FLORE Repository istituzionale dell'Università degli Studi di Firenze Modeling Litz- winding losses in high-frequency power inductorsPESC Record. 27th Annual IEEE Power Electronics Questa è la Versione finale referata (Post print/Accepted manuscript) della seguente pubblicazione:

Original Citation: Modeling Litz-wire winding losses in high-frequency power inductorsPESC Record. 27th Annual IEEE Power Electronics Specialists Conference / M. Bartoli;N. Noferi;A. Reatti;M.K. Kazimierczuk. - ELETTRONICO. - 2(1996), pp. 1690-1696. ((Intervento presentato al convegno Power Electronics Specialists Conference, 1996. PESC '96 Record., 27th Annual IEEE [10.1109/PESC.1996.548808].

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01 October 2021 Modeling Litz-Wire Winding Losses in High-Frequency Power

Massimo Bartoli, Nicola Noferi, Albert0 Reatti, Member, IEEE University of Florence, Department of Electronic Engineering, Via S. Marta 3, 50139, Florence, ITALY Fax +39-55-494569. E-mail: [email protected] Marian K. Kazimierczuk, Senior Member, IEEE Wright State University, Department of Electrical Engineering, 45435, Dayton, OH. Fax 513-873-5009. E-mail: [email protected] .edu

Abstract- The parasitic effects in stranded, conductor is used, the overall cross-sectional area is twisted, and Litz wire windings operating at high spred among several conductors with a small diame- frequencies are studied. The skin and proximity ef- ter. For this reason, stranded, twisted, and Litz wire fects that cause the winding parasitic resistance of results in a more uniform current distribution across an to increase with the operating frequency the wire section. Moreover, Litz are assembled are considered. An expression for the ac resistance so that each single strand, in the longitudinal devel- as a function of the operating frequency is given. opment of the wire, occupies all the positions in the The measured and calculated values of the induc- wire cross section. Therefore, not only the skin ef- tor ac resistance and quality factor are plotted ver- fect but also proximity effect is drastically reduced. sus frequency and compared. The theoretical results However, the following limitations occur when using were in good agreement with those experimentally Litz wires: measured. 1) The utilization of the winding space inside a bob- bin width is reduced with respect to a solid I. INTRODUCTION wire.

2) The dc resistance of a Litz wire is larger than The overall efficiency of power circuits such as that of a solid wire with the same length and power converters and power amplifiers highly de- equivalent cross-sectional area because each pends on the efficiency of inductors used in assem- strand path is longer than the average wire bling the circuit. Therefore, the control of power length. losses in magnetic components is important for in- creasing the efficiency of the overall circuit. One of Consequently, a Litz wire results in a lower ac re- the main problems related to the design of power sistance than a solid wire only if the multistrand inductors is the calculation of the winding ac resis- wire windings are operated in an appropriate fre- tance Rat. Many efforts [1]-[6] have been made to quency range. This range can be predicted only if derive expressions allowing for an accurate represen- an expression for the winding ac resistance of a Litz tation of frequency behavior of Rat. wire as a function of frequency is available. Stranded, twisted, and Litz wires are commonly The purposes of this paper are (1) to derive an thought to result in lower ac resistances than solid accurate analytical expression for Litz wire winding wires. This is because the current through a solid ac resistance suitable for designing of low-loss high- wire concentrates in the external part of the conduc- frequency power inductors, (2) to compare the ac tor at high operating frequencies. If a multistrand reistances of Litz and solid wires, and (3) to ver-

0-7803-3500-7/96/$ 5.00 0 1996 IEEE 1690 ify the theoretical predictions with experimental re- solid wire windings. The twisted wires have the ad- sults. vantage of lower proximity lmses than the stranded The significance of the paper is that the derived ones. Actually, the internal strands are wounded expression for ac resistances allows for an accurate togheter, so that each single strand has an helicoy- prediction of the frequency behavior of an inductor dal movement around the center of the bundle, with Litzrwire windings. Both the expression for R,, and a constant distance from the center itself. Since z the inductor model can be used to design inductors is the longitudinal axis along the direction of the for high frequency operation with reduced winding bundle and X is the twisting pitch, the istantaneus losses. phase of the helix with respect to the center of the bundle J(2) is given by z 11. LITZ WIREAC RESISTANCE 19 (2) = 271. -.x (1) In the expression for the inside the twisted wire, a multipling factor sinJ(z) is con- In [l],the analytical expression for the ac resis- tained. Averaging this effect over one wavelength A, tance of a solid wire winding has been derived under one obtains: the following assumptions:

