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Magnetic Field Evaluation inTransformers and Inductors

By Lloyd H. DixonThis topic reveals, by means of magnetic field plots, many of the problems that occur in magnetic devicestructures, utilizing both conventional cores and planar cores. The understanding gained from fieldplots points the way to reducing leakage inductance and winding losses that can otherwise beprohibitive. The reader is referred to the Unitrode/TI Magnetics Design Handbook TI Literature No.(SLUP003) for supporting information on basic magnetics principles and practical magnetics design.

I. RULES GOVERNING THE MAGNETIC FIELD

The Law of Conservation of Energy dictatesthe pattern of the magnetic fields within atransformer or inductor. This, in turn, determinesthe pattern of current flow within the windings.At the frequencies involved in switching powersupply applications, leakage inductance andwinding losses can be unexpectedly large if therules governing the magnetic field are notunderstood and properly evaluated.

All magnetic fields have two components:Magnetic Force (F, or mmf), and Flux (Φ).

A. Amperes LawThe total magnetic force, F, integrated along

any closed path is equal to the total currentenclosed by that path. In fact, in the SI System ofunits (rationalized MKS) magnetic force, F, isexpressed in Amperes. Wherever there is currentflow there is a corresponding magnetic force.

Magnetic Force (potential) can be describedvectorially, or as a series of equipotentialsurfaces.• Magnetic force equipotential surfaces are not

closed surfaces (like a free-floating soapbubble). These surfaces terminate, and arebounded, by the current flow that producesthe field (like a soap film bounded by a ring).

• The spacing between the equipotentialsurfaces indicates Field Intensity (H, inAmps/meter).

In this paper, the edge view of theequipotential surfaces is shown as a series of finelines which terminate on current flow.

Magnetic Flux (Φ, in Webers) can bedescribed vectorially, or in more physical terms,as lines.• Flux lines always form closed loops. No flux

line ever begins or terminates.• In any homogeneous region, flux lines are

normal to the magnetic force equipotentialsurfaces.

• The spacing between flux lines indicatesFlux Density (B, in Tesla).

• In any homogeneous region, Flux Density, B,is proportional to Field Intensity, H. Theconstant of proportionality is calledpermeability,

HBµµµ r0 =⋅=

The permeability of free space or any "non-magnetic" material,

-70 104πµ ⋅=

The permeability of a magnetic material relativeto free space is µµµµr.

The magnetic field represents energy. Thefield is energy. Energy per unit volume is:

� ⋅= dBHW Joules/m3 (SI system)Looking at a B-H characteristic (Fig.1),

�HdB corresponds to the area between thecharacteristic and the vertical axis.

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H, I

B

Fig. 1. B-H characteristics.When permeability is constant, as in an air

gap or in any non-magnetic region, �HdBsimplifies to:

2µH

2µBBH

21W

22===

Joules/m3

Perhaps the most important rule governingthe magnetic field is based on the Law ofConservation of Energy (LCE):• The magnetic field organizes itself in such a

way as a to minimize the energy stored in thefield (just as a soap film does).

• The resulting pattern of current flow is alsodictated by the LCE. Whenever alternativepaths exist, current will flow in the path(s)that minimize the rate of energy transfer toand from the source. In the frequencydomain, this is observed in the tradeoffbetween current flow through a resistor andthrough a paralleled inductor. This rulesimply means that current will flow in thelowest impedance path(s), resulting in thelowest possible voltage drop and the lowestrate of energy transfer.A solid wire represents within itself an

infinite number of parallel paths. At lowfrequency, the rate of energy transfer in and outof the magnetic field is low. The LCE thendictates uniform current flow within the wire, inorder to minimize resistive losses. The magneticfield inevitably associated with current flow

must then penetrate into the wire because it mustterminate on current, wherever it flows. Thus, atlow frequency, magnetic energy is stored withinthe wire, in addition to the field external to thewire. (The permeability of copper equals µ0.)

