Springer / Electrical EngineeringThe final publication is available at https://doi.org/10.1007/s00202-018-0701-0

Magnetic Equivalent Circuit of MF Transformers:Modeling and Parameters Uncertainties

Thomas Guillod · Florian Krismer · Johann W. Kolar

Received: 9 November 2017 / Accepted: 17 May 2018

Abstract Medium-Frequency (MF) transformers are ex-tensively used in power electronic converters. Accordingly,accurate models of such devices are required, especiallyfor the magnetic equivalent circuit. Literature documentsmany different methods to calculate the magnetizing andleakage inductances of transformers, where, however, fewcomparisons exist between the methods. Furthermore,the impact of underlying hypotheses and parameter un-certainties is usually neglected. This paper analyzes ninedifferent models, ranging from simple analytical expres-sions to 3D detailed numerical simulations. The accuracyof the different methods is assessed by means of Monte-Carlo simulations and linearized statistical models. Theexperimental results, conducted with a 100kHz / 20kW MFtransformer employed in a 400V DC distribution systemisolation, are in agreement with the simulations (below14% inaccuracy for all the considered methods). It isconcluded that, considering typical tolerances, analyticalmodels are accurate enough for most applications and thatthe tolerance analysis can be conducted with linearizedmodels.

Keywords Power Electronics, Medium-FrequencyTransformers, Equivalent Circuit, Leakage, Magnetizing,Finite Element Method, Measurement, Uncertainty,Tolerance, Monte-Carlo Simulation

1 Introduction

Many power electronic converters feature MF transformersfor voltage transformation, impedance matching, andgalvanic isolation [1–4]. MF transformers are combininghigh efficiency, high power density, and fast dynamic. Theanalysis of such devices can be split into several categories:

Thomas Guillod (0000-0003-0738-5823)

E-mail: guillod@lem.ee.ethz.chPhone: +41 44 632 42 67Power Electronic Systems LaboratoryETH Zürich, Switzerland

(a)

Field patterns

|H| [

a.u.

]

|B| [

a.u.

]

Equivalent circuits

vs

is

vp

ip

vs

is

vp

ip

(b)

corecore

Npi p

Npi p

Npi p

Npi p

Nsi s

Nsi s

Nsi s

Nsi s

Fig. 1 (a) Core-type transformer field patterns during rated operatingcondition: magnetic flux density inside the core (magnetizing flux)and magnetic field in the winding window (leakage flux). (b) In-ductance matrix with the corresponding (lossless) linear equivalentcircuits.

– 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 ex-traction of the parasitic capacitances and the insulationcoordination [4, 8].

– Losses: Different methods exist for computing thewinding 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 lossesin 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 intercon-nected and it appears that the extraction of the magneticparameters is particularly important for determining theapplied currents, the losses, parasitic resonances, etc.

Fig. 1(a) illustrates the different magnetic fluxes, whichare defining the flux linkages of a transformer. The re-sulting inductance matrix, cf. Fig. 1(b), can be expressedwith equivalent circuits, which are analyzed in detail in

2 T. Guillod et al.

Appendix A. Typically, the following magnetic parametersare used to characterize a transformer [6, 11]:

– Voltage transfer ratio: The voltage transfer ratio is theratio between the primary and secondary voltages ofa transformer. For a transformer with a high magneticcoupling factor, the voltage transfer ratio is almostequal to the turns ratio and load independent (nearthe nominal load).

– Transformer magnetization: A magnetizing current isrequired for generating the flux inside the core ofthe transformer (cf. Fig. 1(a)). The magnetizing cur-rent is related to the open-circuit inductances (self-inductances) of the transformer, which are measuredfrom the primary and secondary sides. The open-circuitinductances are important for determining the corelosses and the saturation current [12]. The magnetizingcurrent can also be used for the converter operation,typically for achieving Zero-Voltage Switching (ZVS) [4,13].

– Transformer leakage: The leakage flux is the mag-netic flux which does not contribute to the magneticcoupling between the windings (cf. Fig. 1(a)). For atransformer with a high magnetic coupling factor, theleakage flux is related to the short-circuit inductances,which are measured from the primary and secondarysides. The short-circuit inductances are important fordetermining the winding losses and the short-circuitimpedance of the transformer [5]. Moreover, the short-circuit inductances can be used as a series inductor forthe power converter operation [14, 15].

Therefore, the extraction of an accurate equivalent cir-cuit is critical for the design and optimization of transform-ers. Inaccurate values can lead to increased losses [1, 16]or to improper converter operation [17]. The magneticparameters of the transformer can also be used for di-agnosis (production and aging) [18]. This indicates thatthe selection of the most suitable (concerning accuracy,modeling cost, and computational cost) computationmethod for the equivalent circuit is a challenging andimportant 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].

Nevertheless, deviations between the measured valuesand the expected values can arise from model inaccuracies,geometrical tolerances, material parameter tolerances, andmeasurement uncertainties. However, only few compar-

Table 1 MF Transformer Parameters.

Parameter Value

Power P = 20.0kW / S = 22.8kVA

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

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

Insulation Mylar / 1kV (RMS, CM) / 1kV (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 / 1.0dm3 / 2.4kg

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

isons between the methods can be found [25, 27], wherethe impact of tolerances and uncertainties is ignored.

This paper analyzes the impact of tolerances on themagnetic equivalent circuit of MF transformers, whichare employed in higher power (several 1kW to several100kW) converter systems, for different computation meth-ods. Section 2 defines the parameters of the considered100kHz / 20kW MF transformer. Section 3 discusses ninedifferent models for computing the magnetic parameters:standard analytical methods (e.g. Rogowski and McLymanfactors) [6, 12, 22], more elaborate semi-analytical methods(e.g. Schwarz-Christoffel mapping, mirroring method) [5,7, 19], and Finite Element Method (FEM) models (2Dand 3D models with different levels of detail). Section 4summarizes two approaches (Monte-Carlo simulationsand linearized statistical models) for assessing the impactof tolerances on the equivalent circuit. Finally, Section 5presents simulation and measurement results for theconsidered MF transformer, which concludes that simpleanalytical methods are very accurate and that linearizedtolerance analyses are valid.

