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Comparison of Analytical Models of Transformer Leakage Inductance: Accuracy Versus

Computational Effort

Schlesinger R., Biela J.

Power Electronic Systems Laboratory, ETH Zürich Physikstrasse 3, 8092 Zürich, Switzerland

mailto:[email protected]

146 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 36, NO. 1, JANUARY 2021

Comparison of Analytical Models of TransformerLeakage Inductance: Accuracy Versus

Computational EffortRichard Schlesinger , Student Member, IEEE, and Jürgen Biela , Senior Member, IEEE

Abstract—A fast and accurate model of the transformer leakageinductance is crucial for optimization-based design of galvanicallyisolated converters. Analytical models are fastly executable, andtherefore, especially suitable for such optimizations. This articlecompares several analytical leakage inductance per unit lengthmodels with respect to the accuracy and computational effort. Theconsidered models are applicable to E-Core and U-Core transform-ers. 2D FEM simulations are used as a benchmark to evaluatethe model accuracy, whereas the computation time is extractedas an indicator for computational effort. Six different transformerprototypes provide the geometries for the comparison. Based onthe conducted comparisons, Roth’s model is the most accurate. Ro-gowski’s model is the fastest low-error model. Margueron’s modelis the most versatile as it takes the finite permeability of the coreinto account. The conducted comparisons lay the foundation foraccurate and fast Double-2D modeling of the transformer leakageinductance as it is executed for the two main cross sections ofE-core and U-core transformers: inside the transformer window,and outside the transformer window. This article is accompanied bya supplementary document summarizing the equations of Roth’sand Margueron’s model.

Index Terms—Analytical leakage inductance modeling, Double-2D, galvanically isolated converters, optimization, solid-statetransformers, transformer, U-core & E-core.

I. INTRODUCTION

L EAKAGE inductance is an important property of trans-formers in galvanically isolated power electronic convert-

ers as it is crucial for the operation of the converter. Galvanicallyisolated converters can be applied in several applications thatenable a more sustainable energy system such as photovoltaicinverters [1], electric traction systems [2], and power qualitycontrol of the electric grid [3].

Requirements on transformer leakage inductance depend onthe converter topology. In some converter topologies, the leakageinductance should be as small as possible to minimize losses.

Manuscript received November 15, 2019; revised March 6, 2020 and May 5,2020; accepted June 1, 2020. Date of publication June 8, 2020; date of currentversion September 4, 2020. Recommended for publication by Associate EditorX. Ruan. (Corresponding author: Richard Schlesinger.)

The authors are with the Laboratory for High Power Electronic Systems HPE,ETH Zurich, 8092 Zürich, Switzerland (e-mail: [email protected];[email protected]).

This article has supplementary downloadable material available at https://ieeexplore.ieee.org, provided by the authors.

Color versions of one or more of the figures in this article are available onlineat https://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2020.3001056

The flyback converter is an example of these kinds of systems(see [4]). In other topologies, the leakage inductance of thetransformer can be exploited as series inductance to replaceseparate inductors. Thus, the power density is increased andcomponent costs are saved. In these applications, a well-definedleakage inductance is crucial. This concept is frequently appliedin resonant converters (see [5]) and dual-active-bridge (DAB)converters [2]. In DAB converters, the series inductance isrequired for power transfer, power control, and to enable zerovoltage switching.

In the design stage of such a converter, the operating anddesign parameters are usually determined with an optimiza-tion procedure, before the components are physically built [6].Therefore, models of all relevant converter parameters such astransformer leakage inductance, voltages, currents, and lossesare required. During the optimization procedure, the models areexecuted several thousand times (see, e.g., [6]). Consequently,the models have to be computationally efficient to deliver resultswithin a reasonable amount of time.

The transformer leakage inductance can be calculated withnumerical methods, reluctance network modeling (RNM), andanalytical methods. Numerical methods such as the finite-element method (FEM) [7], [8] are versatile and very accuratebut rather time consuming, and therefore, suboptimal for opti-mizations. RNM [9], [10] is faster, but also less accurate thannumerical models. Analytical models such as those proposed byDowell [11] and Rogowski [12] are usually restricted to theirassumed geometry and subject to simplifications, which mayreduce the accuracy of the result. However, analytical modelsare rapidly computable, which makes them very well suited forconverter optimizations.

Analytical modeling of the transformer leakage inductancetypically consists of the following two steps [11]–[18]:

1) calculate the leakage inductance per unit length L′σ from

a 2D transformer cross section;2) scale the per unit length value by the mean length of turns.Recent modeling approaches [19]–[23] pursue a “Double-2D”

concept, i.e., the calculation is based on two cross sectionsof the transformer: inside the transformer window (IW) andoutside the transformer window (OW), as depicted in Fig. 1.

These are the two main cross sections that can be deducedfrom a cut view of E-core and U-core transformers. In theDouble-2D approach, the leakage inductance per unit length is

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SCHLESINGER AND BIELA: COMPARISON OF ANALYTICAL MODELS OF TRANSFORMER LEAKAGE INDUCTANCE 147

Fig. 1. Basic cross sections of “Double-2D” modeling concept. (a) Insidewindow (IW). (b) Outside window (OW).

acquired for both cross sections. The per unit length inductancesare then scaled by the corresponding partial winding length andsummed up to acquire the total leakage inductance. Both of themodeling steps need accurate and fast models to obtain an ac-curate and computationally efficient overall leakage inductancemodel. Therefore, it is necessary to investigate both modelingsteps separately.

Several reviews and comparisons on leakage inductancecalculation have been published. For example, Margueronet al. [15] and Lambert et al. [24] provide a good overview on2D leakage inductance model approaches. Doebbelin et al. [25]compare the accuracy of Rogowski’s model and the mean ge-ometric distance (MGD) model. Fouineau et al. [22] comparethe accuracy and calculation times of Dowell’s model [11] anda Double-2D model with three-dimensional (3D) FEM simu-lations. Kaymak et al. [26] compare a Dowell-based modifiedleakage inductance model with FEM simulations. However, anextensive comparison of leakage inductance models applied tomultiple transformers evaluating computational effort as well asaccuracy is still missing.

Previous Double-2D literature [19]–[21] has focused on com-paring the overall leakage inductance with measurements and3D-FEM simulations. This implies the substantial disadvantagethat the error cannot be attributed to either of the model steps,or both, i.e., inductance per unit length or scaling length.

Hence, this article elucidates the first step of leakage induc-tance modeling in detail, i.e., calculating the leakage inductanceper unit length. Several leakage inductance per unit length mod-els are compared with respect to accuracy and computationaleffort. To exclusively evaluate the accuracy of the per unit lengthmodels, 2D FEM simulations are taken as reference.

In preceding work [18], the comparison has been executedfor the IW cross section. In this article, a separate comparisonfor the OW cross section is added. These two comparisonsallow to attribute the particular error and computational effortto either the model of the IW or OW leakage inductance perunit length. Thus, the best suited model can be selected for eachcross section, which is a crucial precondition for accurate andfast Double-2D modeling.

