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Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding
C. R. Sullivan From IEEE Transactions on Power Electronics, vol. 14, no. 2, pp. 283 291.
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 14, NO. 2, MARCH 1999
Charles R. Sullivan, Member, IEEE
Abstract The number and diameter of strands to minimize loss in a litz-wire transformer winding is determined. With ne stranding, the ac resistance factor can be decreased, but dc resistance increases as a result of the space occupied by insulation. A power law to model insulation thickness is combined with standard analysis of proximity-effect losses to nd the optimal stranding. Suboptimal choices under other constraints are also determined. Index Terms Eddy currents, litz wire, magnet wire, power electronics, power transformers, proximity-effect losses, skin effect, transformer windings.
I. INTRODUCTION SALIENT difculty in the design of high-frequency inductors and transformers is eddy-current effects in windings. These effects include skin-effect losses and proximityeffect losses. Both effects can be controlled by the use of litz wireconductors made up of multiple individually insulated strands twisted or woven together. (Sometimes the term litz wire is reserved for conductors constructed according to a carefully prescribed pattern, and strands simply twisted together are called bunched wire. We will use the term litz wire for any insulated grouped strands.) This paper addresses the choice of the degree of stranding in litz wire for a transformer winding. The number of turns and the maximum winding cross-sectional area are assumed to be xed. Under constraints on maximum number of strands or minimum wire diameter, the best solution may not ll the allocated window space fully. However, as will be shown in Section IV, with those constraints removed, the optimum solution does ll the allocated space. In this case, the crosssectional area of each turn is xed, and as the number of strands is increased, the cross-sectional area of each strand must be decreased. This typically leads to a reduction in eddycurrent losses. However, as the number of strands increases, the fraction of the window area that is lled with copper decreases and the fraction lled with insulation increases. This results in an increase in dc resistance. Eventually, the eddycurrent losses are made small enough that the increasing dc resistance offsets any further improvements in eddy-current loss, and the total losses start to increase. Thus, there is an optimal number of strands that results in minimum loss.
Manuscript received August 28, 1997; revised July 6, 1998. Recommended by Associate Editor, J. Sarjeant. The author is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755-8000 USA. Publisher Item Identier S 0885-8993(99)01828-1.
This paper presents a method of nding that optimum, using standard methods of estimating the eddy-current losses. Optimizations on magnetics design may be done to minimize volume, loss, cost, weight, temperature rise, or some combination of these factors. For example, in the design of magnetic components for a solar-powered race vehicle [1] (the original impetus for this work) an optimal compromise between loss and weight is important. Although we will explicitly minimize only winding loss, the results are compatible with and useful for any minimization of total loss (including core loss), temperature rise, volume, or weight. This is because the only design change considered is a change in the degree of stranding, preserving the overall diameter per turn and overall window area usage. This does not affect core loss or volume and has only a negligible effect on weight. However, the degree of stranding does signicantly affect cost. Although we have not attempted to quantify or optimize this, additional results presented in Section V are useful for cost-constrained designs. The analysis of eddy-current losses used here does not differ substantially from previous work [2][18] ([15] gives a useful review). Although different descriptions can be used, most calculations are fundamentally equivalent to one of three analyzes. The most rigorous approach uses an exact calculation of losses in a cylindrical conductor with a known current, subjected to a uniform external eld, combined with an expression for the eld as a function of one-dimensional (1-D) position in the winding area [17]. Perhaps the most commonly cited analysis [16] uses equivalent rectangular conductors to approximate round wires and then proceeds with an exact 1-D solution. Finally, one may use only the rst terms of a series expansion of these solutions, e.g., [14]. All of these methods give similar results for strands that are small compared to the skin depth [17]. (See Appendix B for a discussion of one minor discrepancy.) The solutions for optimal stranding result in strand diameters much smaller than a skin depth. In this region, the distinctions between the various methods evaporate, and the simplest analysis is adequate. More rigorous analysis (e.g., [17]) is important when strands are not small compared to a skin depth. In this case, losses are reduced relative to what is predicted by the analysis used here, due to the self-shielding effect of the conductor. Previous work, such as [2][9] has addressed optimal wire diameter for single-strand windings. The approach in [2][9] is also applicable for litz-wire windings in the case that the number of strands is xed, and the strand diameter for lowest loss is desired [5], [9]. As discussed in Section V, this can be
08858993/99$10.00 1999 IEEE
equal current in all strands [17], [19], [22]. This assumption is equivalent to assuming that the bundle-level construction has been chosen properly to control bundle-level proximity and skin effects. Note, however, that most of our results remain valid even when there is signicant skin-effect loss at the litzbundle level, for example in a simply twisted bundle. This is because the bundle-level skin-effect loss is independent of the number of strands, and is orthogonal [20] to the strand-level eddy-current losses. We represent winding losses by
Fig. 1. Types of eddy-current effects in litz wire.