2 Considered Transformer

Fig. 2(a) depicts the considered 400V / 100kHz / 20kW DC-DC series-resonant LLC converter. This converter providesa galvanic isolation for a section of a 400V DC distributiongrid, e.g. for the next generation datacenters [28, 29].The converter is operated at the resonance frequencywith 50% duty cycle and acts as a “DC transformer”, i.e.advantageously provides voltage transformation nearlyindependently of the load conditions [14].

Fig. 2(b) shows the considered MF transformer equiva-lent circuit and the corresponding short-circuit and open-circuit inductances. The leakage inductance of the MFtransformer is part of the converter resonant tank [14, 15]and the magnetizing inductance is used for achievingZVS [13, 30]. Fig. 2(c) shows the voltage and current wave-forms applied to the MF transformer. Since the leakageand magnetizing inductances of the MF transformer areactively used for the converter operation, any mismatch

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 3

400

V

(a)

154 mm

−400

+400

Time [μs]

−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-resonant LLC converter. (b) MF 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 transformer prototype (cf. Tab. 1).

(a)

(c) (d)

(b)

Short-circuit

Short-circuit

Short-circuit

Open-circuit

0.0

1.0|H

| / |H

max

| [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 viewside cut view

top cut viewtop cut view

front cut viewfront cut view

front cut viewfront cut view

Fig. 3 (a) Side view, (b) front view, and (c) top view of the magneticfield during short-circuit operation. (d) Front view of the magneticflux density during open-circuit operation. All the cut planes areintersecting at the MF transformer center point.

between the measured and simulated values will leadto non-optimal operating conditions or will require anadaptation of the modulation scheme (e.g. dead times,switching frequency).

Fig. 2(d) depicts the constructed prototype and the keyparameters are listed in Tab. 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, com-plicates the computation of the equivalent circuit. The airgaps also help the flux balancing between the paralleledcore sets. Otherwise, this MF transformer features themost common design choices: E-shaped ferrite core, shell-type windings, and high-frequency (HF) litz wires [1–4].However, the presented methods can also be adaptedto designs employing nanocrystalline cores, core-typewindings, solid wires, foil conductors, etc.

Fig. 3 shows the magnetic field and magnetic fluxdensity during short-circuit and open-circuit operations.For short-circuit operation, the energy is mostly stored inthe winding window and in the cable terminations. For

(a) (b) (c)Ni

R cor

e,l

R gap

,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

e

No 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-element core model with a semi-empirical fringing fieldmodel (McLyman factor) [12], and (c) detailed core model with adetailed fringing field model (Schwarz-Christoffel mapping) [5, 19].The red lines indicate the direction of the magnetic flux density(streamlines).

open-circuit operation, the energy is stored in the air gapsand inside the core. This indicates that the considered MFtransformer features a high magnetic coupling factor. Inaccordance, different methods can be used to extract themagnetic parameters of the transformer.

3 Computation Methods

The extraction of the equivalent circuit depicted in Fig. 1(b)is described in this Section for analytical, semi-analytical,and numerical methods. Additional information about theextraction of the equivalent circuit is given in Appendix B,Appendix C, and Appendix D.

3.1 Analytical Methods

Case: ip = 0∨ is = 0

First, the inductance is computed for the following opera-tion condition: ip = 0∨ is = 0. This computation is similarto the computation of inductors, which implies that the

4 T. Guillod et al.

1D approx. Mirroring methodRogowski factor

(a) (b)

(c)(d)

Termination

Fig. 5 Leakage magnetic field distribution (shown for a 6 : 6 shell-type windings). (a) 1D approximation (Ampère’s circuital law) [6, 7],(b) 2D approximation of the field near the winding tips (Rogowskifactor) [22], (c) 2D computation (mirroring method) [7, 31], and(d) cable termination field (Biot-Savart law). The red lines indicatethe direction of the magnetic field (streamlines).

presented methods can also be applied to inductors. Mostof the analytical methods are based on reluctance circuits:

– The core is modeled with a single reluctance andthe fringing field of the air gaps is neglected (singleequivalent reluctance), cf. Fig. 4(a) [6].

– The core is modeled with a single reluctance andthe McLyman factor (simple semi-empirical model) isused for describing the fringing field of the air gaps,cf. Fig. 4(b) [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. 4(c) [5, 19]. However, the interactionsbetween the different air gaps and the placement of thewires are not considered.

The main hypothesis of these reluctance models isthat a perfect magnetic coupling between the turns com-posing a winding (same flux through all the turns) isgiven (cf. Fig. 3(d)). This implies that these computationmethods are limited to transformers with high magneticcoupling factors. If required, the nonlinearities of the corematerial (flux density and frequency dependences) can beconsidered for the reluctance computations, especially fordesigns with small air gaps [12].

Case: +Npip =−Nsis

In a second step, the inductance is computed for thefollowing excitation: +Npip = −Nsis. With this operatingcondition, most of the energy is stored in the windingwindow (cf. Fig. 3(b)) and the following computationmethods can be used:

– A 1D approximation of the leakage field is computedwith the Ampère’s circuital law, cf. Fig. 5(a) [6, 7]. This

model is inaccurate for windings with a low fill factor,for large distance between the windings and/or thecore, and for modeling the leakage near the windingheads [7, 16].

– The 2D effects occurring near the winding tips,cf. Fig. 5(b) can be approximated with the Rogowkifactor, which is a correction factor for the height of thewindings [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. 5(c) [7, 31, 32]. The equivalent inductance canbe extracted from the energy stored in the field [32], or,more conveniently, directly from the inductance matrixbetween the wires [33, 34]. The boundary conditionsnear the winding heads and the finite core permeabilitycan be considered [7]. However, this method is onlyaccurate if the leakage field can be approximated in 2D(cf. Figs. 3(b) and (c)).

– For a transformer with a reduced leakage inductance,the energy stored inside the cable terminations cannotbe neglected [35]. For calculating the correspondinginductance, the inductance of two parallel wires (Biot-Savart law) can be used, cf. Fig. 5(d) [36]. Nevertheless,the field distribution near the cable terminations isintrinsically a 3D problem, which cannot be perfectlymodeled analytically (cf. Fig. 3(a)).

If required, the aforementioned methods can be ex-tended to other transformer geometries (e.g. core-typewindings, three-phase windings, interleaved windings) [23]or for taking the frequency dependencies of the short-circuit inductances into account (especially for design withsolid wires) [21].

