266 IEEE TRANSACTIONS ON PORER FLbCTROhIC5 V O L 3 NO 7 JULI I Y X X

A Novel Approach for Minimizing High-Frequency Transformer Copper Losses

JEAN-PIERRE VANDELAC AND PHOIVOS D. ZIOGAS, MEMBER, IEEE

Abstract-As power supply frequencies approach the megahertz level, skin and proximity effects become significant factors to consider when calculating copper losses in transformers. Using electromagnetic theory and MMF diagrams in both space and time, a method is pro- posed that provides insight into the mechanism of skin and proximity effect losses and that also yields quantitative results. Using this method, several winding geometries for various topologies are covered. The analysis and optimization process is experimentally verified using an interleaved flyback transformer.

INTRODUCTION NE CRITICAL requirement in designing trans- 0 formers is to guarantee that the power dissipated in

the transformer remains within an acceptable level. Power losses within transformers are generally classified into two types: core and copper losses. In this paper we concen- trate our attention on copper losses.

When a transformer is operated at a low frequency, that is, at a frequency at which the skin depth is much larger than the dimension of the conductors used for the wind- ing, losses can be calculated using the product i$,R, where i,, is the rms value of the current flowing in the conductor and R is the dc resistance of the winding. When operating at higher frequencies, other loss mechanisms, described as skin and proximity effects, come into play. Due to the fact that in transformers these two effects are difficult to separate, we will refer to the losses induced by these phenomena as eddy current losses. Eddy current losses in transformer windings have been previously treated in the literature. The basis for many of these pub- lications is the paper by Dowel1 [l]. The most compre- hensive subsequent publication on the subject is the work by Jongsma [2], [3].

These articles deal with transformers operating with sinusoidal current and voltages. In most switchmode ap- plications, however, waveforms have a broad spectrum and the analysis provided therein is not totally satisfac- tory. The concept of effective frequency has been devel-

Manuscript received June 5, 1987; revised January 14, 1988. This paper was presented at the 1987 IEEE Power Electronics Specialists Conference. Blacksburg, VA, June 16-21. 1987.

J . P. Vandelac is with Unisys Canada Inc.. 3150 Avenue Miller. Dor- Val. PQ, Canada H9P 1K5.

P. D. Ziogaa is with Concordia University, Department of Electrical Engineering. 1455 De Maisonneuve Blvd. West. Montreal, PQ. Canada H3G IM8.

IEEE Log Number 8821629.

oped to compensate that lack [3]. Although a valuable practical contribution, this concept rests on disputable foundations. The effect of the simplifications and assump- tions used to obtain the efecrivefrequency is difficult to evaluate and could lead to erroneous conclusions.

Another approach is to decompose the current into its Fourier components and sum up the losses at each fre- quency. This is done in [4] and [ 5 ] . Both of these papers concentrate on the analysis of the forward converter. However, in many commonly used topologies this ap- proach is not directly applicable as not all windings carry current simultaneously. This is the case in center-tap bridge transformers or in flyback transformers. Also, the fields generating the eddy currents within the conductors do not always have the exact same frequency content as the net current circulating in any individual winding. This occurs in discontinuous mode flyback converters. In ad- dition, other “windings,” for example, electrostatic shields, do not carry the load current, yet nevertheless exhibit eddy current losses induced by the load current.

In this paper a more general method to calculate these losses adequately in high-frequency magnetics is pre- sented. This method of analysis is akin to the previous approach since it degenerates to that presented in [4], IS] when applied to the forward converter transformer.

The approach taken in this article is twofold. First, the graphical part of the proposed analysis method is empha- sized. The field in the interlayer regions of the windings is found by considering the currents of all windings to- gether. This approach removes much of the conceptual difficulty and gives more insight into the phenomenon of eddy current losses in windings. Many important conclu- sions can be arrived at by just considering easily gener- ated sketches. One such conclusion of particular interest is, contrary to what is stated in previous publications, that interleaving can, in fact, be used to minimize copper losses even in flyback transformer design.

The mathematical treatment justifying the use of the field method and which is essential in arriving at any nu- merical result is presented in the Appendix. In this sec- tion, more general equations for the calculation of copper losses than those presented in [ 11 are derived. The relation between the fields in the transformer and copper losses is emphasized. Also, the tools necessary to derive optimi- zation diagrams similar to those given in [SI are provided in the Appendix.

0885-899318810700-0266$01 .OO @ 1988 IEEE

261 VANDELAC A N D 7 1 0 G A S MINIMIZING HIGH-FREQUENCY TRANSFORMER COPPER LOSSES

I . FUNDAMENTALS A. Solenoid Approximation

Transformer and inductor windings can often be ap- proximated by infinitely long solenoids. The presence of a core provides the retum path at infinity for the flux. Consider any layer of the solenoid for Fig. l(a) to be rep- resented by the infinitely long and wide sheet of thickness h of Fig. l(b). It is shown in the Appendix that the losses in any layer of the winding can be calculated if the tan- gential magnetic field on both sides of that layer is known. Equation ( 2 5 ) , given in the Appendix, is of the form

Q = Q ( H , ( h ) , CY, h / 6 ‘ ) ( 1 )

CY = H ; ( O ) / H , ( h ) ( 2 )

where h is the thickness of the layer, 6‘ is the skin depth of the conductor at the frequency considered, H, ( 0 ) is the tangential magnetic field at x = 0, and H:( h ) is the mag- netic field at x = h. Q [6] is the power dissipation per square meter in the z-y plane. In Fig. 2 , Q is normalized and plotted as a function of h / 6 ‘ for various values of CY

and a sinusoidal magnetic field of one A / m peak. The plot is normalized to the dissipation associated with a dc current flowing in the y direction in a layer one skin depth thick. The intensity of this normalizing dc current is 1 A for each meter of the layer in the z direction. Q at a par- ticular frequency U, is found from Fig. 2 and from the following equation:

where CY, is the ratio of the field at the frequency consid- ered.

