Simple Fully Analytical Copper Loss Model of Litz Wire Made of Strands Twisted in Multiple Levels
Kazuhiro Umetani , Jesús Acero , Héctor Sarnago , Óscar Lucia , Eiji Hiraki Graduate School of Natural Science and Technology Okayama University Okayama, Japan
Published in: 2019 IEEE Applied Power Electronics Conference and Exposition (APEC)
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DOI: 10.1109/APEC.2019.8721827 Simple Fully Analytical Copper Loss Model of Litz Wire Made of Strands Twisted in Multiple Levels
Kazuhiro Umetani Jesús Acero Héctor Sarnago Graduate school of natural science and Department of Electronic Engineering Department of Electronic Engineering technology and Communications and Communications Okayama University University of Zaragoza University of Zaragoza Okayama, Japan Zaragoza, Spain Zaragoza, Spain [email protected] [email protected] [email protected]
Óscar Lucia Eiji Hiraki Department of Electronic Engineering Graduate school of natural science and and Communications technology University of Zaragoza Okayama University Zaragoza, Spain Okayama, Japan [email protected] [email protected]
Abstract— The Litz wire has been widely utilized as a wire difficult both for the analytical and numerical [11]–[14] with a low copper loss under high-frequency operation. approaches because multi-level twisting of many thin strands However, design optimization of the Litz wire is difficult generally requires complicated modeling based on the because this wire generally has a complicated structure of thin electromagnetic analysis. strands twisted in multiple levels, which hinders both of the analytical and numerical prediction of the copper loss. To Preceding studies [15]–[25] have proposed analytical overcome this issue, recent studies have proposed the analytical prediction methods of the copper loss in the bundle of twisted models of the copper loss in the bundle of twisted strands, which strands (or the Litz wire with a single level of twisting), which is the basic components constituting the Litz wire. This paper is the basic element of the Litz wire. These methods provide constructs a simple analytical copper loss model of the Litz wire analytical models of the skin and proximity effect of the based on these preceding insights. In addition to these insights, bundle. Some of these models even have attractive features the proposed model further considers the effect of the that they contain only physical constants and geometrical inclination angle of the strands to the Litz wire on the proximity parameters, all of which can be straightforwardly obtained effect loss. The proposed model was tested in comparison with without FEM analysis. Therefore, the copper loss model of the the experimentally measured AC resistance of commercially Litz wire may be constructed by further considering the available Litz wires. As a result, the predicted AC resistance twisting process of these bundles. showed good agreement with the measured ac resistance, suggesting the effectiveness of the proposed model. The purpose of this paper is to construct a simple copper loss model based on these preceding insights. In addition to Keywords—AC resistance, analytical model, copper loss, Litz these insights, the proposed model further considers the effect wire, proximity effect of the inclination angle of the strands to the Litz wire, caused through the twisting process. The proposed model applies I. INTRODUCTION drastic approximations to simplify the equation so that only The Litz wire is widely utilized in industrial applications physical constants and geometrical parameters are required that require high-frequency AC power, such as induction for the copper loss prediction. heating [1]–[3] and high-frequency resonant converters [4]– [8]. The Litz wire is made of a number of thin strands twisted The following discussion comprises 4 sections. Section II in multiple levels, e.g. the twisted bundles each made of analytically derives the copper loss model of the Litz wire. In twisted strands. As pointed out in [9][10], twisting the strands this section, we first briefly review the analytical copper loss can prevent concentration of the ac current in particular model of the strand. Then, we derive the analytical copper loss strands. Therefore, the complicated structure of the Litz wire model of the bundle of the twisted strands. As a result of the with multi-level twisting can ensure equal ac current flows detailed analytical discussions, we propose a slight among all the strands constituting the Litz wire. As a result, modification of the previous copper loss model of the bundle the Litz wire can effectively suppress the copper loss even proposed in [25]. Based on the model of the bundle, we under high-frequency operation. construct the copper loss model of the Litz wire. Section III presents experiments performed to verify the proposed copper However, the too many levels of twisting may result in an loss model of the Litz wire. Finally, section IV gives increase in manufacturing cost and decrease the packing factor, conclusions. which expands the volume occupied by the wire. Therefore, the design of the Litz wire commonly requires seeking II. COPPER LOSS MODEL