The one-dimensional approach proposed in 1 [a] sin28 (z) dz = - and has been followed. 1 1 (2) [3] x 2' 0 A high magnetic permeability has been assumed Twisting allows a 50% reduction of the proximity for the core material. Moreover, it was as- losses. However, there are no evident advantages sumed that the winding layer wires fill the en- of reducing the . Improvements in this tire breadth of the bobbin. Under these as- direction can be achieved by the use of Litz wire. sumptions, the magnetic component behavior is Litz wire is a bundle of strand conductors, where close to that of an ideally infinite long solenoid the strands are twisted with others so that all the winding. strands alternatively occupy all the positions of the wire cross section. With respect to a the twisted The curvature of the windings is neglected in cal- wires, in the Litz wires the strand distance to the culating the magnetic field distribution across center of the bundle is not fixed. This allows the the winding layers. current to flow also in the internal strands even at The orthogonality principle [4],[5] has been ap- higher frequencies and a more unifor current distri- plied. bution is achieved. In this way, an averaging of the overall skin effect is obtained and also at high fre- The main advantage of this approach is that it quencies, the current density due to the mere skin allows for the calculation of the overall ac resistance effect is almost the same in all the strands. Previous of a solid round wire as the sum of an ac resistance considerations allow us to deduce an analitycal ex- due to the skin effect and an ac resistance due to pression for the ac resistance of a Litz wire winding the proximity effect. The ac resistance of multiwire in the following way. conductors, e.g., Litz wires, is due to four different Taking the one-dimensional approach [a],[3], and parasitic effects: using the orthogonality principle, [4], [SI, an analyt- 1) skin effect in each single strand of the bundle. ical expression for the ac resistance of a coil winding 2) skin effect on the overall bundle that causes the has been derived in [l]. For a solid round wire wind- absence of current in the internal strands; ing composed of Nl layers, the procedure given in [l] 3) proximity effect between the single strands; yields: 4) proximity effect between the bundles of the dif- ferent turns. The proximity and skin effects can be assumed to be independent from each other, that is, the orthog- onality principle can be applied to multiwire wind- 4 (Nf- 1) -2v2 +I) (3) ings along with the other assumptions used for the (

1691 ber2 yber'y - beiyybei'y The expression derived for the ac resistance is in ber2y + bei 2y the rectangular coordinates and is deduced ap- proximating a winding layer to a foil conductor where d with the magnetic field assumed to be parallel y=- (4) to the conductor surface. SJZ Assumptions 1) and 2) are guaranteed if the core 7 = porosity factor = material has a high magnetic permeability, and the winding layer wires fill the entire breadth and of the bobbin. Under these assumptions, the magnetic component behavior is close to that 4P Rdc = TLNlT = -N~T (6) of an ideally infinite long solenoid windings. T d2 is the dc winding resistance, S = dmis the The curvature of the windings is neglected skin depth, pr is the relative magnetic permeabil- when calculating the magnetic field distribution ity of the conductor material (pr = 1 for a copper across the layers. conductor), po = 4n10-l' H / mm, d is the solid round conductor diameter in mm, t is the distance These assumptions are the same as those utilized between the centers of two adjacent conductors in in [l]to deduce expression (3): If only the skin effect is considered and a current mm, N is the number of turns of the winding, IT (mm) is the average length of one turn of the wind- Im/ns, where ns is the number of strands, is as- ing, and p is the conductor resistivity at a given sumed to flow through each strand, the power losses of the m-th layer of the winding are given by operating temperature (p = 17.24~10-6 Rmm for a copper conductor at an operating temperature of 20 "C). Wire data sheets usually give values of rL at 20 "C and 100 'C; the latter is typically 1.33 times the former. where Rs strand is given by [I] This approach can be applied also to Litz wire and it allows for calculating the skin effect losses separately from the proximity ones. In Fig. 1, the R.9 strand m cross section of a Litz wire conductor subjected to an external magnetic field He is shown.

