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General Analytical Model for the Thermal Resistance of Windings Made of Solid or Litz Wire

M. Jaritz, A. Hillers, J. Biela

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

mailto:[email protected]

668 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 34, NO. 1, JANUARY 2019

General Analytical Model for the Thermal Resistanceof Windings Made of Solid or Litz Wire

Michael Jaritz , Member, IEEE, Andre Hillers , Student Member, IEEE,and Juergen Biela , Senior Member, IEEE

Abstract—In this paper, an analytical method to model the ther-mal resistance of windings made either of solid or litz wire is pre-sented and validated by measurements. In addition, an extendednumerical approach for litz wire windings, which also considersthe twist pitch, is shown. With the presented model, it is possibleto describe the thermal resistance of arbitrary wire arrangements.This approach can be used in straightforward thermal designs ofmagnetic devices, or it can be integrated in optimization proce-dures to improve the thermal designs of magnetic components.The analytical models are verified by measurements with nonpot-ted and epoxy-potted solid wire test setups and test setups for litzwire, which all show a good matching between the calculated andmeasured values. Besides the models for the thermal resistance be-tween the winding layers, a method to simplify the thermal networkof multilayer windings is presented. Finally, a thermal equivalentcircuit of a test transformer has been calculated, which shows agood match between the measured and the predicted temperaturedistribution.

Index Terms—Inductor windings, litz wire, round solid wire,thermal model, thermal resistance, transformer windings.

I. INTRODUCTION

IN MANY converter systems, the design of magnetic com-ponents is an important step of the overall system design.

Due to the high number of degrees of freedom and geometricparameters, this is often performed with the help of optimiza-tion procedures [see, for example, Fig. 1(a)] [1]. With the coreand the winding losses, the critical operating temperatures arecalculated, which determine if the design is valid or if the ge-ometry has to be modified by the optimizer. Since the windingand core losses are temperature dependent, as shown in [3]–[5],it is of high importance to estimate the occurring winding andcore temperatures properly. For obtaining the temperatures ineach part of the magnetic component, thermal equivalent cir-cuits are required [see Fig. 2(a)]. The thermal model containsall types of heat transfer (conduction, convection, and radiation)represented by equivalent thermal resistors, and the losses aremodeled by equivalent current sources. Thermal capacitances

Manuscript received November 22, 2017; revised February 15, 2018; acceptedMarch 11, 2018. Date of publication March 18, 2018; date of current versionNovember 19, 2018. Recommended for publication by Associate Editor K.Ngo. (Corresponding author: Michael Jaritz.)

The authors are with the Laboratory For High Power Electronic Sys-tems, ETH Zurich, 8092 Zurich, Switzerland (e-mail:, [email protected];[email protected]; [email protected]).

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

Digital Object Identifier 10.1109/TPEL.2018.2817126

Fig. 1. (a) General magnetic component optimization procedure. With thecore and the winding losses, the critical operating temperatures are calculated,which determine if the design is valid or if the geometry has to be modifiedby the optimizer. (b) Example for an optimized transformer geometry (Outputvoltage: 14.4 kV, turns ratio: 20, output power: 9 kW, operation frequency:100–110 kHz, and isolation voltage: 115 kV). A detailed design procedure ofthe transformer is given in [2]. The designators A, B, and C are at the sameposition as in Fig. 12. The transformer is built by AMPEGON AG.

are not considered in this model because, in many designs, onlythe final temperature distribution is of interest, and therefore,the thermal model in Fig. 2(a) represents a steady-state model.One major challenge is the correct estimation of the winding

0885-8993 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

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JARITZ et al.: GENERAL ANALYTICAL MODEL FOR THE THERMAL RESISTANCE OF WINDINGS MADE OF SOLID OR LITZ WIRE 669

Fig. 2. (a) Thermal equivalent circuit of the transformer from Fig. 1(b). Thethermal transition between core parts and windings (Rwp-cc), the heat transferwithin the core (Rcl), between primary and secondary winding (Rwp-ws), respec-tively, as well as within the windings (Rw,x ) is performed by heat conduction.The thermal resistances between core and ambient (Rc-amb) or winding and am-bient (Rws-amb, Rwp-amb) are based on radiation and convection. The losses inthe windings (Pwp, Pws) and the losses in the center leg and the rest of the core(Pcl, Pcr) are modeled by current sources. (b) Cross section of the transformerfrom Fig. 1(b). The circuit is valid for the symmetric half of the transformerin (b).

resistance. For foil windings, this is a relatively straightforwardcalculation, as presented in [6]. However, there is a lack con-cerning fast and accurate analytically models for the thermalwinding resistance Rw,x of round solid windings and litz wirewindings, which can be used in optimization procedures.

In order to obtain a time and a computational effort optimizedresult for the thermal behavior of the windings, the windings aremodeled with the help of thermal lumped resistor models [7]–[11] or simplified finite-element method (FEM) models [12]–[16]. There, winding homogenization techniques are appliedto combine the different thermal conductivities of the complexstructure of a winding (e.g., the randomly distributed copperwires, the insulation material of the wires, and the bondingmaterial between the wires) into a single effective thermal con-ductivity, which is needed for the lumped parameter model, aswell as for the simplified FEM model. The following homoge-nization techniques are given in the literature.

1) The authors in [17] and [18] propose analytical approachesto derive the effective thermal conductivity based on theequations given by Hashin and Shtrikman (H&S) in [19],which are restricted to two components winding amal-gams. In this model, it is assumed that the windingsconsist of copper wires and bonding material. The ther-

mal conductivity of the thin isolation around the wires isassumed to be close to the one of the bonding material andis, therefore, neglected. In order to overcome the restric-tion to two components, the H&S approach is extendedto three-component amalgams in [20]. The H&S and theextended H&S (H&S+) approaches are valid for circularwire profiles and are not applicable to general conduc-tor shapes [20]. In [21], a method is presented, which canbe applied for rectangular conductor profiles. For litz wirewindings, a two-step homogenization based on the H&S+approximation is presented in [22]. First, the equivalentconductivity of a single litz wire bundle is calculated, andin a second step, the equivalent conductivity of the litzwire bundle winding is derived, employing the effectivethermal conductivity of the bundle from step one.

2) The thermal conductivity is extracted by measurements aspresented in [14], [23], and [24].

3) The equivalent conductivity is derived from FEM simula-tions, where parts of the windings are modeled in their fullcomplexity, including all isolation components as givenin [25].

The methods presented in 2) and 3) either make use of mea-surements or are essentially too time consuming to be applied inoptimization routines. Therefore, the focus in this paper is on thederivation of an accurate analytical expression for the thermalwinding resistanceRw,x , which can be used in fast optimizationalgorithms for transformer or inductance design. The resultingexpressions for solid and litz wire windings are derived based onan electrothermal analogy and only rely on datasheet parametersof the wires and the insulation materials. In order to justify thenew contributions of this paper, the determined expressions arebenchmarked with the H&S and the H&S+ approaches for solidwire, as well as for litz wire with the two-step homogenizationfrom method 1).

