Designers Series XIII

1©Copyright 2005 Switching Power Magazine

Most designers are aware of skin effect, and use rudi-mentary calculations to estimate if their designs haveskin effect issues. Skin effect is the tendency of thecurrent in a wire to flow towards the surface of thewire more and more as the frequency of excitation isincreased. Figure 1 shows an isolated wire in freespace carrying an ac current.

The current density is greatest at the surface, and itdecays exponentially inside the wire, reaching a valueof 1/e times the surface current density at the skindepth. This happens because nature tries to minimizethe energy storage due to the high-frequency current. Alarge diameter wire has smaller self-inductance than asmall-diameter wire, and the flow of current in the sur-

We have had many requests over the last few years tocover magnetics design in our magazine. It is a topicthat we focus on for two full days in our design work-shops, and it has always been my intent to start writing acomprehensive design series for high frequency magnetics.

The problem I've always faced is where to start? Thereare hundreds of design rules, equations, and tips thatmake good magnetics work properly. If we started at thebeginning, with the fundamental design rules, it would takemany issues to get to some of the "good stuff".

So I've decided to work backwards– starting with themost complex topic in magnetics design that everyengineer faces with his or her system. This will servetwo purposes– to help you understand the magnitude ofthe problems faced in seemingly simple designs, and topoint out to those who think that magnetics is easy howinvolved and analytical it must be to achieve goodworking designs.

Proximity Loss: What is it?Any time a conductor is inside a varying magnetic field,currents will be induced in the conductor and theseinduced circulating currents are referred to as eddy cur-rents. This is an unavoidable fact of electromagnetictheory. The eddy currents make no net contribution tothe overall current flow through a wire, but merelyserve to redistribute current in a conductor, and increasepower dissipation.

Figure 1: Current distribution in an isolated conductor infree space.

Proximity Loss

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face due to skin effect minimizes the inductance. The skin depth is easily calculated from:

with �0 = 4� x 10-7 , � is the operating radian frequen-cy, and � is the conductivity of the winding material.(copper conductivity = 5.8 x 107 at room temperature.)All units are SI.

This equation is easy to apply. At 60 Hz, the skin depthis 8.5 mm. If you don't use a conductor thicker than 8.5mm, the skin effect on dissipation will be small. At 100kHz, the skin depth is 0.2 mm, or about half the diameterof a #27 wire. If you don't use a bigger gauge than this, theskin depth calculation says you are OK.

But are you? Unfortunately not. Unless you build yourmagnetic with a single strand of wire in free space,which is the situation in which the skin depth equationapplies directly, your actual copper losses will usuallybe much higher than predicted.

This is due to proximity loss–the effect of putting aconductor in a field generated by other conductors inclose proximity to it. Proximity loss is always morethan skin effect loss, and will affect the design and tem-perature rise of almost all high-frequency magnetics.Even if you wind an inductor with just a singleturn, the fact that the turn is shaped around the

bobbin affects the current distribution in the con-ductor, as shown in Figure 2.

This picture shows the cross section of an inductor withthe center leg of the core, and a single layer turn. Theblue shading represents current flowing into the planeof the page, and the red shading the current flow-ing out of the page at any given instant. Theintensity of the shading represents the currentdensity at that part of the conductor.

Notice that the current no longer flows along both sur-faces, but moves towards the inside surface only. Thisagain due to the tendency of the high-frequency currentto arrange itself to minimize the inductance created. Thesmaller the area enclosed by the current, thesmaller the inductance.

The tendency of the current to move towards the surfacecauses an increase in ac resistance, but we can nolonger use a simple equation to calculate this effect.We need to move on to more complex analysis tosolve the problem.

Proximity Loss CalculationAll magnetics structures are complex three-dimensionalobjects that defy any exact analytical modeling. Youcan, if you like, apply Finite Element Modeling soft-ware to try to analyze the complete three-dimensionalfields and currents, but this is a very time-consuming

activity. Plus, it will only give you results fora finished design, not insight into how to cre-ate a design.

As with any aspect of engineering, we don'tjust need numerical solutions, but symbolicanalytical results are needed to yield designrules and procedures. For magnetics, we haveto simplify the three-dimensional structure ofwinding design, all the way down to a modelwith only one dimension of freedom. Inorder to obtain analytical results, we mustassume that the winding layers of an inductoror transformer are continuous sheets of con-ductor, with no variation in the x-y direction,and variation as we go through layers of con-ductors in the z-direction. This is, ofcourse, a gross oversimplification, butthe results yielded with this approach provideus with powerful design guidelines to buildbetter magnetics.

