Published in IET Power Electronics Received on 15th November 2010 Revised on 22nd August 2011 doi 10 1049 iet pel 2010 0359 www ietdl org ISSN 1755 4535 Winding resistance of litz wire and multi strand inductors R P Wojda M K Kazimierczuk Department of Electrical Engineering Wright State University 3640 Colonel Glenn Highway Dayton 45435 OH USA E mail rafal wojda fulbrightmail org Abstract This study presents an approximate model for multi strand wire winding including litz wire winding The proposed model is evaluated using Dowell s equation The model takes into account the existence of proximity effect within the litz wire bundle between the strands and between the bundles as well the skin effect The expressions for optimum strand diameter and number of strands at which minimum winding AC resistance is obtained for the litz wire windings are derived The boundary frequency between the low frequency and the medium frequency ranges are given for both litz wire and solid round wire inductors Hence the low frequency ranges of both wire windings are determined It is shown that litz wire is better than the solid wire only in speci c frequency range The model has been veri ed by the measurements and the theoretical results were in good agreement with those experimentally measured Comparison of the theoretical predictions of the proposed approximate litz wire model with models proposed in other publications and with experimental results is given 1 Introduction Litz wire is used to reduce winding power losses of inductors and transformers Power inductor and power transformer losses consist of winding DC loss winding AC loss and core losses The DC losses are due to a high DC resistance of a winding conductor and they can be simply reduced by increasing a cross sectional area of the conductor The winding AC power loss at high frequencies in an inductor is caused by eddy currents There are two effects of eddy currents the skin effect and the proximity effect 1 45 Both effects are frequency dependent and alter the current density distribution in the conductor through which the current operation the conductor winding resistance is approximately equal to the DC resistance However with increasing the operating frequency increases In order to maintain the low winding AC resistance at higher frequencies it is desirable to use a parallel multi strand wire in German Litzendraht A multi strand wire consists of at two strands and a litz wire consists of many strands up to couple of thousands of strands in a bundle 46 Various approaches to the analysis of multi strand and litz wire windings have been presented 4 6 7 10 12 18 26 31 32 35 including the low frequency the winding litz wire or resistance rapidly ows least For the The objectives of this paper are to introduce a model of litz wire winding to adapt Dowell s equation for the litz wire winding model to develop a relatively simple equation for the resistance of litz wire winding to determine the optimum strand diameter to determine the number of strands in the litz wire bundle at which the minimum winding AC resistance is achieved to determine the boundary frequency between the low frequency and medium frequency ranges for a litz wire winding and a solid round wire winding to compare the resistance of a litz wire winding with that of a solid round wire winding over a wide frequency range for validity of Section 2 introduces an approximate model of litz wire winding including multi strand wire winding and presents assumptions this model Relationships between litz wire and solid round wire windings are given in Section 3 Section 4 includes derivation of a general expression for the boundary frequency between low and medium frequency ranges for solid round wire windings as a function of the number of layers Nl Section 5 adopts Dowell s expression for litz wire windings including multi strand wire windings A low and medium frequency approximation of the modi ed Dowell s equation for litz wire windings is given and expressions for optimum strand diameter and required number of strands in the litz wire bundle at which minimum winding resistance is obtained are derived The general expression for the boundary frequency between low and medium frequency ranges for litz wire windings as a function of the number of layers is given Moreover relation between the boundary frequency of the litz wire windings and the boundary frequency of the solid round wire windings is presented In Section 6 comparison of winding AC resistance of litz wire and winding AC resistance of solid round wire is given This section shows the frequency range where the litz wire windings exhibit lower winding losses than the solid round wire windings Section 7 describes relationship between the the IET Power Electron 2012 Vol 5 Iss 2 pp 257 268 doi 10 1049 iet pel 2010 0359 257 The Institution of Engineering and Technology 2011 www ietdl org inductor winding AC resistance and equivalent series resistance ESR with negligible core losses Additionally the model of the measured inductor as well as the equations for ESR and equivalent series reactance are presented Section 8 presents experimental veri cation and compares several methods to predict the AC resistance of the litz wire windings are conducted in Section 9 conclusions discussions Finally and Proposed model of multi strand and 2 litz wire windings Fig 1 shows a proposed approximate model of the multi strand wire winding including litz wire winding The winding consists of Nl bundle layers Each bundle contains k strands There are different shapes of litz wire bundles such as round square rectangular and hexagonal 46 It is assumed that the net currents in all strands are equal i k is the net current owing through the litz wire where i bundle The number of layers in each bundle is and the number of strand turns per inductor strand layer is Ntl It is also assumed that the individual strands are parallel to the axis of the bundle all strands within the bundle and between the bundles are uniformly spaced and each effective bundle in the model has a square shape Therefore the effective number of layers of a litz wire and multi strand wire inductor is given by cid 2 cid 2 k cid 2 cid 2 k Nl Nll 1 This model will be used in the subsequent analysis from the model proposed in this paper The model of the litz wire winding given in 12 is different is described by Bessel functions in cylindrical coordinates This model is more complex and leads to