The rest of this paper is organized as follows. The derivationof the thermal resistanceRw,x is given in detail in Section II. Af-terwards, a simplified thermal T-equivalent circuit (tTEC) modelfor multilayer windings is derived in Section III. In Section IV,measurement results are presented. First, in Section IV-A,a measurement setup to verify the results of Section II is pre-sented. Then, the results from the proposed calculation are com-pared with measurements and benchmarked with the H&S andH&S+ approximations in Section IV-B. Finally, in Section IV-C, the temperature distribution in a transformer is comparedwith the analytically predicted temperature distribution, usingthe new thermal winding description.

II. DERIVATION OF THE THERMAL RESISTANCE

To describe the heat flow in windings of transformers or in-ductors, a model for the heat flow paths is necessary. This pathcan be represented by the thermal resistance Rw,x , which isderived based on the electrothermal analogy. The thermal resis-tance Rw,x is composed of a parallel connection of a tangentialresistance Rtan, a radial resistance Rrad, and an axial–radial partRax,rad (see Fig. 4). First, the relations of the electrothermalanalogy are recapitulated, and then, the calculation of the tan-gential part and the radial part is presented.


Fig. 3. Comparison of the FEM simulated electrical and thermal field distri-butions. (a) Electrical field lines between two orthogonally arranged solid wireswith potentials p1 and p2 . (b) Thermal heat flux lines between two orthogonallyarranged wires. (c) Electrical field lines between three orthocyclically arrangedwires with potentials p1 , p2 , and p3 . (d) Thermal heat flux lines between threeorthocyclically arranged wires.

Fig. 4. Tangential heat flows Ptan,A-B and Ptan,A’-B’, radial heat flow Prad, andaxial–radial heat flow Pax,rad,A”-B” in multilayer windings [6].

A. Electrothermal Analogy

In the literature, the electrothermal analogy is often describedwith the help of the electrical flow field and the thermal flowfield, which leads to the thermal equivalent circuit as, for ex-ample, depicted in Fig. 2(a) [26]. In this analogy, the thermalresistanceRth,aw directly corresponds to the electrical resistanceRel,aw along a wire and can be determined by replacing thespecific electrical conductivity by the thermal conductivity

Rel,aw =s

γA⇐⇒ Rth,aw =

s

λA. (1)

The variables s andA are the length and the cross section of thewire, respectively. This analogy is applied to the calculation ofthe tangential thermal resistance Rtan.

Another analogy, which is applied for the determination ofthe radial thermal resistanceRrad, is between the static electricalfield and the thermal flow field. Both field problems can bedescribed by elliptical partial differential equations (PDEs). Inthe case of an electrostatic field problem, the PDE [27] is givenas

−div(grad(p)) =ρ

ε(2)

and the steady-state heat conduction PDE [30] is formulated as

−div(grad(T )) =S

λ. (3)

These equations are also called the well-known Poisson’s equa-tions [27], [30]. The material parameters, which are the per-mittivity ε and the thermal conductivity λ, are assumed to beindependent regarding the potential p and the temperature T ,respectively. Furthermore, ε and λ are assumed to be isotropic.Comparing the coefficients in (2) and (3) results in the followingelectrothermal analogies:

p ⇐⇒ T (4)

ε ⇐⇒ λ. (5)

Since the charge density ρ is directly proportional to the electri-cal field strength E · ε and the heat source density S is directlyproportional to the heat flux q, the analogy is given as

E · ε ⇐⇒ q. (6)

Hence, the electrical field lines represent the heat flux lines ina thermodynamic problem. The most important relations be-tween these analogies are given in [6] and [27]–[29] and aresummarized in Table I.

For the calculation of the radial thermal resistance, two basicwinding arrangements are identified, which can be divided intoorthogonally [see Fig. 3(a) and (b)] and orthocyclically arrangedwires [see Fig. 3(c) and (d)]. In orthogonally arranged wires, thewires of the second layer are lying exactly on top or besidethe next layer, whereas, in orthocyclic windings, the wires ofthe next layer are lying in the gaps of the previous layer. Fig. 3(a)shows the electrical field distribution between two orthogonallyarranged wires, and Fig. 3(c) shows the electrical field distribu-tion between three orthocyclically arranged wires, each with anequipotential surface with potentials p1 , p2 , and p3 . The thermalheat flux lines are shown in Fig. 3(b) and (d). There, the conduc-tor surfaces are approximately isothermal surfaces with giventemperatures T1 , T2 , and T3 , respectively. Comparing Fig. 3(a)and (b), as well as Fig. 3(c) and (d), shows a good match betweenthe electrical field and the thermal flux lines. Hence, the calcula-tion of the radial thermal resistance Rrad, between orthogonallyor orthocyclically arranged wires, could be performed by calcu-lating the electrical capacitance between the same arrangementsas presented in [31]. For the sake of completeness, the derivationof the radial electrical capacitance in [31] is summarized andevaluated by FEM simulations in Appendix A. Based on these


TABLE IELECTROTHERMAL ANALOGY [6], [27]–[29]

results, the transition to the thermal resistance is given in thenext section.

The heat flow P in multiple-layer windings can be dividedinto three components [6] (see Fig. 4): first, in a tangential com-ponent Ptan that takes the flow along the winding from layer tolayer into account; second, in a radial component Prad that rep-resents the heat flow between two neighboring winding layersthrough the isolation layer, and, third, in a combined axial andradial componentPax,rad. There, the heat is conducted, e.g., frompoint A′′, in the axial direction through each wire of the innerwinding. Afterwards, it is convectively transmitted into the outerwinding layer and via conduction back to the corresponding wirein point B′′. Due to the high thermal resistance caused by theconvective heat transmission between the winding layers, theheat flow Pax,rad is comparably low and is, therefore, neglected.

The derivation of the tangential and radial thermal resistanceis given in the following.

B. Tangential Thermal Resistance

The tangential heat flow Ptan,A’-B’ from point A′ in Fig. 4has to be conducted first along all wires of the inner windinglayer, in the winding axial direction, and then back down toits consecutive point B′ along all wires of the outer windinglayer, whereas the tangential heat flow Ptan,A-B from the wire in

point A has to be conducted only one round along the inner andthe outer winding to the corresponding wire in point B in thenext layer. Therefore, the tangential resistor Rtan between twowires on two consecutive winding layers is the average of alltangential paths between two consecutive winding layers and isgiven by

Rtan =1Npl

Npl∑

i=1

lw · 2iλCuAwire

=lw(Npl + 1)λCuAwire

≈ lw ·Npl

λCuAwire. (7)

There,Npl is the number of turns per layer, lw is the mean lengthper turn, λCu is the thermal conductivity of copper, and Awire isthe cross sectional area of the wire. In the following sections,the thermal equivalent resistance is derived.

