models) only feature limitedadvantages over the analytical methods. The main advantage

Analyt

ic sim

ple

Analyt

ic ter

minatio

n

2D FEM sim

ple

2D FEM co

mplex

Agilen

t 429

4A

3D FEM sim

ple

3D FEM in

termed

iate

3D FEM co

mplex

Analyt

ic int

ermed

iate

Omicron

Bode 1

00

ed-k

DPG10/10

00A

20

10

0

+10

+20

Short-circuit / comparison / Lsc,ref = 1.566 µH

Dev

iatio

n [%

]L s

c [µH

]

0.0

0.8

1.6

Analyt

ic co

mplex

30

refe

renc

e

{L}lin,nd

{L}lin,wc

{L}meas

L0

Lmeas

Fig. 13 Comparison between measurements (cf. Sect. 5.1) and com-putations (cf. Sect. 5.2) for the short-circuit inductance (Lsc). Thereference value is the measurement obtained with an Agilent 4294Aprecision impedance analyzer [41]

of the 3DFEMmethods is a slight reduction in the tolerances,since the tolerances linked with the model approximationsvanish (“model approx.”, cf. Fig. 11). The method “Analytictermination” (cf. Table 4) represents an interesting trade-offbetweenmodeling complexity, computational cost, and accu-racy.

It also appears that the measurement uncertainties aresmaller than the computation tolerances. Furthermore, thedeviation between the nominal value of the computationmethods is smaller than the tolerances. This implies that theextraction of MF transformer equivalent circuit parameterscan only be marginally improved with new computation andmeasurement methods. Only a reduction in the uncertaintieson the geometry and material parameters will lead to bettermatching between measurements and computations.

6 Conclusion

This paper first investigates various computation methodsfor extracting MF transformer magnetic equivalent cir-cuit, which is a critical parameter for the modeling andoptimization of power electronic converters. Analyticalmod-els are presented for the magnetizing (reluctance circuit,Rogowski factor, and Schwarz–Christoffel mapping) andleakage (1D Ampère’s circuital law, Rogowski factor, 2Dmirroringmethod, and cable terminationmodel) inductances.The advantages and limitations of the analytical methods are

123


discussed and compared to numerical simulations (2D and3D FEM).

In a next step, the uncertainties linked to model inac-curacies, geometrical tolerances, and material tolerancesare analyzed in detail with a statistical analysis applied tothe computation models. Computationally intensive MonteCarlo simulations and simple linearized statistical models,which allow the analysis of the sensitivities for the differentparameters, are considered.

The presented methods are finally applied for calcu-lating the short-circuit and open-circuit inductances of a100kHz/20kW MF transformer. It is concluded that a lin-earized statistical analysis is sufficient, which greatly sim-plifies the tolerance analysis. The deviations between themeasurements and thedifferent computationsmethods canbeexplained with the computed tolerances. For the open-circuitinductance, the measurement uncertainties (0.3–2.3%) aresmaller than the computation tolerances (6–12%). This isalso the case for the short-circuit inductance (2.7–7.1% and5–22%).

A comparison between the different computationmethodsshows that the combination of a reluctance circuit, Schwarz–Christoffel mapping, the mirroring method, and a cabletermination model is a good trade-off between the computa-tional cost and the obtained accuracy. Finally, it is concludedthat the consideration of the uncertainties is required forimproving the modeling of MF transformers. Moreover, theobtained uncertainties are also useful for production diag-nosis and for selecting the right margin of safety during thedesign process.

The presented statistical analysis has been conducted onthe magnetic parameters of MF transformers with compu-tational models. This work could be completed with anexperimental study involving many samples in order to mea-sure the tolerances occurring in a typical production process.Furthermore, the presented methods can be extended to theremaining transformer parameters (e.g., losses and thermalmodels) and components (e.g., switches, inductors) in orderto assess the model uncertainties of complete converter sys-tems.

Acknowledgements This project is carried out within the frame of theSwiss Centre for Competence in Energy Research on the Future SwissElectrical Infrastructure (SCCER-FURIES) with the financial supportof the Swiss Innovation Agency.

Appendices

These appendices review the theoretical background behindthe different transformer equivalent circuits and the corre-sponding pitfalls (cf. “Appendix A”). The procedure forextracting equivalent circuits from analytical (cf. “Appen-

dixB”) and numerical (cf. “AppendixC”) computation is alsopresented. Finally, some approximations are given for trans-formers with high magnetic coupling factors (cf. “Appen-dix D”).