The rest of this article is organized as follows. Section II com-prises an overview of leakage inductance modeling steps andelaborates common approaches of leakage inductance per unit

Fig. 2. IW partial length of windings lw,in and OW partial length of windingslw,out for (a) UR-core and (b) E-core.

length models. Section III introduces the models and comparestheir modeling approach and concept. Section IV is the key partof this article and compares the models from a practical pointof view, i.e., accuracy versus computational effort. Section Velucidates the accuracy of the Double-2D modeling approachcompared to the conventional Single-2D modeling by includingmeasurements and 3D FEM simulations. Section VI clarifiesthe influence of the frequency on the leakage inductance. TheAppendices present the general Double-2D and Single-2D equa-tions and the geometrical parameters of the compared trans-formers. The equations of Roth’s and Margueron’s model arecompactly summarized in the external supplementary documentof this article.

II. ANALYTICAL LEAKAGE INDUCTANCE MODELING

This section clarifies the most important fundamentals ofanalytical leakage inductance modeling that are relevant for thisarticle. For an extensive treatise on fundamentals of leakageinductance modeling, see e.g., [27] and [28].

A. Modeling Steps

Analytical Double-2D leakage inductance modeling is typi-cally performed in the following two steps.

1) The leakage inductances per unit length are calculated forIW and OW cross sections: L′

σ,in and L′σ,out. The values

are calculated based on the depicted geometries in Fig. 1.2) The IW and OW leakage inductances per unit length are

multiplied by the respective partial length of the windingslw,in and lw,out. These are visualized in Fig. 2 for a UR-coreand an E-core. Simple equations for lw,in and lw,out areproposed in Appendix A.

These shares are finally summed up to the total leakageinductance according to

Lσ,Double-2D = L′σ,in · lw,in + L′

σ,out · lw,out. (1)

Both model steps introduce error and computational effort intothe calculation. Hence, the overall error and computational effortresult from both modeling steps. Therefore, it is necessaryto analyze both modeling steps separately to understand andidentify potential error sources and especially time-consumingparts of the overall model. This article focuses solely on thefirst step of calculating the IW and OW leakage inductance per



Fig. 3. Method of images applied to one image plane. (a) OW-constellationwith ferromagnetic material of finite height and thickness. (b) Ideal mirror-constellation with ferromagnetic material of infinite height and thickness.(c) Mathematical equivalent of (b) using the method of images.

unit length. Thus, the most accurate and efficient calculationmodels for the particular leakage inductance per unit length canbe identified.

B. Method of Images

To obtain accurate IW and OW leakage inductance per unitlength values, the presence of the transformer core is taken intoaccount by the method of images. A good treatise on the methodof images can be found in [29]. At this point, it is important toclarify the role of the method of images for the OW and IWleakage inductance per unit length.

The OW cross section represents a current next to the centerleg, i.e., a ferromagnetic material with finite height and thicknessas shown in Fig. 3(a). This is not exactly the constellation that istreated with the method of images. For the ideal mirror constel-lation, as shown in Fig. 3(b) and (c), the ferromagnetic materialhas to be of infinite height and thickness. However, Margueronet al. concluded that the thickness of the core is negligible inpractical cases with a permeability higher than 100 [16]. InSection IV-B, we will show that also the finite height of thecore poses a negligible difference between ideal mirror and OWconstellation. Thus, the ideal mirror-constellation represents avery accurate approximation of the OW-constellation and is,therefore, justified to use.

The IW cross section represents currents within the trans-former window, resulting in four image planes. This leads to aninfinite amount of image layers according to Fig. 4(a). Somemodels approximate this constellation with a finite amount ofimage layers. The accuracy loss of this approximation is ad-dressed in Section IV-A.

C. Common Assumptions of Models

In all models, the turns are unified to rectangular wind-ing blocks. The assumed current densities are through-plane[Jz(x, y)], the resulting magnetic field is in-plane [ �H(x, y),�B(x, y)], whereas the magnetic vector potential �A only has athrough-plane component Az(x, y). To account for arbitrarynumbers of turns of any winding k, the number of turns Nk

is implied in the current density, i.e., Jk = NkIkAwinding,k

, where Jkis the current density of winding k, Ik is the current throughone turn of the winding, and Awinding,k is the surface of thewinding k.

Fig. 4. (a) Method of images applied to the IW cross section of Fig. 1(a).Four image layers lead to an infinite amount of image windings. m is the imagefactor μr,c−1

μr,c+1 and n is the number of the respective image layer. First imagelayer in green. (b) One-dimensional leakage field in case of purely axial leakageflux. Not achievable in real transformers due to required isolation distancesbetween windings and core. (c) Two-dimensional-leakage field with flux fringingconsidered, i.e., radial and axial part of field.

III. CATEGORIZATION OF MODELS

The models introduced in this section have been consideredbecause they are applicable to the considered transformer ge-ometries (E, ER, U, UR-core) and sufficiently compact. Roth’s,Margueron’s, and the MGD model are applicable to windingsof arbitrary height. Kapp’s, Dowell’s, and Rogowski’s modelare only applicable to windings of equal height. Therefore, awinding height transformation is proposed in Appendix D. Theconsidered models are categorized according to Fig. 5 and areshortly summarized as follows.

A. 1D-Models (Axial Leakage Field Assumed, Flux FringingNeglected)

The 1D models assume purely axial leakage flux betweenwindings of equal height as shown in Fig. 4(b). Thus, theleakage field depends only on the x-coordinate, but not on they-coordinate. Furthermore, the field only has a y-componentHy . The assumption of purely axial leakage flux neglects theflux fringing at the top and bottom of the windings due to theinevitable isolation distances between windings and yokes asshown in Fig. 4(c).

Kapp [13]: Kapp derived the DC-leakage field �H(x) fromAmpère’s law assuming axial leakage flux. Next, the magneticenergy density is calculated with wmag(x) =

12μ0H

2 and inte-grated over the transformer window. Finally, the leakage induc-tance per unit length is derived using (4). Kapp’s model leads toa very simple closed-form formula for the leakage inductanceper unit length as follows:

L′σ,Kapp = μ0 N

2

(d+

a1 + a23

)1

h(2)



Fig. 5. Overview of analytical leakage inductance per unit length models (yellow), fundamental assumptions (red), physical and mathematical derivationapproaches (gray), and leakage inductance per unit length equations (green).

where μ0 is the vacuum permeability, N is the number of turnsof the excited winding, and the geometry parameters a1, d, a2,and h are defined in Fig. 4(b).

Dowell [11]: Dowell derived the current density Jz(x, ω)by solving a second-order differential equation derived fromAmpère’s law and the law of electromagnetic induction. Next,the voltage V (ω) across a portion of an arbitrary number ofwinding layers is derived. The impedance of the transformer iscalculated by Ohm’s law. The leakage inductance per unit lengthis finally derived by L′

σ = Im(V (ω)/I)ω .