Circuit Extraction

The equivalent circuit can be extracted from the obtainedvalues (ip = 0∨ is = 0 and +Npip = −Nsis) such that theenergy stored in the equivalent circuit matches with thecomputed 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 ob-tained from numerical simulations, where FEM is typicallyselected [2–4, 37]. The exact geometry, the nonlinearitiesof the core material, and the frequency dependences ofthe current distribution (e.g. skin and proximity effects)

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 5

(b)

(a)

2D FEM simple

2D FEM complexeven sym.

rot.

sym

.

winding heads

odd

sym

.

even sym.

core windowcore window

even sym.

odd

sym

.

core windowcore window

even sym.

rot.

sym

.

winding heads

Fig. 6 (a) 2D FEM model with a simplified winding geometry and(b) with a detailed winding geometry. For both cases, a different modelis used for the core window and the winding heads. The symmetryaxes are also indicated.

can be represented. For extracting the equivalent circuit,which features three degrees of freedom, different simu-lations are required. In this paper, the extraction of theequivalent circuit is done with the following numericallystable procedure. The energy inside the system is extractedfor the following cases: ip 6= 0 ∧ is = 0 (energy in theprimary self-inductance, Lp), ip = 0 ∧ is 6= 0 (energy inthe secondary self-inductance, Ls), and +Npip = −Nsis

(approximately, for high magnetic coupling factors, theenergy in the leakage magnetic field). Then, the equivalentcircuit is determined such that the stored energy matcheswith the simulations (cf. Appendix C). This method is alsoapplicable to transformers with low magnetic couplingfactors.

2D Modeling

Fig. 6(a) depicts a 2D FEM model with a simplified windingstructure and Fig. 6(b) considers the actual placement ofthe wires. For both cases, a planar and an axisymmetricmodel are used for modeling the core window and thewinding heads, respectively (E-shaped core, shell-typewindings). Different symmetry axes are used for reducingthe size of the model.

3D Modeling

Fig. 7(a) shows a simple 3D model with simplified core andwinding structures (single bodies). Fig. 7(b) 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

(a)

(c)

(b)

(b)

3D FEM simple

3D FEM intermediate

3D FEM complex

eveneven

oddodd

odd

odd

odd

odd

oddodd

odd

odd

odd

odd

eveneven

Fig. 7 (a) 3D FEM model with a simplified winding and core geometry,(b) with a more detailed winding and core geometry, and (c) withthe actual transformer geometry (with the cable terminations). Theconsidered symmetry planes are also indicated.

exist. Finally, Fig. 7(c) represents a full 3D model of thetransformer and the cable terminations, which does notfeature any symmetry plane.

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. 2(b) and Section 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 duringthe design of a transformer. Accordingly, in the following,the tolerance analysis is conducted with 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

6 T. Guillod et al.

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

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)

tolerances and the material parameter tolerances. For thestatistical analysis, the following expressions are consid-ered:

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

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

), (2)

where L is the computed inductance, L0 the nominalvalue, and FL the function describing the inductancecomputation method (cf. Section 3). Tab. 2 and Tab. 3summarize the variables used to describe the uncertaininput parameters and the inductances, respectively.

In the following, a simple exemplary function FL isconsidered for explaining the tolerance stacking analysis.Afterward, in Section 5, the defined analysis frameworkis then used for analyzing the tolerances for the open-circuit and short-circuit inductances of the considered

(a)

Normal distributionUniform distribution

p( 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

2δ1 2δ1

2δ2 2δ2

{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.

transformer (cf. Section 2), using the aforementionedcomputation methods (cf. Section 3).

4.2 Worst-Case Analysis

For a worst-case analysis, the variables xi are assumedto be uniformly distributed in the tolerance intervalsand independent of each other. The probability densitydistribution function, fi ,wc, can be set to

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

), (3)

where funiform describes the probability density distribu-tion function of the continuous uniform distribution. Withthe help of FL (cf. (1)), the probability distribution of L canbe computed and the extrema can be extracted, cf. Fig. 8(a).

For the magnetic equivalent circuit, many variablesare present, FL is nonlinear, and FL , for some computa-tion methods, cannot be expressed analytically. For thesereasons, the probability density distribution function isobtained by means of Monte-Carlo simulations [38]. In thecourse of the Monte-Carlo simulations, a large numberof random parameter combinations is generated withrespect to the input probability distributions (cf. (3)) andthe corresponding inductance values are computed. Then,it is straightforward to extract the interval, {L}MC,wc, wherethe inductance value is fluctuating.

However, Monte-Carlo simulations require the com-putation of a large number of designs. Alternatively, thefunction FL can be linearized near the nominal values of

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 7

the different parameters, 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)

whereδL,wc is the obtained tolerance and {L}lin,wc the corre-sponding interval. Beside the reduced computational cost,the linearization allows the analysis of the sensitivity of theinductance value with respect to the different parameters.If the linearization is valid, the ranges {L}MC,wc and {L}lin,wc

should match together. In Fig. 8(a), 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 con-servative. For this reason, normal (Gaussian) distributionsare often used for tolerance stacking analysis [39]. Theprobability density distribution function, fi ,nd, and thestandard deviation, σi ,nd, can be set to

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

), (6)

σi ,nd = δip2erf−1 (

pi ,nd) , (7)

where fnormal is the probability density distribution func-tion of the normal distribution and erf−1 the inverseGauss error function [40]. The confidence intervals, pi ,nd,are describing the percentage of the data lying withinthe accepted tolerances ±δi . With these definitions, theprobability distribution of L can be computed, cf. Fig. 8(b).

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

Alternatively, the linearization of FL can be conducted.A linear combination of normal distributions results in anormal distribution, 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 canbe expressed as

δL,nd =σL,nd

p2erf−1 (

pL,nd)

, (10)

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

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

1.70

Z sc [

dB]

L sc [

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)

Current [A] (e) (f)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 impedancesfor the setup shown in Fig. 2(d) [41]. (c) Open-circuit and (d) short-circuit inductances for different frequencies [41]. (e) Open-circuit and(f) short-circuit small-signal differential inductances for different biascurrents [42]. The primary and secondary windings are the outer andinner windings (with respect to the central limb), respectively. Themeasurement uncertainties (shaded area) and the operating points(“OP”, cf. Fig. 2(c)) are indicated.

where δL,nd is the obtained tolerance and {L}lin,nd thecorresponding interval. In Fig. 8(b), the mismatch betweenthe probability density distribution function obtained withMonte-Carlo simulations and fL,nd (L) results from thenonlinearities of the chosen exemplary function FL .