ATransformer equivalent circuits

The magnetic equivalent circuit of a (lossless) linear trans-former with two windings is fully described by the followinginductance matrix (cf. Fig. 1b) [6,11,43]:

[vpvs

]=

[∂Ψp∂t

∂Ψs∂t

]

=[Lp MM Ls

] [∂ip∂t∂is∂t

]

, (12)

where Lp is the primary self-inductance, Ls the secondaryself-inductance, and M the mutual inductance. The induc-tance matrix features three independent parameters. Theenergy, W , stored in the transformer (quadratic form of theinductance matrix) can be computed as

W = 1

2Lpi

2p + 1

2Lsi

2s + Mipis. (13)

The energy stored in the transformer is always positive (i.e.,the inductance matrix is positive definite), which leads tothe condition M2 < LpLs. Therefore, the mutual inductancecan be expressed with a normalized parameter, the magneticcoupling,

k = M√LpLs

, with k ∈ [0, 1]. (14)

During open-circuit operation, the inductance matrix hasthe following physical interpretation. The self-inductances(Lp and Ls) represent flux linkages of the twowindings them-selves (defined as magnetizing flux linkages). The mutualinductance (M) describes the flux linkage between the wind-ings (defined as coupled flux linkages). The differencesbetween the self- and mutual inductances (Lp − M andLs − M) represent the flux linkage differences (defined asleakage flux linkages), which can, for some designs, be neg-ative (especially if Np �= Ns.

With the aforementioned inductance matrix, the induc-tances, voltage transfer ratios, and current transfer ratios canbe expressed for short-circuit and open-circuit operations(cf. Fig. 2b):

123


T circuit / PI circuit / 4 degrees of freedom

Series-parallel circuit / parallel-series circuit / 3 degrees of freedom(b)

(d)

(a)

(c)

üT

v v

iv

i

v

iL LL

v

iv

i

v

iLL L

i ü

v

iv

i

v

i

L Lü

v

iv

i

v

i

LLü

Fig. 14 Transformer (lossless) linear equivalent circuits. a T circuit,b PI circuit, c series–parallel circuit, and d parallel–series circuit. Thecircuits are referred to the primary side of the transformer

Loc,p = Lp,vs

vp= +k

√Ls

Lp, with is = 0, (15)

Loc,s = Ls,vp

vs= +k

√Lp

Ls, with ip = 0, (16)

Lsc,p = (1 − k2

)Lp,

isip

= −k

√Lp

Ls, with vs = 0, (17)

Lsc,s = (1 − k2

)Ls,

ipis

= −k

√Ls

Lp, with vp = 0. (18)

The terminal behavior (cf. (12)) and the stored energy(cf. (13)) of the transformer can be represented with dif-ferent equivalent circuits. The circuits shown in Fig. 1b aredirectly related to the inductance matrix. On the contrary,Fig. 14 depicts equivalent circuits, which do not provide adirect insight on the magnetic flux linkages. The followingimportant remarks can be given about the different equivalentcircuits of transformers:

– All the presented equivalent circuits (cf. Figs. 1b, 14)model perfectly the terminal behavior and the storedenergy.

– The equivalent circuits with more than three degrees offreedom (cf. Fig. 14a, b) are underdetermined and do nothave a physical meaning, without accepting restrictivehypotheses [11].

– The turns ratio of the transformer, Np : Ns, is not clearlydefined for some transformer geometries (e.g., inductivepower transfer coils) [47,48]. This implies that the turnsratio is not always directly related to the flux linkagesand to the magnetic parameters.

– The magnetizing and leakage fluxes cannot be spatiallyseparated. In other words, it is not always possible to sort

the magnetic field lines into leakage and magnetizingfield lines [48].

– In a transformer, a phase shift is present between theprimary and secondary currents, which originates fromthe modulation scheme, the load, and/or the losses. Thisimplies that the distribution of the magnetic field linesis time dependent [11,48]. Therefore, the leakage andmagnetizing flux linkages only have a clear interpretationfor a lossless transformer during open-circuit and short-circuit operations.

– The magnetizing and leakage inductances are usuallydefined as the parallel and series inductances in theequivalent circuit, respectively. However, the values ofthese inductances depend on the chosen equivalent cir-cuit (cf. Fig. 14) and therefore do not have a clear and/orunique physical interpretation. This also implies that theassociated magnetizing current (im) and leakage volt-age (vσ ) only represent virtual parameters, which are notdirectly measurable [11].

It can be concluded that only the equivalent circuits shownin Fig. 1b feature a clear physical interpretation and thereforeshould bepreferred.The circuitswithmore than three degreesof freedom (cf. Fig. 14a, b) should be avoided since they areunnecessarily complex. The circuits depicted in Fig. 14c, dare interesting for designing transformerswith highmagneticcoupling factors as explained in “Appendix D.”