At high frequency, current flow concentratesnear the surface of the wire. This skin effect isdictated by the LCE in order to reduce oreliminate the magnetic field within the wire. Thereduced total volume of the magnetic fieldreduces the rate of energy transfer in and out ofthe field, even though resistive losses are therebyincreased. (Just as, at high frequency, currentflows through the resistor rather than theparalleled inductor.) The inductance of the wiremeasures slightly less at high frequency because,although the external field (energy) remains thesame, the field (energy) within the wire isreduced.

B. Faraday’s LawA voltage applied to a winding produces a

rate of change of flux within the winding equalto the volts applied per turn. (SI system)

NE

dtdφ −= Webers/sec = Volts/turn

Faraday’s Law is bilateral – that is, anapplied voltage produces a changing flux,likewise a changing flux induces a voltage acrossa coil linked to that flux such that the inducedvolts per turn equals the rate of flux change.

Fig. 2 shows a transformer with a four-turnprimary and a single turn secondary wound on aferrite E-E core. In a switching power supplyforward converter application, assume 6 V isapplied to the primary when the power switchturns on (1.5 Volts/turn). According to FaradaysLaw, this results in flux change at the rate of 1.5Webers per second within the coil. Almost all ofthis flux is confined to the core because itspermeability is much greater than the adjacentregions (µr is very large). Dividing the flux rateof change (Volts/turn) by the core cross-sectionarea (meters2) and multiplying by the switch ONtime gives the total flux density change, ∆B,(Tesla).

ee NAt E∆Edt

NA1∆BEdt;

N1∆φ ≈== ��

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P S

Fig. 2. Transformer.In practice, the flux density change, ∆B, is

limited either by core saturation or by core loss,thereby limiting the volt-seconds per turn thatcan be applied to a specific core cross-sectionarea. To fully utilize any core, the design shouldresult in ∆B close to the saturation or core losslimit, whichever governs, by adjusting either thecore area, the number of primary turns, or theON time.

B�Edt

H, I

Fig. 3 Minor hysteresis loop.In Fig. 3, the B-H characteristic of a ferrite

core material, the minor hysteresis loop showsthe flux excursion, ∆B, limited by core loss.

Because the flux change caused by theVolts/turn applied to the primary is almost allconfined to the core, this flux change is thereforelinked also to the secondary winding. Faraday’sLaw dictates that the secondary must thereforehave the same 1.5 Volts/turn induced — 1.5 Vtotal for the one turn. The primary/secondaryvoltage ratio thus equals the turns ratio.

II. MAGNETIZING INDUCTANCE

Assuming the secondary current is zero (noload), there will nevertheless be a small primarycurrent (magnetizing current) which variesthroughout the switching period. The amount ofmagnetizing current can be determined from thecore B-H characteristic. Referring to Fig. 3,while the power switch is ON, flux density Bincreases at a constant rate (determined byprimary volts per turn) along the minorhysteresis loop. This results in a correspondingchange in field intensity, H. Integrating the fieldintensity along the total magnetic path lengththrough the core defines the total magnetic force,F. According to Faradays Law, magnetic force Fequals the Ampere-turns enclosed by themagnetic path. Thus, the instantaneousmagnetizing current value is calculated from theinstantaneous field intensity:

� ≈=N

HlHdl

N1I e

m

With a high permeability core, field intensityH is quite small. Thus, the magnetizing current,although not negligible, is usually smallcompared to the full load current reflected intothe primary. Magnetizing current representsenergy stored in the magnetic core. When thepower switch turns off, much of this energy(inside the minor hysteresis loop) becomes coreloss. The remaining energy results in a voltagespike across the power switch, requiring asnubber or clamp to absorb this energy andprotect the power switch.

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Note that flux swing and the resultingmagnetizing current is a function solely of volt-seconds per turn applied to the primary,independent of load current. During the powerswitch ON time, flux density increases linearlywith time according to Faraday’s Law becausethe applied primary voltage is constant.However, field intensity and magnetizing currentchange in a nonlinear manner, depending uponthe shape of the B-H characteristic.

When a load is applied to the secondary,current is induced in the primary in addition tothe magnetizing current. The additional primaryAmpere-turns are equal in magnitude to thesecondary Ampere-turns (resulting from theload), but opposite in phase. This occurs for thefollowing reasons:

1. The flux density in the core (which links bothwindings) is determined solely by voltseconds per turn applied to the primary(Faraday’s Law), independent of loadcurrent.