For each value of CY an optimal thickness exists at which power dissipation is minimum. Since the optimal thick- ness and minimum power dissipation are a function of CY,

the layers will, therefore, be classified as CY types. Note that for layers of types CY = 0, 03, and CY = - 1, the min- imum is not as pronounced as for the other layer types and that the curves to the right of the minimum are flat. This fact will be used later.

B. M M F Diagrams MMF diagrams [ l ] are used throughout this paper to

find the H field at the surface of the conductors. The pro- cess to find the MMF diagram is illustrated in Fig. l(a). Outside the solenoid the magnetic field generated by the solenoid itself is zero. Using Ampere’s law, the magnetic field can be found in the regions between each layer. For example, the field between the second and third layer caused by the solenoid current is found to be Hz = 2 * I ’ . I ‘ is the layer net current in the y direction per meter of conductor in the z direction ( A / m ) . To that value we add the field rl’ caused by other current carrying structures. This field is assumed to be in the z direction and is de- noted by He,, in Fig. l (a) . The resulting MMF diagram is shown in Fig. l(c). Note that the sloped lines joining the

Fig. 1. Solenoid approximation. diagram. (a) Solenoid. (b) Solenoid layer. (c) MMF

IO

1.3

Fig. 2 . Power dissipation in layer

value of the field between the layers do not represent the actual field distribution within the conductors. The actual field distribution is a complex function expressed by (20) in the Appendix.

C. Nonsolenoid Windings Although the treatment of transformers with solenoid-

type windings is emphasized throughout this paper, the method presented here is also applicable to other winding geometries. The only requirement is that the field in the interlayer regions be known and be tangential to the con- ductor surface.

For example, ‘‘flat winding” transformers have layers in the p-8 plane. This is illustrated in Fig. 3(b). In this geometry, instantaneous addition and cancellation of the fields generated by the primary and secondary currents produce an MMF distribution similar to the distribution of the solenoid winding. This is illustrated in Fig. 3(a) where H, is plotted across the winding.

The field plot of Fig. 3(b) was produced using a finite- element eddy current solver [9], [lo]. The plot shows a small nontangential field component at the outer layers. This can be explained by the spacing of the primary wind- ing turns (small rectangular cross sections) and also by the fact that the primary turns were modeled as current generators with uniform current density. This simplifica-

268 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL 3. NO 3 . JULY I Y X 8

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'1 r e

(a) (b) Fig. 3. Flat-type winding transformer. (a) Tangential field distribution. (b)

Windings and flux plot.

tion, due to the present limitations of the program, allows the ac flux to permeate the turns of the primary winding.

The secondary windings (large rectangular cross sec- tion) were modeled as copper conductors. Loss calcula- tions using computer analysis were in agreement with the results obtained with the method presented herein. Two- dimensional finite-element analysis also allows study of the effect of fringing at the end of the conductors. This is not covered here.

11. FIELD HARMONIC ANALYSIS In switchmode applications the field in a transformer

may contain components at dc, at the fundamental fre- quency, and at integer multiples of the fundamental fre- quency. Sometimes harmonic components are introduced by the commutation of the current by switches. Harmon- ics can also be caused by the steering of the current from one winding to another as in push-pull transformers or flyback transformers. This distinction is somewhat arbi- trary and not very useful. In this paper, we remove this conceptual difficulty by considering the net resulting field only.

The field analysis method used here will be best ex- plained through an example. Consider the push-pull con- verter primary winding shown in Fig. 4(a), and assume that it is wound with bifilar foil. (A bifilar foil winding can be described as two strips of conductor wound to- gether resulting in the interleaving of the two halves of the winding.) In Fig. 4(b) the turns of P , and Pb therefore alternate.

In a push-pull converter the currents i, and ih alternate as illustrated in Fig. 4(c). We can identify four different time intervals: from to to t l when Po conducts, from t l to t2 when neither P, nor PI, conducts, from t2 to t 3 when P , conducts, and from t 3 to t4 when again neither winding conducts. The instantaneous MMF diagram for the fields can be obtained for each time interval. This is shown in Fig. 4(d). From Fig. 4(d) the field between each winding can be plotted in time as shown in Fig. 4(e). A Fourier analysis can then be performed on the resulting wave- form, and the component of the field on each surface of any given layer can be found at each frequency.