(1) is a factor relating dc resistance to an ac resistance where which accounts for all winding losses, given a sinusoidal As shown in Appendix A, current with rms amplitude we can approximate by (2) where is the radian frequency of a sinusoidal current, is is the number of turns, is the the number of strands, is the resistivity diameter of the copper in each strand, of the copper conductor, is the breadth of the window is a factor accounting for eld area of the core, and distribution in multiwinding transformers, normally equal to one (see Appendix A). For waveforms with a dc component, and for some nonsinusoidal waveforms, it is possible to derive a single equivalent frequency that may be used in this analysis (Appendix C). In an inductor, the eld in the winding area depends on the gapping conguration, and this analysis is not directly applicable [23]. The analysis described here considers the strands of all litz bundles to be uniformly distributed in the window, as they turns of wire the would be in a single winding using diameter of the litz strands. In fact, the strands are arranged in more or less circular bundles. In this sense, the analysis of [20] may be more accurate, but this difference has very little effect on the results. The most important difference between the model used here and the model in [20] is the greater accuracy of [20] for strands that are large compared to a skin depth. The simpler model is used because it is accurate for the small strand diameters that are found to be optimal, and because its simplicity facilitates nding those optimal diameters. Other models (such as [19] and the similar analysis in [22]) also model large strand diameters and circular bundle congurations accurately, but they fully calculate only internal (not external) proximity effect, and so are not useful for the present purposes. III. DC RESISTANCE FACTORS The fraction of the window area occupied by copper in a litz-wire winding will be less than it could be with a solidwire winding. This leads to higher dc resistance than that of a solid wire of the same outside diameter. A cross section of litz wire is shown in Fig. 2, with the various contributions to cross-sectional area marked. In addition to the factors shown in this diagram, the twist of the litz wire also increases the
useful for cost-sensitive applications if the number of strands is the determining factor in cost and the maximum cost is constrained. However, this will, in general, lead to higher loss designs than are possible using the optimal number of strands. II. SKIN EFFECT AND PROXIMITY EFFECT IN LITZ WIRE Skin effect is the tendency for high-frequency currents to ow on the surface of a conductor. Proximity effect is the tendency for current to ow in other undesirable patternsloops or concentrated distributionsdue to the presence of magnetic elds generated by nearby conductors. In litz-wire windings, skin and proximity effects may be further divided into strandlevel and bundle-level effects, as illustrated in Fig. 1. Bundlelevel effects relate to current circulating in paths involving multiple strands, whereas strand-level effects take place within individual strands. Bundle-level effects are controlled by the pattern of twisting or weavingthe construction of the litz wire. Simple twisting is adequate to control bundle-level proximity effect loss, whereas more complex constructions are needed to control bundle-level skin effect. Bundle-level effects are not directly affected by the number or diameter of strands; they are determined by the overall diameter and the choice of twisting pattern. Thus, they need not be considered further for the analysis in this paper. At the strand level, proximity effect dominates skin effect in a winding that has many layers. Since a litz winding has a large effective number of layers as a result of the many strands, strand-level skin effects are negligible. Thus, we need only consider strand-level proximity effect losses for the choice of number of strands. Strand-level proximity effect may be still further divided into internal proximity effect (the effect of other currents within the bundle) and external proximity effect (the effect of current in other bundles) [19], [20]. However, this distinction is useful only as a form of bookkeeping. The actual losses in one strand of a litz bundle are simply a result of the total external eld, due to the currents in all the other strands present. To calculate the total strand-level proximity-effect loss in a litz winding, one can view it as a single winding, made turns of the strand wire, each with current up of owing in it, where is the number of strands, is the number of turns of litz wire, and is current owing in the overall litz bundle. The loss in the litz winding will be the same as in the equivalent single-strand winding as long as the currents owing in all the strands are equal [6], [21]. Other methods of calculating loss in litz wire also assume
SULLIVAN: OPTIMAL CHOICE FOR NUMBER OF STRANDS IN A LITZ-WIRE TRANSFORMER WINDING
Fig. 2. Cross-sectional area of a litz-wire winding showing how area is allocated. Area allocated to anything other than copper increases the resistance in a space-limited winding.