OF LITZ WIRE effective suppression of the copper loss with simple twisting This section derives an analytical copper loss mode based structure. on an example of a typical Litz wire, illustrated in Fig. 1, In this sense, efficient design of the Litz wire may need a carrying sinusoidal ac current. This Litz wire has 3 levels of the twisting. As in many commercially available Litz wires, prediction of the copper loss from the geometrical rd specifications of the Litz wire, such as the twisting structure, this Litz wire (3 level bundle) is assumed to be formed by twisting no more than 5 bundles (2nd level bundle), each of the diameter, and the number of the strands. However, the st copper loss prediction of the Litz wire has been regarded to be which is formed by twisting no more than 5 bundles (1 level 1st level bundle: radius α Current is b Strand: radius αs Radius s
2nd level bundle
Strand Resistivity Permeability 0
External ac
magnetic field Hs
Fig. 2. Strand under consideration. Fig. 1. LitzFigure wire structure 1: Litz underwire consideration structure with under 3 levels of twisting. consideration with 3 levels of twisting. ac current ac current bundle). However, 1st level bundles are assumed to be formed by twisting many (much more than 5) strands, as is common in many Litz wire. The copper loss of a wire is generally dependent on two factors: One is the ac current flowing through the wire, and the ｃ ｃ ｃ other is the external ac magnetic field applied to the wire. Therefore, the copper loss model should be expressed as a External ac External ac function of these two factors. The Litz wire under field H s⊥ field Hs// consideration is assumed to carry the total ac current iL. In (a) i (b) i (c) i addition, the uniform sinusoidal external ac magnetic field HL acs_dens s⊥_dens s//_dens is assumed to be applied in perpendicular to the Litz wire. Fig. 3. AC current distributions of the ac current density iacs_dens, which constitutes i under the absence of the external field, and the eddy current Derivation of the proposed copper loss model is based on s is⊥_dens and is//_dens, which is generated by Hs⊥ and Hs//, respectively. modeling of the copper loss in each level of twisting in the
Litz wire. We first derive the analytical copper loss model of where is⊥_dens and is//_dens is the eddy current generated by a strand. Then, based on this model of the strand, we formulate Hs⊥ and Hs//, respectively. the analytical copper loss model of a bundle made of the strands. Finally, we formulate the analytical copper loss model As depicted in Fig. 3, the current density iacs_dens and is⊥_dens of the Litz wire based on the model of the bundle. Because the are in parallel to the strand, whereas is//_dens is in perpendicular Litz wire is made of non-magnetic material, the linear media to the strand. Furthermore, iacs_dens has the axial symmetry with is assumed in the electromagnetic analysis. respect to the central axis of the strand, whereas is⊥_dens has the anti-symmetry with respect to the line passing through the A. Analytical Copper Loss Model of Strand center of the strand and in parallel to Hs⊥ (dashed line in Fig. We focus on a strand shown in Fig. 2 and formulate the 3(b)). Noting that is⊥_dens and is//_dens must be proportional to the copper loss of the strand. The strand is assumed to carry the magnitude of Hs⊥ and Hs//, respectively, in the linear media, ac current is. In addition, the external ac magnetic field Hs is we can further express (2) as applied to the strand, although we do not limit the direction of Hs to be perpendicular to the strand. 22 The copper loss of the strand is the sum of the Joule loss PdSdSsacs= + densssdenss ii_ ⊥_ SSss generated by the local ac current inside the strand. Therefore, 2 the copper loss of the unit length of the strand P can be s + isdenss//_ dS , () expressed as Ss 2 2 2 =++Rs iGHGH sssss ⊥⊥ //// 2 PdSssdenss= i _ , () Ss where Rs is the strand-level loss coefficient caused by the skin effect; Gb⊥ and Gb// are the proximity effect coefficients,
where ρ is the resistivity of the copper, Ss is the horizontal respectively. cross-section of the strand, and is_dens is the local ac current According to the preceding studies [15]–[25], these density vector inside the strand. coefficients are analytically derived as The local ac current density is the sum of the eddy current density, generated by Hs, and the ac current density iacs_dens, RF=, () constituting is under the absence of the external field. We ss2 ( ) s denote the component of Hs in perpendicular and in parallel to the strand as Hs⊥ and Hs//, respectively. Then, (1) can be rewritten as GKss⊥ =4,( ) ()
2 GK=2, ( ) () Ps= i acs__ dens + i s⊥_ dens + i s// dens dS s , () ss// Ss where s is the radius of the strand. Parameter γs and when no external magnetic field is applied to the bundle. Note functions F and K are defined as that the electromagnetic field of the bundle-level must have the axial symmetry inside the bundle because the current distribution among the strands in the bundle also has the axial 0 ss , () symmetry. In this case, we can approximate the ac current distribution among the strands in the bundle as the ac current distribution inside the solid wire with the same diameter as the xxxxxberbeiberbei− bundle. Fx( ) , () 2 (berbeixx)22+ ( ) We assume that this imaginary solid wire is made of the uniform material with the resistivity ρeff defined as berberbeibeixxxx+ Kxx( ) − 22, () A 22 =s R () (berbeixx) + ( ) effs