It can be seen from Fig. 4(e) that the fields in all inter- layer regions contain odd harmonics and that every sec- ond interlaver region also contains even harmonics. This

2: 2c i ci

H A+ -c; w - -x

(e)

Fig. 4. Field harmonic analysis. (a) Push-pull primary winding. (b) Bifilar foil winding. (c) Push-pull primary current. (d) MMF diagrams in space. (e) MMF diagrams in time. ( f ) Time harmonic MMF diagram for odd harmonics. (8) Time harmonic MMF diagram for even harmonics.

suggests that we can establish two time harmonic MMF diagrams, one for the even harmonics and one for the odd harmonics. These two diagrams are shown in Fig. 4(f) and (g), respectively. The MMF diagram for the odd har- monics is seen to be similar to the MMF diagram of the solenoid of Fig. 1 with r = 0. For the even harmonics each layer is of the type a = 0 or a = 00, which is the same type as seen from Fig. 2. The cosine series coeffi- cients c, of Fig. 4(f) and (g) are given by

-D CO = ~

2 ( 5 )

where D is the duty cycle and is given by

Fig. 4(f) and (g) could have been obtained more di- rectly. Performing a Fourier analysis of the current wave- forms and then usinn Fin. 4(b). we can obtain the MMF " U U \ ,

VANDELAC AND ZIOGAS. M I N I M I / I N G HIGH-FREQUENCY TRANSFORMER COPPER LOSSES 269

diagrams in Fig. 4(f) and (8). The merit of the plots of the field in both space and time (Fig. 4(d) and (e)) is that they provide insight into the mechanism of the trans- former copper losses. These intermediate steps also con- stitute useful tools for evaluating a given winding geom- etry as will be seen in the section on flyback transformers.

Once the MMF diagrams are obtained, the losses can be calculated using the formulas given in the Appendix. The specific calculations for the example of this section are also given in Section C in the Appendix.

111. EXAMPLES

Rather than presenting the method of field harmonic analysis in a formal and abstract framework, it was found preferable to approach these concepts through various ex- amples. The goal is to familiarize the reader with the method such that he can apply it to his own particular transformer design problem. In this section we cover some of the winding and topology types that were mentioned earlier. To facilitate the understanding of the method, the treatment is mostly qualitative. Actual numerical calcu- lations have to be performed using the material given in the Appendix.

A . Eiectrostatic Shield Copper Losses One case of interest that can easily be analyzed using

the method described is the dissipation caused by eddy currents within an electrostatic shield. The electrostatic shield is used between windings in a transformer to min- imize conducted EM1 noise [7]. A shield can be repre- sented as a single-layer solenoid conducting no current and immersed in an external field. For this layer a = 1. The power dissipation in the shield can be found as a function of the shield thickness by using (32) of the Ap- pendix.

The power dissipation has been calculated for two cases: a shield immersed in a sinusoidal magnetic field and a shield immersed in a 0.5 duty cycle full-wave mag- netic field. (The full-wave at 0.5 duty cycle is illustrated by H2 ( t ) and H4 ( t ) of Fig. 4(e).) The peak magnetic field in both cases was taken to be of amplitude H-. The power dissipation has been normalized according to ( 7 ) , where 6 is taken to be the skin depth at the fundamental fre- quency :

( 7 )

The thus normalized power dissipation is plotted in Fig. 5. If the current (“EM1 current”) shunted back by the shields is known and not negligible, it can be included in the analysis as is done in the other sections of this paper for current-carrying conductors. Note that for the calcu- lated losses to reflect the actual losses in the shield, the amount of overlap at the ends of the shield should be min- imized to reduce capacitive currents within the shield it- self [3].

Fig. 5 . Normalized power dissipation in electrostatic shield

B. Bridge Trunsformer Copper Losses

Transformers are often composed of concentric single or multilayer solenoids carrying opposite currents. In most cases there is at least a zero of MMF in the center and one outside the assembly. Other zeros inside the assembly can also exist depending on the primary and secondary winding interleaving. A typical example would be the transformer of a forward converter where all windings carry current at the same time. Zeros of MMF are easily obtained by plotting the instantaneous field. In bridge to- pologies with center-tap secondary or in push-pull topol- ogies, not all windings carry current at the same time, and zeros of MMF are not easily recognized. It has been seen, in the section on field harmonic analysis, that the concep- tual difficulty caused by windings not conducting at the same time can be eliminated through the use of an MMF diagram at each frequency. We will use this technique in the following analysis to evaluate various winding config- urations for bridge transformers with a center-tap second- ary.

1 ) Bridge Transformer Field Analysis: The bridge transformer with center-tap secondary is shown in Fig. 6(a), and its associated typical current waveforms are shown in Fig. 6(b). If we remove the dc component from the secondary current waveforms, we see that each half of the secondary winding carries an ac current which is equal in phase and magnitude. Therefore, as far as eddy current losses are concerned, both halves of the secondary winding are equivalent and carry the same current. This enables us to obtain MMF diagrams readily for various types of winding interleaving. Note that since all harmon- ics of the current in the different windings are in phase, the MMF diagram will be of the same shape at all fre- quencies. Furthermore, these waveforms contain no even harmonics.

2) Effect of Winding Interleaving on Copper Losses: To evaluate the effect of winding interleaving on losses, three different transformer winding constructions are compared. Their respective merits are qualitatively eval- uated. In Fig. 7(a) the foil windings S , and S, are wound separately. The resulting ac MMF diagram is shown in

270 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3. NO. 3. JULY 19RX

(h) Fig. 6. Center-tap secondary bridge transformer. (a) Transformer sche-

matic. (b) Primary and secondary currents.

I

(b) Fig. 7 . Bridge transformer with noninterleaved primary.(a) Winding. (b)

AC time harmonic MMF diagram.