Fig. 3. The cross section of strands becomes elliptical when the bundle is twisted. In this extreme case of lay (length per twist) equal to 4.7 times the bundle diameter, a total of six strands t where seven would have t untwisted.
dc resistance. In order to nd the optimal number of strands for a litz winding, it is necessary to quantify how the factors affecting dc resistance vary as a function of the number of strands. A. Serving Typically, litz bundles are wrapped with textile to protect the thin insulation of the individual strands. This serving adds about 0.06 mm (2.5 mil) to the diameter of the bundle. For a given number of turns lling a bobbin, or a section of a bobbin, the outside diameter of the litz wire must be xed. The area devoted to serving will then also be xed, independent of the number of strands. B. Strand Packing Simply twisted litz wire comprises a group of strands bunched and twisted into a bundle. More complex constructions begin with this step, and then proceed with grouping and twisting the subbundles into higher level bundles. Particular numbers of strands (1, 7, 19, 37, etc.) pack neatly into concentric circular arrangements. However, with large numbers of strands (e.g., 19), and/or very ne strands [e.g., 4450 American Wire Gauge (AWG)], it is difcult to precisely control the conguration, and the practical packing factor becomes an average number approximately independent of the number of strands. Since the optimal strand diameter is typically much smaller than a skin depth, but the lowest level bundle can be near a skin depth in diameter, in most cases we can assume that there is a large number of strands in the innermost bundle. Thus, this packing factor is independent of the number of strands. C. Bundle Packing and Filler The way the strands are divided into bundles and subbundles is chosen based on considerations including bundle-level skineffect losses, exibility of the overall bundle, resistance to unraveling, and packing density. In some cases, a nonconducting ller material may be used in the center of a bundle in place of a wire or wire bundle that would, in that position, carry no current because of skin effect. A typical conguration chosen to avoid signicant bundlelevel skin-effect losses should have a carefully designed and
potentially complex construction at the large-scale level where bundle diameters are large compared to a skin depth. However, because the optimal strand diameter will be small compared to a skin depth, a simple many-strand twisted bundle may be used at the lowest level. If the overall number of strands is increased, the number of strands in each of these low-level bundles should be increased, but the diameter of each low-level bundle should not be changed, nor should the way they are combined into the higher level construction be affected. Thus, for our purposes, the bundle packing factor is independent of the number of strands. D. Turn Packing The way turns are packed into the overall winding is primarily a function of winding technique, and it is assumed not to vary as a function of the stranding. However, note that loosely twisted litz wires can deform as the winding is constructed, allowing tighter packing. Another option providing tight turn packing is rectangular-cross-section litz wire. In addition to its turn-packing advantage, it has tighter strand and bundle packing, as a result of the mechanical compacting process that forms it into a rectangular cross section. E. Twist The distance traveled by a strand is greater in a twisted bundle than it would be if the strands simply went straight, and so the resistance is greater. An additional effect arises from the fact that a cross section perpendicular to the bundle cuts slightly obliquely across each strand. Thus, the cross section of each strand is slightly elliptical. This reduces the number of strands that t in a given area, and so effectively increases the resistance. An extreme case of this is illustrated in Fig. 3. The choice of the pitch of the twist (lay or length per twist) is not ordinarily affected by the number of strands in the lowest level bundle, and so for the purpose of nding the optimal number of strands, we can again assume it is constant. F. Strand Insulation Area Thinner magnet wire has thinner insulation. However, the thickness of the insulation is not in direct proportion to the wire diameter. Thinner wire has copper in a smaller fraction of the overall cross-sectional area and insulation in a larger fraction.
