© 2017 IEEE

Proceedings of the 18th IEEE Workshop on Control and Modeling for Power Electronics (COMPEL 2017), Stanford, California, USA, July 9-12, 2017

Litz Losses: Effects of Twisting Imperfections

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

This material is published in order to provide access to research results of the Power Electronic Systems Laboratory / D-ITET / ETH Zurich. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the copyright holder. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. Litz Wire Losses: Effects of Twisting Imperfections

Thomas Guillod, Jonas Huber, Florian Krismer, and Johann W. Kolar

Power Electronic Systems Laboratory (PES) ETH Zurich 8092 Zurich, Switzerland

Abstract—High-Frequency (HF) litz are extensively used for the windings of Medium-Frequency (MF) magnetic compo-

nents in order to reduce the impact of losses that longitudinal originate from skin and proximity effects. Literature documents radial different methods to calculate eddy current losses in HF litz wires, however, most of the computation methods rely on perfect azimuthal twisting of the strands, which is often not present in practice. This paper analyzes the implications of imperfect twisting on (a) the current distribution among the different strands of HF litz wires and the corresponding losses by means of a fast 2.5D Strand PEEC (Partial Element Equivalent Circuit) method. The effects Strand-level twisting of different types of twisting imperfections (at the bundle-, sub- Mid-level twisting bundle-, or strand-level) are examined. It is found that imperfect (b) Top-level twisting (c) twisting can lead to increased losses (more than 100 %). However, perfect twisting of the strands, which is difficult to achieve, Fig. 1. (a) Projection and (b) 3D view of the twisting scheme (azimuthal is often not required, i.e. suboptimal twisting is sufficient. and radial) of a single strand. The depicted path depends on the structure of the bundle, the pitch lengths of the bundles, and the used twisting scheme. Analytical expressions are given for distinguishing between (c) Packing structure of a HF litz wire with bundles and sub-bundles. critical and uncritical imperfections. The experimental results, conducted with a 7.5kHz / 65kW , reveal a reduction The permutation in radial direction prevents eddy of the error on the predicted losses from 52 % (ideal HF litz wire • model) to 8 % (presented model) and, thus, confirm the accuracy currents in case of an azimuthal magnetic field. An improvement achieved with the proposed approach. azimuthal field is created by the currents that are Index Terms—Power Electronics, Litz Wires, , present in the considered HF litz wire itself. Proximity Effect, Twisting Imperfections, Medium-Frequency, The permutation in azimuthal direction suppress the , , PEEC, Numerical Simulations. • oriented area spanned between the strands with respect I.INTRODUCTION to an homogeneous external field. The cancellation of The computation of winding losses represents an essential the magnetic flux between the strands, prevents the part of the modeling process of inductors and trans- formation of circulating eddy currents. formers. Power electronic converters are usually operating In addition to the number of strands and the diameter at Medium-Frequency (MF), which can be defined with of a single strand, the pitch (or length of lay), which is the power/frequency product (usually 0.1 1.0GHzW for the spatial period length of the twisting (in longitudinal − medium-power systems). At MF, the impacts of eddy current direction), denotes a third important parameter of a HF litz losses cannot be neglected any more. Usually, eddy current wire. losses are split up into losses due to skin and proximity For HF litz wires with a high number of strands, simulta- effects: the skin effect describes the eddy currents due to neous azimuthal and radial twisting is difficult to achieve. the magnetic field created by the current in the conductor Such HF litz wires are often constructed in a recursive itself and the proximity effect characterizes eddy currents manner, i.e., the total HF litz wire is composed of twisted that originate from an external magnetic field. bundles of HF litz wires and each bundle itself can again In order to obtain efficient MF inductors and transformers, be composed of twisted bundles or single strands. In this the impacts of losses due to skin and proximity effects need paper, the first twisting level (twisting of the strands) and to be mitigated. This is usually achieved with foil conductors the last level (forming the complete HF litz wire) are called or High-Frequency (HF) stranded conductors, e.g. Roebel strand-level and top-level twisting, respectively. Additional bars or HF litz wires. In this regard, HF litz wires are very twisting levels between the top- and the strand-level are popular since they allow for low losses and great flexibility denoted as mid-levels, cf. Fig. 1(c). in the design of the magnetic component [1]–[5]. With high numbers of strands and complex twisting HF litz wires are composed of multiple insulated strands, schemes, the accurate computation of losses in such as shown in Fig. 1. Each single strand is subject to skin stranded conductors is a challenging task. Over the years, and proximity effects and thus, the single strand diameter different calculation methods have been proposed for the is selected with regard to acceptable losses [6]. The strands calculation of the losses, which rely on analytical and/or are twisted together in both radial and azimuthal directions: numerical calculations [6]–[11]. However, it is found that the measured losses are often larger than the values computed Current Excitation: Iwire = 1 A / Hext = 0 A/m 200 with the aforementioned methods [1], [2], [12]. Different effects can explain such deviations:

