* University of Limerick, Ireland

Transcription

1 Optiizing the AC Resistance of Multilayer Transforer Windings with Arbitrary Current Wavefors' W.G. Hurley, Senior Meber, IEEE +, E. Gath +, and J.G. Breslin, Meber, IEEE + + Power Electronics Research Centre, National University of Ireland, Galway, Ireland * University of Lierick, Ireland Abstract - AC losses due to non-sinusoidal current wavefors have been found by calculating the losses at haronic frequencies when the Fourier coefficients are known. An optiized foil or layer thickness in a winding ay be found by applying the Fourier analysis over a range of thickness values. This paper presents a new forula for the optiu foil or layer thickness, without the need for Fourier coefficients and calculations at haronic frequencies. The new forula requires the rs value of the current wavefor and the rs value of its derivative. It is siple, straightforward and applies to any periodic waveshape. NOMENCLATURE Thickness of foil or layer. Duty cycle. Fundaental frequency of current wavefor in Hz. Average value of current. rs value of the n* haronic. rs value of the current wavefor. rs value of the derivative of the current wavefor. Ratio of the ac resistance to dc resistance at n* haronic frequency. Nuber of turns per layer. Haronic nuber. Nuber of layers. Radius of bare wire in wire-wound winding. ac resistance of a winding with sinusoidal excitation. dc resistance of a winding. Effective ac resistance of a winding, with arbitrary current wavefor. dc resistance of a winding of thickness 6,. Rise tie (0-100%). T Period of the current wavefor. 0=27cf.. Skin depth at fundaental frequency, 6, Skin depth at the n* haronic frequency. A d6,. q Porosity factor, see Fig. 1. I. INTRODUCTION T ransforers are operated at high frequencies in order to reduce their size [l]. Switching circuits and resonant circuits have greatly iproved the efficiencies of power supplies. These power supplies have non-sinusoidal current wavefors and give rise to additional ac losses due to haronics. AC resistance effects due to sinusoidal currents were treated by Bennett and Larson [2] and this work was tailored specifically for transforers by Dowel1 [3]. These works are based on a one-diensional solution of the diffusion equation as applied to conducting parallel plates. Dowell's forula has been found to reliably predict the increased resistance in cylindrical windings where the foil or layer thickness is less than 10% of the radius of curvature. The forula has been utilized in any applications such as planar agnetics by Kassakian [4] and Sullivan [5], atrix transforers by Willias [6], toroidal inductors by Cheng [7], distributed air-gaps by Evans [8] and slot bound conductors by Hanselan [9]. With the advent of switch ode power supplies, attention switched to non-sinusoidal current wavefors. These currents were decoposed into Fourier coponents; the haronic coponents are orthogonal so that the total loss is equal to the su of the losses calculated by Dowell's forula for the aplitude and frequency of each haronic in turn. Venkatraan [ 101 showed that for a pulsed wavefor typical of a forward converter, there is an optiu layer thickness to iniize ac losses. Carsten [ll] extended the analysis to square wavefors, which are encountered in full bridge ' This work was supported by PE1 Technologies, Dublin, Ireland /99/$ IEEE 580

2 converters and to triangular wavefors, which occur in filter chokes. Vandelac [12] extended the analysis to flyback converters. The optiu layer thickness is found as follows: 0 Calculate the Fourier coefficients Calculate the losses at each haronic frequency Calculate the total losses for each thickness in a range of values Read the optiu thickness fro a graph of ac resistance versus layer thickness Typically this ight involve loss calculations at up to 30 haronics for up to 10 thickness values in order to find the optiu value. Furtherore Fourier coefficients are only available for a few wavefors. This paper presents a new forula for ac resistance and the optiu layer thickness for any current wavefor. The forula only requires knowledge of the rs value of the current wavefor and the rs value of its derivative. Both these quantities can be easily easured or calculated with siulation progras such as PSPICE. The results are just as accurate as the cubersoe ethod based on Fourier analysis. 11. THE AC RESISTANCE The real part of Dowell's forula gives the ac to dc resistance factor: --I Rac -A dc r sinh(2a) + sin(2a) cosh(2a) - COS(~A) +- 2(p2-1) sh(a) - sin(a) 1 3 cosh(a) + cos(a) 1 where A is the ratio of the layer thickness d to the skin depth 6,. This is a very good approxiation to the original cylindrical solution, particularly if the layer thickness is less than 10% of the radius of curvature. Windings which consist of round conductors, or foils which do not extend the full winding window, ay be treated as foils with equivalent thickness d and effective conductivity a,=qa. This calculation is shown graphically in Fig. 1, a detailed treatent of wire conductors is given by Ferreira [ 131 and Jongsa [14]. The orthogonality of skin and proxiity effects in wire windings is described by Ferreira [ 131. The trigonoetric and hyperbolic functions in (1) ay be represented by the series expansions: sinh2a+sin2a = -+-A - -A7 + O(A1'), (2) COS~~A-COS~A A sha-sina 1 %-A3 - L A 7 +O(A1'). cosha+cosa Fig. 1. Porosity factor for foils and round conductors. If only ters up to the order of A3 are used, the relative error incurred in (2) is less than 1.2% for A4.2 and the relative error in (3) is less than 4.1% for A 4 and is less than 8.4% if A<1.2. The asyptotic values of the functions on the left hand side of (2) and (3) are 1 for A>2.5. Ters up to the order of A3 are sufficiently accurate to account for the Fourier haronics which are used to predict the optiu value of A which is norally in the range Thus (1) becoes R,, - w A Rdc where w=- 5pz An arbitrary periodic current wavefor, ay be represented by its Fourier Series i(t) = Id, + a, cos n at + bn sin nwt. (6) The sine and cosine ters ay be cobined to give an alternative for (4) i(t) = Idc + c, cos (n cot + qn (7) where I, is the dc value of i(t) and c, is the aplitude of the n* haronic with corresponding phase cp,. The rs value of the n"' haronic is I,=c,ld