B. 2D Models (Flux Fringing Considered)

In 2D models, the flux density �B(x, y) and the magneticpotential Az(x, y) are functions of both space coordinates x andy. Additionally, the magnetic field has x- and y-components.Consequently, the flux fringing at the bottom and the top of thewindings illustrated in Fig. 4(c) is taken into account. All con-sidered 2D-models assume homogeneous current distribution inthe windings. Rogowski’s, Roth’s, and Margueron’s model arebased on a solution of the Poisson equation for the magneticpotential Az . These three models compute the magnetic energyper unit length W ′

mag according to

W ′mag =

1

2

∫∫AAz · Jz dA. (3)

Using the magnetic energy per unit length, the leakage induc-tance per unit length L′

σ can be calculated with

L′σ =

2W ′mag

I2(4)

where I is the current of the actively excited winding.A detailed and comprehensive explanation on the magnetic

potential Az can be found in [30, ch. 5]. The considered 2Dmodels are listed as follows.

Rogowski [12]: Rogowski mirrored the original windings ofequal height across the transformer legs. Referring to Fig. 4(a),there is only one horizontal image series in the x-direction. Theremaining spatial current distribution is expressed as Fourierseries dependent on the x-coordinate (single-space harmonics).Rogowski’s model is geometrically constrained by the followingassumptions:

a) windings of equal height;b) no gaps between windings and legs [i.e., dx,i = dx,o = 0

in Fig. 8(a)];c) interleaved windings: Constant distance between wind-

ings;d) interleaved windings: Windings of constant width.With the current distribution, a magnetic potential Az(x, y)

is derived from Poisson’s and Laplace’s equation. Rogowski’ssolution consists of several functions valid in either winding do-main or air domain and are coupled by intercontinuity boundaryconditions. The leakage inductance per unit lengthL′

σ is derivedusing the magnetic energy per unit length W ′

mag according to (3)and (4).

Rogowski expressed the infinite Fourier series as a finiteexpression leading to the same equation that Kapp had derived;only with a correction factor K taking the flux fringing at topand bottom of the windings into account shown as

L′σ,Rogowski = K · L′

σ,Kapp (5)

K = 1− 1− e−kh

kh·[1− 1

2e−2kdy,b

(1− e−kh

)

×(1 + e−k(dy,t−dy,b) − e−k(2dy,b+2dy,t+h)

)] (6)

k =π

a1 + d+ a2.



Rogowski’s model can be simplified by assuming winding-yokedistances dy,b and dy,t to be infinite (see Fig. 8(a) for geometricalparameter definition). In this case, the expression in squaredbrackets in the Rogowski factor (6) becomes 1 as in [21] and[24]. This model is referred to as “Rogowski simple” in thisarticle.

Roth [14], [31]: Edouard Roth used a double Fourier se-ries dependent of both, x- and y-coordinates (double spaceharmonics) to express the current density distribution of eachsingle winding [14]. This leads to the constellation in Fig. 4(a),only with the approximation of μr,c → ∞. Next, a potentialAz,k(x, y) for each windingk is derived with Poisson’s equation.Unlike Rogowski’s solution, Roth’s potential of each windingAz,k(x, y) is valid for the total computational domain. The totalpotential Az(x, y) is calculated by superposing the potentialsAz,k of all windings (Az =

∑Ck=1 Az,k), whereC is the number

of windings within the transformer window. Analogously toRogowski, the leakage inductance is derived via the magneticenergy with (3) and (4). Roth’s final equation for the leakageinductance per unit length still contains the double infiniteseries, however, the series converge rapidly with ∝ 1

n3 . Theequations of Roth’s model are compactly summarized in thesupplementary document.

A compact derivation of the leakage inductance from Roth’spotential vector was published in [31]. An insightful and com-pact summary of Roth’s work and a comparison to Rogowski’sapproach can be found in [32]. Further publications using Roth’spotential vector can be found in [33] and [34].

Margueron [15], [16]: Margueron’s model is based on thesuperposition of magnetic potentials of single windings. Themethod of images is used to mirror these windings one by one.This leads to a discrete amount of image layers for the IW crosssection in Fig. 4(a). Margueron’s model is the only consideredmodel that takes the finite permeability of the magnetic core intoaccount.

Margueron derived the potential of each winding Az,k(x, y)from a function commonly used in partial element equivalentcircuit (PEEC) modeling. Note that the same potential functionhas already been derived in 1926 in [35]. A detailed derivationof the potential function can be found in [30, ch. 5.2.2].

Margueron used this potential function to calculate the leak-age inductance of transformers. In the 2007 publication [15],the leakage inductance is obtained by numerically integratingthe energy density. In the 2010 publication [16], Margueronimproved his model by carrying out the integration analyticallyusing a primitive function for the potential Az,k(x, y), whichdrastically reduces the computational effort. The equations ofMargueron’s model are compactly summarized in the supple-mentary document.

Note that Lambert [24], [36] proposed the same model with aconstant offset in the potential (which drops out when calculatingthe magnetic field), a multiwinding geometry, and an inductancematrix referred to each winding couple. The model in [22] is alsomathematically related to Margueron’s model.

Mean Geometric Distances (MGD) [17], [37], [38]: Themodel simplifies the winding blocks to winding filaments. Thesewinding filaments carry filamentary currents that cause circular

magnetic fields. Self-inductance per unit length of the primarywinding L′

p, self-inductance per unit length of the secondarywinding L′

s, and mutual inductance per unit length M ′ ofthe resulting filamentary coils in air are directly derived fromAmpère’s law. The inductances are calculated using geometri-cally averaged distances between windings (MGD). The coreis taken into account by adding mutual inductances per unitlength M ′

original-image between original and discretely mirroredimage windings. The core permeability is assumed to be infiniteμr,c → ∞.

The model was first introduced by J. C. Maxwell in his famoustreatise [37, ch. 691] and expanded to leakage inductance byPetrov [17]. The equations of the MGD model can be foundin [38].

The aforementioned categorization is illustrated in Fig. 5.Each model has properties that imply advantages and disad-vantages. These qualitative properties are compared in Table I.

IV. ACCURACY VERSUS COMPUTATIONAL EFFORT OF MODELS

This section shows the results of the comparison of the leakageinductance per unit length models with respect to the accuracyand computational effort. The model error indicates accuracy,whereas the calculation time indicates computational effort. Toexclusively evaluate the accuracy of the per unit length models,the leakage inductance per unit length calculated by 2D FEMis taken as reference to determine the error. The models areapplied to the geometries of six transformer prototypes listed inTable V. A comparison is presented for both, IW and OW crosssection.

A. Inside-Window (IW)

Fig. 6(a) and (b) shows the model error (accuracy) versusthe calculation time (computational effort) of the comparedmodels applied to the particular IW geometries of the comparedtransformers in Table V.