where ω is the angular frequency, μ0 is the permeability of where is the porosity factor of the bundle, and As is the the air. cross-sectional area of the strand. This solid wire model gives B. Analytical Copper Loss Model of 1st Level Bundles the solution of the ac current distribution that ensures 1. the direction of the ac current to be in parallel to the bundle and 2. A 1st level bundle is a group of twisted strands. Therefore, the ac electric potential to be the same at any point inside the the strands placed at the same distance from the center of the cross-section of the solid wire. If we assume the same ac bundle rotationally exchange their position along the bundle. current distribution inside the bundle as the solid wire, we can This indicates that these strands must carry the same ac current. find that these features satisfy the requirement of the ac current Hence, the current distribution among the strands of the 1st distribution among the strands in the bundle because level bundle must have the axial symmetry. 1. The former feature prohibits the ac current flowing As mentioned in the previous subsection, the external outside the strand, (We assume that the inclination magnetic field applied in perpendicular to the bundle must angle of the strands to the bundle is small because the induce the ac current with the anti-symmetry distribution with diameter of the 1st level bundle is far smaller than that respect to the line passing through the center of the bundle and of the Litz wire and therefore twisting the strands does in parallel to the field. However, this anti-symmetry current not cause large inclination angle to the bundle.) distribution cannot be achieved in the 1st level bundle because of the requirement of the axial symmetry current distribution 2. The latter feature prohibits the ac current circulating among the strands. As a result, this external magnetic field through any pair of the strands. does not induce the ac current circulating through any pairs of the strands, although this field still induces the local eddy Furthermore, the total copper loss generated in this solid wire model is the same as the copper loss in the bundle. In fact, current confined inside the strand, i.e. is⊥_dens mentioned in the previous subsection. (We neglected the bundle-level by determining the strand current is according to proximity effect [25] because we assume the Litz wire much longer than the twisting pitch.) As iisbdens= _ , () Similarly, the external magnetic field applied in parallel to the bundle must induce the ac current of the direction in perpendicular to the bundle. However, the bundle is made of where ib_dens is the ac current density vector component thin isolated strands; and therefore, this requirement cannot parallel to the wire in the solid wire model (The component also be achieved in the bundle. As a result, this external perpendicular to the wire is zero.), we have magnetic field does not induce the circulating ac current through the strands, although this field still induces the local A i= i =s i i dS , () eddy current inside the strands, i.e. is//_dens. b s b__ dens b dens b jjS Consequently, the copper loss per unit length of the 1st b level bundle Pb has the following form. 2 A R i2== R i 2 Rs i 2 PR=++ iG222 HG H () b b s s s b_ dens bb bss jss j⊥⊥ //// jj jj () As 22 = effi b__ dens eff i b dens dS b , where Rb is the bundle-level loss coefficient caused by the j Sb strand-level and bundle-level skin effect; ib is the total ac current flowing through the bundle; j is the index of the strands st Equation (13) indicates that the total ac current flowing contained in this 1 level bundle; Hs⊥j and Hs//j are the RMS through the bundle is the same as that in the solid wire model. value of the local external magnetic field Hs⊥ and Hs//, In addition, (14) indicates that the total copper loss inside the respectively, at the strand j. bundle is the same as that in the solid wire model. The coefficient Rb can be regarded as the ac resistance of Therefore, the copper loss in the solid wire model can be the bundle under the absence of the external magnetic field. analytically calculated, similarly as in the previous section. As Therefore, we can determine Rb by calculating the copper loss a result, inferred from (4), Rb can be obtained as eff AF( ) Direction from the center RFR== sb, () Internal magnetic bbs 22( ) of the Litz wire to the bb field Hint// center of 1st level bundle Current density vector st where b is the radius of the 1 level bundle. Parameter γb is defined as Internal magnetic Direction field Hint⊥ of HL 0 bb , () eff
Substituting (4) into (15), Rb can be finally determined as External magnetic Direction of the field HL θ 2nd level bundle RFFFF== , () bsbsb 22( ) ( ) ( ) ( ) Direction of bss n the Litz wire
st where ns is the number of the strand in a 1 level bundle. Fig. 4. Internal magnetic field genreated as a result of twisting.