Fig. 7(b). In Fig. 8 the secondary is wound with bifilar foil, so its layers alternately carry i, and i,. In Fig. 9 the two halves of the secondary winding are wound on op- posite sides of the primary. The corresponding MMF dia- gram is given in Fig. 9(b). In all three MMF diagrams c, is given by

where D is given by (6). DC field distribution is omitted from the MMF diagrams as it can be shown that dc losses are independent of interleaving and depend only on the conductor surface area.

Comparing the first two constructions, we see that the ac MMF diagrams are identical. Both constructions are, therefore, deemed equivalent with respect to copper losses. Note that in both cases the layer thicknesses of S, and S, can be made equal or can be optimized separately.

At this point we can make an interesting observation. Consider that a bifilar foil winding is to be constructed with two conductor strips of the same thickness. The op- timal thickness of the strips of that N turn bifilar winding carrying i, and ib will be the same as the optimal thickness of a 2N-turn foil winding carrying i,. This can be easily verified by replacing ih by i, in either Fig. 7 or Fig. 8. This leaves a four-layer secondary winding carrying i,.

/ a ib

x

X

(h ) Fig. 8. Bridge transformer with bifilar foil secondary. (a) Winding. (b) AC

time harmonic MMF diagram.

11 . .

(b) Fig. 9. Bridge transformer with interleaved primary. (a) Winding. (h) AC

time harmonic MMF diagram.

For this new winding we obtain the same ac MMF dia- gram as in Figs. 7 and 8. This implies that the same op- timization procedure can be used for both cases by letting p = 2N, for the bifilar design.

Considering Fig. 9(b), we see that the construction of Fig. 9(a) yields half of the field intensity in about half of the interlayer areas. Since losses are proportional to H’, ac losses will be lower for that design. (DC losses can also be made to be lower as thicker conductors can be used.) If close coupling is desired between the secondar- ies, the design of Fig. 9 is the worst of all three designs. A trade-off has to be made between minimizing losses and improving secondary coupling. The design of Fig. 9 will also yield the best primary to secondary coupling (low leakage inductance).

3) Part of Windings with a Half-Layer: One key ob- servation has to be made about Fig. 9(b). The symmetry of the field on each side of the primary winding reveals that this winding is formed of two portions with equiva- lent MMF diagrams with respect to its center. The layer at the center is considered to be formed of two half-layers. For the case illustrated, each portion of the primary wind- ing consists of 1.5 layers. Therefore, the ac field at the center of the primary winding is zero, even though there never is an actual instantaneous field cancellation there.

The interested reader can easily verify this conclusion by performing all the steps described in the section on field harmonic analysis (Section 11). The resulting plots will show equal ac fields of opposing phase riding on a common dc offset for the two surfaces of the center layer.

VANDELAC A N D ZIOGAS: MINIMIZING HIGH-FREQUENCY TRANSFORMER COPPER LOSSES 27 1

4) Benejts of One-Layer Winding Design: The power- loss reduction of the design of Fig. 9 compared to the designs of Figs. 7 and 8 would have to be calculated on a case-by-case basis by using the formulas of the Appen- dix. Nevertheless, a case of practical importance exists for which the improvement can be readily evaluated: in high-frequency designs each winding section will be often composed of only one layer ( a = 0 or a = 1 ). Assuming that the thickness of the layer is chosen such that h / 6 ’ at the fundamental frequency is equal to the optimum thick- ness shown in Fig. 2, it is seen that for harmonics of higher frequency A/&*’ will lie to the right of the minima in the region where the curves are flat. Any small devia- tion from the optimum thickness will not result in any reduction of losses at the high harmonic frequencies, while this will increase the losses at the fundamental frequency. Therefore, for the one-layer case the optimal thickness for minimum ac losses is the same for the current waveforms of Fig. 6(b) as for sinusoidal current.’ This is explained by the fact that losses at the higher harmonic frequencies, being read from the flat part of the curve which is basi- cally at the same level as the minimum, are automatically minimized.

As the current flowing in the primary of the winding has no dc component, the foregoing reasoning can be used directly to estimate the power saving in the primary wind- ing attributed to interleaving as in Fig. 9. For the one- layer design corresponding to Figs. 7(a) and 8(a), the pri- mary layer will be of type a = 0, and for the one-layer design corresponding to Fig. 9(a) the primary layer will be of type a = - 1. The power dissipation in the primary winding is, therefore, reduced by a factor of two in the design of Fig. 9. Power savings in the secondary are not as readily found as the dc component as to be included. Detailed calculations based on the formulas of the Appen- dix show that a power loss reduction factor of two can also be achieved in the case where each half of the sec- ondary winding is composed of one layer.

C. ‘ ‘Nonoptimal ” Windings Dowell [ 11 has classified windings into two categories:

optimal and nonoptimal windings. Optimal windings have their zeros of MMF in the regions between the layers or exactly in the middle of a layer. Windings not satisfying that requirement were called nonoptimal. Dowell and all authors of subsequent articles have restricted their study to “optimal” windings.

In practical situations, there are rimes where because of physical constraints “nonoptimal” winding transformers become a design alternative. It would be desirable to be able to evaluate these designs.

Consider a bridge transformer built with the winding interleaving of Fig. 10(a). In this design all primary lay- ers carry the same current. The secondaries carrying i, are connected in parallel as are the secondaries carrying ib.

‘Mathematically, this is strictly true only for the cases where the fun- damental is dominant, that IS. for D ? 0.5.