Figure 2: Current distribution with single-layer inductor winding

Proximity Loss

3©Copyright 2005 Switching Power Magazine

Figure 3 shows the cross section of an inductor withfive layers of rectangular winding around the center legof an E-E core. Electric fields will be relatively uniformfor the part of the winding away from the edges, and weextend the uniform field assumption all the way to theedge in order to find an analytical solution to the cur-rents in the winding layers.

Figure 4 shows a cross-section of one side of the bobbinof a five-layer structure. The windings are truncated inthe horizontal plane, but are assumed to have no endeffects, leading to uniform fields, H1 through H6

between the layers of windings. The windings areassumed to be a continuous uniform cross-section ofcopper, as would be obtained with a flat foil. (In manydesigns, this layer may be formed with individual wires,and these must be approximated to a foil sheet, a stepoutside the scope of this article.)

The simplification to a one-dimensional variation onlyis needed in order to arrive at an analytical solution toMaxwell's equations. It is not possible to arrive at closed-form solutions of a situation more complex than this.

Dowell's EquationEven this simple situation is a formidable problem to

solve. Fortunately for modern engineers, ithas already been done, so all we have toworry about is understanding and applyingthe results.

At the heading of this article is a form ofDowell's equation. This gives the power dissi-pation Pd, of a winding by summing up thepower loss through each of the layers from 1to n. It is a crucial analytical result that lets uscalculate the dissipation of a magnetics wind-ing at frequencies other than dc. It is a veryintimidating equation, even more so whenyou examine the terms in it such as:

These terms are combinations of sinusoidal functions,which we use all the time, and hyperbolic sine andhyperbolic cosine functions which, most likely,you haven't seen since college calculus.

The term � is the ratio of the height of a windinglayer, divided by the skin depth for the given fre-quency of analysis.

where hi is the height of layer i and the skin depth, � , isgiven earlier in this article. In other words, if the wind-ing layer height is equal to 2 skin depths, � = 2.

At this point, you may be feeling overwhelmed by themathematical expressions. You should be– these are noteasy concepts to think about, or easy equations to grasp.Like many power supply engineers, I spent a large partof my career aware of the problems of proximity loss,but unable to apply the results of analysis due to a lackof time to spend to try and understand it all. The textsand papers containing this material are very advanced,and you need to set aside a substantial amount oftime with no interruption to make good progress.

Figure 3: Cross-section of an inductor with a five-layer winding

Figure 4: Multilayer winding structure with variation in thez-direction only.

Proximity Loss

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The dedicated engineers amongst you will want to presson and try to apply this equation. For this, you willneed the following additional information:

The field at the boundary of each layer i is given by:

where N is the number of turns forming the layer(which may be formed of N multiple adjacent conduc-tors), and I is the current through each turn. The widthof each layer of conductor is denoted by bw, and thelength of a turn on layer i is given by li.

The final term you need to know about is �i, which isthe ratio of the fields on either side of layer i. The con-vention for calculating this is to order the fields suchthat the smaller one is in the numerator. For the 5-layerinductor example, the calculations would be:

(Note: one indication of how "effective" a winding iscan be assessed from the value of this term. As itapproaches unity, that means that the fields on eitherside of the winding are minimally affected by the cur-rent through that winding. That means you have a pieceof conductor sitting in a field with induced eddy cur-rents in it, and the winding will be inefficient.

The most effective windings have an associated �value of -1. This can only be achieved in transformerwindings, and occurs when the current through a layerreverses the field from one side to the other. This issimilar to an isolated conductor in free space whichexperiences only skin effect.)

With this information, you can now work on solving theequations. The simplest way to set up the solution tothis equation is in either MathCad or Excel. The com-plicated terms G1 and G2 are merelyconstants for a given layer heightand skin depth. (The results can besimplified if the result is normalizedfor a 1 A current in terms of the dcresistance of the wire.)

If you don't have the few days nec-essary to experiment with this, theresults have all been programmed

into POWER 4-5-6, and you can use this as a handycalculator to find the ac resistance of a windingvery quickly.

Proximity Loss Results for 5-Layer WindingOnce you have programmed equation 1 into yourfavorite software, the input variables for calculating theac resistance in terms of the dc resistance are the fre-quency of operation, the height of the winding layers,and the number of layers of winding used. From this,you can calculate the resistance ratio for each layer of thewinding, as shown in Table 1.