is the overall diameter, including the insulation where thickness, is the diameter of the copper only, and is an arbitrarily dened reference diameter used to make the constants and unitless. The parameters found for singlebuild insulation wire are and for chosen to be the diameter of AWG 40 wire (0.079 mm). For and Note that heavy-build insulation, although (4) provides an accurate approximation for wire in the range of 3060 AWG, its asymptotic behavior for largestrand diameters is pathological. Insulation thickness goes to zero around 6 AWG and is negative for larger strands. IV. NUMBER OF STRANDS FOR MINIMUM LOSS With no constraints on number of strands or strand diameter, the minimum-loss design will be with a full bobbin. Any design that does not ll the bobbin can be improved by increasing the number of strands by a factor and decreasing the strand diameter by This keeps the dc resistance constant and decreases ac resistance, as shown by (2). This improvement can be continued until insulation area increases result in a full bobbin. Thus, the minimum-resistance solution lls the bobbin, and we can nd this solution by analyzing a full bobbin. For a full bobbin, the outside diameter of the complete litz bundle is (5) where is the breadth of the bobbin, is the height allocated is the for the particular winding under consideration, is a turn-packing number of turns in that winding, and factor for turns in the winding, expressed relative to perfect the litz bundle would occupy square packing (for of the window area). accounting for serving area, bundle Assuming a factor packing, any ller area, strand packing, and the effect of twist on diameter, we can nd the outside diameter of a single strand (6) where is the number of strands in the overall litz bundle. The diameter of the copper in a single strand can then be written using (4) (7) We now dene a total resistance factor ac resistance of litz-wire winding dc resistance of single-strand winding (8) is the ratio of dc resistance of the litz wire to the where dc resistance of a single strand winding, using wire with the same diameter as the litz-wire bundle. Using (6) and (7), we can show (9)
Fig. 4. Insulation build (twice the insulation thickness) for AWG single-build magnet wire. The dashed curve is the minimum build according to the equation provided by [24]. The lower staircase curve is the tabulated data provided by [24] for minimum build. The points marked by are nominal build from a wire manufacturers catalog obtained by subtracting an exact theoretical nominal wire diameter from the tabulated nominal overall diameter. The approximation described by (4) comes closest to these points.
Of all of the dc-resistance factors considered, this is the only one that varies with the size or number of strands used at the lowest level of the construction. Thus, quantifying this effect on dc resistance gives a good approximation of the total variation in dc resistance as a function of the size or number of strands. The other factors can be lumped into an overall dc resistance multiplying factor which is a constant for the present purposes. One approach to quantifying the relationship between the insulation area and strand diameter would be to store tables in computers, and use them to nd the optimal strand diameter by calculating the losses for different strand diameters until the optimum was found, similar to [11]. However, an analytical description of the variation of insulation thickness with wire size can facilitate an analytical solution for the optimal number of strands. An equation describing magnet wire insulation thickness is provided by [24] AWG (3) is the minimum insulation build in mils ( in, where 1 mil m), for single-build insulation for heavy (double) build, and AWG is the and American Wire Gauge number.1 However, this only applies to wire sizes between 14 and about 30 AWG. For smaller wire sizes, it does not correlate with the tabulated data in [24] (Fig. 4). For wire in the range of 3060 AWG, we nd a better t to manufacturers tabulated nominal insulation build by using (4)
= 0:0050(92)(36 AW G)=39 :
1 The
American wire gauge denes nominal wire diameter in inches as
0 Fig. 5. Total resistance factor Fr as a function of number of strands (solid line) for the example discussed in the text at 375 kHz. Also shown are the ac resistance factor Fr (dashed) and the dc resistance factor Fdc (dotted). The minimum total resistance factor is at the point where increases in Fdc balance decreases in Fr with an increasing number of strands.
0 Fig. 6. Total resistance factor Fr as a function of number of strands for the example discussed in the text at 1 MHz. The solid line indicates resistance factor for a full bobbin. The dashed line shows the lower resistance that is possible by choosing the strand diameter for minimum loss, with the number of strands xed. Where this optimal diameter results in a full bobbin, the two curves are tangent. For larger numbers of strands, the optimal strand diameter, shown as a dotted line, would overll the bobbin and so is not possible.