where y, = d,/S& and d, = 2 r, is the single strand conductor diameter in mm, and Rdc strand is the internal strand dc resistance. The skin effect losses may be approximated by the sum of the effects in the n, Litz wire internal strands. Moreover, thanks to the Litz wire geometry the overall skin effect in the bundle can be nglected. As a consequence, power losses in the m-th layer of the winding can be calculated as

Fig. 1. Cross section of a Litz wire.

The analitycal expression for the ac resistance of the Litz wire winding has been derived under the following assumptions: Lets now evaluate the contributions of the two 1) The one-dimensional approach proposed in [2] proximity effect components (internal and external) and [3] has been followed to the power losses. It has been demonstrated in [5]

1692 that the overall proximity losses can be calculated of two different factors is necessary in the analysis as the sum of the contributions due to the internally of Litz wires because of the geometrically different generated magnetic field (Hi) and to the external operating influences of the external (He)and inter- one (He): nal (Hi) magnetic field on the single strand of the bundle. The two proximity components of the ac Pp m = Pp m int + ppm ext. (9) resistance, for the m-th layer of a Litz wire winding, are then given by Under the assumption that all the strands in the bundle behave in the same way with respect to the 7s external field, Pp ezt is given by [l]: m ext = Rdc strand m- % 2 ber2"ysber'ys - beizysbei'ys -2Tns771 [ ber2ys + bei 2~s and

where 7S Rpm int = Rdc strand m- 2 p berzy, ber'y, - bei2ys bei'y, -2Tr72 - 21r ber2ys + bei 2^/s where the packing factor of a Litz wire is defined as

77, = external porosity factor = 58 (12) and do = 2 ro is the Litz wire diameter excluding t0 the insulating layer. where to is the distance between the centers of two Fkom (7), (15) and (16) and applying the orthog- adjacent Litz wire conductors. onality principle as done in [l]to obtain (3), the A good approximation of the overall internal prox- overall ac resistance for the m-th layer is given by imity losses is given by [l],[5]:

Rw-m = Rdc-m7r(- bery, bei'y, - beiy, ber'y, 72, '$ KZstrand I:n I Pp m int = (1.3) 8 ~2 rg 2 ns bert2ys+ beit2yS where

berzySber'ys - beizysbei'ys q2 = internal porosity factor = (14) ber2y, + bei 2ys "8tS As a consequence, if an Nl-layer Litz wire winding ro is the Litz wire radius excluding the insulating is considered, where all layers have the same number layer and t, is the distance between the centers of of turns and then the same Rdcstrandm, its overall two adjacent strands, taking into account the ef- ac resistance can be calculated as follow: fective turn-to-turn, layer-to-layer, and strand-to- strand distances and the non-uniformity of the mag- netic field. In [l],the importance of the introduction in (3) of -2T [ 4 (Nf - 1) ill the porosity factor, q2,has been underlined. Actu- ally, it takes into account the effective turn-to-turn and layer-to-layer distances and the absence of uni- formity of the magnetic field. For these same rea- sons, coefficients 771 and 772 have been introduced berzy,ber'y, - bei2ySbei'ys also in (10) and (13), respectively. The introduction ber2ys + bei2ys

1693 where In Fig. 3, the resitanceas of a solid round wire, and Litz wire windings with a different number of strands are compared. The wire overall cross- sectional areas were chosen to result in the same dc is the Litz wire dc resistance, N is the number of resitance. The plots of Fig. 3 lead to the following turns of the winding, IT (mm) is the average length conlcusions: of one turn of the winding. 1) The Litz and solid wire conductors have the same The expression for the ac resistance given in (19) resistance up to = depending provides an accurate prediction of the Litz wire ac f 10~15kHz, on the number of strands. resistance over a wide frequency range. Moreover, it also allows for choosing the Litz wire with the lowest 2) The ac resistances of Litz wires are lower than resistance for any given application. The theoretical the solid wire resistances in a frequency range results are in good agreement with the experimental depending on the number of strands as follows tests. The results of this paper can be effectively used in the design of high quality factor, low-loss, 15 + 800 kHz for a 10 strand Litz wire; high-frequency power inductors. 15 kHz c 1 MHz for a 20 strand Litz wire; 15 kHz + 1.7 MHz for a 60 strand Litz wire.