The (external) magnetic field is inaccurately com- 100 [A/m] • puted [13]. Moreover, certain computational methods RMS H presuppose the external magnetic field to be homoge- 1 MHz 10 MHz 100 MHz neous across the cross section of the HF litz wire [11]. 0 (a) The terminations of the HF litz wire are generating Field Excitation: Iwire = 0 A / Hext = 1 A/m • 2.0 losses. Moreover, terminations can add an impedance mismatch between the strands which leads to undesired circulating currents [2], [12], [14]. 1.0 [A/m] Due to twisting, the length of the strands is greater • RMS than the length of the HF litz wire [15]. For typical HF H 1 MHz 10 MHz 100 MHz litz wire, however, this effect is often negligible (less 0.0 (b) than 10%). The packing (hexagonal, square, etc.) of the strands is Fig. 2. (a) Low frequency and high frequency magnetic field patterns • for current and (b) external field excitation. A 343 100µm HF litz wire is not homogeneous and/or not perfectly modeled [8], [9]. considered (2.5mm outer diameter). × The exact packing in a HF litz wire, which is often not clearly defined, has only a limited impact on the losses effect inside the strands. The corresponding losses, P, are (less than 10%). expressed as [11] A non-integer number of pitches is used, leading to • P ¡f ¢ R ¡K ¡f ¢ I 2 K ¡f ¢ H 2 ¢, (1) circulating currents inside the HF litz wire [7], [16]. = DC I wire + H ext Alternatively, local variations (along distances that are where KI and KH are frequency-dependent parameters and similar to the pitch length) of the external magnetic field RDC denotes the low frequency resistance. The losses can also create additional losses [12]. Such local variations be split up into four categories: are typically produced by the fringing fields of air gaps. DC losses (represented by RDC): conduction losses An imperfect twisting scheme is used, leading to an • • which are not linked to eddy currents. inhomogeneous current distribution in the strands. This Skin effect (included in KI ): eddy current losses in includes improper azimuthal and/or radial twisting (at • a strand that originate from the field created by the the top-, strand-, or mid-level), poor choices of the current in the same strand. ratios of the pitch lengths between twisting levels, etc. Internal proximity effect (included in KI ): eddy current • The last two effects can lead to significant additional losses losses in a strand that originate from the field created and commonly occur in practice. In particular, commercially by the currents in the other strands. available HF litz wires are frequently found to be subject External proximity effect (included in KH ): eddy current • to imperfect twisting [2]. Therefore, this paper analyzes the losses in a strand by reason of an externally applied impact of the twisting scheme on the losses, which, to the homogeneous magnetic field. knowledge of the authors, has not been studied in details The parameters RDC, KI , and KH are figures of merit in the literature. Section II summarizes loss computation for the HF litz wire itself. In a magnetic component methods for perfectly twisted HF litz wires. Section III (e.g. , transformer), the relevant part of the external analyzes the implications of twisting imperfections on the magnetic field, Hext, depends on the conductor current, losses. Finally, Section IV, compares calculated results to Iwire, and the geometrical properties of the component (e.g. experimental results, using a MF transformer that employs selected core, air-gap length, number of turns). This allows a HF litz wire with imperfect twisting. the definition of the equivalent AC resistance, which is a figure of merit for the complete winding of a defined II.PERFECT TWISTING magnetic component and not only for the HF litz wire. A perfectly twisted (radial and azimuthal directions) With (1), the equivalent AC resistance can be calculated HF litz wire with an infinitesimally small pitch length is according to considered (ideal HF litz wire). The HF litz wire conducts ¡ ¢ 2 P RAC f Iwire, Hext Iwire. (2) the current Iwire and is subject to a homogeneous external = ∼ magnetic field, Hext. Fig. 2 depicts the magnetic field Different methods exist for the computation of the determined with FEM simulations for a 343 100µm HF litz losses, which feature different hypotheses, high frequency × wire at different frequencies. At 1MHz, the eddy currents accuracies, and computational costs. The most common generate losses but have a limited impact on the magnetic methods are listed below: field distribution. For f 1MHz the skin depth is less than Asymptotic approximation (“Asymp”): The losses by À • the strand diameter of a single strand and the eddy currents reason of skin effect are neglected (the skin depth is alter the magnetic field distribution, leading to a shielding assumed to be larger than the diameter of strands) and the losses due to proximity effects (internal and Current loss factor Field loss factor 103 Asymp. 10-1 Asymp. external) are modeled with a low-frequency asymptotic Bessel fct. Bessel fct. approximation. The packing pattern is only modeled Biot-Savart 10-2 Biot-Savart FEM hom. FEM hom. with the fill factor [2], [6], [10]. 102 FEM disc. ] -3 FEM disc. Bessel functions (“Bessel fct.”): The losses due to skin 2 10