3 The total power loss due to all the haronics is P = R + R dc C k pn In2 (8) where kp, is the ac resistance factor at the n"' haronic frequency, which ay be found fro (1) 1 Sinh(2&A) +Sin(2&A) - - Cosh(24nA) - Cos(2JnA) kp =&A 2(p2-1) Sinh(&A)-Sin(&A) Cosh(&A) + Cos(&A) Kff is the ac resistance due to i(t) so that P=KffI2, I, being the rs value of i(t). Thus the ratio of effective ac resistance to dc resistance is -- Re, - dc Idc2 +Ckpn1n2 2 Irs The skin depth at the n"' haronic is 6,=6,& and, fro (4), the ac resistance factor at the n"' haronic frequency is kp+-n 3 w 2A 4 Substituting (1 1) into (10) yields 1 This is a straightforward expression for the effective resistance of a winding with an arbitrary current wavefor and it ay be evaluated without knowledge of the Fourier coefficients of the wavefor THE OPTIMUM CONDITIONS There is an optiu value of d, which gives a iniu value of effective ac resistance. Define & as the resistance of a foil of thickness 6, such that which iplies that --A-. Re, - Reff Rdc R8 Evidently, a plot of Kff I & versus A has the sae shape as a plot of Kn versus d at a given frequency. A 3-D plot of K, I & versus A with p, the nuber of layers in the winding, on the third axis is shown in Fig inia p=10 dc - 2 Lns The rs value of the current in ters of its haronics is e, R The derivative of i(t) in (7) is di -=-Cncn sin(nwt+@,> (14) dt and the rs value of the derivative of the current is [ 151 which, upon substitution into (12) using (13), yields A 1.5 Fig. 2. Plot of ac resistance versus A and nuber of layers p. For each value of p there is an optiu value of A where the ac resistance of the winding is iniu. These optiu points lie on the line arked inia in the graph and the corresponding value of the optiu layer thickness is dopt = Aopt6O. Fro (1 8), using (1 6) 582

4 The optiu value of A is found by talung the derivative of (20) and setting it to zero. The optiu value of A is I Substituting this result into (16) yields the optiu value of the effective ac resistance with an arbitrary periodic current wavefor: Jongsa [ 141 and Snelling [ 161 have already established this result for sinusoidal excitation. The corresponding value for wire conductors with sinusoidal excitation is 312 [14,16]. We ay also write (16) in ter of A, / \4 We now have a set of siple forulas with which to find the optiu value of the foil or layer thickness of a winding and its effective ac resistance, these forulas are based on the rs value of the current wavefor and the rs value of its derivative. IV. VALIDATION Consider the pulsed current wavefor in Fig. 3 along with its derivative, which is typical of a forward converter. This wavefor has a Fourier series: The rs value of I(t) and the the rs value of its derivative are tr Fig.3. Pulsed current wavefor and its derivative. The optiu value of A given by the Fourier series (25), for p=6 and t,/t=4% is and the value given by the proposed forula (22) is which represents an error of 7.4%. Wavefor 5 in Table I1 is an approxiation to the pulse in Fig. 3 and the optiu value of A using Fourier analysis is which represents an error of 7.2% when copared to the Fourier analysis of the wavefor given by (25). Evidently wavefors with known Fourier series are often approxiations to the actual wavefor and can give rise to errors which are of the sae order as the new forula, which is sipler to evaluate. At 50 lchz the skin depth in copper is With AOpt=0.418, dopt=0.295xo.418= The new forula ay be validated by coparing the value of Aopt obtained with (22) and the value obtained with Fourier analysis by plotting (lo), using (9), over a range of values of A and finding the optiu value. The results are shown in Table I for the wavefors in Table 11. In general the agreeent is within 3%, with the exception of wavefor 5 where the error is 6.5%. For the Fourier analysis 19 haronics were evaluated and was calculated for 20 values of A. This eans that (9) was coputed 380 ties for each wavefor in order to find the optiu layer thickness, (22) was coputed once for the sae result. For 1 to 3 layers the accuracy of the proposed forula is not very good, however, as evidenced by Fig. 2, the plot of k&, is alost flat in the region of the optiu value of A, and therefore the error in the ac resistance is negligible. I, =Io \i ZT 11, = 1.p- 3t,T 583