111. COMPARISON OF LITZ AND SOLID ROUND 3) For frequencies higher than those shown above WIRERESISTANCES the Litz wire ac resistances are higher than the solid wire resistances.

Two different windings with a nearly equal dc re- Therefore, if the number of strands of the Litz sistance are considered as follows: wire is increased, its ac resistance is lower and is lower than that of a solid round wire over a wider 1) solid round wire AWG#28, with d = 0.32 mm, frequency range. The limitation of a high strand transversal conducting area A = 0.081 mm2, number Litz wire is that it has a large winding dis- and dezt = 0.37 mm; placement.

2) Litz wire 20~0.08,with a number of strands n, = 20, conducting diameter d, = IO5104 0.08 nim, a total transversal conducting area r A = 0.1005 mm2, and dezt = 0.554 mm. 103 I Rac Both wires have been wound on the same E25 102 Micrometals core with a number of turns N = 114 (0) in order to provide the same . The solid 10' round wire results in a 3-layer winding and Litz in more than a 4-layer winding. In Fig. 2, two winding ac resistances calculated lO-'l , , , , I from (19) are plotted as functions of the inductor operating frequency. These plots show that the ac Frequency (Hz) resistance of a wire winding is lower than that Litz Fig. 2. Resistances R,, of a solid and Litz wire of a solid round wire in a frequency range from 20 versus frequency. kHz to 800 kHz. Above this frequency, the Litz wire resistance becomes much higher than the solid wire resistance. This demonstrates that Litz wire pro- vides lower losses only in a limited frequency range. IV. EXPERIMENTALRESULTS Since at low frequencies the dc resistances of the two wires are nearly the same, the solid round wire is preferred because of its better utilization of the To verify the accuracy of expression (19), the winding width. lumped-parameter equivalent circuit of inductors

1694 shown in Fig. 4 was used. In Fig. 4(a) , L is permeability core materials. Tests have been per- the nominal inductance, R,, is the winding resis- formed on an inductor assembled with 114 turns in tance, R, is the magnetic core resistance, and C is 4-layers of Lit2 20~0.08wire wound on a Micromet- the inductor self-capacitance. Both resistances are als E25 material mix #52 with p,=75 core with a frequency dependent and, if the core resistance R, gapped central leg of 9 mm and an effective magnetic is much lower than the winding ac resistance R,,, permeability pe= 6. An inductance of L = 168 pH the inductor model simplifies to the form depicted and a resonance frequency fr = 2.620 MHz were in Fig. 4(b). Most LCR meters measure an equiv- measured. alent series reactance X, and an equivalent series resistance R, of two-terminal devices as shown in Fig. 4(c). The impedance of the equivalent circuit of Fig. 4(c) is

R,, i-jwL (1 - w2LC - z,= = R, +jX,. (1 - w2LCI2+ w2C2R;, (21) For frequencies f much lower than the first self- resonant frequency f,. , the equivalent series reac- tance X, has an inductive character and can be ex- press as X, = wL,. Hence, the equivalent series inductance is L, = X,/w as shown in Fig. 4(d). The inductor quality factor at a given frequency is

jwL (I - w2LC - )X,) *) 1 . (22) &=-=R, R,, Fig. 4. Equivalent circuit of inductors. (a) Lumped parameter equivalent circuit. 20 (b) Simplified equivalent circuit for R, << Rac. 10 (c) Series equivalent circuit. (d) Equivalent circuit assumed by many LCR meters. 5 RLitz

0.5 10

0.1 10 io2 io 10 IO lo6 lo7 Rs 102 Frequency (Hz) Fig. 3. Resistances R,, of 10 strands Litz wire, 20 strands Litz wire, and 60 strands Litz wire normalized with respect to a solid round wire 10 O 1 resistance. I I lo-' ' i __1 To verify the accuracy of expression (19), tests lo2 lo3 lo4 lo5 lo6 lo7 were performed using an HF'4192A "LF Impedance Frequency (Hz) Analyzer" equipped with an HP16047A test fixture to achieve a higher accuracy in minimizing resid- Fig. 5. Equivalent series resistance R, of the ual parameters and contact resistances. The induc- experimentally tested inductor. -- calculated; tor with negligible core losses was built using low- o-o- measured.