• [1] [m I f H c,HF

K -4 and proximity effects are modeled with Bessel functions. fc,HF K 10 HF effects are considered for every strand but the 101 shielding effect of the eddy currents (cf. Fig. 2) is not 10-5 modeled. The packing pattern is only modeled with the 100 10-6 fill factor [2], [11]. 104 105 106 107 108 104 105 106 107 108 Biot-Savart discrete modeling (“Biot-Savart”): The losses • Frequency [Hz] (a) Frequency [Hz] (b) by reason of skin and proximity effects are modeled with

Bessel functions. The exact packing pattern is modeled Fig. 3. (a) Loss factor KI and (b) KH computed with different methods and the magnetic field is computed with a Biot-Savart (asymptotic approximation, Bessel functions, Biot-Savart summation, FEM homogenization, and FEM discrete modeling). A 343 100µm HF litz wire summation [5]. is considered (2.5mm outer diameter). × FEM homogenization (“FEM hom.”): The packing pat- • tern of the HF litz wire is assumed to be periodic. A III.TWISTING IMPERFECTIONS unit cell of the pattern is solved with FEM simulation and virtual material parameters (conductivity and A. Computation Method permeability) are extracted for the packing pattern. Subsequently, the virtual conductor parameters are This section investigates the implications of different used to compute the losses. All high frequency effects twisting schemes (e.g. radial, azimuthal, combined) and are modeled. The limitations of this approach are the numbers of pitches (integer or non-integer) on the expected presupposed periodicity of the packing pattern and AC losses of HF litz wires. In such HF litz wires, the inaccuracies at the boundary of the HF litz wire [8], [9]. assumption of equal current distribution among the strands FEM discrete modeling (“FEM disc.”): All strands are is, in general, incorrect and the aforementioned methods • modeled with FEM. The obtained solution contains all are not applicable. high frequency effects and considers the exact packing In the literature, an analytical method for the simulation pattern [14]. Alternatively the PEEC (Partial Element of imperfect HF litz wires has been proposed [16], especially Equivalent Circuit) method can be used [14]. with regard to the impact of the pitches. However, this method, which does not compute the current sharing Fig. 3 depicts a comparison between the methods for a between the strands, cannot be used for simulating complex 343 100µm HF litz wire (cf. Fig. 2). The corner frequency × twisting schemes. Further references propose 3D simulation fc,HF characterizes the maximum frequency up to which methods, based on integral equations (e.g. PEEC), differential the low frequency asymptote of the proximity losses is equations (e.g. FEM) [7], [17], [18], or both [13]. However, considered to be valid and can be expressed as 3D methods are complex to implement and demand for 2 32 3 very high computational cost, which limit the number of fc,HF (ds) , (3) strands to less than hundred [7], [18]. For larger HF litz wires, ≈ πσµ d 2 0 s massive parallel computing is required [17]. Furthermore, the where σ is the conductivity and ds the diameter of the 3D methods require a complete parametric representation strands. For f fc,HF, it can be concluded that the simplest of all the strands, which is difficult to obtain for HF litz < method, the asymptotic approximation, features sufficient wires with a high number of strands (e.g. consideration of accuracy for most applications. For f fc,HF, the solutions deformations of the bundles) [10], [19]. > based on Bessel functions or, if higher accuracy is required, The 2.5D PEEC method proposed here and briefly de- the FEM-based methods need to be considered (cf. Fig. 2). scribed in the following offers a trade-off between modeling Most commonly, however, f f is avoided in order > c,HF accuracy, modeling effort, and computational cost. Fig. 4 to avoid excessive HF losses and a HF litz wire with a presents the main steps of the method, which is based on a smaller single strand diameter is considered [2]. Therefore, custom-built PEEC solver. First, the geometry (packing and in this paper, f f is presupposed. Furthermore, the < c,HF twisting) of the HF litz wire is generated and the HF litz comparative evaluation given in Fig. 3 reveals that an exact wire is sliced into different sections (about ten per pitch). In model of the packing pattern is not required. a second step, the and resistance matrices are For f f the results obtained with all considered calculated using a 2D model. For taking the twisting scheme < c,HF computational methods are matching. This indicates that into account, the matrices are permuted (according to the potential deviations between computed and measured losses slicing of the HF litz wire) and the total impedance matrix of originate from phenomena that violate the hypothesis of a the HF litz wire is generated. The current sharing problem perfectly twisted HF litz wire with an infinitesimally small is solved by equalizing the voltage drops (computed with pitch length. the impedance matrix) between the strands [19]. Finally, the Wire geometry Packing Twisting & slicing 4×5 : Perfect twisting × Perfect twisting Packing (2D) Twisting (3D) Permutation matrix (a) 42 : Bunched wires Equivalent circuit Res. matrix Ind. matrix Imp. matrix Inductance matrix Resistance matrix R L Z Impedance matrix (b) 7×7 : Bunched wires × Bunched wires Current sharing Equation system Current sharing Equalize the voltage drops Set the total current J Solve the linear system (c) Losses Mag. field Skin losses Proximity losses 4×12 : Perfect twisting × Bunched wires Magnetic Field Skin effect losses H P P Proximity effect losses