(b) Fig. 10. “Nonoptimum” center-tap secondary bridge transformer. (a)

Winding. (b) Time harmonic MMF diagram.

The resulting ac time harmonic MMF diagram is shown in Fig. 10(b). Taking the ratio of the magnetic field on both sides of each layer, we see that there are two layers of type cy = 0, two of type CY = - 1 / 5 , two of type CY = -1/4, two of type CY = - 1/2 , two of type (Y = -2 /3 , and one of type CY = - 1. If the equivalent curves were shown in Fig. 2 , they would lie in between the curves with a = 0 and CY = - 1. Therefore, for a quick estimate of the ac losses for such a design, an average value for the layer type could be selected. Alternately, losses in each layer could be calculated with the help of the for- mulas given in the Appendix.

D. Flyback Transformer Copper Losses

In this section we will use the method of field harmonic analysis to study the case of the copper losses in flyback converters. The field distribution used is based on the as- sumption that the windings can be approximated by so- lenoids. This results in a field distribution which is ex- actly opposite to what is given in [ 3 ] . Given the solenoid approximation, we conclude that interleaving can be used to minimize copper losses in flyback converters. This conclusion is then verified experimentally.

1) Analysis of Continuous-Mode Flybacks: The basic flyback transformer is represented in Fig. 1 l(a). Neglect- ing the ripple, the current in each winding of a continu- ous-mode flyback is shown in Fig. 1 l (b) . We can identify two periods, from io to t l when the primary conducts and from t , to t2 when the secondary conducts. Assuming that the primary and secondary are wound as in Fig. 1 l(d), we can plot the MMF distribution for both periods as shown in Fig. 1 l(e). (Refer to Section 111-D3 for the meaning of the dashed arrows.) Using these MMF diagrams, we can plot the time variation of the magnetic field at the bound- ary of each layer. This is done in Fig. 1 l ( f ) . Subtracting the dc component from the various signals leaves the new axis of Fig. 1 I(f). We can now find the ratio of the ac fields: al = & / H I = 0, a2 = 2, CY^ = 1/2 , and a4 = 00. Therefore, it is seen from Fig. 2 that two of the layers will be of the type CY = 0. Their thicknesses can thus be

272 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3, NO. 3, JULY 1988

Fig. 11. Flyback transformer analysis. (a)-(c) Flyback currents. (a) Flyback transformer. (b) Continuous-mode flyback cur- rents. (c) Critical conduction flyback currents. (d)-(g) Noninterleaved flyback. (d) Noninterleaved flyback winding. (e) Non- interleaved MMF diagrams in space. ( f ) Noninterleaved continuous conduction MMF diagrams in time. (9) Noninterleaved critical conduction MMF diagrams in time. (h)-(k) Interleaved flyback. (h) Interleaved flyback winding. (i) Interleaved M M F diagrams in space. ( j ) Interleaved continuous conduction MMF diagrams in time. (k) Interleaved critical conduction M M F diagrams in time.

selected to yield lower losses than can be achieved in the two remaining layers of type CY = 2.

Interleaving the windings as in Fig. l l (h ) , we again plot the MMF distribution in space and time, as shown in Fig. 1 l(i) and (j) . Taking the ratio of the ac fields we now obtain: CY^ = a2 = c y 3 = a4 = 0. All layers are therefore of the type CY = 0. Referring to Fig. 2, it is seen that the thickness of two more layers can be optimized to yield lower losses than in the previous example.

2) Experimental Verijication of Copper Loss Reduction with Interleaved Flyback Transformers: To verify that in- terleaving can be used to minimize flyback copper losses, an experimental flyback converter has been assembled. Two transformers were designed such that core losses would be minimal compared to copper losses. Both pri- maries were wound with 66 turns of AWG22 wires and the secondaries with two turns of foil 0.020 in thick in a CC-50 core. The converter operated at 300 kHz with a duty cycle of 0.5. Based on the formulas given in the Ap- pendix, detailed calculations, including dc, the funda- mental, and 25 harmonics, showed that the noninter- leaved design should present 2.7 times more losses than

the interleaved design. The measured winding tempera- ture rise for the noninterleaved winding was 2.3 times the temperature rise of the interleaved design. Owing to the nonlinearity of the thermal impedance in natural convec- tion and the presence of some constant core losses, the agreement was estimated to be more than satisfactory.

Note that neither of the designs was really optimized to yield the lowest possible losses. This is especially true of the noninterleaved winding, which could have been made to have lower losses by using thinner conductors. The ex- periment was designed with two goals in mind. The first goal was to show that attention should be given to eddy currents losses in the design of switchmode magnetics. The second goal of the experiment was to demonstrate that interleaving can be used to minimize copper losses in the continuous-mode flyback converter.

3) Analysis of Continuous-Mode Flybacks with Large Ripple and Discontinuous-Mode Flybacks: The method of field harmonic analysis can also be applied to contin- uous-mode flybacks with large ripple and to discontin- uous-mode flybacks. As a representative example, the current of a flyback converter in critical conduction is

VANDELAC A N D ZIOGAS M I N I M l Z l K G HIGH-FREQUENCY TRANSFORMER COPPER LOSSES 273

shown in Fig. 1 I(c). The MMF distributions for the non- interleaved and the interleaved design are shown in Fig. 1 l(e) and (i), respectively. The dashed arrows represent the linear variation of the field during each interval. The H fields are then sketched in time in Fig. l l ( g ) and (k). The impact of interleaving can now qualitatively be eval- uated. Although the amplitude and harmonic content of H2 has been dramatically reduced, H,, H I , H,, and H4 remain unchanged. The reduction in losses will not be as significant as with the continuous-mode flyback. It may, in fact, be hardly be measurable in optimized designs. The reduction in losses may not justify in itself the added complexity of the interleaved design. The actual loss re- duction could be calculated by first performing a Fourier analysis on the waveforms and then using the formulas of the Appendix.