Roth’s model is the most accurate in every considered ge-ometry. The error tends toward 0%, as Roth’s model representsthe full analytical solution to the originally posed field problem.Roth’s model takes both, the spatially distributed current densityand the magnetic core (with μr,core → ∞) into account. Thecomputational effort of the model is remarkably small consider-ing the calculation times of around 0.2 ms. As the double Fourierseries converges rapidly (∝ 1

n3 ), the amount of sum terms in thehorizontal and vertical Fourier series are set to 25. These valuescan be increased/decreased to increase/decrease the accuracyresulting in higher/lower computational effort.

Rogowski’s model is a viable solution for optimizations whenit comes to geometries with rather small winding-leg distances.Transformers No.1–3 are exemplary for this, as the error isbelow 1% in these scenarios. The small computational effortof Rogowski’s formula is the biggest advantage of the model,which is due to the compactness of the model’s closed-formleakage inductance formula (5). Calculation times are as low asapproximately 5–10 μs.

Margueron’s model yields below 5% error in all consideredgeometries, which makes the model versatile in geometry. Note



TABLE IQUALITATIVE PROPERTIES OF MODELS

x Constraint: Axial leakage field (1D-field) assumed, i.e., flux fringing neglectedxx Constraint: Windings of equal heightxxx Constraints: • Constant distances between windings (interlayer gaps) • Fixed width of windings* Trick: Set core-yoke distances and core-outer leg distances sufficiently far away** Originally: μr,core of yokes considered. However μr,core of whole core required, to increase accuracy according to 2D FEM results*** by: Increasing the number of sum terms in Fourier series**** by: • Adding more layers of images

that the analytical integration of the energy integral accordingto the Margueron 2010 model [16] decreases the computationaleffort of the model significantly by about three orders of magni-tude. The Margueron 2007 model [15] is much slower becausevalues of the magnetic potential all over the windings need to becalculated explicitly to numerically compute the energy integral(3). Margueron’s model has an important advantage over theother models: The finite magnetic permeability of the core μr,c

is taken into account. This is essential, when it comes to lowpermeability cores.

For this comparison, only one image layer was used. Theaccuracy of Margueron’s model can be further increased byadding more layers of images by the cost of higher calculationtime.

The MGD model is the second fastest model and yieldserrors below 3% in geometries with similar winding dimensions(i.e., a1 ≈ a2, resp., h1 ≈ h2). Transformers No.1 and 4–6 areexamples for this. The MGD model is not suitable for geometrieswith high relative height differences h1−h2

harithm. mean, in which the

error becomes relatively high (No.3: 9%, No.2: 6%). This isdue the fact that the approximation of circular fields is notaccurate in this case. This circumstance makes the model quiteunreliable as the error very much depends on the geometry.Similar to Margueron’s model, one image layer was specifiedfor this comparison and the accuracy of the MGD model can be

increased by adding more layers of images by the cost of highercomputational effort.

Kapp’s model is the most inaccurate because it neglects thenonaxial part of the leakage field between the windings, i.e., theflux fringing effect. The error reaches 16% in the most criticalcase (tr. No. 5). The flux fringing effect is mainly caused by thepractically unavoidable isolation distances between windingsand core [geometry parameters dy,b and dy,t in Fig. 8(a)]. Kapp’smodel is the model with the least computational effort due toits very compact closed formula (2). The computation time isaround 1–2 μs.

Dowell’s model yields the same result as Kapp’s model for theconsidered static leakage inductance. Therefore, Dowell’s modelis equally inaccurate as Kapp’s model due to the neglected fluxfringing. The computational effort is remarkably high due to theconsideration of frequency. As the magnetostatic inductance isalready erroneous due to the 1D-approximation of the field, theconsideration of the frequency does not yield great benefits.

B. Outside Window (OW)

First, proof is required that the approximation of applying anideal mirror [core with infinite height and thickness, Fig. 3(b)and (c)] to the OW constellation [core with finite height andthickness, Fig. 3(a)] is justified, as elaborated in Section II-B.



Fig. 6. Error versus calculation time of the models applied to the IW and OW cross sections of the transformers listed in Table V. Reference: 2D FEM leakageinductance per unit length L′

σ . (a) and (b) Inside window (IW). (c) and (d) Outside window (OW). To get a rough impression of implementation complexity andeffort, the number of code lines of the MATLAB-implementation are given in the legend.

To prove that the introduced error of finite height and finite thick-ness of the core material is negligibly small for the consideredtransformers, FEM simulations have been performed. Theseshow that the difference in leakage inductance between theideal mirror-constellation and the OW-constellation is negligiblysmall. Table II shows the error introduced by this approximationfor all considered transformers.

Fig. 6(c) and (d) shows the error versus the calculation timeof the models applied to the considered transformer geometriesfor the OW cross section. Fewer models are compared for theOW cross section because Kapp’s model, Dowell’s model, andRogowski’s model are not applicable to this cross section.

Margueron’s model is very accurate with an error range of0%–0.2%. This error is only caused by approximating the OWconstellation [see Fig. 3(a)] with the ideal mirror constellation[see Fig. 3(b) and (c)]. Mathematically, Margueron’s modelrepresents the full analytical solution to the ideal-mirror con-stellation in Fig. 3(b) and (c). This can be seen in Table IIIthat shows the errors of the leakage inductances per unit length

TABLE IILEAKAGE INDUCTANCE PER UNIT LENGTH L′

σ OF THE OW CONSTELLATION

[SEE Fig. 3(A)] COMPARED TO THE IDEAL-MIRROR

CONSTELLATION [Fig. 3(B) AND (C)]

The introduced error by applying an ideal mirror to the OW cross section isnegligibly small. This simplification is, therefore, justified. Values have beencalculated with 2D FEM.

calculated with the Margueron 2010 model and the ideal mirrorcalculated by 2D FEM, which are very close to 0%.

Margueron’s model is accurate in the OW cross sectionbecause Margueron’s solution fully takes the core with finite



TABLE IIILEAKAGE INDUCTANCE PER UNIT LENGTH L′

σ OF THE MARGUERON 2010MODEL (OW) COMPARED TO THE IDEAL-MIRROR SCENARIO

The error is in the permille range, which indicates the high accuracy of themodel concerning the OW scenario.

permeability and the spatially distributed current distributioninto account.

Note that the Margueron 2010 model [16] is much faster thanthe Margueron 2007 model [15]. This is due to the explicitfield calculation required for the version of 2007 (just as forthe IW cross section). The Margueron 2010 model requiresapproximately 0.2 ms of calculation time.

Roth’s model is also quite accurate in the OW cross section.The error is only slightly higher in comparison to Margueron’smodel. However, the computational effort exceeds the Mar-gueron 2010 model, making it a worse choice than the Mar-gueron 2010 model.

Roth’s model is tailored to an IW cross section. The trickto apply Roth’s model to the OW cross section is to set theyokes and the outer leg sufficiently far away from the windings.In the particular case, the distances dy,b, dy,t, and dx,o [seeFig. 8(a)] have been set to 10 hw+ww

2 . This parametrization wasempirically found to yield good results.