Fig. 4. Workflow for the computation of losses in HF litz wires with (d) arbitrary twisting schemes and numbers of pitches. The analysis is based on a 2.5D PEEC method. Fig. 5. (a) Perfect twisting, (b) bunched wires, (c) bundle in the center, and (d) hybrid twisting (perfect and bunched). Each title line specifies the magnetic field and the losses (due to skin and proximity corresponding number of bundles and the number of strands per bundle. effects) are extracted. TABLE I Compared to a full 3D method, the 2.5D PEEC method TWISTINGSCHEMES (343 100µm). neglects the current flows in azimuthal and radial directions, × which are, however, much smaller than the longitudinal Name Twisting scheme Number of wires “343/PT/PT/PT” PT PT PT 7 7 7 currents [19]. This implies that the method is not able to × × × × “343/PT/PT/BW” PT PT BW 7 7 7 simulate configurations with a large external field applied × × × × “343/PT/BW/BW” PT BW BW 7 7 7 in longitudinal direction, which are, anyway, uncommon for × × × × “343/BW/BW/BW” BW BW BW 7 7 7 magnetic components. The main advantage of the 2.5D PEEC × × × × “343/PT/BW” PT BW 7 49 method is the reduced computational cost (and memory × × requirements), which enables the simulation of a HF litz “343/BW” BW 343 wire with thousands of strands in less than a minutes on a personal computer. Moreover, the method does not require and is commonly found in commercially available HF a complete 3D full parametric representation of the HF litz wires [2], [6], [12]. litz wire and, hence, substantially reduces the modeling These twisting schemes (“PT” and “BW”) feature correct complexity. Finally, the fact that the PEEC method is based twisting in azimuthal direction since imperfect azimuthal on an equivalent circuit allows the combination of the HF twisting is uncommon for HF litz wires. Fig. 5 illustrates dif- litz wire model with external models, such as terminations ferent combinations of the aforementioned twisting schemes [2], [14]. Furthermore, the method is also applicable to high (many levels of twisting). Fig. 5(c) shows an imperfect frequencies (f f ), even if this case is not considered twisting scheme which feature an untwisted bundle in the > c,HF in this paper. center of the wire [2]. Fig. 5(d) depicts a hybrid twisting B. Considered Twisting Imperfections scheme, which is also commonly found: the “BW” twisting scheme is used at the strand-level and all other levels (e.g. Many different twisting schemes exist for HF litz wires [6], top-level) employ perfect twisting (“PT”) [6], [12]. [14], [18], featuring different bundle structures, radial and/or All simulations consider the HF litz wire introduced in azimuthal permutations, etc. However, most litz wires are Section II (343 100µm) and the six different combinations of × constructed with the following twisting schemes: the aforementioned twisting schemes (“PT”, “BW”, cf. Fig. 5) Perfect twisting (“PT”): the strands are perfectly twisted • listed in Tab. I. The pitch length of the top-level twisting (radial and azimuthal direction). This ideal construction is 30mm. The ratio of the pitch between two successive is difficult to realize with a large number of strands twisting levels is set to 2.0 (given that the strand-level (more than six). twisting pitch is the shortest). Bunched wires (“BW”): the strands are only twisted in • azimuthal direction. The strands located in the center C. Simulation: Twisting Schemes of the wire, stay in the center and are not twisted. This The six twisting schemes listed in Tab. I are considered represents the easiest and cheapest twisting scheme with an integer number of pitches. Fig. 6(a) illustrates the 33B”i osdrd oee,terslsaesimilar are results the However, considered. is “343/BW” ie r orcl wse naiuhldrcinadan and direction azimuthal in twisted correctly are wires nme ftitn eid)o h Flt ie o the For wire. litz HF the of periods) twisting of (number ( than (more ( xenlmgei edectto n urn excitation). current (no 7(b) excitation Fig. field circulating magnetic in to external shown leading is as strands currents, the between asymmetry strand-level [16]. the and critical of mid- less (e.g. impacts are