Combining (2), (8), and (9) results in (10) where (11) Equation (10) can now be minimized with respect to the optimal number of strands to nd is large, only a close to one. Note that although the factor factor of 1.18 is due to the change in wire insulation thickness. The remaining factor of 1.95 is due to the dc resistance factors that do not vary with number of strands, such as serving area and strand packing. The optimization leads to choosing a large number of ne strands, which will often mean a high cost, and will sometimes require ner strands than are commercially available. From Fig. 5, one can see that a decrease from the optimum of 130 to about 50 strands entails only a small increase in ac resistance. Consideration of the cost tradeoff for a particular application becomes necessary. Given a suboptimal number of strands, chosen to reduce costs, a full bobbin may no longer be best. The problem of choosing the optimal strand diameter for a xed number of strands has been addressed by many authors [2][5], [7][9], [14]. Although this is typically only used for single strands, the analysis also can be applied for more than one strand by simply using the product of the number of turns and the number of in place of the number of turns The result strands holds, and [7][9], [14] that (13) In many practical cases, cost is a stronger function of the number of strands than of the diameter of the strands. In the range of about 4246 AWG, the additional manufacturing cost of smaller wire approximately offsets the reduced material cost. Thus, designs using the diameter given by (13) often approximate the minimum ac resistance for a given cost. Fig. 6 shows total resistance factor as a function of the number of strands for the same example design, but at 1 MHz, where the optimal stranding is a difcult and expensive 792
(12) This will give nonintegral numbers of strands; the nearest integral number of strands can be chosen to minimize ac resistance. V. DESIGN EXAMPLES AND SUBOPTIMAL STRANDING For a design example, we used a 14-turn winding on an RM5-size ferrite core. The breadth of the bobbin is 4.93 mm, and the breadth of the core window 6.3 mm. A height of 1.09 mm is allocated to this winding. Based on an experimental and litz packing factor hand-wound packing factor , unserved, plus a 32- m (1.25 mil) layer of serving, the above calculation indicates that, for a frequency of 375 kHz, 130 strands of number 48 wire gives minimum , ac resistance, with a total resistance factor of , and a dc resistance factor ac resistance factor Fig. 5 shows the total calculated resistance factor and its components as a function of number of strands. The gure and the numbers conrm the intuition that because is close to one and the dc resistance increases only very slowly, the decrease in resistance using ner strands outweighs the decreased crosssectional area until the ac resistance factor is brought very
diameter, unless this is too many strands to t in the available window area. The above analysis shows how to nd the optimal stranding, given a constraint on either strand diameter or number of strands, both of which are important in determining cost. More explicit analysis of cost is addressed in [25]. VI. EXPERIMENTAL RESULTS The designs specied in the preceding section were constructed with two types of litz wire: 130 strands of 48 AWG and 50 strands of 44 AWG. The primary and secondary windings were made from a single length of litz wire, wound on the bobbin in opposite directions. This is magnetically equivalent to having a shorted secondary, but it reduces potential problems with interconnect resistance. In order to evaluate skin effect in the absence of external proximity effect, litz wire was also measured outside of a winding. The resistance was measured with an HP 4284A LCR meter, using a custom-built test jig for low-impedance measurements. The measurements are shown in Fig. 8. Although the overall litz-wire diameter was small enough to limit bundle-level skin-effect losses to a few percent, the ne strands in the optimal solution also limit proximity-effect losses to similar levels, so it is necessary to separate the two effects in order to judge the accuracy of the proximityeffect calculations. Fortunately, the losses are orthogonal [20], and the skin effect losses (for a litz wire outside of the winding) can simply be subtracted from the measured losses in the transformer in order to isolate proximity-effect losses. Accurate prediction of the bundle-level skin effect was found to be difcult, in part because the details of the bundle constructions were not well known. However, if the experimentally measured bundle-level skin effect is subtracted from the total measured losses, the result matches the proximityeffect losses predicted by (2) very closely, as can be seen in Fig. 8. This conrms the validity of the model used in the optimization. VII. CONCLUSION The number of strands for a minimum-loss litz-wire winding may be found by evaluating the tradeoff between proximityeffect losses and dc resistance. Of the factors leading to increased dc resistance in a litz-wire winding, only the space allocated to strand insulation varies signicantly with the number of strands in a well designed construction. A power law can be used to model insulation thickness in the region of interest. Combining this with standard models for eddy-current loss results in an analytic solution for the optimal number of strands. The simplest model for loss, using only the rst terms of a series expansion, can be used because good designs use strands that are small compared to a skin depth. Experimental results correlate well with the simple model. Stranding for minimum loss may lead to many strands of ne wire and thus excessive expense. Minimum loss designs constrained by minimum strand size or maximum number of strands have also been derived.