Note that higher values ofww and hw hamper the convergenceof the double Fourier series in Roth’s model. This results in ahigher number of sum terms in both of the Fourier series requiredfor achieving satisfactory accuracy. For the OW cross section,250 sum terms are specified in horizontal and vertical Fourierseries, which naturally increases the calculation time.

The MGD model is the fastest of the compared models.However, the error depends very much on the geometry, as canbe seen from the transformer No.2 (error≈ 5%) and transformerNo.3 (error ≈ 8%). Just as for the IW cross section, this is dueto the considerable relative height difference h1−h2

harithm. meanand the

approximation of using circular fields.Further details concerning the time extraction and calculation

routine can be found in [18, Sec. 3.1]. Each model relies onspecific simplifications that inevitably introduce error into thecalculation. The errors of the particular models were attributedto geometrical circumstances and model simplifications in [18,Sec. 3.2].

V. ACCURACY OF DOUBLE-2D MODELING

The main focus of this artice is to identify the fastest and mostaccurate leakage inductance per unit length models for IW andOW cross sections. However, leakage inductance per unit length

TABLE IVMEASURED LEAKAGE INDUCTANCE OF THE TRANSFORMERS COMPARED TO

DOUBLE-2D ROTH, SINGLE-2D ROTH, SINGLE-2D DOWELL, AND 3D-FEM

The nRMSE represents the rough scale of error to expect from the particular model.

nRMSE =

√16

∑6

Tr.No.=1(Lσ,model,Tr.No.−Lσ,measured,Tr.No.)

2

Lσ,measured,max−Lσ,measured,min.

values cannot be compared to measurements as there is no wayto measure them. Therefore, the scaling length models proposedin Appendix A are used to obtain the total calculated leakageinductance. These total leakage inductances can be compared tothe measured transformer leakage inductance.

To assess the accuracy potential of the Double-2D approach,the static leakage inductance of each transformer is measured.Furthermore, the leakage inductances per unit length calculatedby Roth’s model and Dowell’s model are scaled using theSingle-2D approach (7) as well as the Double-2D approach(1). Furthermore, 3D-FEM simulations are conducted. Table IVshows the measured leakage inductances and the error of theparticular models applied to each of the compared transformers.Moreover, the normalized root mean square error (nRMSE) ofeach model is given. The nRMSE indicates the rough scale oferror independently of the transformer geometry.

The Double-2D modeling approach (1) yields considerablylower error than the Single-2D modeling approach (7). Thiscan be observed when comparing the nRMSE values of Roth-Double-2D (8.5%) and Roth-Single-2D (13.9%). Roth’s leakageinductance per unit length model is selected for assessing theaccuracy of Double-2D compared to Single-2D, as Section IVshowed that Roth’s model yields errors close to 0% for IW andOW cross section.

Section IV already showed that 2D leakage inductance perunit length models are significantly more accurate than 1D leak-age inductance per unit length models. This error is also reflectedin the overall leakage inductance. This conclusion can be drawnfrom comparing Roth’s Single-2D model to Dowell’s Single-2Dmodel. Here, the nRMSE is 13.9% and 24.6%, respectively.

It has to be clarified that obtaining 0% error is practically notpossible with any model. The main reason is that geometrical andmaterial properties will never be exactly the same as specifiedin the model. Other reasons are unavoidable measurement in-accuracies and measurement influences. This circumstance canbe observed when looking at the 3D-FEM error. With equalgeometry parameters of the model and real transformer, themeasured value and 3D-FEM value should coincide very well as3D-FEM is very accurate. However, the error is in a range of up toapproximately 5% with an nRMSE of 3.3%. This is roughly the



Fig. 7. Measured static short-circuit inductance Lsc,0 and frequency-dependent short-circuit inductance Lsc(ω) of the transformer No. 2 of Table V.The difference is as low as 4% in the relevant frequency range.

benchmark of the accuracy that models can achieve in practice.The remaining difference in nRMSE between 3D-FEM (3.3%)and the Double-2D model (8.5%) suggests that there is still roomof improvement considering the partial winding length modelsof Appendix A.

VI. FREQUENCY DEPENDENCE OF LEAKAGE INDUCTANCE

Galvanically isolated power converters operate in the mediumfrequency range—typically at a few tens of kilohertz (e.g., [1]and [2]). At these frequencies, skin and proximity effect play arole in the redistribution of the current density within a conductorand must be considered for winding resistance calculations.However, leakage inductance depends on the stored magneticenergy—which is a global physical quantity. As global quantity,magnetic energy is not heavily affected by local current densityredistributions within a conductor because the conductor crosssection is designed to be only a small fraction of the total windingcross section in a technical transformer. In other words, localcurrent density redistributions within a proportionally smallconductor represent only a minor macroscopical fluctuation.Therefore, the effect of frequency on leakage inductance isnot significant for technical transformers within the relevantfrequency range.

To prove this statement, the short-circuit inductances of thetransformers were measured statically and as a function offrequency. Fig. 7 shows the measured static short-circuit induc-tanceLsc,0 and the frequency-dependent short-circuit inductanceLsc(ω) of the transformer No. 2 (see Table V) in the relevantfrequency range of 10 kHz up to 1 MHz. Both windings of thetransformer No.2 are foil windings. These are typically heavieraffected by high-frequency effects than Litz wire windings due totheir considerable conductor height. Therefore, the transformerNo.2 reflects a worst-case scenario when it comes to frequencydependence. Still, the figure shows that the difference in leakageinductance due to frequency effects is as low as 4% in the relevantfrequency range.

As frequency effects play a minor role in leakage inductance,only one frequency-dependent model (Dowell’s model [11])was considered in the comparison. The big disadvantage withDowell-based 1D models is that their solution of the magnetic

field is one-dimensional, i.e., the leakage flux is assumed purelyaxial as shown in Fig. 4(b). Section IV showed that this as-sumption of neglecting the fringing flux introduces significantlyhigher error than the error introduced by frequency effects.Additionally, 1D models are not applicable to windings ofdifferent height, which is another substantial disadvantage. As aconsequence, the conducted comparison is focused on static 2Dmodels instead of frequency-dependent 1D models due to theirconsideration of fringing leakage flux and higher geometricalversatility.

VII. CONCLUSION

Several analytical 1D and 2D leakage inductance per unitlength models have been compared and assessed with respect toa tradeoff between accuracy and computational effort. The com-parison has been executed for both, IW and OW cross section.Depending on the requirements on accuracy and computationtime, the best suited model can be chosen for IW and OW leakageinductance per unit length.

Regarding the IW cross section, Roth’s model delivers the bestaccuracy combined with rather low computational effort. Theerror referred to the 2D FEM simulation is negligibly small as themodel represents the complete analytical solution to the posedleakage inductance problem. The calculation times are as lowas 0.2 ms on a standard notebook. The model is geometricallyversatile as it is applicable to all geometries with rectangularwindings within a rectangular transformer window, given thatwinding and window edges are parallel.