5 Results

The MF transformer depicted in Section 2 is considered.First the magnetic parameters of the MF transformer aremeasured. In a second step, the aforementioned computa-tion methods (cf. Section 3) and tolerance stacking analysis(cf. Section 4) are applied. Finally, the measurement andcomputation results are compared.

5.1 Measurements

The open-circuit (Loc,p and Loc,s) and short-circuit(Lsc,p and Lsc,s) inductances (cf. Fig. 2(b)) are ex-

8 T. Guillod et al.

tracted from small-signal impedance measurements [43].Figs. 9(a) and (b) depict the measured impedances, wherethe resonance frequencies of the MF transformer canbe seen. Figs. 9(c) and (d) show the extracted nominalinductances (Lmeas) with the corresponding measurementuncertainties ({L}meas). The inductances are extracted witha resistive-inductive series equivalent circuit.

These measurements are obtained with an Agi-lent 4924A precision impedance analyzer [41]. The un-certainties are composed of the measurement devicetolerances and an absolute error. The absolute error, whichis estimated from measurements, is set to 20nH for short-circuit measurements (interconnection inductances) andto 100nH for open-circuit measurements (interconnectioninductances and slight variation of the length of the airgaps in case the core halves are pressed together). Thesemeasurements have been successfully reproduced (lessthan 1% deviation) with an Omicron Bode 100 networkanalyzer [44, 45].

Fig. 9(c) and (d) indicate that the frequency depen-dences of the inductances are negligible due to the usedferrite cores and HF litz wires [6, 21]. The slight increaseof the open-circuit inductances near 1MHz is due to thecapacitive currents (resonance frequency of 3.2MHz) [4].The slight decrease of the short-circuit inductances near10MHz is explained by high-frequency effects (inducedcurrents in the windings and permeability change ofthe core) [9, 21]. However, near the operating frequency(100kHz), the impact of these high-frequency effects is neg-ligible (constant open-circuit and short-circuit inductancesbelow 1MHz).

However, the MF transformer is not operated withsmall-signal excitations. For this reason, the MF trans-former has also been measured with a ed-k DPG10/1000Apower choke tester [42]. This device applies a current rampto the MF transformer and measures the induced voltage.This enables the extraction of the small-signal differen-tial inductance (L = ∂Ψ/∂i ) for different bias currents,cf. Figs. 9(e) and (f). The uncertainties are composed ofthe measurement device tolerances and an absolute error.The absolute error is set to 50nH and 120nH, respectively,for short-circuit (interconnection inductances) and open-circuit measurements (interconnection inductances andvariation of the length of the air gaps).

Fig. 9(e) and (f) shows that the open-circuit induc-tances are also linear below 40A (magnetizing current),which means that the MF transformer is operated wellbelow the saturation (cf. Fig. 2(c)). As expected, the short-circuit inductances are independent of the current. Due tothe high magnetic coupling factor of the MF transformer(k = 0.986), the inductances measured from the primaryand secondary sides are very similar (Np = Ns).

Table 4 Computation Methods.

Name Model CL nx

Analytic simple Figs. 4(a) / 5(a) 50FLOPs 12

Analytic intermediate Figs. 4(b) / 5(b) 150FLOPs 14

Analytic complex Figs. 4(c) / 5(c) 200kFLOPs 16

Analytic termination Figs. 4(c) / 5(c) / 5(d) 200kFLOPs 20

2D FEM simple Figs. 6(a) / 5(d) 8kDOFs 19

2D FEM complex Figs. 6(b) / 5(d) 12kDOFs 20

3D FEM simple Figs. 7(a) / 5(d) 0.9MDOFs 17

3D FEM intermediate Figs. 7(b) / 5(d) 1.4MDOFs 17

3D FEM complex Fig. 7(c) 7.2MDOFs 16

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

5.2 Computations

The computation methods described in Section 3 arecombined, as shown in Tab. 4, for extracting the equivalentcircuit of the MF transformer. For the analytical models, thecomputational cost, CL , is measured in Floating Point Op-erations (FLOPs) [46]. For FEM models, the computationalcost, CL , is expressed in Degrees Of Freedom (DOFs) [37],given that three operating points are solved for extractingthe equivalent circuit (cf. Subsection 3.2).

For the statistical analysis, the methods describedin Section 4 are used. The confidence intervals for theparameters (pi ,nd) and for the inductances (pL,nd) are set to95%. The number of uncertain parameters, nx , is indicatedin Tab. 4. The uncertain parameters can be classified intodifferent categories:

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

– Some geometrical parameter uncertainties are alsolinked to the geometrical approximations included inthe modeling. The fact that a 3D geometry is modeledin 1D/2D (e.g. winding heads and cable terminations)introduces additional uncertainties since some geomet-

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 9

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

sc )

[1/µ

H]

Lsc [µH] (b)

Loc [µH]

p( L

oc )

[1/µ

H]

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-circuit inductances. Worst-case and normal distribution analyzes areperformed by means of Monte-Carlo simulations and linearization(cf. Section 4). The model “Analytic termination” is considered(cf. Tab. 4).

rical parameters in the 1D/2D models do not exist inthe 3D models.

– The simplifications used in the different computationmethods (e.g. 1D/2D modeling, no fringing field model,no cable terminations model) are intrinsic inaccuraciesof the methods and, therefore, are not considered asuncertainties.

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”) neglect many parameters(e.g. placement of the wires, cable terminations). Therefore,the full statistical analysis is performed with the model“Analytic termination”, where 2×103 Monte-Carlo samplesare considered.

Fig. 10 shows the obtained tolerances for the induc-tances, which are not negligible. It appears that the lin-earized statistical analysis is valid (compared to Monte-Carlo simulations). The linearized analysis allows theextraction of the sensitivities of the different parameters(cf. (4)), which is shown in Fig. 11. The main uncertaintiesare originating from the air gaps length, the size of the core,the permeability of the core, the insulation distances, andthe simplification of the 3D geometry (depicted as “modelapprox.”).