2. A specific flux density in the core equates toa specific magnetic force according to thecore B-H characteristic. According toAmpere’s Law, this defines the specificmagnetizing Ampere-turns required(independent of load current).

3. This specific requirement for magnetizingAmpere-turns applies in total to all windingslinked to the core. Thus, additional current inany winding (load current in the secondary)must be offset by equal and opposingAmpere-turns in other windings (reflectingthe load into the primary).

The resulting primary current equals thesecondary current divided by the turns ratio(Ampere-turns cancel), plus the magnetizingcurrent.

III. LEAKAGE INDUCTANCE

There is one location within the transformerstructure where the primary and secondaryampere turns do not cancel and that is in theregion between the two windings. A closed looppath between the two windings and around theouter core leg encircles only the outer winding.

According to Amperes Law, the magnetic forcealong this path equals the Ampere-turns of theouter winding (the secondary in Fig. 2). Thismagnetic force can be quite large, especially atfull load. The magnetic force, F, appears almostentirely along that portion of the path betweenthe windings because the permeability in thatnon-magnetic region is much less than in theremainder of the path through the core outer leg.The magnetic force through the outer leg is, bycomparison, negligible.

The flux resulting from this magnetic forcebetween the windings completes its path throughthe core outer leg. Thus, this "leakage" flux linksthe outer winding but not the inner winding.From a circuit point of view, the energy storagecapability of the magnetic field between thewindings is called leakage inductance. Leakageinductance energy is proportional to load currentsquared (W = ½ LI2). When the power switchturns off, current in the windings collapses. Asnubber is required to absorb the leakageinductance energy and prevent damage to thepower switch.

Leakage flux density in the core outer leg ismuch less than magnetizing flux density anddoes not make a significant contribution to coreloss or core saturation. Even though leakage fluxdensity is much less than magnetizing fluxdensity, leakage inductance energy at full load isusually much greater than magnetizinginductance energy. This is because the leakagefield exists in the region between the windingswhere permeability is very low. Thus, eventhough flux density is low, field intensity is verygreat.

Leakage inductance is minimized by keepingthe windings as close together as possible, whichminimizes the volume of the magnetic field, andby using a core with a long, narrow window. Themagnetic force is thereby spread over a greaterdistance, reducing field intensity, H, whichreduces flux density, B. Thus, energy density(½ B·H) is greatly reduced.

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IV. PROXIMITY EFFECT

High frequency AC current flowing througha round wire which is isolated from all otherconductors results in a perfectly radial,symmetric magnetic field surrounding the wire.The radial magnetic force equipotential surfacesterminate on the current, which flows along theentire surface of the wire to a skin depth whichdepends upon the frequency. Because the currentflows only near the surface of the wire, the highfrequency AC resistance is greater than theresistance at low frequency where the current isuniformly distributed.

When the wire is brought into proximity withother conductors, the radial fields around eachwire add vectorially. The field is no longerradially symmetric around each wire. In thetransformer windings of Fig. 3, the result is alinear field concentrated between the windings.Since the field must terminate on the currentwhich produces the field, current flows only onthose portions of the conductor surfaces that areadjacent to the field, facing the opposingwinding. Because only a portion of the wiresurface conducts current, the AC resistance iseven greater than with simple skin effectportrayed in the previous paragraph.

V. INDUCTORS AND FLYBACK TRANSFORMERS

Unlike a true transformer, in an inductor orflyback transformer, primary and secondaryAmpere-turns do not cancel. (In a simpleinductor, there is no secondary, and obviously nocancellation.) The sum of all of the current, in allof the windings, comprises the magnetizingcurrent. (In a flyback transformer, magnetizingcurrent switches back and forth between primaryand secondary, and does not cancel.) Theresulting large magnetic force appears across anon-magnetic gap (low permeability, highreluctance), which must exist along the magneticpath in series with the lower reluctance (highpermeability) core magnetic material. It is in thisnon-magnetic gap that the inductive energy isstored.