C. Radial Thermal Resistance

In order to determine the radial thermal resistance, the anal-ogy between the electrostatic field and the thermal flow fieldfrom Table I is applied. By simply replacing all permittivitieswith the equivalent thermal conductivities and inverting (63)and (64), the radial thermal resistance can be obtained as

Rorth =

[2λairlwα

(Y +

λair

8λiso

(2δro

)2Z

α

)]−1

(8)

and

Rcyc =[4λairlw

(Mair+Miso

δλair

λisor2o

(ro − δ

2

))]−1

(9)

with

εo ⇐⇒ λair (10)

εlay ⇐⇒ λlay

λair(11)

εiso ⇐⇒ λiso

λair(12)

where λair is the thermal conductivity of air, λlay is the ther-mal conductivity of the isolation layer, and λiso is the thermalconductivity of the wire isolation.

The derivation of the thermal resistance of multilayer wind-ings will be investigated in the next section.

D. Multilayer Thermal Resistance Model of Round Solid WireWindings

For an either pure orthogonal or pure orthocyclic multilayerwinding, the thermal resistance can be written as

Rw,x = (Rtan||Rrad)Nl

Npl. (13)

Since Rtan and Rrad are per-turn resistances, the thermal per-turn parallel connection (Rtan||Rrad) in (13) must be divided byNpl. Nl is the number of layers in series and Npl is the numberof turns per layer. In general, every winding is a combinationof both thermal radial resistances (orthogonal and orthocyclic).Due to manufacturing reasons, the overall thermal resistanceRw,x of a solid wire winding, as shown for example in Fig. 5(a),


Fig. 5. (a) Cross section of a solid wire winding with orthogonal and or-thocyclic layers. (b) Resistive thermal circuit model of Rw,x formed by thetangential and radial thermal components Rtan, Rorth, and Rcyc. The thermalresistance inside the wire is neglected due to the high thermal conductivity ofthe copper.

is usually a combination of orthogonal and orthocyclic layers.At least both halves of the outer layers of the winding, whichrepresents an orthogonal transition to the next medium, can bemodeled as one orthogonal layer [see Fig. 5(a) and (b)]. Thisleads to the general form

Rw,x = (Rtan||Rcyc)Nl −North

Npl+ (Rtan||Rorth)

North

Npl(14)

where North is the number of assumed orthogonal layers. Thethermal resistance inside the wire is neglected due to the highthermal conductivity of the copper λCu (see Table II).

E. Multilayer Thermal Resistance Model of Litz WireWindings

As litz wire is often used to reduce high-frequency losses,two thermal models for a single litz wire bundle are presentedin the following. First, a straightforward analytical model is de-termined, and second, an extended numerical model, which alsoconsiders the twist pitch, is derived. Afterwards, the transition tothe multilayer thermal resistance model is given. Fig. 6(a) showsthe cross section of a litz wire winding with three orthocycliclayers and one orthogonal layer. The zoomed view shows a non-circular shaped litz wire bundle. The transition from the physicallitz wire winding to its thermal model is shown in Fig. 6(b) andcan be divided into three steps. Step (A) starts with the trans-formation of the noncircular litz wire bundle with Ns strandsinto a square litz bundle with side length

√Ns · 2 · ro,litz. Due to

the loose positions of the strands inside the litz bundle and theunknown force, which is applied to the litz wire bundle duringthe winding process, the exact litz strand positions inside thebundle are not available, and the bundle shape usually differsfrom a circular shape, as can be seen in Fig. 6(a). Therefore,to overcome this problem, in the new proposed model, the litzstrands are rearranged to an equivalent squared pattern, in orderto uniform the shape inside of each bundle, which reduces thecomplexity for deriving the equivalent resistance of the bun-dle. In step (B), the equivalent resistance of a litz bundle Rlitz

TABLE IIMATERIAL PROPERTIES AND SOLID OR LITZ WIRE WINDING PARAMETERS

is calculated, and, finally, in step (C), the thermal circuit of amultilayer winding is derived. This model is based on the ther-mal network in Fig. 5(b), but additionally considers the thermalresistance inside the litz wire bundle Rlitz.

1) Analytical Thermal Resistance Litz Wire Bundle Model:One way to model the thermal resistance of the litz wire bundleRlitz is the series connection of the orthocyclic contributionsRl,cyc between each litz strand and the orthogonal part Rl,orth atthe edges [see Fig. 6(b)]. Both resistors are determined by usinglitz strand parameters and (9) for Rl,cyc and (8) for Rl,orth. Thethermal resistance of the litz wire is

Rlitz = Rl,cyc

(1 − 1√

Ns

)+Rl,orth√Ns

. (15)

2) Extended Numerical Thermal Resistance Litz Wire BundleModel: Litz wire consists of bundles and subbundles of strands,which are twisted to minimize the high-frequency losses. Usu-ally, the exact twisted bundle structure is not available, andtherefore, a modeling approach, which takes the twist pitch ofthe litz wire bundles into account, is presented in the following.


Fig. 6. (a) Cross sections of a litz wire winding and a noncircular litz wirebundle. (b) Transition from the noncircular litz wire bundle into a square litzbundle (A). There, the litz strands of the doted red square are rearranged toa squared pattern and the outer isolation is removed. (B) Equivalent thermalnetwork of one rearranged litz bundle. (C) Thermal setup of a litz-wire-basedmultilayer winding, which also considers the thermal resistance inside the litzwire bundle.

The extended model of the litz wire bundle is depicted in Fig. 7.There, the heat flow is modeled by the already known radial partPl,rad and, in addition, a part along a single litz strandPlay causedby the lay. The thermal path along the strand can be described by

Rlay =llay/2

λCur2o,litzπ

(16)

where llay is the so-called twist pitch, which is the length of layof the litz wire and results in the network shown in Fig. 7(b).The thermal resistors R

′l,cyc and R

′l,orth are also calculated for

llay/2 with

R′l,cyc =

lwRl,cyc

llay/2(17)

Fig. 7. Extended model of the litz wire, which takes the twist pitch of the litzwire bundles into account. (a) Litz wire with two considered heat flow pathsPl,rad and Play. (b) Equivalent thermal network of one horizontal litz strandlayer from A to B and A’ to B’. One horizontal litz strand layer consists ofn = �√Ns� lay layers.

and

R′l,orth =

lwRl,orth

llay/2(18)

where the thermal resistorsRl,orth andRl,cyc are derived with litzstrand parameters and (8) and (9), respectively. Because thereis no compact equation for the resistance R

′litz, this network is

solved numerically. The thermal resistor R′litz represents one

horizontal layer of litz wires from point A to B and in parallelfrom point A’ to B’. One horizontal litz strand layer consists ofn = �√Ns� lay layers. The resulting thermal resistance Rlitz ofa litz wire bundle with length lw is given as

Rlitz =R

′litz · llay/2√Nslw

. (19)

Finally, using (15) or (19) with litz strand parameters and (7),(8), as well as (9) with litz bundle parameters gives the generalform of the thermal resistance of litz wire windings as

Rw,x = [Rtan||(Rcyc +Rlitz)]Nl −North

Npl

+ [Rtan||(Rorth +Rlitz)]North

Npl. (20)

The derivation of the thermal equivalent circuit model formultilayer windings will be investigated in the next section.