Therefore, the linearized statistical analysis is appliedto all the computation methods described in Tab. 4. Thisgreatly reduces the computational cost since only 2nx +1equivalent circuit computations are required. The obtained

Open-circuit Short-circuit

(b)(a)

δ L,w

c = 6

.87

µH

δ 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. Tab. 4) and a linearized worst-case analysis is applied (cf. (4)).

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 µH

Dev

iatio

n [%

]L o

c [µH

]

0

30

60

Analyt

ic co

mplex

refe

renc

ere

fere

nce

{L}lin,nd

{L}lin,wc

{L}meas

L0

Lmeas

Fig. 12 Comparison between measurements (cf. Subsection 5.1) andcomputations (cf. Subsection 5.2) for the open-circuit inductance(Loc). The reference value is the measurement obtained with anAgilent 4294A precision impedance analyzer [41].

values (L0, {L}lin,wc, and {L}lin,nd) are used for comparingthe computations and the measurements.

5.3 Comparison: Measurements and Computations

Fig. 12 and Fig. 13 show the comparison between measure-ments and computations for the open-circuit and short-circuit inductances, respectively. The most accurate mea-surements, i.e. the values obtained with the Agilent 4294Aprecision impedance analyzer [41], are selected as refer-ence values for the extraction of the relative errors. Thedeviations between the nominal values and the referencevalue are smaller than 14% for all the methods.

For the open-circuit inductance, the method “Analyticsimple” (cf. Tab. 4) underestimates the inductance, which

10 T. Guillod et al.

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

ere

fere

nce

{L}lin,nd

{L}lin,wc

{L}meas

L0

Lmeas

Fig. 13 Comparison between measurements (cf. Subsection 5.1) andcomputations (cf. Subsection 5.2) for the short-circuit inductance(Lsc). The reference value is the measurement obtained with anAgilent 4294A precision impedance analyzer [41].

is due to the neglected fringing field of the air gaps. Forall the 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”, “Analyticintermediate”, and “Analytic complex”) underestimate theinductance. For all the other methods, the tolerances({L}lin,wc and {L}lin,nd) match with the measurement un-certainties. The method “Analytic termination” is also ingood agreement with the “3D FEM complex” method (5%error).

Therefore, it can be concluded that the complex com-putation methods (2D and 3D FEM models) only featurelimited advantages over the analytical methods. The mainadvantage of the 3D FEM methods is a slight reductionof the tolerances, since the tolerances linked with themodel approximations vanish (“model approx.”, cf. Fig. 11).The method “Analytic termination” (cf. Tab. 4) representsan interesting trade-off between modeling complexity,computational cost, and accuracy.

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 thatthe extraction of MF transformer equivalent circuit param-eters can only be marginally improved with new computa-

tion and measurement methods. Only a reduction of theuncertainties on the geometry and material parameterswill lead to better matching between measurements andcomputations.

6 Conclusion

This paper first investigates various computation methodsfor extracting MF transformer magnetic equivalent circuit,which is a critical parameter for the modeling and opti-mization of power electronic converters. Analytical modelsare presented for the magnetizing (reluctance circuit,Rogowski factor, and Schwarz-Christoffel mapping) andleakage (1D Ampère’s circuital law, Rogowski factor, 2D mir-roring method, and cable termination model) inductances.The advantages and limitations of the analytical methodsare discussed and compared to numerical simulations (2Dand 3D FEM).

In a next step, the uncertainties linked to model inaccu-racies, geometrical tolerances, and material tolerances areanalyzed in detail with a statistical analysis applied to thecomputation models. Computationally intensive Monte-Carlo 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 ofa 100kHz / 20kW MF transformer. It is concluded that alinearized statistical analysis is sufficient, which greatlysimplifies the tolerance analysis. The deviations betweenthe measurements and the different computations meth-ods can be explained with the computed tolerances. For theopen-circuit inductance, the measurement uncertainties(0.3−2.3%) are smaller than the computation tolerances(6−12%). This is also the case for the short-circuit induc-tance (2.7−7.1% and 5−22%).

A comparison between the different computation meth-ods shows that the combination of a reluctance circuit,Schwarz-Christoffel mapping, the mirroring method, anda cable termination model is a good trade-off between thecomputational cost and the obtained accuracy. Finally, itis concluded that the consideration of the uncertainties isrequired for improving the modeling of MF transformers.Moreover, the obtained uncertainties are also useful forproduction diagnosis and for selecting the right margin ofsafety during the design process.

The presented statistical analysis has been conductedon the magnetic parameters of MF transformers withcomputational models. This work could be completedwith an experimental study involving many samples inorder to measure the tolerances occurring in a typicalproduction process. Furthermore, the presented methodscan be extended to the remaining transformer parame-ters (e.g. losses and thermal models) and components

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 11

(e.g. switches, inductors) in order to assess the modeluncertainties of complete converter systems.

Appendices

These appendices review the theoretical background behind the dif-ferent transformer equivalent circuits and the corresponding pitfalls(cf. Appendix A). The procedure for extracting equivalent circuitsfrom analytical (cf. Appendix B) and numerical (cf. Appendix C) com-putation is also presented. Finally, some approximations are given fortransformers with high magnetic coupling factors (cf. Appendix D).

A Transformer Equivalent Circuits

The magnetic equivalent circuit of a (lossless) linear transformer withtwo windings is fully described by the following inductance matrix(cf. Fig. 1(b)) [6, 11, 43]:[

vp

vs

]=

[ ∂Ψp

∂t∂Ψs∂t

]=

[Lp M

M Ls

][ ∂ip

∂t∂is∂t

], (12)

where Lp is the primary self-inductance, Ls the secondary self-inductance, and M the mutual inductance. The inductance matrixfeatures three independent parameters. The energy, W , stored inthe transformer (quadratic form of the inductance matrix) can becomputed as

W = 1

2Lpi 2

p +1

2Lsi 2

s +Mipis. (13)

The energy stored in the transformer is always positive (i.e. theinductance matrix is positive definite), which leads to the conditionM 2 < LpLs. Therefore, the mutual inductance can be expressed witha normalized parameter, the magnetic coupling,

k = M√LpLs

, with k ∈ [0,1]. (14)

During open-circuit operation, the inductance matrix has thefollowing physical interpretation. The self-inductances (Lp and Ls)represent flux linkages of the two windings themselves (defined asmagnetizing flux linkages). The mutual inductance (M) describes theflux linkage between the windings (defined as coupled flux linkages).The differences between the self and mutual inductances (Lp − Mand Ls −M) represent the flux linkage differences (defined as leakageflux linkages), which can, for some designs, be negative (especially ifNp 6= Ns.