VI. PRINCIPLES SUMMARIZED

According to Faraday's Law, the total changein magnetic flux through any winding is equal tothe Volt-seconds per turn applied to the winding(SI system of units). Thus, in a switching powersupply application, knowing the pulse amplitudeand pulse width applied to a winding, andknowing the number of turns, the total fluxchange can be precisely calculated. (It is alsoclear from Faraday's Law that the Volt-secondsintegrated over several cycles of operation mustaverage zero, otherwise the flux change willcontinue cumulatively until the core saturates).

According to Ampere’s Law, the totalmagnetic force along a closed path (loop) isequal to the total Ampere-turns linked by thatpath (SI system of units). Thus, taking a paththrough the core, linking all of the windings, thetotal magnetic force along that path equals thealgebraic sum of the Ampere-turns of all of thewindings.

In a true transformer, the primary andsecondary Ampere-turns are proportional to loadcurrent, and almost completely cancel, regardlessof load. An additional small primary current,independent of load, is called “magnetizingcurrent”. This small Ampere-turn inequalityproduces the small magnetic force required topush the flux through the low reluctancemagnetic core. How much flux? Faraday's Lawtells us precisely the amount of flux at anyinstant of time. How much force is required topush this flux through the core? The B-Hcharacteristic from the core materialmanufacturer provides this information.

VII. WAVEFORM ANALYSIS

Before proceeding with the magnetic fieldplots, the current waveforms encountered mustbe evaluated. The first step in the waveformanalysis is to separate each waveform into its DCand AC components. Figs. 4, 5, and 6 showtypical waveforms for a true transformer, aflyback transformer operating in the continuousmode, and a flyback transformer operatingdiscontinuously.

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A. Forward Converter TransformerFig. 4(a) and (b) show waveforms typical of

a forward converter transformer, operating at40% duty cycle.

During the ON time of the power switch,current flows in both the primary and secondarywindings. The secondary current equals theoutput filter inductor current, whose DC valueequals the load current. When the power switchturns off, voltage reverses across the windings inorder to reset the core. The free-wheeling diodein the output circuit carries the filter inductorcurrent. Current through both primary andsecondary windings thus becomes essentiallyzero during the OFF time of the power switch..

Secondary A-T

AvgDC

0

b

Differential A-T

0

AvgDCc

Primary A-T

0

AvgDCa

Fig. 4. True transformer waveforms.Except for a very small primary magnetizing

current (not shown), the primary ampere turnsare equal and opposite to the secondary ampereturns. The total ampere turns enclosed by aclosed path through the core (which links bothwindings) is essentially zero throughout theswitching period. Except for the smallmagnetizing current, all of the current (both DCand AC components) appears differentiallybetween the windings as shown in Fig. 4(c). Thisresults in a magnetic field located in this regionas shown in Fig. 7. The energy in this field isleakage inductance energy. In a true transformer,any energy storage is undesirable, although it

inevitably occurs in the form of leakageinductance and magnetizing inductance.

Leakage inductance is minimized by bringingthe windings as close together as possible, andincreasing the winding breadth by using a corewith a wide window opening. Stretching thewinding out across a wide window opening alsoincreases the conductor surface area, whichreduces current density and resistive losses.

B. Flyback Transformer —Continuous ModeFig. 5 shows current waveforms typical of a

flyback transformer operating at 40% duty cycle,in the continuous inductor current mode. Aflyback transformer is not a true transformer, butis actually a coupled inductor. Unlike a truetransformer, its main purpose is to store energy.

0

Combined A-Tacross gap

0

AvgDC

Secondary A-T

0

AvgDC

Differential A-Td

c

b

Primary A-T

0

AvgDCa

Fig. 5. Flyback transformer waveforms.The primary and secondary waveforms

shown in Fig. 5 look deceptively similar to theforward converter waveforms of Fig. 4. Butnotice the difference in the zero current locationin Fig. 5(b) vs. Fig. 4(b).