III. tTEC MODEL OF MULTILAYER WINDINGS

In the following, a simplified thermal T-equivalent model thatallows us to calculate the hotspot temperature and the hotspot


Fig. 8. (a) Simplified layer stack of the winding. (b) Corresponding thermalmodel with one heat flux source per winding layer. (c) Left (PL, i ) and right(PR, i ) components of the heat flux generated by the ith source as well as thetemperature contribution ΔTi,k of the ith source at the kth node. (d) Tempera-ture contribution ΔTL,k of all sources to the left of the kth node, including thesource at the node. (d) Temperature contribution ΔTR,k of all sources to theright of the kth node. (f) Derived tTEC model with a lumped heat flux source.(g) Special case for Rb = ∞. (h) Special case for Rb = Ra.

location in multilayer windings as the one shown in Fig. 8(a) ispresented. In this model, each layer-to-layer transition is de-scribed with a separate thermal resistor. The losses in eachwinding are represented by a heat-flux source. In the follow-ing, it is shown how the tTEC model illustrated in Fig. 8(f)can be used to predict the worst-case temperature based on the

thermal resistances previously calculated for solid wire and litzwire.

At first, the contribution of the losses from each layer issuperimposed to calculate the temperature Tk at all possiblepoints k between two layers. Afterwards, an analytic expressionfor the location of the worst-case temperature point is derivedby letting the number of winding layers go to infinity, while theoverall losses stay the same. The so-obtained result is used toderive a closed-form solution with a single lumped heat source,which provides a simplified way of calculating the worst-casetemperature for arbitrary choices of Ra, Rb, and Rw. It is worthnoting that this approach is mathematically accurate and is notequal to simply splitting the overall thermal resistance of thewinding stack in half and placing a single lumped heat sourcein between.

A. Modeling Assumptions

In the following, it is assumed that the interlayer thermalresistance is

Rl =Rw

N − 1. (21)

In a first approximation, the losses P in the windings are as-sumed to be homogenous and evenly contribute to the heat fluxin each layer

Pw1 = · · · = PwN =P

N. (22)

The lumped thermal resistors Ra and Rb denote the effectivethermal resistances from the winding surfaces to ambient. Theseresistors are usually representing convection or radiation, andtheir determinations are given in [32]. It is assumed that heat isonly transferred through the faces of the winding1 and that thetemperature distribution is homogenous. The equivalent circuitof such a layer stack is shown in Fig. 8(b).

B. Temperature Contribution of Each Layer

As can be inferred from the equivalent circuit shown inFig. 8(b), each heat source Pwi generates a heat flux

PL,i =P

N

Rb + (N−i)Rl

Ra +Rb +Rw(23)

to its left and a heat flux

PR,i =P

N

Ra + (i−1)Rl

Ra +Rb +Rw(24)

to its right. This situation is illustrated in Fig. 8(c). Consequently,the sources P1 . . . Pwk to the left of node k contribute

ΔTi,k |i≤k = PR,i ·RR,k = PR,i · ((N−k)Rl +Rb) (25)

to the temperature rise at node k, which is illustrated in Fig. 8(d),while the sourcesPwk+1 . . . PN on the right of node k contribute

ΔTi,k |i>k = PL,i ·RL,k = PL,i · ((k−1)Rl +Ra) (26)

1The edges are thermally isolated by the bobbin.


Fig. 9. (a) Temperature profile for finite number of winding layers N inrelation to the prediction using the tTEC model and the simplified tTEC model,

where P = P . The hottest point is always closest to k. For the calculations,Rw = Rws,Ra = Rwp-amb, andRb = Rws-amb have been chosen. (b) Accuracyof the tTEC approach when Ra + Rb � Rw is not fulfilled. The dashed-and-dotted lines show the respective hotspot temperature as calculated with the P =P tTEC approach. For the calculations, Rw = Rws, Ra = Rws

2 , and N = 13have been chosen.

which is illustrated in Fig. 8(d). According to the principle ofsuperposition, the overall temperature at point k is equal to thesum of the individual contributions and the ambient temperature

Tk = Tamb +k∑

i=1

ΔTi,k |i≤k︸︷︷︸

:=ΔTL, k

+N∑

i=k+1

ΔTi,k |i>k︸︷︷︸

:=ΔTR, k

. (27)

The two sums in (27) can be rewritten in a closed form as

ΔTL,k =P

N

(N−k)Rw+Rb

Ra +Rb +Rw

(kRa

N+

(k−1)kRw

2

)(28)

ΔTR,k =P

N

(k−1)Rw+Ra

Ra +Rb +Rw

(kRb

N+

(N−k−1)kRw

2

).

(29)

C. Location of the Worst-Case Temperature

In order to find the worst-case temperature, the relative loca-tion k = k

N at which the temperature is worst has to be deter-mined. This is done analytically by letting N in (28) and (29)go to infinity

limN→∞

ΔTR,k =kP

2(2Ra−kRw)(Ra + (1−k)Rw)

Ra +Rb +Rw(30)

limN→∞

ΔTL,k =(1−k)P

2(2Ra−kRw)(Ra+(1−k)Rw)

Ra +Rb +Rw. (31)

The relationship described in (22) ensures that the result staysfinite. Fig. 9(a) shows that this approximation is sufficientlyaccurate, even for low numbers ofN . Adding (30) and (31) andmaximizing the result with respect to k leads to the worst-case

value for k

kwc =12

2Rb +Rw

Ra +Rb +Rw. (32)

D. Closed-Form Solution and tTEC Model

By inserting (32) into (30) and (31) and adding the results,a closed-form solution of the maximum temperature inside thewinding is obtained

Twc = Tamb +

(Ra + Rw

2

) (Rb + Rw

2

)

Ra +Rb +RwPRa +Rb + Rw

2

Ra +Rb +Rw.

(33)The result corresponds to the tTEC-model shown in Fig. 8(f),where the individual heat flux sources have been replaced bythe lumped heat flow source

P = PRa +Rb + Rw

2

Ra +Rb +Rw. (34)

Because P ≤ P , embedding the tTEC model directly into alarger overall thermal network such as the one shown in Fig. 2(a)leads to lowered heat fluxes in the overall model and therewitha reduced accuracy of the calculations. To overcome this draw-back, the P = P approximation is proposed, which delivers anupper bound for the winding temperature inside the stack with-out altering the overall heat fluxes in the rest of the network.As shown in Fig. 9(a), the accuracy is high for Ra +Rb � Rw,which is the case for the transformer discussed in this analysis.The location of the hotspot temperature can be calculated asshown in (32).

E. Special Cases

For the case that eitherRa orRb is large, the tTEC model canbe simplified to the model shown in Fig. 8(g). In this case, theP ≈ P approximation becomes an equality. Thus, the model canbe directly embedded into a superordinate thermal network. Thelocation of the hotspot temperature can again be calculated asshown in (32) without modifications. For cases whereRa = Rb,the tTEC model described by (33) can be simplified to the circuitshown in Fig. 8(h)

limR a→Rb

Twc = Tamb + P

(Ra + Rw

4

) (Ra + Rw

4

)

Ra +Ra + Rw2

= Tamb +P

2

(Ra +

Rw

4

). (35)

Notice how the lumped resistors adjacent to the heat source arenow Rw

4 instead of Rw2 in the regular tTEC model. The sensitivity

analysis shown in Fig. 9(b) reveals that, without this adjustment,the model would be inaccurate for cases where Ra and Rb arewithin the same order of magnitude asRw. For the sake of clarity,the hotspot temperatures calculated with this approach are notexplicitly shown in Fig. 9(b), but are in perfect correspondencewith the peaks of the colored curves.