4) Discussion: It is interesting to point out that al- though the MMF diagrams for the flyback converter given here differ diametrically from those given in [ 3 ] , Fig. I l (g ) would tend to support the design rule given there. It is recommended that the outer winding be wound with a thinner conductor (one step in the wire table). It is seen that although H I and H2 are of smaller average amplitude than H3 and H4, their harmonic content is much higher. High harmonics dictate the use of thinner conductors in multilayer designs. In discontinuous-mode flyback con- verter transformers the optimum thickness of the outer winding may very well be thinner than if this same wind- ing was wound as the inner winding. This conclusion is speculative on our part and has not been verified.

CONCLUSION

Attempts to calculate eddy current losses in transformer windings using previously published material on the sub- ject can lead to some conceptual problems. In some in- stances the calculations cannot even be performed. In this paper a graphical and numerical method of calculating and minimizing losses in windings, that generalizes previous findings, has been introduced. Using MMF diagrams in both space and time, several common geometries where these conceptual problems are encountered were covered. In particular, it has been shown that winding interleaving can, in fact, be used to reduce copper losses even in fly- back transformers.

The method is based on Maxwell’s equations. Using these equations, it can be shown that losses within a con- ductor can be evaluated from the field at the boundary of that conductor. A complete derivation of the formulas re- lating the losses to the boundary field is given in the Ap- pendix. These formulas can be used to calculate the losses in a transformer or inductor windings. The mathematical tools necessary to generate optimization charts are also given in the Appendix. A list of symbols is given in the Nomenclature.

The strength of this method is that it provides insight into the process of minimizing copper losses in high-fre- quency magnetics. Moreover, once understood, it is ap-

plicable to many other topologies not covered in this ar- ticle.

b b,. B c, D D E g, Gr H , H h i I I ’

J

k m n N

i

.i

NI 1, P Pd e r Rb S t U,, U p , U0

P

6, 61

CY

6’

rl PO P U

w

NOMENCLATURE

Breadth of a conductor. Winding width. Magnetic flux density. Fourier series coefficient. Duty cycle. Electric displacement vector. Electric field intensity. rms factor for the ith harmonic. Loss normalization factor ( 5 1) . Magnetic field intensity. Thickness of a conductor. Current. Peak current. Peak current density (A/m). Harmonic number. peak current density (A/m2).

Wave number. Integer value of p . Layer number. Number of layers in the part of winding un-

der study. Number of turns in a layer. Length of a turn or mean length turn. Number of layers in a winding. Power dissipation. Power dissipation per square meter. Relative strength of the external field. Normalizing resistance (38). s = p - m . Time. Base vectors in cylindrical coordinates. Real part of the field ratio. Imaginary part of the field ratio (equal to

zero in all examples). Skin depth at the fundamental frequency. Equivalent skin depth for spaced conduc-

Conductor spacing factor. Permeability of free space. Radius. Conductivity. Angular frequency.

( - I ) ” *

tors.

APPENDIX MATHEMATICAL TREATMENT

A . Electromagnetic Analysis In this section, we present a derivation of the formulas

to be used in the calculations of losses in a winding. This derivation is fundamental to the method of MMF dia- grams in space and time used throughout this paper since it relates the losses in a transformer conductor to its tan- gential magnetic field.

214 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3 , NO 3. JULY 19x8

It is shown in [8] that the magnetic field inside a long one-layer solenoid is

NI I b,,

(9) H = - * U, + 0 * U p + 0 * UO.

The field outside the solenoid is

For thin layers, 2wp = I, at the external boundary. In the following derivation it is assumed that

2 p] >> 1.

The magnetic field generated by a layer of the solenoid of Fig. 1 is, therefore, neglected outside the region en- closed by the layer as it is small and falls off as 1 / p . The magnetic field inside the region enclosed by the layer and generated by the layer itself has only a component in the z direction. The magnitude of that component is given by (9). The magnetic field in any interlayer region of the multilayer solenoid is found by summing the contribution of the concentric layers as in Fig. l (c) .

Three types of conductor shapes are covered: the foil conductor which extends the whole width of the bobbin, the rectangular, and the round conductors. Round con- ductors can, for simplicity, be approximated by square conductors of the same cross section as shown in Fig. 12(a) and (b). As long as the spacing between the con- ductors is not excessive, we can assume, as in [ I ] , that the H field in the interlayer region of the winding is uni- form when moving in the z direction. With only a z com- ponent to the H field, this assumption is equivalent to re- placing the conductors of Fig. 12(b) with a single foil

(a) (b) (C) (d)

Fig. 12. Equivalent foil layer. (a) Round wire layer. (b) Equivalent square wire layer. (c) Equivalent foil layer. (d) Rectangular conductor layer.