sub-bundles the twisting) the since of schemes lengths five pitch scheme remaining twisting the the for only brevity, and simplicity of sake Pitches of Number Simulation: D. chosen. is pitches of number integer factor [12]. [11], [6], losses respect proximity with high-frequency distributions the current even to feature better wires due litz losses, HF lower twisted imperfectly frequencies and higher disappear At II. a Section in that presented with ideal showing methods perfectly the the matches Finally, losses, (“343/PT/PT/PT”) required. the scheme not twisting actually on is impact twisting perfect negligible than (“343/PT/PT/BW”) a (less strand-level has the smaller at much “343/PT/BW”), imperfection already twisting and is (“343/PT/BW/BW” increase the mid-level the twisting imperfect at For “343/BW”). (“343/BW/BW/BW”, level than (less small neglected. is resistance DC the according losses increased to flowing leads is to This current bundles. more other that the implies in This current. respect (no total with which phase-shifted the excitation bundles is to twisted current the radially in a perfectly current not of are The excitation). case field the external for sharing current, current total the of phase the to respect (with currents strand the of 343 differences Phase (a) 6. Fig. I f wire fannitgrnme fpthsi sd h impedance the used, is pitches of number non-integer a If pitches of number the by affected further are losses The field external the on schemes twisting the of impact The pronounced particularly are increases loss the expected, As ≈ 343/PT/PT/BW × i.6(b) Fig. 343/PT/PT/PT 100 1MHz = 1A K µ H , Flt iei osdrd(.m ue diameter). outer (2.5mm considered is wire litz HF m lutae h mato h ubro ice on pitches of number the of impact the illustrates H (cf. f i.6b) h fet ftitn imperfections twisting of effects the 6(b)), Fig. cf. , ext 100% for = (1) 0A/m K sngiil ic h osdrdH litz HF considered the since negligible is ) 343/PT/BW/BW I o wsigipretosa h top- the at imperfections twisting for ) .()Factor (b) ). 343/PT/BW (cf. (1) i.7(a) Fig. .Teefc ftetitn on twisting the of effect The ). K I (cf. o 5 for (1) -40 -30 -20 -10 0 +10 +20 o ifrn wsigipretos(f a.I,cmae oapretytitdH izwr c.Fg ) A 3). Fig. (cf. wire litz HF twisted perfectly a to compared I), Tab. (cf. imperfections twisting different for ) 5% . ice n an and pitches 5 n,teeoe is therefore, and, ) Current phase [°] 343/BW/B/BW 343/BW 40% .A ). 33B”wt 5 with “343/BW” ihrdcdnme ftrs sepce,teimpacts the expected, As turns. of number reduced with c.Fg (),wihlast iiaino h losses the of mitigation a to leads which 7(a)), Fig. (cf. 100% h mat ftitn mefcin ntelse.For losses. the of on assessment imperfections rapid twisting a of for required impacts still the are formulas ical Recommendations Design & Approximations Analytical shownE. as currents, circulating to 7(a). lead Fig. also in fringing gaps, the air by of produced Such fields typically direction). are longitudinal which variations, the external local in non-homogeneous (variable a field in magnetic placed pitches of number become pitches of frequencies. number higher non-integer at implies a negligible This of frequency. effects over factor the rapidly the that increases [11]. 3, wire strands Fig. litz in the HF shown between as currents contrast, circulating By the to change due currents frequencies, circulating higher the At of wire. distributions litz spatial HF for the the significantly of reduce lengths pitches increased of numbers non-integer of considered. not therefore and smaller much factor field external 343 the A 3). diameter). Fig. pitches outer (cf. of (2.5mm wire scheme considered number litz twisting is different HF the wire twisted for litz for perfectly and a currents “343/BW” to strand scheme compared the twisting of the for Amplitude (a) 7. Fig. -180 +30 -150 -120 -90 -60 -30 0 1 MHz/n 1 kHz/n vntog h .DPE iuain r at analyt- fast, are simulations PEEC 2.5D the though Even integer an with winding for appears effect same The than (more pronounced particularly is increase loss The (a) o hr Flt ie hc sciia o windings for critical is which wires litz HF short for ) Current phase [°] p p = =