With the aforementioned inductance matrix, the inductances,voltage transfer ratios, and current transfer ratios can be expressedfor short-circuit and open-circuit operations (cf. Fig. 2(b)):

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,is

ip=−k

√Lp

Ls, with vs = 0, (17)

Lsc,s =(1−k2)

Ls,ip

is=−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 different equivalentcircuits. The circuits shown in Fig. 1(b) are directly related to theinductance matrix. On the contrary, Fig. 14 depicts equivalent circuits,

T circuit / PI circuit / 4 degrees of freedom

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

(d)

(a)

(c)

üT

vσ,T,p vσ,T,s

im,Tvp

ip

vs

isLσ,T,p Lσ,T,sLm,T

vσ,PI

im,PI,pvp

ip

vs

isLσ,PILm,PI,p Lm,PI,s

im,PI,s üPI

vσ,SP

im,SPvp

ip

vs

isLσ,SP Lm,SP

üSP

vσ,PS

im,PSvp

ip

vs

isLσ,PSLm,PS

üPS

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

which do not provide a direct insight on the magnetic flux linkages.The following important remarks can be given about the differentequivalent circuits of transformers:

– All the presented equivalent circuits (cf. Fig. 1(b) and Fig. 14)model perfectly the terminal behavior and the stored energy.

– The equivalent circuits with more than three degrees of freedom(cf. Figs. 14(a) and (b)) are underdetermined and do not have aphysical meaning, without accepting restrictive hypotheses [11].

– The turns ratio of the transformer, Np : Ns, is not clearly definedfor some transformer geometries (e.g. inductive power transfercoils) [47, 48]. This implies that the turns ratio is not always di-rectly related to the flux linkages and to the magnetic parameters.

– The magnetizing and leakage fluxes cannot be spatially separated.In other words, it is not always possible to sort the magnetic fieldlines into leakage and magnetizing field lines [48].

– In a transformer, a phase shift is present between the primaryand secondary currents, which originates from the modulationscheme, the load, and/or the losses. This implies that thedistribution of the magnetic field lines is time-dependent [11,48].Therefore, the leakage and magnetizing flux linkages only have aclear interpretation for a lossless transformer during open-circuitand short-circuit operations.

– The magnetizing and leakage inductances are usually definedas the parallel and series inductances in the equivalent circuit,respectively. However, the values of these inductances dependon the chosen equivalent circuit (cf. Fig. 14) and, therefore, donot have a clear and/or unique physical interpretation. This alsoimplies that the associated magnetizing current (im) and leakagevoltage (vσ) only represent virtual parameters, which are notdirectly measurable [11].

It can be concluded that only the equivalent circuits shown inFig. 1(b) feature a clear physical interpretation and, therefore, shouldbe preferred. The circuits with more than three degrees of freedom(cf. Figs. 14(a) and (b)) should be avoided since they are unnecessarilycomplex. The circuits depicted in Figs. 14(c) and (d) are interestingfor designing transformers with high magnetic coupling factors asexplained in Appendix D.

B Equivalent Circuit from Analytical Computations

The analytical methods are based on several assumptions (cf. Subsec-tion 3.1). The inductances are accepted to scale quadratically withthe number of turns. The inductances are extracted for ip = 0∨ is = 0

12 T. Guillod et al.

(the energy is confined inside the core and air gaps) and for +Npip =−Nsis (the energy is confined inside the winding window). Then, thefollowing inductances can be extracted for a virtual 1 : 1 transformer:

L′m = 2

W

i 2pN 2

p= 2

W

i 2s N 2

s, with ip = 0∨ is = 0, (19)

L′σ = 2

W

i 2pN 2

p= 2

W

i 2s N 2

s, with +Npip =−Nsis. (20)

The equivalent circuit (cf. (12) and (14)) of the transformer is extractedsuch that the stored energy (cf. (13)) matches, which leads to

Lp = N 2pL′

m, Ls = N 2s L′

m, (21)

M = NpNs

(L′

m − 1

2L′σ

), k = 1− 1

2

L′σ

L′m

. (22)

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

Loc,p = N 2pL′

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,

is

ip=−k

Np

Ns, with vs = 0, (25)

Lsc,s = N 2s

(1+k

2

)L′σ,

ip

is=−k

Ns

Np, with vp = 0. (26)

C Equivalent Circuit from FEM Simulations

Different methods exist for extracting the magnetic equivalent circuitof a transformer from numerical simulations (e.g. FEM): integration ofthe magnetic flux, computation of the induced voltages, extraction ofthe energy, etc. The energy represents a numerically stable parameterwhich is easy to extract. Therefore, the energy is extracted for thefollowing cases: ip 6= 0∧is = 0, ip = 0∧is 6= 0, and +Npip =−Nsis. Thislast solution can be obtained by the superposition of the two firstsolutions. This leads to

Lp = 2W

i 2p

, with ip 6= 0∧ is = 0, (27)

Ls = 2W

i 2s

, with ip = 0∧ is 6= 0, (28)

M = W

ipis− 1

2Lp

ip

is− 1

2Ls

is

ip, with +Npip =−Nsis. (29)

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

D Approximations for High Magnetic Coupling Factors

For a transformer with a high magnetic coupling factor (k > 0.95), theparameters of the equivalent circuits shown in Fig. 14 are convergingtogether. Then, it is possible to define the transformer with thefollowing parameters:

Lσ ≈ N 2pL′

σ ≈ Lsc,p ≈ Lsc,s

N 2p

N 2s

,

≈ 2Lσ,T,p ≈ 2Lσ,T,s ≈ Lσ,PI ≈ Lσ,SP ≈ Lσ,PS,

(30)

Lm ≈ N 2pL′

m ≈ Loc,p ≈ Loc,s

N 2p

N 2s

,

≈ Lm,T ≈ 1

2Lm,PI,p ≈ 1

2Lm,PI,s ≈ Lm,SP ≈ Lm,PS,

(31)

−400

+400

v [V

]

0

Terminal voltage

Time [μs]

−70

+70

0 5 10

i [A

]

0

−80

0

+80

−15

0

+15

(a)

Terminal current

Leakage voltage Magnetizing currentTime [μs]

0 5 10(b)

Time [μs]0 5 10

(c) Time [μs]0 5 10

(d)

v [V

]

i [A

]

vpvsip

is

vp-(Np/Ns)vs

vσ,SP

vσ,PSip+(Ns/Np)is

im,SP im,PS

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

u ≈ Np

Ns≈ uT ≈ uPI ≈ uSP ≈ uPS, (32)

where Lσ is the leakage inductance, Lm the magnetizing inductance,and u the voltage transfer ratio. These three parameters, which arenearly independent of the chosen equivalent circuit, are typicallyused for the design process of transformers.