Fig. 10. (a) Cross section of the thermal test setup cross section. (b) Manu-factured thermal test setup, here of single solid wire.

IV. MEASUREMENT RESULTS

A. Measurement Setup

For validating the models depicted in Figs. 5 and 6, a test setupas shown in Fig. 10 has been developed. Three plastic layers areused for thermal isolation through which a negligible amount ofheat is dissipated and the generated heat will be forced to flow inthe y-direction through the winding. The inner isolation layer ismade of polyethylenterephthalate (PET) because this materialwithstands continuous temperatures up to 100 ◦C but unfortu-nately has a relatively high thermal conductivity. To overcomethis problem, two additional layers of polyvinylchloride (PVC)are attached on the top and bottom respectively.

Two power resistors were mounted to the inside of a squarecross sectional aluminum tube, which is also used as the windingbobbin. The tube can be seen as an isothermal interface due tothe high thermal conductivity of aluminum (see Table II), whichleads to a uniform heat distribution on the surface of the tube.To determine the inner and the outer winding temperature, twotemperature sensors (K-type thermo couples) are mounted, as

Fig. 11. Simplified thermal equivalent circuit of the test setup.

shown in Fig. 10(a) and (b). All material properties are listedin Table II. The thermal resistance of the test setup in the x-direction of the thermal isolation layer is

RPVC,PET = 2(2RPVC +RPET) = 18.5 ◦C/W (36)

where

RPVC =4l

d2AπλPVC

and RPET =4l

d2AπλPET

. (37)

The photo of the test setup is shown in Fig. 10(b), on the lefthand side without the winding and with mounted heating resistorand completely assembled on the right side.

B. Winding Measurement Results

The comparison between the measurements and the analyticalmodel is given in the following for the single solid wire windingexemplarily [see Fig. 10(b)].

In Table II, TSS,air and TSL,air,1-2 are the test setups (TS) withsolid wire (S), or litz wire (L) and an air gap. TSS,epoxy is thetest setup (TS) with solid wire (S) and an epoxy potted winding(epoxy). TWL,1 and TWL,2 are the litz wire (L) transformerwinding (TW ) parameters of the high-voltage high-frequencytransformer, which is investigated in Section IV-C. Using (14)and the winding parameters from the solid wire test setup TSS,air

given in Table II results in Rw,x = 1.63 ◦C/W. Concerning thatthe isolation of the wire is a combination of a polyetherimideand a polyamidimide material, the mean value of both of themwill be used for the thermal conductivity λiso. Due to the fact thatthe ratio between RPVC,PET and Rw,x is about 11, the heat flowthrough the isolation layer could be neglected, and the thermalsetup could be described with the simplified thermal circuit, asshown in Fig. 11. The measured thermal resistance Rw,x ,m canbe calculated by

Rw,x ,m =ΔTP

=T1 − T2

P. (38)

The comparison between the measured and the calculated ther-mal resistances of the tested winding setups is given in Table III.Generally, the number of orthogonally (North) and orthocycli-cally arranged layers is usually not available. Therefore, in prac-tical setups, North is chosen to be 1 because at least both halvesof the outer layers can be regarded as one orthogonal transition,and all other transitions are assumed as orthocyclic transitions.This results in a very low resistance in the case of solid wire(see Table III, TSS,air and TSS,epoxy). Using the correct num-ber of orthogonal transitions (North = 6) results in a relativelysmall error between the calculated and the measured resistance,as can be seen in Table III. In the case of a moulded/potted


TABLE IIICOMPARISON OF THE PROPOSED APPROACH WITH WINDING MEASUREMENTS AND BENCHMARK APPROACHES

winding (TSS,epoxy), λair is replaced by λepoxy. A polyester foil(Mylar, λiso,litz,1) is used for the outer isolation of the litz wirewindings. Due to the flexible outer shape of the litz wire bundle,the deviation of the thermal resistance in the case of litz wireis higher. This can be explained by the fact that the analyticalapproach assumes that the heat transition points toward the bob-bin and between the litz wires bundles are pure contact points,not, as seen in Fig. 6(a), some kind of contact areas. There-fore, the calculated resistance is too high and can be seen as aworst-case approximation, if it is used as, e.g., in the lumpedthermal resistance model in Fig. 12, whereas, if the twist pitchis considered, the error is low. However, the proposed approachachieves a significantly better accuracy compared to the analyt-ical H&S benchmark approach. In the case of solid wire TSS,air,H&S leads to a huge error (+319%) if the thermal conductiv-ity of the wire insulation material differs strongly from that ofthe winding impregnation material (compare λair with the meanvalue of λiso,1 and λiso,2 in Table II). In the case that the thermalconductivities are similar (TSS,epoxy), the error is in the range ofthe proposed method (+28.3%). The H&S+ approach achievesa higher accuracy for both solid wire setups (+9.7% for TSS,air

and +19.6% for TSS,epoxy) compared to the proposed approachin the case of North = 1 (−20.7% for TSS,air and −24.8% forTSS,epoxy). If the exact number of North is known, the proposedapproach leads to more accurate results (+2.5% for TSS,air and+15.2% for TSS,epoxy) than the H&S+ approach. In the caseof litz wire, the H&S, as well as the H&S+ approach, leads inthe best case to deviations, which are still larger than 76.1%,whereas the proposed analytical method results in +51.4% forTSL,air,1 and +38.57% for TSL,air,2 and the proposed numericalmethod results in −20.7% for TSL,air,1 and −0.5% for TSL,air,2.

Fig. 12. Cross section of the considered test transformer, which is rotatedby 90◦ compared to Fig. 1(b). The designators A, B, and C are at the sameposition as in Fig. 1(b). The simplified thermal equivalent circuit is valid for thesymmetric half of the transformer (compare Fig. 2). The green region representsa convective channel and the blue regions represent convective closed gaps.

Consequently, the new approach results in a similar accuracy,in the case of solid wire test setups, but gives a significantly moreaccurate estimation of the thermal winding resistance in the case


TABLE IVVALUES OF THERMAL RESISTORS AND MATERIAL PARAMETERS

TABLE VOPERATION POINT OF THE TEST TRANSFORMER

of litz wire windings and is, therefore, a serious alternative forfast thermal resistance estimations in optimization algorithms.The analysis of the thermal resistance of the H&S and H&S+approaches is given in [20] and are summarized in Appendix B.

The verification of the thermal resistance of the primary (Rwp)and the secondary (Rws) litz wire winding of a high-voltagehigh-frequency transformer is given in the next section.