5.5

5.5 . ice ( pitches 5 KI / KI,ideal [%] 100 140 180 220 260 0 12 22 0 130 260 10 3 (a) 343/PT/PT/PT 343/PT/PT/BW 343/PT/BW/BW 343/PT/BW 343/BW/BW/BW 343/BW I

Current [µA] Current [nA] wire H Current lossfactor:twistingschemes ext at ) K

= KH / KH,ideal [%] 100 155 210 265 320 H 1A/m 100kHz 5 (cf. 10 Field lossfactor:numberofpitches Frequency [Hz] 4 , I (1) wire o ifrn wsigschemes twisting different for .Teipc on impact The ). = Number ofpitches 10 0A .()Factor (b) ). 10 5 K H 15 fa ideal an of × K 100 H 1 MHz 100 kHz 10 kHz 1 kHz (cf. µ K m (b) 10 I 6 (1) HF (b) 20 is ) 33B” b osfactors, Loss (b) “343/BW”. ol epeeal for preferable be would where considered. are 6), Fig. (cf. cases extreme the represent which ieis wire efcl wse Flt ie h onrfeunisfor (complete) frequencies the corner of The wire. factor litz loss HF the twisted and perfectly bundles single for (“ is wire bundle litz each HF twisted perfectly a “ scheme twisting (for the avoided be should and the on imperfections given of losses. in impacts expected required the is assess expression to an order imperfections Therefore, twisting acceptable. certain results be however, the can 6, process to Fig. According the in expensive. makes depicted and which many difficult required, strands, twisting are single of of bundles numbers of HF high of levels very case In with reduced instead. wires preferred, with litz is bundles strands on single of based numbers construction a and strands [6]. [2], required also are is wires direction litz at HF strands However, operated all usually [6]. the properties of frequency bunching a high that seen be can It factor. fill strands, as derived be can frequency, intersect, the curves for asymptotes, these expression With an asymptotes. different the presents 8(a) Fig. approach used. following the be of 5), evaluation can Fig. the (cf. For schemes used. twisting be different can analytical [16] the in presented field, of formulas external numbers non-homogeneous non-integer or due pitches increases loss the computing ( considered 343 is A shown. wire also litz are HF intersections and asymptotes corresponding (diameter wire for litz considered HF are twisted perfectly the and factors, Loss (a) 8. Fig.

ic mefcin ntetpms ee r otcritical most are level top-most the in imperfections Since of numbers large for achieve to difficult is twisting Radial “343/BW”, and “343/PT/PT/PT” schemes twisting the First KI [1] 10 10 10 10 10 0 1 2 3 Perfect Twisting vs.bunching 3 d σ Intersections Asymptotes 343/BW 343/PT/PT/PT outer d 10 stecnutvt and conductivity the is outer eit h orsodn osfactors, loss corresponding the depicts 4 Frequency [Hz] f . c,BW,PT i.8(b) Fig. 10 h xenlH izwr imtr and diameter, wire litz HF external the d 5 d bundle bundle 10 ( f c,HF K d 6 f I f outer K PT c,BW,PT o h wsigshms“4/TP/T and “343/PT/PT/PT” schemes twisting the for , < d (from n h ue imtro h Flitz HF the of diameter outer the and I hw h osfactors, loss the shows 10 outer o h uce ude (diameter bundles bunched the for , f × 7 c,BW,PT , f d BW = s 0.3mm (a) 10 > f ) 2.5mm). 8 < ≈ sslce n oprdto compared and selected is ” f c,BW,PT f πσµ uhta wsigi radial in twisting that such 10 c,BW,PT KI [1] PT 2 1 5 10 to Choice ofthebundlediameter 4 f d f × c,BW c,BW,PT 0 1.6mm 1024 Intersections Asymptotes BW bundles PT s d F u otesuperior the to due , PT d