Fig. 15 illustrates the leakage voltage and magnetizing currentobtained with the equivalent circuits depicted in Figs. 14(c) and (d)for the considered MF transformer (k = 0.986, series-resonant LLCconverter, cf. Fig. 2). It can be seen that the aforementioned approxi-mations (cf. (31), (30), and (32)) are valid.

With these assumptions, a clear and unique definition of theleakage and magnetizing fluxes is achieved. The leakage field, whichis linked to the magnetizing inductance (Lσ and vσ), is located insidethe winding window and is related to the load current (series induc-tance). The magnetizing field, which is linked to the magnetizinginductance (Lm and im), is located inside the core and air gaps andis related to the applied voltage (parallel inductance). During ratedoperating condition, the leakage and magnetizing magnetic fieldsfeature 90° phase shift (cf. Fig. 15).

Acknowledgements This project is carried out within the frame ofthe Swiss Centre for Competence in Energy Research on the FutureSwiss Electrical Infrastructure (SCCER-FURIES) with the financialsupport of the Swiss Innovation Agency.

References

1. M. Leibl, G. Ortiz, and J. W. Kolar, “Design and ExperimentalAnalysis of a Medium Frequency Transformer for Solid-StateTransformer Applications,” IEEE Trans. Emerg. Sel. Topics PowerElectron., vol. 5, no. 1, pp. 110–123, 2017.

2. F. Kieferndorf, U. Drofenik, F. Agostini, and F. Canales, “ModularPET, Two-Phase Air-Cooled Converter Cell Design and Perfor-mance Evaluation with 1.7kV IGBTs for MV Applications,” in Proc.of the IEEE Applied Power Electronics Conf. and Expo. (APEC),Mar. 2016, pp. 472–479.

3. S. Zhao, Q. Li, and F. C. Lee, “High Frequency Transformer Designfor Modular Power Conversion from Medium Voltage AC to 400VDC,” in Proc. of the IEEE Applied Power Electronics Conf. andExpo. (APEC), Mar. 2017, pp. 2894–2901.

Magnetic Equivalent Circuit of MF Transformers: Modeling and Parameters Uncertainties 13

4. T. Guillod, F. Krismer, and J. W. Kolar, “Electrical Shielding ofMV/MF Transformers Subjected to High dv/dt PWM Voltages,”in Proc. of the IEEE Applied Power Electronics Conf. and Expo.(APEC), Mar. 2017, pp. 2502–2510.

5. J. Mühlethaler, “Modeling and Multi-Objective Optimization ofInductive Power Components,” Ph.D. dissertation, ETH Zürich,2012.

6. V. C. Valchev and A. Van den Bossche, Inductors and Transformersfor Power Electronics. CRC press, 2005.

7. J. A. Ferreira, Electromagnetic Modelling of Power ElectronicConverters. Springer Science & Business Media, 2013.

8. T. Guillod, R. Färber, F. Krismer, C. M. Franck, and J. W. Kolar,“Computation and Analysis of Dielectric Losses in MV PowerElectronic Converter Insulation,” in Proc. of the IEEE EnergyConversion Congr. and Expo. (ECCE), Sep. 2016, pp. 1–8.

9. T. Guillod, F. Huber, J. Krismer, and J. W. Kolar, “Wire Losses:Effects of Twisting Imperfections,” in Proc. of the Workshop onControl and Modeling for Power Electronics (COMPEL), Jul. 2017,pp. 1–8.

10. K. Venkatachalam, C. R. Sullivan, T. Abdallah, and H. Tacca,“Accurate Prediction of Ferrite Core Loss with NonsinusoidalWaveforms using only Steinmetz Parameters,” in Proc. of theIEEE Workshop on Computers in Power Electronics, Jun. 2002,pp. 36–41.

11. H. Kleinrath, “Ersatzschaltbilder für Transformatoren und Asyn-chronmaschinen (in German),” e&i, vol. 110, no. 1, p. 68–74, 1993.

12. W. T. McLyman, Transformer and Inductor Design Handbook.CRC Press, 2004.

13. M. Kasper, R. M. Burkart, G. Deboy, and J. W. Kolar, “ZVS of PowerMOSFETs Revisited,” IEEE Trans. Power Electron., vol. 31, no. 12,pp. 8063–8067, 2016.

14. F. C. Schwarz, “A Method of Resonant Current Pulse Modulationfor Power Converters,” IEEE Trans. Ind. Electron., vol. 17, no. 3,pp. 209–221, 1970.

15. R. W. A. A. De Doncker, D. M. Divan, and M. H. Kheraluwala, “AThree-Phase Soft-Switched High-Power-Density DC/DC Con-verter for High-Power Applications,” IEEE Trans. Ind. Appl.,vol. 27, no. 1, pp. 63–73, 1991.

16. N. Dai and F. C. Lee, “Edge Effect Analysis in a High-FrequencyTransformer,” in Proc. of the IEEE Power Electronics SpecialistsConf. (PESC), Jun. 1994, pp. 850–855.

17. M. Meinhardt, M. Duffy, T. O’Donnell, S. O’Reilly, J. Flannery,and C. O. Mathuna, “New Method for Integration of ResonantInductor and Transformer-Design, Realisation, Measurements,”in Proc. of the IEEE Applied Power Electronics Conf. and Expo.(APEC), Mar. 1999, pp. 1168–1174.

18. L. M. R. Oliveira and A. J. M. Cardoso, “Leakage InductancesCalculation for Power Transformers Interturn Fault Studies,” IEEETrans. Power Del., vol. 30, no. 3, pp. 1213–1220, 2015.

19. J. Muhlethaler, J. W. Kolar, and A. Ecklebe, “A Novel Approach for3D Air Gap Reluctance Calculations,” in Proc. of the IEEE EnergyConversion Congr. and Expo. (ECCE Asia), May 2011, pp. 446–452.