As long as the distances between windings and core are smallcompared to the transformer window dimensions, Rogowski’smodel yields satisfactory accuracy (less than 1% in the simulatedcases). Rogowski’s closed-form leakage inductance formula israpidly executable with calculation times around 5–10 μs.

The model of Margueron is the only considered model thattakes the finite permeability of the core μr,core into account.Transformers with a low-permeability core are, therefore, bestexamined with this model. Furthermore, the error is below5% in all simulated transformer geometries. The accuracy ofthe model can be further improved by increasing the num-ber of image layers (by the cost of additional computationaleffort). These properties make Margueron’s model the mostversatile.

The 1D-models such as Dowell’s and Kapp’s model result inconsiderably large errors up to 16% for the leakage inductanceper unit length as they neglect the flux fringing caused by thedistances between windings and core yokes.

Regarding the OW cross section, Margueron’s model is thebest performing model. Here, the error is below 0.2% in allsimulated cases. It was shown that the remaining error doesnot result from the model itself but from the simplification ofapplying an ideal mirror (center leg with infinite height andthickness) to the OW cross section (center leg with finite heightand thickness). The computation time of Margueron’s modelapplied to the OW cross section is approximately 0.2 ms on astandard notebook.



TABLE VPARAMETERS OF CONSIDERED TRANSFORMERS, SEE Fig. 8 FOR PARAMETER DEFINITIONS

APPENDIX APARTIAL LENGTH OF WINDINGS EQUATIONS

For conventional Single-2D modeling, the leakage inductanceper unit length of the IW cross section L′

σ,in is multiplied by themean length of turns lm according to

Lσ,Single-2D = L′σ,in · lm. (7)

The mean length of turns lm and partial length of the windingslw,in and lw,out depend on the shape of the center leg. In case of acircular center leg, the lengths are curved, in case of a rectangularleg, the lengths are straight. To take the core shape into account,the factor score is introduced according to

score = 1 . . . for U- and UR-core

score = 2 . . . for E- and ER-core.(8)

The mean length of turns lm is not uniquely defined in theliterature. In this article, the definition according to Hurley [28]is used.

A. Circular Center Leg

According to [28], the mean length of turns lm is calculatedwith (9). For geometry definitions, see Fig. 8.

lm = 2π

(r1 + r2

2

)= 2π(dleg + 2dx,i + a1 + d+ a2) (9)

where r1 is the inner diameter of the inner winding and r2 is theouter diameter of the outer winding.

To perform Double-2D modeling, the IW and OW partiallengths of the windings lw,in and lw,out are required. The proposedequations are as follows:

lw,in = score · α

2π· lm (10)

lw,out =2π − score · α

2π· lm. (11)

The angle α = 2 arcsin( dcdleg+2dx,i+2a1+2d+2a2

) represents theshare of the length that belongs to the IW cross section [seeFig. 8(c)].

B. Rectangular Center Leg

The mean length of windings in case of a rectangular centerleg can be calculated according to

lm = 2 bleg + 2 aleg + 4 (dx,i + a1 + d+ a2). (12)

Fig. 8. Definition of required geometrical parameters.

The partial length equations for Double-2D modeling of trans-formers with rectangular center leg are proposed as follows:

lw,in = score · aleg (13)

lw,out = (2− score) aleg + 2 bleg + 4 (dx,i + a1 + d+ a2).(14)

APPENDIX BDEFINITION OF GEOMETRICAL PARAMETERS

The geometrical parameters of an arbitrary transformer areillustrated in Fig. 8.

APPENDIX CPARAMETERS OF COMPARED TRANSFORMERS

The geometrical parameters of the six compared transformersare listed in Table V.

APPENDIX DWINDING HEIGHT TRANSFORMATION (REQUIRED FOR KAPP,

DOWELL, AND ROGOWSKI)

For transforming windings of different height into windingsof equal height, it is proposed to proceed as following: Thegeometric mean height is calculated according to (15). Thegeometric mean was found to yield more accurate results thanthe arithmetic mean. The distances from windings to yokes dy,b

and dy,t are corrected according to (16), such that the windowheight hw is the same as before

hgeom = h =√h1 · h2 (15)

dy,b,corr = dy,b +h1 − h

2; dy,t,corr = dy,t +

h1 − h

2. (16)



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[34] E. Billig, “The calculation of the magnetic field of rectangular conduc-tors in a closed slot, and its application to the reactance of transformerwindings,” Proc. IEE, IV, Instit. Monographs, vol. 98, no. 1, pp. 55–64,1951.

[35] M. Strütt, “Das magnetische Feld eines rechteckigen, von Gleichstromdurchflossenen Leiters,” Archiv für Elektrotechnik, vol. 17, no. 5, pp. 533–535, 1926.

[36] M. Lambert, M. Martinez-Duro, J. Mahseredjian, F. De Leon, and F. Sirois,“Transformer leakage flux models for electromagnetic transients: Criticalreview and validation of a new model,” IEEE Trans. Power Del., vol. 29,no. 5, pp. 2180–2188, Oct. 2014.

[37] J. C. Maxwell, A Treatise on Electricity and Magnetism, 1st ed. Oxford,U.K.: Clarendon, 1873.

[38] H. Mecke, “Betriebsverhalten und Berechnung von Transformatoren fürdas Lichtbogenschweissen,” Ph.D. dissertation, Technische HochschuleOtto von Guericke Magdeburg, Magdeburg, Germany, 1979.

Richard Schlesinger (Student Member, IEEE) re-ceived the Dipl.-Ing. degree in energy and envi-ronmental technology with distinction from the TUWien, Vienna, Austria, in 2018. He is currently work-ing toward the Ph.D. degree with the Laboratoryfor High Power Electronic Systems, Swiss FederalInstitute of Technology (ETH), Zurich, Switzerland.

In his master’s thesis, he modeled and evaluatedthree-terminal impedance measurement setups forsolid oxide fuel cell electrodes. During his studies,he spent exchange semesters at UPC Barcelona and

at the Beuth University of Applied Sciences, Berlin, Germany, and worked withthe Austrian Institute of Technology. His research interests include modelingand design of magnetic components.

Jürgen Biela (Senior Member, IEEE) received theDiploma (hons.) degree from Friedrich-AlexanderUniversität Erlangen-Nürnberg, Erlangen, Germany,in 1999, and the Ph.D. degree from the Swiss FederalInstitute of Technology (ETH), Zürich, Switzerland,in 2006.

In 2000, he joined the Research Department,Siemens A&D, Erlangen, and in 2002, he joined thePower Electronic Systems Laboratory, ETH Zürich,as a Ph.D. Student focusing on electromagneticallyintegrated resonant converters, where he was a Post-

doctoral Fellow from 2006 to 2010. Since 2010, he has been an AssociateProfessor, and since 2020, a Full Professor of high-power electronic systemswith ETH Zurich.