h imtro the of diameter the × ,aH izwr with wire litz HF a ), outer

PT wire outer ) h imtrof diameter The ”). 3 2 Frequency [Hz] f 10 1 3 c,PT .Frbt lt,the plots, both For ). hr h two the where , .Dfeetdiameters Different ). 6 d s K 3 4 (4) , I calculated , 10 5 K × d d I bundle 100 and , F bundle the ↓ µ (b) 10 m ) 7 where eg o da wsig ubro ice)it naccurate an into pitches) of number twisting, ideal non (e.g. hstasomri sdi CD ovre opsn a composing converter DC-DC a in used is transformer This hnteoeaigfeuny(yapoiaeyafactor a greater approximately be (by to frequency needs operating bundles losses, the the single overall of than the the frequency in of increase corner losses considerable current the a eddy to the lead that bundles avoid to order In bundles, the and wire litz HF twisted perfectly the Flt ie.Almaueet r odce ihthe with conducted are measurements employed All the and wires. windings litz the HF of parameters relevant [21]. the [20], (SST) Transformer Solid-State component. real a of windings the of model (for used be can procedure design ( diameter (7). not with should computed “BW”) value (e.g. the scheme exceed twisting resulting imperfect bundles largest an of the from numbers of arbitrary diameters with The wires levels. litz twisting HF to generalized be also the for derived: condition is bundle following each the of this, diameter With two/three). of f c,BW A effects aforementioned the combine to possible is it Now outer • given a with wire litz HF a of construction the For • • 7.5kHz wsigshm diinlmdlvltitn steps “PT “ (e.g. twisting a added for mid-level be large should additional too is scheme bundles twisting of number the If extracted. is bundle factor. diameters fill the the From twisting reduce the potentially of and losses complexity scheme the the increase reduces significantly marginally a but with only bundles diameter Using is smaller strands. bundle bunched strand-level of This composed twisting. strand-level the ( diameter bundles The a eepesdas: expressed be can , V M IV. f d c M urn 0 95A 6.0 Rectangular 700 winding LV strands Rectangular of 6.0 Number size wire winding Litz 70A HV profile wire Litz size Winding Current RMS Voltage Name ubro ie 7 BW wires of Number Scheme Twisting bundle steslce onrfeuny hs eut can results These frequency. corner selected the is d AUEET OF EASUREMENTS / outer 65kW f c,PT ¡ f c,BW f .kz/6k FT MF 65kW / 7.5kHz n taddaee ( diameter strand and ) c ¢ ( < d Ftasomri osdrd cf. considered, is transformer MF ( outer d p bundle πσµ , d d 4 ± AL II TABLE s outer × 1.1kV ) ) 0 d f F ≈ ≈ T × × 5 × × bundle × WISTING .m 8.1 6.0mm 8.1 72mm PT 200 πσµ πσµ c × 07 20 and , RANSFORMER × PT µ WBW BW 1000 m cf. , 0 0 p f f d F d F × c 16 d 128 < < bundle BW’). I outer bundle 2 (7) f MPERFECTIONS f ± a.II Tab. c,BW c,BW,PT × 0.8kV d scmue for computed is ) × × [10,11] s × d . h ubrof number the , ,tefollowing the ), × 8.1mm 70mm BW s (6) . < 200 (5) , ): f × summarizes c,PT × µ BW m 14 PT f (7) . c,PT i.10 Fig. × BW and ” . Litz wire / twisting Current sharing / amplitude Current sharing / phase Losses / HV/LV windings 20 0 Simulations Ideal litz wires 3 Measurements 2.5D PEEC ) [°]

) [%] Measurements wire I

wire [p.u.] / I

/ 10 -90 DC 2 R

center / I

center I

AC Simulations R abs( angle( 1 0 Measurements -180 1k 10k 100k 1k 10k 100k 1k 10k 20k 10 mm (a) Frequency [Hz] (b) Frequency [Hz] (c) Frequency [Hz] (d)

Fig. 9. (a) HF litz wire structure for the HV winding where the untwisted bundle in the center has been removed. (b) Amplitude and (c) phase of the current in the center bundle compared to the total current in the HV winding. (d) Ratio RAC/RDC (cf. (2), for both windings). Also shown are the simulated losses for hypothetical perfectly twisted HF litz wires and for the actually present imperfect twisting.