The analytical solution of Rb, shown in (17), is similar to current generates the magnetic field parallel to the bundle, as the model of the preceding study [25]. However, there is a illustrated in Fig. 4, which does not appear without twisting. slight difference that ρeff contained in b was set at ρ/η in [25], whereas ρeff is defined as (11) in the proposed model. This This effect takes place in all levels of the twisting. difference may have a subtle effect on the calculation of Rb in However, considering this effect for all levels of bundles may many actual cases. Nonetheless, as discussed above, (17) has lead to the extremely complicated calculation. Therefore, we the theoretical basis on the electromagnetism. simply consider this effect only for twisting the 2nd level bundles to form the Litz wire because its largest scale of C. Analytical Copper Loss Model of Litz Wire twisting tends to be the major cause of the inclination angle of The 2nd and 3rd level bundles are assumed to be formed by the strands. Hence, all the strands are assumed to be twisted to twisting no more than 5 bundles. Therefore, the bundles of the form the 1st level bundle with a negligible inclination angle to same twisting group rotationally exchange their position along this bundle; and similarly, all the 1st level bundles are assumed the Litz wire. Because these bundles have the axial symmetry, to be twisted to form the 2nd level bundle with a negligible thus carrying the same ac current. As a result, all of the 1st inclination angle to this 2nd level bundle. Nonetheless, all the level bundles in the Litz wire carry the same ac current. 2nd level bundles are twisted with the constant inclination angles to the Litz wire. As a result, all the strands contained in Noting that the 1st level bundles have a greater length than a 2nd level bundle (and therefore all the strands of the Litz wire the Litz wire as a result of the twisting process, the total copper because of the axial symmetry among the 2nd level bundles) loss of the Litz wire P can be obtained as L are assumed to be inclined at the same angle as the 2nd level bundle. 2 iL 2 Because all the 1st level bundles carry the same ac current, Pmn=+ n RmGH Lbss kj 21 nn ⊥⊥ we approximate the magnetic field inside the Litz wire as the 21 kj, () sum of the external field HL and the internal field Hint 2 +mGHss//// kj , generated by the uniform current density in the Litz wire kj, cross-section, flowing at the constant inclination angle . Certainly, the current distribution inside the 1st level bundles where m is the length ratio of the 1st level bundle to the may not be uniform in high frequency due to the bundle-level Litz wire, iL is the total ac current of the Litz wire, n1 is the skin effect, i.e. the ac current distribution of the solid wire st nd number of the 1 level bundles in a 2 level bundle, n2 is the model in subsection 2.B. However, we consider this small- number of the 2nd level bundles in a Litz wire, and k is the scaled fluctuation of the magnetic field caused by this non- st st index of the 1 level bundles contained in the Litz wire, Hs⊥kj uniformity inside the 1 level bundle to be small compared and Hs//kj are the RMS value of the local external magnetic with large-scale magnetic field generated by uniform current st st field Hs⊥ and Hs//, respectively, at the strand j in the 1 level distribution among all the 1 level bundles. bundle k. Under this approximation, the internal magnetic field Hint