SUPPLEMENTARY DOCUMENT: IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 36, NO. 1, JANUARY 2021

Supplementary Document of Paper:Comparison of Analytical Models of Transformer

Leakage Inductance:Accuracy vs. Computational Effort

Richard Schlesinger, Student Member, IEEE and Jurgen Biela, Senior Member, IEEE

Abstract—This supplementary document provides the equa-tions of Roth’s and Margueron’s model in a compact andcomprehensible way. The parameter definitions concur with thedefinitions of the main paper in Fig. 8a.

EQUATIONS OF ROTH’S MODEL [14], [31]

Applying the image method to the inside-window crosssection leads to a configuration according to Fig. 4a. Rothassumes infinite permeability of the core µr,c → ∞. Hence,the image current densities are equal to the original currentdensity due to Jimage = Joriginal

µr,c−1µr,c+1 . Thus, the z-component

of the current density distribution of each winding k Jz,k(x, y)becomes a two-dimensional rectangular signal in space thatcan be expressed as double Fourier series according to (17)

Jz,k(x, y) =

∞∑m=0

∞∑n=0

Jk,mn cos

(m2π

Txx

)cos

(n2π

Tyy

)(17)

where Jk,mn are the Fourier coefficients and Tx and Ty arethe periods in x- and y-direction.Fig. 9 shows the current distribution in space. The figure showsthat the periods of the signal in x- and y-direction can beexpressed according to (18)

Tx = 2ww ; Ty = 2hw (18)

where ww is the window width and hw is the window height.This leads to the Fourier series in (19)

Jz,k(x, y) =

∞∑m=0

∞∑n=0

Jk,mn cos

(mπ

wwx

)cos

(nπ

hwy

)(19)

Next, the Fourier coefficients have to be calculated accordingto (20). The Fourier coefficients vary depending on whether

R. Schlesinger and J. Biela are with the Laboratory for High PowerElectronic Systems HPE, ETH Zurich, Switzerland.

y

y

x

x

Windowx-Image Planes

y-ImagePlanes

0 Tx = 2ww

Ty = 2hw

ww

hw hk

wk

Jz(x)Jz(y)

a-

h-

h+

a+

ww

Nk.Ik

wk.hk

Nk.Ik

wk.hk

Fig. 9: Mirroring winding k with Nk turns across the trans-former window gives a spatial 2D rectangular signal. Thissignal is expressed as a spatial 2D Fourier series accordingto Roth. Image layers in green

m or n, or both equal zero. Note that the even symmetry ofthe function is exploited to shorten the integrals

Jk,00 =2

Tx

2

Ty

∫ Ty/2

0

∫ Tx/2

0

Jk dxdy =1

wwhw

∫ h+k

h−k

∫ a+k

a−k

Jk dxdy

Jk,m0 =4

Tx

2

Ty

∫ Ty/2

0

∫ Tx/2

0

Jk cos

(mπ

wwx

)dxdy

=2

wwhw

∫ h+k

h−k

∫ a+k

a−k

Jk cos

(mπ

wwx

)dxdy

Jk,0n =2

Tx

4

Ty

∫ Ty/2

0

∫ Tx/2

0

Jk cos

(nπ

hwy

)dxdy

=2

wwhw

∫ h+k

h−k

∫ a+k

a−k

Jk cos

(nπ

hwy

)dxdy

Jk,mn =4

Tx

4

Ty

∫ Ty/2

0

∫ Tx/2

0

Jk cos

(mπ

wwx

)cos

(nπ

hwy

)dxdy

=4

wwhw

∫ h+k

h−k

∫ a+k

a−k

Jk cos

(mπ

wwx

)cos

(nπ

hwy

)dxdy

(20)


where a+k is the upper x-coordinate, a−k the lower x-coordinateof the winding, h+k and h−k are the upper resp. lower y-coordinates of winding k (see Fig. 9). The number of turns ofthe windings are taken into account by implying them in thecurrent density (21)

Jk =NkIkwkhk

(21)

where Ik is the current, Nk is the number of turns, wk is thewinding’s width, and hk is the height of winding k. This way,the number of turns of the windings are taken into account. Ifthe leakage inductance should be referred to the primary side,the values for the primary side should be inserted in (21), i.e.k = 1. To refer the result to the secondary side, simply insertk = 2. The amplitude of the current of the referred side Ik isarbitrary and can for example be set to I1 = 1A.Four different terms for the Fourier coefficients follow from(20)

Jk,00 = Jkwwhw

(h+k − h

−k

) (a+k − a

−k

)(22)

Jk,m0 = 2Jkmhwπ

(sin(mπa+kww

)− sin

(mπa−kww

))(h+k − h

−k

)(23)

Jk,0n = 2Jknwwπ

(sin(nπh+

k

hw

)− sin

(nπh−

k

hw

)) (a+k − a

−k

)(24)

Jk,mn = 4Jkmnπ2

(sin(mπa+kww

)− sin

(mπa−kww

))(sin(nπh+

k

hw

)− sin

(nπh−

k

hw

))(25)

The net Fourier coefficients of the current density Jmn(x, y)can be acquired by superposing the current density coefficientsof all original windings according to (26)

Jmn =

C∑k=1

Jk,mn (26)

where C is the number of all windings in the transformerwindow. This leads to the overall current density Jz(x, y)according to (27)

Jz(x, y) =

C∑k=1

∞∑m=0

∞∑n=0

Jk,mn cos

(mπ

wwx

)cos

(nπ

hwy

)=

∞∑m=0

∞∑n=0

Jmn cos

(mπ

wwx

)cos

(nπ

hwy

)(27)

Next, the magnetic potential Az,k(x, y) of each winding k isalso expressed by a 2D-Fourier series according to (28)

Az,k(x, y) =

∞∑m=0

∞∑n=0

Ak,mn cos

(mπ

wwx

)cos

(nπ

hwy

)(28)

The overall potential coefficients Amn are calculated withsuperposition according to (29)

Amn =

C∑k=1

Ak,mn (29)

The potentials of each winding are superposed accordingly tothe current density, resulting in (30)

Az(x, y) =

C∑k=1

∞∑m=0

∞∑n=0

Ak,mn cos

(mπ

wwx

)cos

(nπ

hwy

)=

∞∑m=0

∞∑n=0

Amn cos

(mπ

wwx

)cos

(nπ

hwy

)(30)

Next, equations (27) and (30) are set into the Poisson equationfor the magnetic potential in 2D cartesian coordinates Az(x, y)

∂2Az(x, y)

∂x2+∂2Az(x, y)

∂y2= −µ0Jz(x, y) (31)

This delivers the relation between the Fourier coefficients ofthe magnetic potential and the current density according to(32)

Amn =µ0 Jmn(

mπww

)2+(nπhw

)2 (32)

Next, the magnetic energy per unit length W ′mag is acquired.