The results are very surprising if you are not familiarwith the severe effects of proximity loss. The first rowshows the results for a layer with a foil 0.3 mm inheight (about the same as the diameter of a 30 AWGwire), and with a frequency of 100 kHz. The ratio ofthe layer height to skin depth is 1.46, certainly a rea-

sonable choice according to skindepth design rules.

Notice how rapidly the ac resist-ance increases from one layer to

the next. The innermost winding layer has 27 times theresistance that would be calculated or measured at dc.The average increase over the 5 layers is 11.6 times.

It is common when an inductor is running too hot toincrease the thickness of the copper used. If it is dou-bled to 0.6 mm (24 AWG wire diameter), the averageincrease in resistance is 51 times. The inner winding(which is also the most difficult to extract heat from)has 123 times increased resistance. Increasing furtherto 1.1 mm almost doubles these numbers again.

The rearrangement of current in the windings for thiscase is more complex than for a single layer. Figure 5shows the current distribution in the winding layers.On the outermost layer, the current moves toward theinside of the conductor. The net current through thisturn is normalized to 1 A.

Table 1: AC to DC resistance ratios at 100 kHz for 5-layer wind-ings with different thickness

Proximity Loss

5©Copyright 2005 Switching Power Magazine

The second layer has two current compo-nents– one on the outside of the conductorwhich mirrors the adjacent current densityin the outermost turn, and flows in theopposite direction to the net currentthrough the turn. This is Mother Naturearranging the currents to cancel high-fre-quency magnetic fields. The second com-ponent of the current is on the inside of thelayer, which has twice the amplitude of theouter current. The net current through theturn must, of course, be the same as for the out-ermost layer since they are series turns of thesame inductor.

The third layer again has two components–an outer reverse current with a normalizedamplitude of 2 A to mirror the turn outside it, and aninner current with a normalized value of 3 A. This con-tinues through the remaining layers.

The most powerful way to reduce the proximity lossresistance in an inductor winding is to reduce the num-ber of layers. The final row of the table shows that theoverall ac resistance is only 2.6 times the dc resistancewith just two layers, versus 27 times with five layers. Inthis case, it is much better to use a finer gauge wire thatcan fit the needed number of turns in two layers, versus abigger wire that requires five layers.

Choosing the Right Conductor SizeBefore assessing which thickness copper is best for agiven application, you must also assess the currentwaveforms which will be applied though the wind-ing. For an output inductor on a dc-dc converter,there will be a strong dc component of current, asshown in Figure 6.

Despite the relatively large amount of ripple shown inthe waveform, the dc portion of the current accounts for6.26 A, but the ac portion is only 0.81 A rms. For mostproximity loss analysis, it is common to split the cur-rent into just two parts– the dc and the ac. The com-bined ac component is the rms value of all of theFourier components other than dc. The dc current isthen used to calculate the dc resistive loss. The ac cur-rent is assumed to all be at the switching frequency,and used with the calculated ac resistance at that fre-quency to estimate the ac conduction loss. In manyinductor design cases, the ac resistance can be veryhigh, but the overall contribution to loss can be quite

small due to the largedc component. Thisallows us to designmulti-layer inductorseffectively despite thelarge proximity effectresistances.

For more precise proximity loss estimation, the currentwaveform should be split properly into its full spectrumof frequency components, and each of these used withthe corresponding ac resistance calculation to find themore accurate result. For converters with reasonableduty cycles close to 50%, combining the ac currents intoa single expression is usually sufficient. If your convert-er has very narrow current pulses, the more involvedcalculation will yield significantly higher and moreaccurate results.

Figure 5: Current distribution with five-layer inductor winding

Figure 6: Inductor current waveform and components

Proximity Loss

6©Copyright 2005 Switching Power Magazine

Transformer Winding Proximity LossTransformer windings typically have much stronger accomponents in the current waveforms, as shown inFigure 7. For this reason, we avoid the use of multiple lay-ers of primary and secondary windings in transformers.

Fortunately, we have more design flexibility when mak-ing high-frequency transformers than we do when mak-ing inductors. In an inductor, all of the current flows inthe same direction, and our only design freedoms arethe number of layers and conductor thickness.