Equation (18) still contains the unknown parameters Hs⊥kj can be determined as and Hs//kj. The next subsection discusses the derivation of these parameters to determine the copper loss model of the Litz wire. (ar−) tan Hi= L , () int// 2 L D. Magnetic Field in Litz Wire aL The multi-level twisting structure of Litz wire makes complicated the estimation of Hs⊥kj and Hs//kj because the r strands have the inclination angle to the Litz wire. For Hiint⊥ = L , () 2a2 example, the ac current flowing inside a bundle forms a L circulating component around the center of the bundle. This st where Hint// and Hint⊥ are the RMS values of the parallel Because all the 1 level bundles are regarded to have the and perpendicular components of Hint with respect to the Litz constant inclination angle θ to the Litz wire, the following wire, aL is the radius of the Litz wire, and r is the distance from relation is obtained between θ and m, i.e. the length ratio of the center of the Litz wire. the 1st level bundle to the Litz wire: Because all the strands are assumed to be inclined to the 1 Litz wire at the constant angle θ, the magnetic field at the m = . () strands can be obtained as c o s
2 Substituting (27) into (25) and (26), we have HHHskjLint// =−( +sinsinsinkj ⊥ () 2 2 +Hitn // cos, ) 2 HL 1 mGHGnmsskjstotal⊥⊥⊥ + 2 m kj, () 2 HHH =− + sincoscos 2 skjLint⊥⊥( kj m3 11 i () ++−m L , 2 22 312 m 22 −HHint// sincos,) + Lkj 2 aL
2 where HL is the RMS value of the external field and kj is mGHsskj//// the angle between HL and the vector drawn from the center of kj, st () the Litz wire to the center of the strand j of the 1 level bundle 22 k. HiLL111 +−Gnmstotal// . 224 22a m Therefore, the second and third terms of the right-hand L side of (18) can be obtained by utilizing (19)–(22). First, we Finally, analytical copper loss model of the Litz wire is approximate the second and third terms by replacing the obtained by substituting (17), (28), and (29) into (18) summation by the areal integration:
n n n 2 mGHmGHdS22 s 12 ss⊥⊥⊥⊥ kjss kjL iL A PmnLsb= nFF21 ( ) ( ) kj, L S 2 L n nn21 () ss 2 aL n H 2 1 =mGdrHdrtotal 2 , L skjs⊥⊥ kj +4 +Knm( stotal) AL 2 m kj ==00r 3 2 m 11 iL n n n ++−m 22s 12 312 m 222a mGs//////// H s kj mG s H s kj dS L L kj, AL SL 22 () HiLL111 2 aL +2, +−Knm( ) () n stotal 22 total 2 224 aL m =mGs//// d kj rH s kj dr, AL kj =00r = m 2 = FFi( sbL) ( ) 2n st s total where ns is the number of the strands in the 1 level 3 bundles, and ntotal is the total number of the strands contained 4nKtotals ( ) 413m 11 2 +−+ m iL inside the Litz wire (Hence, ntotal=nsn1n2.), AL is the cross- 22 8 aL 36 6m section area of the Litz wire, SL is the horizontal cross-section of the Litz wire. 31m 2 +4, +nKHtotalsL ( ) Substituting (19)–(22) into (23) and (24), we have 44m As can be seen in (30), the copper loss of the Litz wire can 2 2 2 be expressed in the form of PL=RLiL +GLHL , where RL and GL 22 HL are the loss coefficient of the skin effect and that of the mGHmGss⊥⊥⊥ kjs total n + (1 cos ) 2 proximity effect. These coefficients are given as kj, () 111cos 2 i2 ++−1,L m 22 2 RFFLsb= ( ) ( ) 3cos212 aL 2 sn total 3 () 2 4nKtotals ( ) 41311m mGs//// H s kj +−+ m , 22 kj, 8 a 366 m () L Hi2211 mG n LL +1 − cos2 . s// total 2 24 22 ( ) 31m aL GL=4, n total K ( s ) + () 44m 0.07 Experiment TABLE I. SPECIFICATIONS OF EXPERIMENTAL LITZ WIRES 0.06
) Simulation Wire A Wire B 0.05
Strand diameter s 0.05mm 0.071mm (m L 0.04 Litz wire diamter L mm mm R 0.03 Number of strands n total 1050 800 Wire length 3 3 0.02
Parameters n s, n 1, n 2 42, 5, 5 40, 5, 4 0.01
Parameter m 1.055 1.077 AC resistance 0 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Frequency (Hz) Bobbin (a) Wire A 0.14 Litz wire Experiment 0.12
) Simulation
0.1 (m
L 0.08 R 100 0.06 (a) Solenoidal coil for evaluation of RL 0.04
Litz wire 0.02 AC AC resistance 0 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Frequency (Hz) (b) Wire B
Fig. 6. Comparison result between the theoretically predicted RL and experimentally measured RL.