C. Measured Temperature Distribution Results of aHigh-Voltage High-Frequency Transformer

In the following, the measured temperatures of a high-voltagehigh-frequency transformer are investigated and compared tothe calculation results based on the thermal network shown in

Fig. 13. (a) Section of the transformer under test. The entire transformer isdepicted in Fig. 1(b). (b) Temperature distribution of the transformer after aheat run test of 7.3 h. All transparent oil gap barriers are removed during themeasurements with the thermal camera. The white rectangles and the circledefines the area of averaging for the temperature measurements.

Fig. 12. The simplified thermal equivalent circuit is valid forthe symmetric half of the transformer. Due to the high operationfrequency (100–110 kHz), ferrite is used as core material andLitz wire is used for the primary and the secondary windingto reduce the high-frequency losses. The detailed design of thetransformer is given in [2]. The thermal network consists of theabove-derived thermal resistors of the primary and the secondarywindings (Rwp and Rws, parameters see Table II, TWL,1 andTWL,2) and the thermal resistors that model the conductive andconvective heat transfer for the rest of the transformer. Theformulas for these types of heat transfer are taken from [32]and are just summarized in the following. The resistance for theconductive heat transfer is [32]

Rcond =l

λA(39)


where l is the effective length of the heat transfer and A is theeffected cross section of the material. The thermal resistanceRcl through the center leg and the thermal resistance of the pri-mary and the secondary bobbins (Rbwp, Rbws) are, for example,belonging to that type of heat transfer.

The resistance for the convective heat transfer is givenin [32] as

Rconv =l

NuλA(40)

with

Nu = f(Ts − Tamb). (41)

The Nusselt number Nu is an empirical number, which is ameasure of the improvement of the heat transfer compared tothe case with static fluid [41]. Since it depends on the differenceof the surface temperature Ts and the ambient temperature Tamb,thermal networks, which contain convective resistors, have to besolved iteratively until a stable surface temperature is reached.The convective heat resistance of the surface of the core tothe ambient (Rc-amb) is a parallel connection of the horizon-tal surfaces with heat emission on the top and bottom planes(Rhtop,c, Rhbot,c) and the vertical surface (Rv,c) of the core

Rc-amb =(

1Rhtop,c

+1

Rhbot,c+

1Rv,c

)−1

. (42)

The convective heat resistance of the secondary winding sur-face to the ambient (Rws-amb) is a parallel connection of thehorizontal surfaces with heat emission on the top and bottomplanes (Rhtop,ws, Rhbot,ws) and the vertical surface (Rv,ws) of thewinding

Rws-amb =(

1Rhtop,ws

+1

Rhbot,ws+

1Rv,ws

)−1

. (43)

The convective heat resistance of the primary winding surfaceto ambient (Rwp-amb) is approximated as a convective channel[32] (the green region in Fig. 12), which is formed by the part ofthe primary surface outside the core window and the secondarybobbin. The thermal resistance between primary and secondarywindings (Rwp-ws) is formed by the series connection of thesecondary bobbin resistor (Rbws), the convective heat resistanceof the primary winding surface to the secondary bobbin insidethe core window (blue regions in Fig. 12) (Rhcgap,wp-bws), andthe convective heat resistance related to the air layer betweensecondary bobbin and secondary winding (Rws-bws)

Rwp-ws =Rbws+Rhcgap,wp-bws+Rws-bws. (44)

The resistor Rhcgap,wp-bws represents a convective horizontalclosed gap and the resistorRws-bws is formed by the parallel con-nection of the thermal resistors Rhcgap,ws-bws and Rvcgap,ws-bws,which represent the convective horizontal and vertical closedgaps between the secondary winding and the secondarybobbin [32]

Rws-bws =(

1Rhcgap,ws-bws

+1

Rvcgap,ws-bws

)−1

. (45)

Fig. 14. Averaged measured temperature sequences (T1 ,T2 , andT3 ) recordedover 7.3 h and their according best fit curves (T1,bf, T2,bf, and T3,bf), which areused for extrapolation until a stable steady-state temperature is reached. Themeasured signals are related to the measurement positions in Fig. 13(b).

The heat transfer between the center leg core and primary wind-ing (Rcl-wp) is formed by the series connection of the thermalresistors of the primary bobbin (Rbwp) and the air layer betweenthe center core and the primary winding bobbin (Rcl-bwp)

Rcl-wp = Rbwp +Rcl-bwp (46)

with

Rcl-bwp =(

1Rhcgap,cl-bwp

+1

Rvcgap,cl-bwp

)−1

. (47)

The numerical values of the thermal resistors, which are used inthe simulations, are given in Table IV. Additionally, the thermalconductivities for calculating the conductive heat resistancesand the equations for the Nusselt numbers, in the case of con-vective heat transfer, are listed in Table IV. Table V shows theoperation point for the heat run test in air and the calculatedlosses in the center leg of the core Pcl, the rest of the corePcr, and the windings Pwp, Pws. All resistance and loss valuesare given for the symmetric half of the transformer. The mainfrequency of the input voltage is f = 105.8 kHz and the trans-former is operated in the pulse mode (Tp = 3.5 ms) with a pulserepetition rate of frr = 14 Hz [1]. The high-frequency losses arecalculated as presented in [2].

Fig. 13 shows a section of the transformer from Fig. 1(b)during the heat run test. The thermal distribution in Fig. 13(b)clearly shows the hotspots after 7.3 h operation around the pri-mary winding. The white rectangles and the circle define thearea of averaging for the temperature measurements (T1 , T2 ,and T3), which are depicted in Fig. 14. Best fit curves (T1,bf,T2,bf, and T3,bf) are generated out of the measurement data andare used for extrapolation to estimate when a stable steady-state temperature is reached. The error is between +1.3% and+12.5%, comparing the extrapolated temperature values at a


TABLE VICOMPARISON BETWEEN THE MEASURED AND THE CALCULATED

TEMPERATURES OF THE PROTOTYPE TRANSFORMER

Measured temperature Calculated temperature Error

T1,bf(20 h) 34.18 °C T1,calc 34.61 °C +1.3%T2,bf(20 h) 40.52 °C T2,calc 42.97 °C +6%T3,bf(20 h) 35 °C T3,calc 39.38 °C +12.5%

time after 20 h (T1,bf(20 h), T2,bf(20 h), and T3,bf(20 h)) withthe calculated values (T1,calc, T2,calc, and T3,calc) in Table VI.

V. CONCLUSION

In this paper, the derivation of the thermal resistance of mul-tilayer windings with solid or litz wires has been presentedand validated by experimental measurements. In addition, theproposed approach has been benchmarked by two analytical ap-proaches from the literature regarding accuracy. This analyticalapproach can be used in straightforward designs of magneticdevices or could be integrated in optimization procedures. Withthe determined approach, moulded windings can also be con-sidered. The relative error between the analytical solutions andmeasurements, in the case of noncasted solid wire, is in therange of −20.7% to +2.5% and −24.8% to +15.2% for epoxycasted test setups. These results are similar, compared to theresults of the benchmark approaches. In the case of litz wire,the derived formulas show a matching between the calculatedand the measured values in the range of +51.4% to −20.7%,which is remarkable better than the results of the benchmarkapproaches, where the deviation is in the range of +76.1% to+397.6%. In addition, a tTEC model for multilayer windingshas been derived. An analytical temperature distribution predic-tion of a high-frequency high-voltage test transformer, based onthe new thermal winding model, has been compared to mea-sured temperature values and shows a highly accordance withthe measured temperature distribution. The error between thecalculated and the measured temperatures is between +1.3%and +12.5%.