In the flyback transformer, when the powerswitch turns off, the voltage across the windings

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reverses and the instantaneous primary Ampere-turns transfer to the secondary. Thus, the primaryand secondary Ampere-turns do not cancel as ina true transformer, but combine to provide alarge magnetic force along a closed path throughthe core. But the high permeability core cannotsupport such a large magnetic force. A non-magnetic gap placed in series with the core isrequired to support the magnetic force. Thedesired energy storage takes place within thisgap.

Fig. 5(c) shows how the primary andsecondary ampere turns combine to provide alarge DC magnetic force across the gap, with arelatively small AC component. Flux in the coreis proportional to this force. Core loss is afunction of flux swing. In continuous modeoperation, with a small AC flux swing,maximum flux density is usually limited by coresaturation rather than by core loss.

As shown in Fig. 5(d), there is verysignificant AC field between the primary andsecondary windings as a result of the currentswitching back and forth between the windings.This differential leakage inductance field is verysimilar to the AC component of the differentialfield in the forward converter transformer, shownin 4(c).

C. Flyback Transformer —DiscontinuousMode

Waveforms for the discontinuous modeflyback are shown in Fig. 6.

In the discontinuous mode, by definition,current is zero for a portion of each switchingperiod. Thus, for a given power level, peakcurrents must be considerably greater than withcontinuous mode operation. As shown in Fig.6(b), the combined ampere-turns applied to thegap has a very large AC component. Theresulting large flux swing will almost certainlybe limited by core loss to much less thansaturation flux density. In addition, the leakageinductance between the windings causesdifficulty when the high peak ampere turns areswitched from the primary to the secondarywinding.

Primary A-T

0

AvgDC

Secondary A-T

0

AvgDC

Combined A-Tacross gap

AvgDC

a

b

Fig.6. Flyback converter discontinuous modewaveforms.

III. FIELD PLOT EXAMPLES

A. General ConsiderationIn order to make sense of the magnetic field

patterns within a transformer or inductor, it isnecessary to break the waveforms down intotheir AC and DC components. The assumptionof linearity enables each of these components tobe dealt with independently. It has been shownthat some of these waveform components actdifferentially between windings, causing leakageinductance fields, other components maycombine to cause magnetic force along a paththrough the core.

Rigorous evaluation requires Fourier analysisof the current waveform in each winding.Penetration depth (skin depth) and thus theeffective resistance is different for everyharmonic frequency. But for the sake ofsimplicity, only the fundamental AC frequencyis considered in the field plot examples whichfollow.

The examples used in this paper aredeliberately simplified and show only singlelayer windings, in order to provide bettervisibility. The dramatically adverse effects thatcan occur with multiple layer windings isdiscussed extensively in the Unitrode/TIMagnetics Design Handbook.

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The examples used in this paper show themagnetic field and current flow patterns revealedby FEMLAB finite element analysis software [1].

In these drawings, the magnetic force (equalto the Ampere-turns enclosed by the closed looppath) is shown by a series of fine linesrepresenting the edge view of magnetic forceequipotential surfaces. Magnetic field intensity,H, is shown by the spacing between thesesurfaces. Close spacing indicates high fieldintensity. Note that the magnetic forceequipotential surfaces terminate on the currentcreating the field. Flux lines (usually not shownin these drawings) are normal to theequipotential surfaces.

Conductor thickness in these drawings isapproximately four times the skin depth at 100kHz. This is certainly not the recommendedpractice. The conductors are made thick in orderto reveal the high frequency current distribution.

B. Forward Converter TransformersIn the simple forward converter transformer

shown in Fig. 7, the volts per turn applied to theprimary winding cause a flux change through thecore (not shown), in accordance with Faraday'sLaw. This changing flux performs the vitalfunction of linking the windings to each other.Because this coupling flux passes through thehigh permeability ferrite core, almost negligiblemagnetic force (ampere-turns) is required. Thus,with small magnetizing current, the ampere-turnsin the primary and secondary windings verynearly cancel each other. These currents are adirect function of load (Fig. 4(a) and (b)). Aload-dependent magnetic field representingleakage inductance energy is therebyconcentrated between the two windings.