76 was 100mm. Then, we measured the frequency dependence 86 of the ac resistance below 2MHz using the LCR meter (b) Toroidal coil for evaluation of GL (Agilent E4980A). The measured ac resistance was divided by Fig. 5. Photographs of the experimental setup for evaluation of the loss the total wire length to obtain the ac resistance per unit length, coefficient RL and GL. which equals RL. Consequently, the frequency dependence of RL was obtained as a result of this experiment. The proposed copper loss model of the Litz wire contains For the evaluation G , the Litz wire to be tested was wound only physical constants and geometrical parameters of the Litz L on the core-less toroid with 20 turns, as shown in Fig. 5(b), to wire except for the external field H , which is dependent on L generate the uniform ac magnetic field perpendicular to the the magnetic structure outside of the Litz wire and therefore wire. The inner and outer diameter of this toroid was 76mm should be given by the external condition. Therefore, the proposed model is a fully analytical copper loss model of the and 86mm, respectively; and the height of the toroid was Litz wire. 50mm. Then, the frequency dependence of the ac resistance was measured below 2MHz using the LCR meter. The measured ac resistance was divided by the total wire length to obtain the ac resistance per unit length. The difference of the XPERIMENT III. E resultant ac resistance per unit length from RL, which was A simple experiment was carried out to verify the measured using the non-inductive solenoidal coil, is the ac proposed copper loss model. In this experiment, we evaluated resistance per length contributed by the external ac magnetic the copper loss coefficients RL and GL of two different Litz field. Therefore, we compared this ac resistance difference wires and compared the results with the theoretical values with the theoretical value to verify GL of the proposed model. analytically calculated based on (31) and (32). These We denote this ac resistance difference per unit length as experimental Litz wires are formed with three levels of R . The theoretical value of R can be obtained as follows. twisting. Specifications of these Litz wires are presented in GL GL The ac magnetic field inside the toroidal Litz wire coil Htroid Table I. can be approximately calculated as For the evaluation of RL, the Litz wire to be tested was wound on the core-less bobbin to form a non-inductive NiL solenoidal coil with four clockwise turns and four counter- Htroid = , () 2r clockwise turns, as shown in Fig. 5(a), so that the wire eq experience no external field HL. The diameter of the bobbin Skin effect (1st term) 0.012 120 Experiment Internal proximity effect loss (2nd term) 0.01
100 ) Simulation 0.008
80 (m
[%]
L
GL R 60 R 0.006
40 0.004 Ratio toRatio 20 0.002
0 AC resistance 0 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Frequency [Hz] Frequency (Hz) (a) Wire A (a) Wire A 0.035 Skin effect (1st term) Experiment 120 0.03 Internal proximity effect loss (2nd term) ) Simulation
100 0.025 (m
GL 0.02
80
R [%]
L 0.015 R 60 0.01
40 0.005 Ratioto 0
20 AC resistance -0.005 0 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Frequency (Hz) Frequency [Hz] (b) Wire B (b) Wire B Fig. 8. Comparison result between the theoretically predicted RGL and Fig. 7. Conposition ratio of the 1st and 2nd terms of equation (31) in the experimentally measured RGL. simulation result of RL.