0 Fig. 7. Contour lines of total resistance factor Fr as a function of number of strands and diameter of strands for the example discussed in the text at 1 MHz. The diagonal dashed line indicates a full bobbin. The valley at the upper right is the minimum loss. The minimum loss without overlling the bobbin is marked by an . Contour lines are logarithmically spaced.
strands of AWG 56 wire, and so analysis of alternatives is more important. The solid line is for a full bobbin, and the dashed line is for the same number of strands, but with the diameter chosen for minimum losses rather than to ll the bobbin. Where the two lines meet, the optimal diameter just lls the bobbin. Beyond that point it would not t, and the line is shown dotted. The example can be understood more completely by examas a function ining contour lines of total resistance factor of both strand diameter and number of strands (Fig. 7). The minimum resistance is in the valley at the upper right (a large number of ne strands). To t on the bobbin, designs must be below the dashed diagonal line. Minimum loss designs for a xed number of strands can be found by drawing a horizontal line for the desired number of strands, and nding the point tangent to contour lines. One could also consider a constraint for minimum wire diameter. Many manufacturers cannot provide litz wire using strands ner than 48 or 50 AWG. In Fig. 7, the minimum resistance for 50 AWG stranding is with a full bobbin, but for 40-AWG wire, the minimum ac resistance can be seen to occur with fewer than the maximum number of strands. This situation can be analyzed by considering (2) with all parameters xed except for the number of strands, such that (14) where is a constant obtained by equating (2) and (14). The total resistance factor is then (15) is the dc resistance factor with a full bobbin, for where the xed strand diameter. The value of that minimizes this , such that This will be expression is the optimal number of strands, given a xed minimum strand
eld is not zero at one edge of the winding, a factor is used to account for the resulting change [14]. in losses, where APPENDIX B EXPANSION OF DOWELL SOLUTION
Equation (2) is similar to the expression for the rst terms of a series expansion of the exact one-dimension solution (17) where is the number of layers and is the ratio of effective conductor thickness to skin depth. For a large number of layers (equivalent to the assumption, given above, of a trapezoidal The eld distribution), this reduces to is usual expression for
(18) and are the height and breadth where is the number of an equivalent rectangular conductor and of turns per layer. Based on equal cross-sectional area, This results in (19) is the height of the bobbin area allocated to this where winding. The number of layers is Substituting these expressions for and into the simplied version , we obtain of (17) and using (20) for and the same as (2), except for the substitution of This discrepancy, which was the addition of a factor of rst noted in [6], can be explained by comparing (16) to the equivalent expression for a rectangular conductor (21) where is the side of a square conductor. Equating these Thus, it appears that two, we obtain using an equivalent square conductor with sides equal to for proximity-effect loss calculations would be a more accurate approximation than the equal area approximation that is usually used [16]. APPENDIX C NONSINUSOIDAL CURRENT WAVEFORMS Nonsinusoidal current waveforms can be treated by Fourier analysis. The current waveform is decomposed into Fourier components, the loss for each component is calculated, and the loss components are summed to get the total loss (22)
(b) Fig. 8. Experimental ac resistance factor Fr as a function of frequency. (a) Graph is for litz wire with 50 strands of 44 AWG wire. (b) Graph is 130 strands of 48 AWG wire. Both are in the example transformer described in the text. Total measured resistance factor in the transformer is marked with stars. Measured skin effect in a straight piece of litz wire is marked with . The difference, equal to proximity-effect losses, is marked with circles. These correspond closely to the predicted proximity-effect losses (solid line).