drical coordinates. From [6] it is seen that this is the case when

[ yz % 1 (17)

where p, is the inner radius of the layer considered and po is its outer radius ( po = p , + h ) . Each layer is therefore considered to be represented by the infinitely high and long conductor of Fig. l(b). Since the H field is entirely in the z direction, the Helmholtz equation degenerates to a one-dimensional problem. From (16) we get

a2J,,/ax2 = k : J,, (18)

k : = jw,poqa. (19)

where

From (13) and (15) we get

aJ,., / ax H ; * ( x ) = -~

kf The solution to ( 1 8) is of the form

conductor of conductivity vu, carrying the same ampere turns. y is a linear spacing factor defined by The boundary conditions to that problem require that

the tangential component of the magnetic field be contin- uous across the conductor surfaces. The expressions for J , and J2 are found from H 2 ( 0 ) , H , (h ) , and (20). Letting (12)

The same equivalence can be obtained for the layer of H z , ( 0 ) = (a , + j P 4 ) . f & , ( h ) , (22) rectangular conductors of Fig. 12(d).

To find the power dissipation in a winding layer we first find the current density distribution in the layer. Relevant Maxwell's equations are

we get

-k""(h) (cosh ( k , x ) - ( a , + j f i , ) J b l ( x ) = sinh ( k , x )

v x E, = -aB,/at ( 1 3 ) cosh ( k , ( x - h ) ) ) . (23)

V x H, = J , + aD,/dt.

Letting

J , = VUE,,

To obtain the power dissipated per square meter in the y-z plane, we integrate ( 14)

ignoring displacement currents, and considering a sinu- H m Q , ( h ) = 7 [ ( I + a? + Of) FI(P,)

g 4 w 6 soidal steady state, we get the following Helmoltz equa- tion:

As long as the curvature of a layer is small, the problem can be solved in Cartesian coordinates rather than cylin-

V'J, = jw .povJ , . (16)

275 VANDELAC AND ZIOGAS: MINIMIZING HIGH-FREQUENCY TRANSFORMER COPPER LOSSES

( 2 9 ) sinh ( 2 p , ) + sin ( 2 p , )

F 1 ( p ' ) = cosh ( 2 p , ) - cos ( 2 p , )

sinh (p,) - cos (p,) + cosh (p,) . sin (p,)

cosh ( 2 p , ) - COS (2p,) F2(P,) =

6 is the skin depth of the actual conductor material and 6' is the skin depth of the equivalent material of Fig. 12(c). Note that although the derivation for Q; is not valid at dc (i = 0) , it can easily be verified that the correct numerical result will always be obtained with go = 1. (The dc field can be considered as a very slow sinusoidal wave which is at its peak for the time considered. The rms factor go is therefore equal to 1 .)

B. Transformer and Inductor Windings

To get the power dissipation in a winding layer, we have to add the power dissipation per square meter at each harmonic and multiply by the area (I, * b,) of the layer. In most practical transformers magnetic fields are in phase or 180" out of phase. The imaginary part of the field ratio, 0, is therefore zero. The power dissipation in any one layer becomes

In most cases the losses in a winding have to be related to the current it carries. Consider a winding to be formed of m layers. Let the magnetic field H,,(h) and the mag- netic field ratio be related to the current flowing in the winding by the following relations:

NI I , b,

HZc(h) = MMF (r , , n, i ) -, n = 1, m (33)

where

I, = c, * I (34)

( 3 5 ) MMF ( r* , n - 1, i) a ( r , , n, ;) =

MMF (r , , n , i ) '

I is the peak value of the current and c; is the Fourier coefficient at the frequency considered. The power dissi-

each layer. Thus

. Q,'(v,, a ( r , , n , 4)) . ( 3 6 ) 1 Assuming that the length of a turn does not vary sig-

nificantly from layer to layer, we let 1, be the mean length turn, and we obtain

6* Pd = R612 - cf p. MMF2( r,, n , i )

bh m n g,

where

R6 is the dc resistance of a winding that would be wound with a conductor of area equal to one skin depth square.

Some MMF distributions reccur in many practical ap- plications. The MMF distribution of Fig. 1 is the most common, and it is therefore convenient to obtain the expression for the sum of the powers in all layers of a winding. Let the solenoid in Fig. 1 be composed of m layers, each carrying I f , that is, Z A for each meter of the layer in the z direction. Assume that the winding is im- mersed in an external H field of intensity rl ' . Letting n be the layer number, counting from the position of minimum MMF, we get the MMF diagram of Fig. 1 with

MMF (r, , n , i) = n + r, (39)

a ( r , , n , i ) = n - l + r , n + r,

The power dissipation will therefore be

h2 Pd = R612 ~ c cf fi bh m n ' g,

[ ( 2 n 2 + 4r,n - 2n + 2 r f - 2r, + 1) F , ( p , )

- 4(n2 + 2r,n - n + rf - r,) F2(p,)]. ( 4 1 )

The interested reader can verify that the term in the brackets is identical to Perry's expression [6] when r, = 0. Summing for the m layers we get

2m2 + 6r,m + 6 r f + 1 3

h 2 bh g,

Pd = R612 - c c: 'p.

m 2 + 3r,m + 3 r f - 1 3 F,(cp,) - 4

( 4 2 )

One important case of nonzero external field is the half- layer geometry described in Section 111-B3. In this case,

pated in a winding is the sum of the power dissipated in the external field is of intensity rI' = 0.51' and is gener-

276 IEEE TRANSACTIONS ON POWER ELECTRONICS. VOL. 3. NO. 3. JULY 1988

ated by a half-layer of thickness h / 2 , where 1‘ /2 A / m are flowing. The losses in the half-layer are equivalent to half the losses in a layer of thickness h , with Hzh = 0.51’ and CY = - 1. Adding these losses to the losses of the winding, letting p be the total number of layers, m be the integer value of p , and s be the difference between p and m ( s = p - m ) , we obtain the total losses in the new solenoid:

* (2m2 + 6sm + 6s2 + 1)/3

-s3 + in * ( m 2 + 3sm + 3s2 - 1)/3

FI (Pa) . r2 + P

- 4 .