Equivalent circuit from analytical computations

The analytical methods are based on several assumptions(cf. Sect. 3.1). The inductances are accepted to scale quadrat-icallywith the number of turns. The inductances are extractedfor ip = 0 ∨ is = 0 (the energy is confined inside the coreand air gaps) and for+Npip = − Nsis (the energy is confinedinside thewindingwindow). Then, the following inductancescan be extracted for a virtual 1 : 1 transformer:

L ′m = 2

W

i2pN2p

= 2W

i2s N2s, with ip = 0 ∨ is = 0, (19)

L ′σ = 2

W

i2pN2p

= 2W

i2s N2s, with + Npip = − Nsis. (20)

The equivalent circuit (cf. (12) and (14)) of the transformeris extracted such that the stored energy (cf. (13)) matches,which leads to

Lp = N 2p L

′m, Ls = N 2

s L′m, (21)

M = NpNs

(L ′m − 1

2L ′

σ

), k = 1 − 1

2

L ′σ

L ′m. (22)

123


The following expressions can be extracted for the open-circuit and short-circuit operations of the transformer:

Loc,p = N 2p L

′m,

vs

vp= +k

Ns

Np, with is = 0, (23)

Loc,s = N 2s L

′m,

vp

vs= +k

Np

Ns, with ip = 0, (24)

Lsc,p = N 2p

(1 + k

2

)L ′

σ ,isip

= −kNp

Ns, with vs = 0, (25)

Lsc,s = N 2s

(1 + k

2

)L ′

σ ,ipis

= −kNs

Np, with vp = 0. (26)

Equivalent circuit from FEM simulations

Different methods exist for the extraction of the mag-netic equivalent circuit of a transformer from numericalsimulations (e.g., FEM): integration of the magnetic flux,computation of the induced voltages, extraction of the energy,etc. The energy represents a numerically stable parameterwhich is easy to extract. Therefore, the energy is extractedfor the following cases: ip �= 0 ∧ is = 0, ip = 0 ∧ is �= 0,and +Npip = − Nsis. This last solution can be obtained bythe superposition of the two first solutions. This leads to

Lp = 2W

i2p, with ip �= 0 ∧ is = 0, (27)

Ls = 2W

i2s, with ip = 0 ∧ is �= 0, (28)

M = W

ipis− 1

2Lp

ipis

− 1

2Ls

isip, with +Npip = − Nsis. (29)

It should be noted that these expressions are general since noassumptions are required for the geometry, the turns ratio,the coupling factor, etc.

Approximations for highmagnetic coupling factors

For a transformer with a high magnetic coupling factor(k > 0.95), the parameters of the equivalent circuits shown inFig. 14 are converging together. Then, it is possible to definethe transformer with the following parameters:

Lσ ≈ N 2p L

′σ ≈ Lsc,p ≈ Lsc,s

N 2p

N 2s,

≈ 2Lσ,T,p ≈ 2Lσ,T,s ≈ Lσ,PI ≈ Lσ,SP ≈ Lσ,PS, (30)

Lm ≈ N 2p L

′m ≈ Loc,p ≈ Loc,s

N 2p

N 2s,

≈ Lm,T ≈ 1

2Lm,PI,p ≈ 1

2Lm,PI,s ≈ Lm,SP ≈ Lm,PS, (31)

u ≈ Np

Ns≈ uT ≈ uPI ≈ uSP ≈ uPS, (32)

400

+400

v [V

]

0

Terminal voltage

Time [ s]

70

+70

0 5 10

i [A

]

0

80

0

+80

15

0

+15

(a)

Terminal current

Leakage voltage Magnetizing currentTime [ s]

0 5 10

(b)

Time [ s]0 5 10

(c)Time [ s]

0 5 10

(d)

v [V

]

i [A

]

vpvsip

is

vp-(Np/Ns)vs

v

vip+(Ns/Np)is

i i

Fig. 15 Voltage and current waveforms for the considered MF trans-former (k = 0.986, series-resonant LLC converter, cf. Fig. 2).a Terminal voltages, b terminal currents, c voltage across the leak-age inductance, and d current through the magnetizing inductance. Theequivalent circuits shown in Fig. 14c, d are considered and compared

where Lσ is the leakage inductance, Lm the magnetizinginductance, and u the voltage transfer ratio. These threeparameters, which are nearly independent of the chosenequivalent circuit, are typically used for the design processof transformers.

Figure 15 illustrates the leakage voltage and magnetiz-ing current obtained with the equivalent circuits depicted inFig. 14c, d for the considered MF transformer (k = 0.986,series-resonant LLC converter, cf. Fig. 2). It can be seen thatthe aforementioned approximations (cf. (31), (30), and (32))are valid.

With these assumptions, a clear and unique definition ofthe leakage and magnetizing fluxes is achieved. The leakagefield, which is linked to the magnetizing inductance (Lσ andvσ ), is located inside the winding window and is related tothe load current (series inductance). The magnetizing field,which is linked to the magnetizing inductance (Lm and im),is located inside the core and air gaps and is related to theapplied voltage (parallel inductance). During rated operat-ing condition, the leakage and magnetizing magnetic fieldsfeature 90◦ phase shift (cf. Fig. 15).

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