APPENDIX A LOSS CALCULATION used here may The expression for ac resistance factor be derived by rst calculating loss in a conducting cylinder in a uniform eld, with the assumption that the eld remains constant inside the conductor, equivalent to the assumption that the diameter is small compared to a skin depth. This results in a wire of length in power dissipation (16) where is the peak ux density. This is equal to the rst term of an expansion of the exact Bessel-function solution [26]. Combining this with the assumption of a trapezoidal eld distribution results in (2). For congurations in which the
is the rms amplitude of the Fourier component at where From (2), it can be seen that frequency (23) Dening by leads to
A square wave with nite-slope edges leads to a nite value of , which can be found from (29) to be (30) where is the transition time as a fraction of the total period. , the waveform becomes triangular and (30) gives For as calculated above. This expression the same value of (30) and the calculation and minimization of loss based on (2) is valid as long as there is not signicant harmonic current for which the wire diameter is large compared to a skin depth. Based on the rule thumb that the highest important harmonic [27], a rough check on this number is given by would be to calculate skin depth for a maximum frequency and compare this to the wire diameter. If there are signicant harmonics for which the skin depth is small compared to wire diameter, then the analysis in [27] can facilitate 1-D analysis of nonsinusoidal waveforms, or for more accuracy Bessel-function analysis [17] with a Fourier decomposition of the waveform can be used. ACKNOWLEDGMENT The author thanks New England Electric Wire Corporation for litz wire samples, Magnetics Division of Spang and Co. for ferrite core samples, and A. M. Tuckey for a careful reading of an earlier version of this paper. REFERENCES
[1] C. R. Sullivan and M. J. Powers, A high-efciency maximum power point tracker for photovoltaic arrays in a solar-powered race vehicle, in 23rd Annu. IEEE Power Electronics Specialists Conf., 1993, pp. 574580. [2] S. Butterworth, Effective resistance of inductance coils at radio frequencyPart I, Wireless Eng., vol. 3, pp. 203210, Apr. 1926. [3] , Effective resistance of inductance coils at radio frequencyPart II, Wireless Eng., vol. 3, pp. 309316, May 1926. [4] S. Butterworth, Effective resistance of inductance coils at radio frequencyPart III, Wireless Eng., vol. 3, pp. 417424, July 1926. [5] , Effective resistance of inductance coils at radio frequencyPart IV, Wireless Eng., vol. 3, pp. 483492, Aug. 1926. [6] G. W. O. Howe, The high-frequency resistance of multiply-stranded insulated wire, in Proc. Royal Society of London, vol. XCII, Oct. 1917, pp. 468492. [7] J. Jongsma, Minimum loss transformer windings for ultrasonic frequencies, Part 1: Background and theory, Phillips Electronics Applications Bull., vol. E.A.B. 35, pp. 146163, 1978. [8] , Minimum loss transformer windings for ultrasonic frequencies, Part 2: Transformer winding design, Phillips Electronics Applications Bull., vol. E.A.B. 35, pp. 211226, 1978. [9] , High frequency ferrite power transformer and choke design, Part 3: Transformer winding design, Tech. Rep. 207, 1986. [10] P. S. Venkatraman, Winding eddy current losses in switch mode power transformers due to rectangular wave currents, in Proc. Powercon 11, 1984, pp. 111. [11] N. R. Coonrod, Transformer computer design aid for higher frequency switching power supplies, in IEEE Power Electronics Specialists Conf. Rec., 1984, pp. 257267. [12] J. P. Vandelac and P. Ziogas, A novel approach for minimizing high frequency transformer copper losses, in IEEE Power Electronics Specialists Conf. Rec., 1987, pp. 355367. [13] L. H. Dixon, Jr., Review of basic magnetics. theory, conceptual models, and design equations, in Unitrode Switching Regulated Power Supply Design Seminar Manual, pp. M41M411. Unitrode Corp., 1988. [14] E. C. Snelling, Soft Ferrites, Properties and Applications, 2nd ed. London, U.K.: Butterworth, 1988.