where N is the number of turns wound on the toroid, and Figure 6 shows the evaluation results of RL. The frequency req is the representative distance from the center of the toroid dependence predicted by the proposed model agreed well with to the coil. In this experiment, we simply set req=40.5mm, the experiment in both of the two experimental Litz wires. st which equals the average of the inner and outer radius of the According to (31), the coefficient RL is the sum of the 1 term, coreless toroid. which is related to the skin effect inside the Litz wire, and the 2nd term, which is related to the proximity effect caused by the The ac magnetic field outside the toroidal Litz wire coil is internal magnetic field Hint. As can be seen in Fig. 7, which zero. Therefore, the part of the Litz wire that faces the outside st presents the ratio of these two terms to RL, the 1 term is of the toroidal coil has the zero ac magnetic field, whereas the dominant at low frequencies, whereas the 2nd term is dominant part that faces the inside of the toroidal coil has the ac at high frequencies. Therefore, the consistency of RL between magnetic field Htroid. However, we approximate that the the theory and the experiment for wide frequency range uniform representative external ac magnetic field HL_rep was implies appropriateness of both of the 1st and 2nd terms of (31). applied to the Litz wire. We set HL_rep at the RMS average of the ac magnetic field, simply assuming that the ac magnetic Figure 8 shows the evaluation result of RGL. The frequency field increases linearly from the outer side to the inner side of dependence predicted by the proposed model also agreed well the Litz wire toroidal coil. Hence, we set with the experimental result in both of the two experimental Litz wires, although drastic approximation was applied for calculating RGL. As a result, Fig. 8 supported the NiL HL_ rep = , () appropriateness of (32). Consequently, the experiment 23req supported the appropriateness of the proposed model.
Consequently, the theoretical value of RGL can be IV. CONCLUSIONS calculated as The copper loss prediction of the Litz wire is important for design optimization of the Litz wire. However, the Litz 2 wire has the complicated structure with multi-level twisting, GN R i2== G H 2L i 2 , which hinders the analytical and numerical analysis of the GL L L L_ rep22 L 12 req copper loss. In spite of the difficulty of the fully analytical () model of the Litz wire, the preceding studies have long GN2 =R L . accumulated analytical insights for the loss of the strands and GL 22 12 req the bundle of twisted strands. Based on these insights, this paper constructed a simple fully analytical copper loss model of the Litz wire. Appropriateness of the proposed copper loss [12] R. Y. Zhang, J. K. White, and J. G. Kassakian, “Fast simulation of model was evaluated by the experiment. As a result, the complicated 3-D structures above lossy magnetic media,” IEEE Trans. proposed copper loss model successfully predicted the Magn., vol. 50, no. 10, pp. 7027416, Oct. 2014. [13] A. Roßkopf, E. Bär, C. Joffe, and C. Bonse, “Calculation of power frequency dependence of the ac resistance of the commercial losses in litz wire systems by coupling FEM and PEEC method,” IEEE Litz wires, supporting the appropriateness of the proposed Trans. Power Electron., vol. 31, no. 9, pp. 6442-6449, Sept. 2016. model. [14] S. Hiruma and H. Igarashi, “Fast 3-D analysis of eddy current in litz wire using integral equation,” IEEE Trans. Magn., vol. 53, no. 6, pp. REFERENCES 7000704, Jun. 2017. [1] I. Lope, J. Acero, and C. Carretero, “Analysis and optimization of the [15] J. A. Ferreira and J. D. van Wyk, “A new method for the more accurate efficiency of induction heating applications with litz-wire planar and determination of conductor losses in power electronic converter solenoidal coils,” IEEE Trans. Power Electron., vol. 31, no. 7, pp. magnetic components,” in Proc. 3rd Intl. Conf. Power Electron. 