APPENDIX A

In the following, the analysis of the basic capacitance cells ofmultilayer windings is summarized based on [31] and comparedto FEM simulations.

The presented approach of the derivation of the radial thermalresistance is a 1-D approach, which assumes that the temper-ature in a single layer is equally distributed and the heat flowis directed only in one direction. For that reason, the electricalpotential in a single layer is the same for each turn, and theelectrical capacitances between nonadjacent turns in the samelayer are considered to be zero. The capacitances to turns innonadjacent layers are also neglected because of the compara-ble high distance between the turns. Fig. 15 shows the electricalfield distribution of a four-layer winding. The height of thelayers lL = 53 mm, the radius ro is given in Table II, the testsetup is TSS,air , and the gap h = 0.1 mm. The minor influence

Fig. 15. FEM simulation of the electrical field E distribution of a four-layerwinding for showing the minor influence of the capacitance between turns innonadjacent layers. The winding arrangement is located in an infinite air box,whereas layer L1 is set to potential P1 = 0 V, layer L2 is set to potentialP2 = 1 V, layer L3 is set to potential P3 = 2 V, and layer L4 is set to potentialP4 = 3 V.

is clearly shown because the electrical field in the regions be-tween nonadjacent layers is lower than 10%, compared to thefield between two adjacent layers. Based on these assumptions,electrical multilayer windings can be described with the helpof independent basic orthogonal cells [see Fig. 16(b)] and/ororthocyclic cells [see Fig. 17(b)].

A. Orthogonal Winding Capacitance Model

To derive the capacitance model for an orthogonal winding,two wires are considered, which are arranged next to each other,separated by an isolation layer [see Fig. 16(a)]. In a first step, theelectrical field inside the rectangular basic cell [see Fig. 16(a)]is calculated, and then, the energy inside the basic cell is derivedby integration. By equating this energy with the energy storedin the equivalent capacitor, the value for the capacitance couldbe derived.

In Fig. 16(a), the electrical field lines cross the boundary be-tween copper surface and the wire insulation perpendicularly.Furthermore, the transition of the field lines at the boundarybetween wire insulation surface and the air is also perpendicu-lar, and afterwards, they are directed toward the other electrode.There, they hit the surface of the isolation layer h nearly per-pendicularly and the symmetry line exactly perpendicularly. Fordetermining the electrical field in the basic cell in Fig. 16(a), thefollowing assumptions are made.

1) The thickness of the wire insulation is usually small, sothe deviation inside the wire insulation toward the otherelectrode can be neglected, and the field lines are assumedto run in the radial direction with respect to the wire center.

2) After their perpendicular transition from the wire insula-tion into air, the field lines immediately bend toward theother electrode and hit the isolation layer exactly perpen-dicularly.


Fig. 16. (a) Orthogonal basic cell (solid rectangle), which is formed by twoorthogonally arranged solid wires with assumed simplified electrical field lines.(b) Electrical field lines between two layers of an orthogonally arranged windingsimulated with FEM and the corresponding independent basic cells.

Hence, all electrical field components are normal compo-nents. Comparing Fig. 16(a) and (b), the deviations of the fieldlines have their maximum at ϕ = 0 and a minimum at ϕ = π/2.Due to the fact that most of the field lines will run proximallyto ϕ = π/2, the resulting error is low. The derivation of theelectrical field is given in the following.

According to the boundary conditions between wire insula-tion and air, it follows that:

εisoEiso = εairEair sinϕ (48)

where εiso is the relative permittivity of the wire insulation andεair is the relative permittivity of air, which is considered witha value of 1. Eiso is the electrical field strength of the wireinsulation and Eair is the field strength in air. The perpendicularingress of the field lines from air into the isolation layer leads to

εlayElay = εairEair (49)

where εlay is the relative permittivity and Elay is the electricalfield strength of the isolation layer. Performing the line integralof the voltage between the two conductors gives

V = 2Eisoδ + Elayh+ 2Eairσ. (50)

Fig. 17. (a) Three orthocyclically arranged wires with assumed simplifiedelectrical field lines and basic cell (solid rhombus). (b) Electrical field linesbetween two layers of an orthocyclically arranged winding simulated with FEMand the corresponding independent basic cells.

Using (48), (49), (50), and

sinϕ =

√r2

o − ξ2

roas with at σ = ro −

√r2

o − ξ2 (51)

and by substituting

α = 1 − δ

εisoroand β =

1α

(1 +

h

2εlayro

)(52)

all field strengths can be calculated by

Elay =V

2εlayα(βro −

√r2

o − ξ2) (53)

Eiso =V

√r2

o − ξ2

2εisoroα(βro −

√r2

o − ξ2) (54)

Eair =V

2α(βro −

√r2

o − ξ2) . (55)

With the electrical field, the electrical energy stored in the iso-lation layer is

Wlay =εoεlay

2lwh

∫ +ro

−ro

Elay(ξ)2dξ. (56)


The energy in the wire insulation in polar coordinates with theorigin in the center of the left wire is

Wiso = εoεlaylw

∫ π

0

∫ +ro

ro− δ2

Eiso(ϕ)2rdrdϕ (57)

and the energy in the air is

Wair = εolw

∫ +ro

−ro

∫ σ (x)

0Eair(x, y)2dydx. (58)

Consequently, the total electrical energy in a basic cell is then

Wall = Wlay +Wiso +Wair. (59)

After solving the integrals and several steps of simplification,the following expression results:

Wall =εolwV

2

α

{Y +

18εiso

(2δro

)2Z

α

}(60)

with

Y = arctan

(√β + 1β − 1

)β√β2 − 1

− π

4(61)

and

Z =β

(β2 − 2

)

(β2 − 1)3/2 arctan

(√β + 1β − 1

)− β

2β2 − 2− π

4.