W ′mag =1

2

∫∫window

Az(x, y) · Jz(x, y) dA

=1

2

∞∑m=0

∞∑n=0

µ0J2mn(

mπww

)2+(nπhw

)2·∫ hw

0

∫ ww

0

cos2(mπ

wwx

)cos2

(nπ

hwy

)dx dy

=1

2hwwwµ0 ·

1

2

∞∑m=1

J2m0(mπww

)2 +1

2

∞∑n=1

J20n(nπhw

)2+

1

4

∞∑m=1

∞∑n=1

J2mn(

mπww

)2+(nπhw

)2

(33)

Note that the Fourier coefficient J00 = 0 does not contributeto the magnetic energy due the balanced magnetomotive force(MMF ) of primary and secondary side (MMF1 = N1I1 =−MMF2 = −N2I2). The product Az(x, y) · Jz(x, y) doesnot yield mixed sum terms due to the orthogonality relation(34).∫ T

0

cos

(m2π

Tx

)· cos

(n2π

Tx

)= 0 . . . for m 6= n (34)

where T is the period of the signal and m and n are arbitraryintegers.The cos2 functions were integrated using (35). This integra-tion is essential for computational efficiency, as the constantprimitive function spares the explicit calculation of the fieldinside the windings.∫ T

0

cos2(m2π

Tx

)= 2

∫ T/2

0

cos2(m2π

Tx

)=T

2(35)

where T is the period of the signal and m is an arbitraryinteger.Finally, the leakage inductance per unit length L′σ is obtained


using the correlation between inductance (per unit length) andmagnetic energy (per unit length) from (36).

L′σ =2W ′mag

I2ref(36)

where Iref is the current of the side the inductance is referredto (primary or secondary).The series in (33) are very fast convergent ∝ 1

m3 . However,higher values of ww and hw hamper the convergence of theseries. The results of this paper were produced with a numberof series terms of 25 (inside-window), resp. 250 (outside-window). For the outside-window cross section, a highernumber of sum terms is required due to the very high valuesof ww and hw.

EQUATIONS OF MARGUERON’S MODEL [15], [16]

Margueron’s model is based on a formula for the vectorpotential in space due to a current-carrying winding k embed-ded in a homogeneous medium with the permeability µ0 as afunction of its location and dimensions. The z-component ofthe vector potential Az,k(x, y) due to the current in windingk can be calculated according to (37)

Az,k(x, y) =−µ0

4πJk[F(x− xc,k − wk

2 , y − yc,k −hk

2

)−F

(x− xc,k + wk

2 , y − yc,k −hk

2

)−F

(x− xc,k − wk

2 , y − yc,k +hk

2

)+F

(x− xc,k + wk

2 , y − yc,k +hk

2

)](37)

where Jk = NkIkwkhk

is the homogeneous current density ofwinding k. Just as in Roth’s model, the number of turns Nk isalready implied in the current density. The variables xc,k andyc,k are the coordinates of the center of the winding, wk isthe width of the winding, and hk is the height of the winding,as shown in Fig. 10. The function F (X,Y ) can be calculatedaccording to (38)

F (X,Y ) = XY ln(X2 + Y 2)

+X2 arctan

(Y

X

)+ Y 2 arctan

(X

Y

)(38)

The singularities at X = 0 and Y = 0 are avoided by settingthe respective arctan-term to 0, which is justified due to theprefactor X2 resp. Y 2.

hk

y

xxc,k

yc,khk

2hk

2

wk

2wk

2

- akak

+

hk-

+

Nk.Ik

Fig. 10: Relevant geometry parameters of winding k requiredfor Margueron’s model

The transformer core is taken into account by discrete imagesof the windings. In the outside-window cross section, the

windings are mirrored across the edge of the center trans-former leg according to section II-B of the main document.In the inside-window cross section, the windings are mirroredaccording to Fig. 4a, only with a discrete amount of imagelayers. The amount of image layers has to be assumed.Assuming one image layer is the fastest and the least accurateway. Adding image layers leads to higher accuracy but alsohigher computational effort. Note that the finite permeabilityof the core µr,c is taken into account by scaling the mirroredcurrents according to Jimage = Joriginal

µr,c−1µr,c+1 .

Margueron’s model also uses the magnetic energy integral(39).

W ′mag =1

2

∫∫window

Az,tot · Jz dA (39)

The total potential Az,tot(x, y) is a superposition of the po-tential due to all original and mirrored currents according to(40)

Az,tot(x, y) =

D∑k=1

Az,k(x, y) (40)

where D is the amount of all windings, including the imagewindings. The number D depends on the number of images,i.e. the particular cross section and the amount of image layers.Note that the integral (39) is only nonzero in the winding-domain since the rest of the space is without current.

Margueron 2007-model [15]

In the Margueron 2007-model [15], the integration in (39) isexecuted numerically. I.e. several points in space are selectedand the values are calculated. Between these points, thepotential is linearly interpolated. The points without currentcan be neglected, since they don’t contribute to the integral(39).

Margueron 2010-model [16]

In the Margueron 2010-model [16], the integral (39) isexecuted analytically. The contribution to the magnetic en-ergy of each original and mirrored conductor k is separatelycalculated. This is done by computing (41)

W ′cont,k =1

2

∫∫window

Az,k · Jz dA (41)

Assuming a 2-winding transformer as depicted in Fig. 8a(indices 1 resp. 2), (41) splits up in two integrals

W ′cont,k =1

2

(J1

∫∫conductor 1

Ak(x, y) dx dy

+J2

∫∫conductor 2

Ak(x, y) dx dy

) (42)


where J1 and J2 are the current densities of conductor 1 andconductor 2, respectively.The integrals in (42) can be solved analytically using (43)∫∫

conductor iF (x− xc,k ± wk

2 , y − yc,k ±hk

2 ) dA =

=

∫ h+i

h−i

∫ a+i

a−i

F (x− xc,k ± wk

2 , y − yc,k ±hk

2 ) dxdy =

= G(a−i − xc,k ±wk

2 , h−i − yc,k ±

hk

2 )

−G(a+i − xc,k ±wk

2 , h−i − yc,k ±

hk

2 )

−G(a−i − xc,k ±wk

2 , h+i − yc,k ±

hk

2 )

+G(a+i − xc,k ±wk

2 , h+i − yc,k ±

hk

2 )

(43)

where the index i represents the particular original conductor(range of integration) and the index k represents the conductorthat causes the field.The primitive function G(X,Y ) is given in (44)

G(X,Y ) =− 1

24(X4 − 6X2Y 2 + Y 4) ln(X2 + Y 2)

+1

3XY

[X2 arctan

(Y

X

)+ Y 2 arctan

(X

Y

)]− 7

24X2Y 2

(44)

Again, the singularities at X = 0 and Y = 0 are avoided bysetting the respective arctan-term to 0, which is justified dueto the prefactor X2 resp. Y 2.Next, the contributions to the magnetic energy of all conduc-tors are superposed.

W ′mag, tot =

D∑k=1

W ′cont,k (45)

where D indicates the amount of all conductors, i.e. originaland mirrored conductors. Finally, the leakage inductance perunit length L′σ can be computed by (46)

L′σ =2W ′mag, tot

I2ref(46)

where Iref is the current of the side the inductance is referredto (primary or secondary).

comparison of analytical models of transformer leakage inductance… · 2020. 12. 9. ·...

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