5089-5101, Jul. 2016. Variable-Speed Drives, 1988, pp. 184-187. [2] M. Hataya, Y. Oka, K. Umetani, E. Hiraki, T. Hirokawa, and M. Imai, [16] J. A. Ferreira, “Analytical computation of AC resistance of round and “Novel thin heating coil structure with reduced copper loss for high rectangular litz wire windings,” IEE Proc. B – Elect. Power Appl., 1992, frequency induction cookers,” in Proc. IEEE Intl. Conf. Elect. Mach. vol. 139, no. 1, pp. 21-25. Syst. (ICEMS2016), Chiba, Japan, Nov. 2016, pp. 1-6. [17] M. Bartoli, N. Noferi, A. Reatti, and M. K. Kazimierczuk, “Modeling [3] M. Hataya, K. Kamaeguchi, E. Hiraki, K. Umetani, T. Hirokawa, M. Litz-wire winding losses in high-frequency power inductors,” in Proc. Imai, and S. Sadakata, “Verification of the reduction of the copper loss IEEE Power Electron. Specialists Conf., 1996, vol. 2, pp. 1690-1696. by the thin coil structure for induction cookers,” in Proc. IEEE Intl. [18] X. Nan and C. R. Sullivan, “An improved calculation of proximity- Power Electron. Conf. (IPEC2018), Niigata, Japan, May 2018, pp. effect loss in high-frequency windings of round conductors,” in Proc. 410–415. IEEE Power Electron. Specialists Conf., 2003, vol. 2, pp. 853-860. [4] S. Inoue and H. Akagi, “A bidirectional isolated DC–DC converter as [19] J. Acero, R. Alonso, J. M. Burdio, L. A. Barragan, D. Puyal, a core circuit of the next-generation medium-voltage power conversion “Frequency-dependent resistance in litz-wire planar windings for system,” IEEE Trans. Power Electron., vol. 22, no. 2, pp. 535-542, Mar. domestic induction heating appliances,” IEEE Trans Power Electron., 2007. vol. 21, no. 4, pp. 856-866, Jul. 2006. [5] W. Shen, F. Wang, and D. Boroyevich, C. W. Tipton IV, “High-density [20] J. Acero, R. Alonso, J. M. Burdio, L. A. Barragan, C. Carretero, “A nanocrystalline core transformer for high-power high-frequency model of losses in twisted-multistranded wires for planar windings resonant converter,” IEEE Trans. Ind. Appl., vol. 44, no. 1, pp. 213- used in domestic induction heating appliances,” in Proc. IEEE Appl. 222, Jan./Feb. 2008. Power Electron. Conf. and Expo., 2007, pp. 1247-1253. [6] W. Water and J. Lu, “Improved high-frequency planar transformer for [21] X. Nan and C. R. Sullivan, “An equivalent complex permeability line level control (LLC) resonant converters,” IEEE Magn. Lett., vol. model for litz-wire windings,” IEEE Trans. Ind. Appl., vol. 45, no. 2, 4, 6500204, Nov. 2013. pp. 854-860, Mar./Apr. 2009. [7] E. L. Barrios, A. Ursua, L. Marroyo, and P. Sanchis, “Analytical design [22] D. Sinha, P. K. Sadhu, N. Pal, and A. Bandyopadhyay, “Computation methodology for litz-wired high-frequency power transformers,” IEEE of inductance and AC resistance of a twisted litz-wire for high Trans. Ind. Electron., vol. 62, no. 4, pp. 2103-2113, Apr. 2015. frequency induction cooker,” in Proc. Intl. Conf. Ind. Electron. Control [8] M. Leibl, G. Ortiz, and J. W. Kolar, “Design and experimental analysis Robotics, 2010, pp. 85-90. of a medium-frequency transformer for solid-state transformer [23] R. P. Wojda and M. K. Kazimierczuk, “Winding resistance of litz-wire applications,” IEEE J. Emerg. Sel. Topics Power Electron., vol. 5, no. and multi-strand inductors,” IET Power Electronics, vol. 5, no. 2, pp. 1, pp. 110-123, Mar. 2017. 257-268, Jan. 2012. [9] C. R. Sullivan and R. Y. Zhang, “Simplified design method for litz [24] V. Väisänen, J. Hiltunen, J. Nerg, and P. Silventoinen, “AC resistance wire,” in Proc. IEEE Appl. Power Electron. Conf. Expo. (APEC2014), calculation methods and practical design considerations when using litz pp. 2667-2674, Mar. 2014. wire,” in Proc. Annu. Conf. IEEE Ind. Electron. Soc. (IECON2013), [10] B. A. Reese and C. R. Sullivan, “Litz wire in the MHz range: modeling 2013, pp. 368-375. and improved designs,” in Proc. IEEE Workshop Control Modeling [25] C. R. Sullivan and R. Y. Zhang, “Analytical model for effects of Power Electron. (COMPEL2018), pp. 1-8, Jul. 2017. twisting on litz-wire losses” in Proc. IEEE Workshop Control [11] A. Roßkopf, E. Bär, and C. Joffe, “Influence of inner skin- and Modeling Power Electron. (COMPEL), 2014, pp. 1-10. proximity effects on conduction in litz wires,” IEEE Trans. Power Electron., vol. 29, no. 10, pp. 5454-5461, Oct. 2014.