In a transformer, primary and secondary windings con-duct in opposite directions. If we arrange a single layerof transformer primary next to a single layer of trans-former secondary, the currents flow as shown in figure8. At high frequencies, the currents flow close to thesurface, in an effort to cancel the magnetic field fromadjacent windings. The calculations for proximity loss ofthis single layer design are the same as for an inductor.

When we add more layers to a primary or secondary,we should always try to avoid making multiple layers,as for the inductor, since this will lead to higher con-duction losses. Figure 9 shows how the currents are dis-tributed in a five layer primary winding. This is identi-cal to the inductor current windings, with the exceptionof the current in the secondary, which is in the oppositedirection. The secondary winding acts as a single layerwinding. It does not make any difference to this wind-ing whether the primary is constructed of single ormultiple layers.

An interesting side note arises out of this particulararrangement of currents in multi-layer transformerwindings. At low frequencies, the currents are distrib-uted evenly through the layers. Leakage inductance,which is determined by the separation of primary and sec-ondary currents, is large when measured at low frequencies.

At high frequencies, all of the primary fields are can-celled at each of the surfaces until a current of 5 A isbuilt up on the inner surface, close to the secondarywinding. The separation of primary and secondary cur-rents is now very small, and the leakage inductancemeasurement will drop substantially as we increase themeasurement frequency up to this point. This frequencydependence of leakage inductance is very important to

consider when designing and charac-terizing high-frequency magnetics.

Also, if you see a substantial change inleakage inductance measurements asyou approach the switching frequencyof your converter, it is a good indica-tion that you will have significantproximity loss in your transformer.

Figure 7: Transformer current waveform and components

Figure 8: Current distribution with single-layer pri-mary and secondary transformer windings

Proximity Loss

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Interleaved TransformerWindings In some low-cost designs, we are forced to keep pri-mary and secondary windings separate due to safetyand spacing requirements that do not allow multipleinsulation systems. However, an ideal transformer delib-erately introduces multiple primary and secondary wind-ings, interleaved with each other, to reduce field strengths andminimize proximity loss effects.

Figure 10 shows how splitting the primary into twoparts, one on either side of the secondary, can lower theloss in a transformer.

Each of the primary layers now carries just half the netcurrent, and has proximity loss according to the single-layer solution. This is a dramatic improvement overbuilding the two primaries in succession, which couldend up with as much as five times the loss.

Even the secondary winding, which remains unchangedin terms of its construction in the transformer, experi-ences a drop in proximity loss. The current in the sec-ondary now flows equally in both surfaces, and muchbetter utilization of the conductor is achieved.

ConclusionsOut of all the detailed papers on magneticsdesign and proximity loss, one simple rulealways emerges: keep conductors out of stronghigh-frequency magnetic fields. There are manyways to achieve this. The simplest inductors areconstructed of single-layer designs where the aclosses are minimized. This is essential for induc-tors that have strong ac components of currentrelative to dc currents. If you must have morethan one layer, due to the number of turnsrequired, they should be the right thickness toprovide the optimum balance of dc and ac resist-ance for your particular waveforms. There areno closed-form formulas for optimizing this dueto the wide variety of variables involved. Werecommend returning to the basic proximity lossequation to find the right balance for yourdesign.

Transformers are designed the same way– keepto no more than a single layer design, and if youneed more layers to accommodate turns, try toFigure 9: Current distribution with five-layer primary and single secondary

transformer windings

Figure 10: Current distribution with single-layer secondary andsplit primary transformer winding

Proximity Loss

8©Copyright 2005 Switching Power Magazine

build them interleaved with secondaries to minimize theaccumulated magnetic fields. This is not always practi-cal with conventional isolation techniques due to theexpense of multiple safety boundaries.

Planar transformers lend themselves naturally to multi-ple interleaving. Each layer of a pc board has limitedturns count and current-carrying capability, and it iscommonplace to have numerous primary and secondarylayers to build power transformers. While planars canwork well, they are not always the most cost-effectivesolution. Many power supply manufacturers cannoteven afford to go beyond a single layer PC board, and

conventional wound magnetics are still the lowest costsolution. Prices on standalone planars have dropped sig-nificantly recently, however, and should be consideredfor custom designs. Manufacturers such as Payton mag-netics can provide custom quick-turn designs for yourapplications.

There are many more aspects to proximity loss that arenot covered in this article, such as end effects, core gapeffects, and different wire shapes, that complicatedesign further. We will cover some of these effects inupcoming issues of Switching Power Magazine.

Proximity Loss

9©Copyright 2005 Switching Power Magazine