5 r TABLE I VALIDATION RESULTS, p=6, D=0.4, t,lt=4% Wavefor No. I Fourier Analvsis 1 New Forula I V. CONCLUSIONS A new forula has been presented to find the optiu foil or layer thickness in a ultilayer winding. The forula applies to any arbitrary periodic current wavefor. It is coputationally easier to use than Fourier analysis while enjoying the sae level of accuracy. It has a wider range of application than the Fourier approach by virtue of its siplicity. ACKNOWLEDGEMENT Prof. Hurley would like to thank Prof. J.G. Kassakian of the Massachusetts Institute of Technology for the use of the facilities at the Laboratory for Electroagnetic and Electronic Systes, where this paper was written. The coents of Rudy Severns of Springtie Enterprises Inc. are greatly appreciated. REFERENCES W.G. Hurley, W.H. Wolfle, J.G. Breslin, Optiized Transforer Design: Inclusive of High Frequency Effects, IEEE Trans. Power Electron., vol. 13, no. 4,pp , July E. Bennett, S.C. Larson, Effective Resistance to Alternating Currents of Multilayer Windings, Trans. ofaiee, vol. 59, pp , P.L. Dowell, Effect of Eddy Currents in Transforer Windings, IEE Proc., vol. 113, no. 8, pp , August J.G. Kassakian, M.F. Schlecht, High-Frequency High-Density Converters for Distributed Power- Supply Systes, Proc. of the IEEE, vol. 76, no. 4, pp , April [131 [ 141 r151 [I61 C.R. Sullivan, S.R. Sanders, Design of Microfabricated Transforers and Inductors for High- Frequency Power Conversion, IEEE Trans. Power Electron., vol. 11, no. 2, pp , March 1996 R. Willias, D.A. Grant, J. Gowar, Multieleent Transforers for Switched-Mode Power-Supplies: Toroidal Designs, IEE Proceedings, Pt. B, vol. 140, no. 2, pp , March 1993 K.W.E. Cheng, P.D. Evans, Calculation of Winding Losses in High-Frequency Toroidal Inductors Using Multistrand Conductors, IEE Proc.- Electr. Power Appl., vol. 142, no. 5, pp , Septeber P.D. Evans, W.M. Chew, Reduction of Proxiity Losses in Coupled Inductors, IEE Proc. Pt. B, vol. 138, no. 2, pp , March D.C. Hanselan, W.H. Peake, Eddy-Current Effects in Slot-Bound Conductors, IEE Proc.-Electr. Power Appl., vol. 142, no. 2, pp , March P.S. Venkatraan, Winding Eddy Current Losses in Switch Mode Power Transforers Due to Rectangular Wave Currents, Proc. of Powercon 11, section A-1, pp. 1-11, B. Carsten, High Frequency Conductor Losses in Switchode Magnetics, HPFC Proc., pp , May J. Vandelac, P.D. Ziogas, A Novel Approach for Miniizing High-Frequency Transforer Copper Losses, IEEE Trans. Power Electron., vol. 3, no. 3, pp , July J.A. Ferreira, Iproved Analytical Modelling of Conductive Losses in Magnetic Coponents, IEEE Trans. Power Electron., vol. 9, no. 1, pp , January J. Jongsa, Miniu Loss Transforer Windings for Ultrasonic Frequencies, Part 3: Transforer Winding Design, Phillips Electronics Applications Bulletin, vol. E.A.B. 35, pp , Charles R. Sullivan, Optial Choice for Nuber of Strands in a Lib-Wire Transforer Winding, in 28Ih Annual IEEE Power Electronics Specialists Conference, pp , E.C. Snelling, Soft Ferrites, Properties and Applications, Butterworths, second edition,

6 L Current Wavefor 1. TABLE I1 FORMULAS THE OPTIMUM THICKNESS OF A WINDING FOR VARIOUS WA FORMS, Y=(5p2-1)/15, p=no. OF L 'ERS and I,' Fourier Series, i(t) 1 =I, Sin(wt),A s 1 =- u 2n TA'' 2. ~+~T{-~)cos(nt) 2DI 4D10 Cos(nrD) = 0=I 1-4n2D I, =I0E 4. I*=Io 1-- J ":T I' = I0d+ 41,. 10(2D -1) + C-Sin(nnD) nn 5. I,=Io D-'E I 9 xsinc nn- ( xsinc nn' Cos(not) Cos(not) 1- Aopt = Y lop+: 2 2 2I,Sin(nnD) 8 xzn2d(1 - D) Sin(nt) I, = I,/+ 9. 'p/$qr-at: In wavefor 2 for n=k=l, I, =I./; I' =A 41-6 T the {expression in curly brackets} is replaced by n2/16. ]EN (the set of natural nubers), and in wavefor 3 for n=k=l/den, 5 85