MF transformer HV/LV windings for the simulations and the measurements conducted for the present HF litz wires and, in addition, includes the simulation results for hypothetical perfectly twisted HF litz wires. At 10kHz (near the operating frequency), the resistance ratio for hypothetical perfectly twisted HF litz wires is 1.27, the simulated ratio for the non-ideal HF litz 230 mm 250 mm (a) (b) wires is 1.79, and the measurement returns 1.93. Hence, the mismatch between expected losses with perfectly twisted Fig. 10. (a) Considered 7.5kHz / 65kW MF transformer. (b) HV/LV windings HF litz wires and the measurements (52%) can be explained without the magnetic cores. with the presented simulation method. magnetic (nanocrystalline) cores being installed in order V. CONCLUSION to obtain a realistic distribution of the magnetic field. The This paper investigates the implications of imperfections simulations employ the method presented in Fig. 4 and the in HF litz wires (e.g. number of pitches, twisting) on the external magnetic field is calculated with FEM. The number losses and particularly focuses on imperfect twisting which of pitches is very large (more than 70 for each winding), is commonly observed in commercially available HF litz hence, an integer number of pitches has been assumed in wires. A fast 2.5D PEEC solver, which is able to simulate HF the simulations. litz wires with thousands of strands, has been implemented In a first step, the current distribution among the bundles to simulate the current distribution among the strands and of the HV winding has been measured at different frequen- determine the losses. cies, with a short-circuit being applied to the LV winding. According to the results obtained from the analysis of Fig. 9(a) shows the structure of the HF litz wire of the HV different types of imperfect twisting schemes it is found that winding, for which an untwisted bundle is present in the imperfect twisting, in particular of the top-level bundles, can center (“BW” at the top-level twisting, cf. Fig. 5(c)). This lead to a significant increase of losses due to eddy currents center bundle has been removed in Fig. 9(a) to highlight (up to 100% increase). It is also found that imperfections the present construction of the HF litz wire. Therefore, an may be tolerated at sub-bundle-levels. In this context, unequal current distribution among the bundles is expected simple analytical expressions are derived and the results are (cf. Fig. 6 and (7)). Figs. 9(b)-(c) depict the ratio of the summarized in a design guideline for the construction of currents in the center bundle to the total current, with suitable HF litz wires. Furthermore, an increase of the losses respect to amplitude and phase. A good agreement between is to be expected for windings with low number of pitches simulations and measurements is achieved. Furthermore, it and/or non-homogeneous external magnetic field. appears that measuring the amplitude of the current in the The presented method is finally applied to a center bundle is insufficient since almost only the phase is 7.5kHz / 65kW MF transformer, which employs imperfectly affected by the imperfect twisting. Furthermore, it is found twisted HF litz wires, for experimental verification. Both, that the presence of the magnetic cores has only a minor the measured current distribution among the bundles and impact on the current distribution in the HF litz wire (in the losses, are in good agreement with the simulations. The both, simulations and measurements). imperfections of the HF litz wires lead to an increase of In a second step, the winding losses have been measured the winding losses by 52%. with a “Yokogawa 3000WT” power analyzer (LV winding short-circuited). In the course of this measurement, the ACKNOWLEDGMENT ratio RAC/RDC is determined, since it is a figure of merit This work is part of the Swiss Competence Centers for for the losses of the windings (for both HV and LV, cf. (2)). Energy Research (SCCER) initiative which is supported by Fig. 9(d) presents a comparative evaluation of the results the Swiss Commission for Technology and Innovation (CTI). REFERENCES [11] H. Mühlethaler, “Modeling and Multi-Objective Optimization of Inductive Power Components,” Ph.D. dissertation, ETH Zurich, 2012. [1] S. Zhao, Q. Li, and F. C. Lee, “High Frequency Transformer Design for [12] H. Rossmanith, M. Doebroenti, M. Albach, and D. 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