20. A. Van den Bossche, V. C. Valchev, and R. Filchev, “Improvedapproximation for fringing permeances in gapped inductors,”in Proc. of the IEEE Industry Applications Conf., Oct. 2002, pp.932–938.

21. Z. Ouyang, J. Zhang, and W. G. Hurley, “Calculation of Leakage In-ductance for High-Frequency Transformers,” IEEE Trans. PowerElectron., vol. 30, no. 10, pp. 5769–5775, 2015.

22. A. L. Morris, “The Influence of Various Factors upon the LeakageReactance of Transformers,” Journal of the Institution of ElectricalEngineers, vol. 86, no. 521, pp. 485–495, 1940.

23. R. Doebbelin, M. Benecke, and A. Lindemann, “Calculationof Leakage Inductance of Core-Type Transformers for PowerElectronic Circuits,” in Proc. of the IEEE International PowerElectronics and Motion Control Conf. (PEMC), Sep. 2008, pp. 1280–1286.

24. Z. Ouyang, O. C. Thomsen, and M. A. E. Andersen, “The Analysisand Comparison of Leakage Inductance in Different WindingArrangements for Planar Transformer,” in Proc. of the IEEE Conf.Power Electronics and Drive Systems (PEDS), Nov. 2009, pp. 1143–1148.

25. A. M. Urling, V. A. Niemela, G. R. Skutt, and T. G. Wilson,“Characterizing High-Frequency Effects in Transformer Windings- A Guide to Several Significant Articles,” in Proc. of the IEEEApplied Power Electronics Conf. and Expo. (APEC), Mar. 1989, pp.373–385.

26. J. Mühlethaler and J. W. Kolar, “Optimal Design of InductiveComponents Based on Accurate Loss and Thermal Models,” inTutorial at the IEEE Applied Power Electronics Conf. and Expo.(APEC), Feb. 2012.

27. R. Doebbelin, C. Teichert, M. Benecke, and A. Lindemann,“Computerized Calculation of Leakage Inductance Values ofTransformers,” Piers Online, vol. 5, no. 8, pp. 721–726, 2009.

28. G. AlLee and W. Tschudi, “Edison Redux: 380 Vdc Brings Relia-bility and Efficiency to Sustainable Data Centers,” IEEE PowerEnergy Mag., vol. 10, no. 6, pp. 50–59, 2012.

29. A. Pratt, P. Kumar, and T. V. Aldridge, “Evaluation of 400V DCDistribution in Telco and Data Centers to Improve EnergyEfficiency,” in Proc. of the IEEE Telecommunications EnergyConference (INTELEC), Oct. 2007, pp. 32–39.

30. R. M. Burkart and J. W. Kolar, “Comparative η-ρ-σ Pareto Opti-mization of Si and SiC Multilevel Dual-Active-Bridge Topologieswith Wide Input Voltage Range,” IEEE Trans. Power Electron.,vol. 32, no. 7, pp. 5258–5270, 2017.

31. K. J. Binns and P. J. Lawrenson, Analysis and Computation ofElectric and Magnetic Field Problems. Elsevier, 1973.

32. D. Leuenberger and J. Biela, “Accurate and ComputationallyEfficient Modeling of Flyback Transformer Parasitics and theirInfluence on Converter Losses,” in Proc. of the European Conf. onPower Electronics and Applications (EPE), Sep. 2015, pp. 1–10.

33. M. Eslamian and B. Vahidi, “New Methods for Computation ofthe Inductance Matrix of Transformer Windings for Very FastTransients Studies,” IEEE Trans. Power Del., vol. 27, no. 4, pp.2326–2333, 2012.

34. M. Lambert, F. Sirois, M. Martinez-Duro, and J. Mahseredjian,“Analytical Calculation of Leakage Inductance for Low-FrequencyTransformer Modeling,” IEEE Trans. Power Del., vol. 28, no. 1, pp.507–515, 2013.

35. G. R. Skutt, F. C. Lee, R. Ridley, and D. Nicol, “Leakage Inductanceand Termination Effects in a High-Power Planar Magnetic Struc-ture,” in Proc. of the IEEE Applied Power Electronics Conf. andExpo. (APEC), Feb. 1994, pp. 295–301.

36. I. J. Bahl, Lumped Elements for RF and Microwave Circuits, ser.Artech House Microwave Library. Artech House, 2003.

37. S. S. Rao, The Finite Element Method in Engineering. ElsevierScience, 2011.

38. I. M. Sobol, A Primer for the Monte Carlo Method. CRC press,1994.

39. F. Scholz, “Tolerance Stack Analysis Methods,” Research andTechnology: Boeing Information, pp. 1–44, 1995.

40. L. C. Andrews, Special Functions of Mathematics for Engineers, ser.Oxford Science Publications. SPIE Optical Engineering Press,1992.

41. Agilent Technologies, “Agilent 4294A Precision Impedance Ana-lyzer, Operation Manual,” Feb. 2003.

42. ed-k, “Power Choke Tester DPG10 – Series,” Feb. 2017.43. J. G. Hayes, N. O’Donovan, M. G. Egan, and T. O’Donnell,

“Inductance Characterization of High-Leakage Transformers,”in Proc. of the IEEE Applied Power Electronics Conf. and Expo.(APEC), Feb. 2003, pp. 1150–1156.

44. omicron lab, “Bode 100, User Manual,” Feb. 2010.45. ——, “B-WIC & B-SMC, Impedance Test Adapters,” Feb. 2014.46. A. Quarteroni and F. Saleri, Scientific Computing with MATLAB,

ser. Texts in Computational Science and Engineering. SpringerBerlin Heidelberg, 2012.

47. W. G. Hurley, M. C. Duffy, J. Zhang, I. Lope, B. Kunz, and W. H.Wölfle, “A Unified Approach to the Calculation of Self- andMutual-Inductance for Coaxial Coils in Air,” IEEE Trans. PowerElectron., vol. 30, no. 11, pp. 6155–6162, 2015.

48. R. Bosshard, T. Guillod, and J. W. Kolar, “Electromagnetic FieldPatterns and Energy Flux of Efficiency Optimal Inductive PowerTransfer Systems,” Electrical Engineering (Springer), vol. 99, no. 3,pp. 969–977, 2017.