1695 100 by the same authors, can be used to design power inductor with reduced power losses. Therefore, this 80 paper represents an useful tool for designing high- efficiency power converter. The extension of the presented results to induc- tor winding operated at non sinusoidal current such e 6o as in inductors used in PWM DC-DC converters is 40 highly recommended. REFERENCES 20 [I] M. Bartoli, N. Noferi, A. Reatti, and M. K. Kaz- imierczuk, “Modelling winding losses in high- 0 frequency power inductors,” World Scientific I lo3 lo4 lo5 lo6 lo7 Journal of Circuits, Systems and Computers, Frequency (Hz) Special Issue on Power Electronics, Part 11, Fig. 6. Quality factor Q of the experimentally vol. 5, NO. 4, pp. 607-626, Dec. 1996. tested inductor. -- calculated; measured. o-o- [a] E. Bennet and S. C. Larson, “ERective resistailcc of alternating currents of multilayer windings,” Fig. 6 shows the plots of the theoretical and mea- sured equivalent series resistances of the inductor Transactions of American Institute of Electrical modelled as in [6]-[8]. Note that R, is approximately Engineering, vol. 59, pp. 1010-1017, 1940. one hundred times Rdc at 1 MHz and R, increases at [3] M. P. Perry, “Multiple layer series connected higher frequencies. The plots of the inductor qual- winding design for minimum losses,” IEEE ity factor Q is shown in Fig. 6. Theoretical and Trans. Power Application Systems, vol. PAS-98, experimental results were in good agreement. pp. 116-123, Jan./Feb. 1979. [4] J. A. Ferreira, Electromagnetic Modeling of V. CONLCUSIONS Power Electronic Converters, Kluwer Academic Publishers: Boston-Dordrecht-London, 1989. [5] J. A. Ferreira, “Improved analytical modeling An expression for the ac resistance of inductor of conductive losses in magnetic components,” winding wound by using Litz wire has been derived. IEEE Trans. Power Electronics, vol. 9, No. 1, The accuracy of this expression has been verified pp. 127-131, Jan. 1994. by comparing theoretical results with experimental results derived from testing an iron-powder core in- [6] M. Bartoli, A. Reatti, and M. K. Kazimier- ductor. A relatively large air gap was introduced czuk, “Modeling iron-powder inductors at high to minimize the core losses. The theoretical results frequencies ,” Proc. of 1994 IEEE-IAS Annual were in agreement with measured results. Meeting, Denver, CO, Oct. 2-7, 1994, vol. 2, The expression for the Litz wire winding ac resis- pp. 1225-1232. tance was used to compare the frequency behavior [7] M. Bartoli, A. Reatti, and M. I<. Kazimierczuk, of Litz wire and solid wire windings. This compar- ison has shown that the resistances of the windings “Predicting the high-frequency ferrite-core in- ductor performance,” Proc. of Electrical Manu- remains nearly the same up to about 15 kHz. For facturing & Coil Windings Association Meeting, frequencies higher than 12-18 kHz, the ac resistance Rosemont, Chicago, IL, USA, Sept. 27-29, 1994, of Litz wire winding is much lower than that of solid pp. 409-413. wire windings. At frequencies higher than some hun- dreds of kilohertz, the solid wire results in a lower [8] M. Bartoli, A. Reatti, and M. K. Kazimierczuk, ac resistance than the Litz wire. “High-frequency models of ferrite cores induc- The expression for the ac resistance for Lztz wire tors,” Proc. of IECON’94, Bologna, Italy, Sept. winding shown in this work, along with the expres- 5-9, 1994, pp. 1670-1675. sion for the solid wire winding shown in other papers

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