(43 1 This equation is valid for s = 0 or s = 0.5. Equation (43) can, through some manipulations, be found to equal that given in [ l] , [ 2 ] . Note that for sections of winding with half layers, N in the expression for R, is the number of turns in the portion considered. For example, in a 27-turn winding composed of two sections of 1.5 layers, N = 13.5 for each half-winding section.

C. Calculation Example

on field harmonic analysis. For that bifilar foil winding, Consider the push-pull primary winding of the section

N = 2 * N p (44)

N , = r = l (45)

p = m (46)

s = p - m = 0. (47)

In Fig. 4(f) the time harmonic diagram for the odd har- monics is similar to the MMF diagram of Fig. 1. Equation (43) is, therefore, directly usable, and the power loss as- sociated with the odd harmonics is found to be

(48) 4 = 1 , 3 . . . For the even harmonics including dc, each layer has a

zero of MMF on one of its two surfaces. Each of the p layers is therefore of type CY = 0. The total dissipation associated with the even harmonics is p times the dissi- pation in one layer. Therefore, from (37) and (25) we get

i = 0 , 2 , 4 * * - (49)

where the coefficients c, are given by (4) and ( 5 ) in the main part of the text. The total dissipation in the bifilar winding is

D. G, Factor To optimize the design of windings, it is very conve-

nient to normalize the losses in a winding to the rms cur- rent flowing into that winding. To do so, we define the G, factor:

where

i b s = I 2 c:, i = 0, 1, 2 . . . . ( 5 2 )

For foil windings, b = b,v and the K , [ 5 ] factor may be thought to be more convenient:

bn, K, = - * G,. 6 (53)

For rectangular and round conductor windings the G, factor is preferable as b is variable. For rectangular con- ductor windings, b is a design choice. For round conduc- tors b is equal to h , and h cannot be optimized indepen- dently of b .

Note that for bifilar windings i,, is the rms value of the current flowing in one-half of the winding ( i , or i h ) . Since Pd is the total dissipation in the bifilar winding, the dis- sipation in one-half of the winding is simply obtained from

. 2

(54) ‘ unns

P h a l f = G, * R6 * ~

2 .

REFERENCES

[ l ] P. L . Dowell, “Effects of eddy currents in transformer windings,” Proc. Insr. Elec. E n g . , vol. 113, no. 8 , Aug. 1966.

[2] J. Jongsma, “Minimum-loss transformer windings for ultrasonic frc- quencies,” Philips Electron. Appl. Bull., vol. 35, pp. 146-163 and 211-226, 1978.

[3] -, “Transformer and winding design,” part 3 of High-Frequency Ferrite Power Transformer and Choke Design, Philips Tech. Pub. 207, Philips, The Netherlands, 1986.

[4] P. S . Venkatraman, “Winding eddy current losses in switch mode power transformers due to rectangular wave currents,” in Proc-. P o w ercon I I , 1984, Power Concepts Inc. , Ventura, CA.

[5] B. Carsten, “High frequency conductor losses in switchmode mag- netics,” in High Frequency Power Conversion ConJ Proc., May 1986, lntertec Communications Inc. , Ventura, CA.

[6] M. P. Perry, “Multiple layer series connected winding design for minimum losses.” [€€E Trans. Po%%*er App . Sysf., vol. PAS-98. no. 1, Jan.-Feb. 1979.

[7] L. E. Jansson, “Radio frequency interference suppression in switched- mode power supplies,” Mullard Tech. Commun., vol. 12, no. 120. 1973.

San Francisco, CA: Freeman, 1970, pp. 315, 316.

[8] P. Lorrain and D. Corson, Elecrromagneric Fields and W U ~ ~ U S .

[9] MagNer User’s Manual, Infolytica Corp., Montreal, PQ, Canada. [ 101 D. A. Lowther and P. Silvester, Computer-Aided Design in Magner-

ics. New York: Springer-Verlag. 1986.

VANDELAC AND ZIOGAS. MINIMIZING HIGH-FREQUENCY TRANSFORMER COPPER LOSSES 277

Jean-Pierre Vandelac received the B.Eng. de- gree from McGill University, Toronto, ON, in 1982. He is a part-time student in the master’s program at Concordia University, Montreal, PQ.

He has been with Unisys Dorval Power Supply Operations since 1983, where he now holds a po- sition in Research and Development.

Phoivos D. Ziogas (S’75-M’78) received the B.S., M.S., and Ph.D. degrees from the Univer- sity of Toronto, Toronto, ON, in 1973, 1974, and 1978, respectively.

Since 1978 he has been with the Department of Electrical Engineering of Concordia University in Montreal, PQ, where he is engaged in teaching and research in the area of static power convert- ers. He has also participated as consultant in sev- eral industrial projects.