(62)The capacitance can be found by comparison of the coefficients(60) with W = CorthV

2/2, which results in

Corth =2εolwα

{Y +

18εiso

(2δro

)2Z

α

}. (63)

1) Orthocyclic Winding Capacitance Model: In case of or-thocyclic layers, the basic capacitance describes the energy,which is stored between the two neighboring wires in the leftlayer and two wires in the right layer as shown in the dashedrhombus basic cell in Fig. 17(a). Again, the paths of the elec-trical flux lines depicted in Fig. 17(a) and (b) match very well.Taking the same assumptions into account for the electrical fieldlines like in the orthogonal case leads to [31]

Ccyc = 4εolw

[Mair +Miso

(δ

εisor2o

) (ro − δ

2

)](64)

with

Mair =∫ π

6

0

cos2 ψ − cosψ√

cos2 ψ − 0.75 − 0.5[cosψ − α(

√cos2 ψ − 0.75 + 0.5)

]2 dψ (65)

and

Miso =∫ π

6

0

sin2 ψ + cosψ√

cos2 ψ − 0.75[cosψ − α(

√cos2 ψ − 0.75 + 0.5)

]2 dψ. (66)

In order to evaluate the accuracy of the analytically derived basiccell equations, a comparison with FEM simulations is carriedout. The two setup models are surrounded by air and the usedFEM software is COMSOL. The simulation model setup of the

TABLE VIISINGLE BASIC CELL CAPACITANCE EVALUATION

orthogonal basic cell is shown in Fig. 16(a), whereas the leftconductor is set to the potential P1 = 0 V and the right conduc-tor is set to potential P2 = 1 V. The simulation model setup ofthe orthocyclic basic cell is shown in Fig. 17(a), whereas bothconductors on the left-hand side are set to potential P1 = 0 Vand both conductors on the right-hand side are set to potentialP2 = 1 V. After the calculation of the electrical fieldE with theFEM solver, the electrical energy density W inside the areas Aof the basic cells is derived by

W =∫E2dA. (67)

Finally, the capacitance per length is given as

Corth,cyc =2W

(P2 − P1)2 . (68)

The comparison between the analytical results and the FEMsimulations of the orthogonal and the orthocyclic basic cell ca-pacitances per length are given along with the used permittivitiesof the insulation materials in Table VII. For the evaluation, thegeometric values δ and ro are taken from the winding test setupTSS,air given in Table II. The width of the isolation layer h, incase of the orthogonal basic cell, is chosen as h = 0.1 mm. Theerror in both cases is sufficiently low: −1.8% for the orthogonalbasic cell and −2.7% for the orthocyclic basic cell.

APPENDIX B

A summary of the H&S approach [19] for the calculation ofthe effective thermal conductivity ke of two components wind-ing amalgams and the H&S+ approach [20], which considersthree component winding amalgams, is given in the following.Both approaches are valid for round conductor shapes [21], andthe same notation as in [20] is used. The variables v and k are thenotations of the volume ratios and the thermal conductivities.

A. Effective Thermal Conductivity Based on the H&SApproach

The effective thermal conductivity ke of round solid wirewindings is given as

ke = kp(1 + vc)kc + (1 − vc)kp(1 − vc)kc + (1 + vc)kp

(69)


TABLE VIIIVOLUME RATIOS (−) AND THERMAL CONDUCTIVITIES (W/K·M) USED FOR THE BENCHMARK EVALUATIONS

where the subscript expressions c and p represent the conductorand the insulation/impregnation material between the conduc-tors. The conductor volume ratio is calculated by

vc =(ro − δ)2πlwNl ·Npl

awbwlw(70)

where aw is the width and bw is the height of the winding, ascan be seen in Fig. 10(a).

The same equations can be used for litz wire windings, wherethe insulation around the strands is neglected and the wholebundle is considered as solid copper conductor with outer insu-lation.

B. Effective Thermal Conductivity Based on the H&S+Approach

The effective thermal conductivity ke of round solid wirewindings is given as

ke = kp(1 + vc)kc + (1 − vc)kp(1 − vc)kc + (1 + vc)kp

with kp = ka (71)

ka = kiivii

vii + vci+ kci

vcivii + vci

. (72)

The subscript expressions c, ci, and ii represent the conductor,the conductor insulation, and the insulation/impregnation mate-rial between the conductors, respectively. The conductor volumeratio vc is calculated by (70), and the conductor insulation vol-ume ratio by

vci =(2roδ − δ2)πlwNl ·Npl

awbwlw. (73)

Finally, the insulation/impregnation material between the con-ductors volume ratio vii is given as

vii = 1 − (vc + vci). (74)

In the case of round litz wire windings, the effective thermalconductivity ke is calculated in two steps. First, (70)–(74) areapplied with strand parameters, resulting in k∗e . The parametersro and δ are replaced by ro,litz and δlitz, and Nl ·Npl and awbw

are substituted by Ns and (ro − δ)2π, respectively. In a secondstep, (70)–(74) are used with litz bundle parameters, where in(70), kc is replaced by k∗e .

Finally, the thermal resistance, which is used for the bench-mark test in Section IV-B, is defined as

Rw,x =aw

bwlwke. (75)

All wire and material parameters, as well as all volume ratios,are listed in Tables II and VIII, respectively.

ACKNOWLEDGMENT

The authors would like to thank the project partners CTIand AMPEGON AG very much for their strong support of theCTI-research project 13135.1 PFFLR-IW.

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Michael Jaritz (S’13–M’17) was born in Graz,Austria, in 1984. He received the Dipl.-Ing. degreein electrical engineering from the Graz University ofTechnology, Graz, in 2010. Since September 2011,he has been working toward the Ph.D. degree at theLaboratory for High Power Electronic Systems, ETHZurich, Zurich, Switzerland.

His diploma thesis dealt with dc voltage linkinverters in a power range of 500 kW. His current re-search interests include series–parallel-resonant con-verters, which are used in long-pulse modulators gen-

erating highly accurate voltage pulses.

Andre Hillers (S’12) received the B.S. and M.S.(with distinction) degrees (majored in power elec-tronics and minored in power systems) from theSwiss Federal Institute of Technology in Zurich (ETHZurich), Zurich, Switzerland, in 2009 and 2012, re-spectively. Since 2012, he has been working towardthe Ph.D. degree at the Laboratory for High PowerElectronic Systems, ETH Zurich.

His research interests include the optimal de-sign and hardware realization of highly efficient andhighly compact modular power converters for battery

energy storage systems connected directly to the medium-voltage distributiongrid.

Juergen Biela (S’04–M’06–SM’16) received theDiploma (Hons.) degree from Friedrich-Alexander-Universitat Erlangen-Nurnberg, Erlangen, Germany,and the Ph.D. degree from the Swiss Federal Instituteof Technology (ETH Zurich), Zurich, Switzerland, in1999 and 2006, respectively.

He dealt, in particular, with resonant dc-link in-verters with the University of Strathclyde, Glasgow,U.K., and the active control of series-connected inte-grated gate-commutated thyristors with the TechnicalUniversity of Munich, Munich, Germany, during his

studies. In 2000, he joined the Research Department of Siemens Automation andDrives, Erlangen, where he was involved in inverters with very high switchingfrequencies, silicon carbide (SiC) components, and electromagnetic compati-bility. In 2002, he joined the Power Electronic Systems Laboratory (PES), ETHZurich, for Ph.D. research, focusing on optimized electromagnetically inte-grated resonant converters. From 2006 to 2007, he was a Postdoctoral Fellow atPES and a Guest Researcher at the Tokyo Institute of Technology, Tokyo, Japan.From 2007 to 2010, he was a Senior Research Associate at PES. Since 2010,he has been an Associate Professor at the Laboratory for High Power Elec-tronic Systems, ETH Zurich. His current research interests include the design,modeling, and optimization of power factor correction, dc–dc, and multilevelconverters with an emphasis on passive components, and the design of pulsed-power systems and power electronic systems for future energy distribution.

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