This can also be written as where
(24) -
One may calculate this effective frequency for a nonsinusoidal current waveform and use it for analysis of litz-wire losses, or for other eddy-current loss calculations. Note that this applies to waveforms with dc plus sinusoidal or nonsinusoidal ac components. The results will be accurate as long as the skin depth for the highest important frequency is not small compared to the strand diameter. A triangular current waveform with zero dc component , where is the results in an effective frequency of fundamental frequency. Once the effective frequency of a pure ac waveform has been calculated, the effective frequency with a dc component can be calculated by a reapplication of (26) (27)
Finding Fourier coefcients and then summing the innite series in (26) can be tedious. A shortcut, suggested but not eshed out in [9], can be derived by noting that rms so that rms (29) (28)
The primary limitation of effective-frequency analysis is that it does not work for waveforms with more substantial harmonic content. For instance, the series in (26) does not converge for a square wave. Similarly, the derivative of a square wave in (29) results in an innite rms value. A Bessel-function-based description of loss may be necessary. However, in practice leakage inductance prevents an inductive component from having perfectly square current waveforms.
[15] A. M. Urling, V. A. Niemela, G. R. Skutt, and T. G. Wilson, Characterizing high-frequency effects in transformer windingsA guide to several signicant articles, in APEC 89, Mar. 1989, pp. 373385. [16] P. L. Dowell, Effects of eddy currents in transformer windings, Proc. Inst. Elect. Eng., vol. 113, pp. 13871394, Aug. 1966. [17] J. A. Ferreira, Improved analytical modeling of conductive losses in magnetic components, IEEE Trans. Power Electron., vol. 9, pp. 127131, Jan. 1994. , Electromagnetic Modeling of Power Electronic Converters. [18] Norwell, MA: Kluwer, 1989. [19] A. W. Lot and F. C. Lee, A high frequency model for litz wire for switch-mode magnetics, in Conf. Rec. 1993 IEEE Industry Applications Conf. 28th IAS Annu. Meeting, vol. 2, Oct. 1993, pp. 116975. [20] J. A. Ferreira, Analytical computation of ac resistance of round and rectangular litz wire windings, Proc. Inst. Elect. Eng., vol. 139, pp. 2125, Jan. 1992. [21] B. Carsten, High frequency conductor losses in switchmode magnetics, in Tech. Papers of the 1st Int. High Frequency Power Conversion 1986 Conf., May 1986, pp. 155176. [22] M. Bartoli, N. Noferi, A. Reatti, and M. K. Kazimierczuk, Modeling litz-wire winding losses in high-frequency power inductors, in 27th Annu. IEEE Power Electronics Specialists Conf., vol. 2, June 1996, pp. 16901696. [23] J. Hu and C. R. Sullivan, Optimization of shapes for round-wire high-frequency gapped-inductor windings, in Proc. 1998 IEEE Industry Applications Society Annu. Meeting, 1998, pp. 907912. [24] National Electrical Manufacturers Association, Magnet Wire, MW1000-1997.
[25] C. R. Sullivan, Cost-constrained selection of strand wire and number in a litz-wire transformer winding, in Proc. 1998 IEEE Industry Applications Soc. Annu. Meeting, pp. 900906. [26] J. Lammeraner and M. Sta, Eddy Currents, G. A. Toombs, Ed. Boca Raton, FL: CRC, 1966. [27] J. G. Breslin and W. G. Hurley, Derivation of optimum winding thickness for duty cycle modulated current waveshapes, in 28th Annu. IEEE Power Electronics Specialists Conf., vol. 1, pp. 655661, 1997.
Charles R. Sullivan (M93) was born in Princeton, NJ, in 1964. He received the B.S. degree in electrical engineering with highest honors from Princeton University, Princeton, NJ, in 1987 and the Ph.D. degree in electrical engineering from the University of California, Berkeley, in 1996. From 1987 to 1990, he was with Lutron Electronics, Coopersburg, PA, developing highfrequency dimming ballasts for compact uorescent lamps. He is presently an Assistant Professor at the Thayer School of Engineering, Dartmouth College, Hanover, NH. He has published technical papers on topics including thin-lm magnetics for high-frequency power conversion, dcdc converter topologies, energy and environmental issues, and modeling, analysis, and control of electric machines. Dr. Sullivan received the AIME Ross N. Tucker Electronics Materials Award in 1995.
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