The Law of Conservation of Energy causesthe field to spread out linearly across theavailable winding breadth, as shown by themagnetic force equipotential surfaces in Fig. 7.Ampere’s Law requires that magnetic forceequipotentials must terminate on current flow,thus current flow is spread out uniformly acrossthe full breadth of the single turn secondary. Onecould argue that current is forced to spread outbecause of the primary side 14 turns in series.But even if the primary winding was a singleturn copper strip, like the secondary, the currentand the field would still spread out uniformlyacross both strips.

Fig. 7. Forward converter transformer.In the planar forward converter transformer

structure shown in Fig. 8, the windings are not ina cylindrical form, but spiral outward from thecenter. Current is distributed uniformly acrossthe conductor surfaces facing each other, becausethe primary turns are series-connected. Thus, themagnetic force equipotentials (which mustterminate on current) are likewise distributeduniformly between the primary and secondarywindings.

Fig. 8. Planar forward converter transformer.

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C. Flyback TransformersIn flyback transformers, there is always a

large differential AC field between thewindings, regardless of operating current mode,because current is switched back and forthbetween the windings. But unlike the forwardconverter transformer, there is no differentialDC field between the windings. Instead, the DCcurrents in the windings combine to produce astrong DC field across the core gap, where theenergy storage inherent in the flybacktransformer is located. DC currents aredistributed uniformly throughout the windings,and the DC field has no adverse effects.

The differential AC field between theflyback transformer windings is very similar tothe forward converter transformer – see Fig.4(c) and 5(d). These differential fields areshown in Figs. 7 and 8 for the forward convertertransformer, and are not shown in the flybackillustrations which follow.

In flyback transformers, there is also an ACcomponent to the field across the core gap. ThisAC component is usually small in transformersoperated in the continuous mode, as shown inFig. 5(c), and is often negligible in its effect.But in transformers designed for discontinuousmode operation, the AC component is verylarge, as shown in Fig. 6(b). The resulting coreloss limits the maximum flux density at whichthe core can operate. Also, the large ACcomponent in the fringing field adjacent to thegap can cause severe problems with windinglosses and reliability if not properly dealt with.

The flyback transformer illustrations whichfollow show the AC field in and adjacent to thegap. These drawings apply to continuous ordiscontinuous operating modes, but it is only inthe discontinuous mode that the associatedproblems become severe.

In Fig. 9 the field is concentrated in a single,centrally located centerleg gap. The magneticforce equipotential surfaces shown in thedrawing must terminate on current flow. Tosatisfy the minimum energy condition, thefringing field pulls the secondary current intowards the center of the copper strip, resultingin increased copper losses.

Fig. 9. Single gap, secondary inside.In Fig. 10 this problem is remedied

somewhat by swapping the primary andsecondary winding locations. With the 14 turnprimary adjacent to the centerleg gap, currentcannot concentrate toward the center of thewinding, because the turns are in series. But ifthe primary winding had more than the onelayer shown, the intense AC fringing field willpenetrate through the turns adjacent to the gap,causing very high losses in that vicinity.

Fig. 10 Single gap, primary inside.In Fig. 11, the copper strip secondary is

located adjacent to the gap, as in Fig. 9, but withthe centerleg gap split into two smaller gaps.This significantly reduces the fringing field, andthe current concentration is much less than inFig. 9.

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Fig. 11. Two gaps, secondary inside.In Fig. 12, the primary is located adjacent to

the two small gaps in centerleg, with furtherreduction of the fringing field effects.

Fig. 12. Two gaps, primary inside.In Fig. 13, the entire centerleg of the ferrite

core is removed and replaced by a compositecenterleg with low relative permeability (MolyPermalloy Powder or Kool-Mu with relativepermeability in the range of 5-20). The gap isthus distributed along the entire centerleg. Theresult is nearly ideal – the fringing field iscompletely eliminated. High permeability ferriteis retained for the outer portion of the core, in

order to minimize core loss and “short out” theexternal field to minimize EMI.

The disadvantages of the compositecenterleg are: (1) High cost, and (2) Compositematerials such as MPP and Kool-Mu aregenerally lossier than ferrite.

D. Gapped Planar CoreFig. 14 could represent a filter inductor or a

flyback transformer built on a planar core. TheAC field component through the centerleg gapcauses the AC current flow in the windings tobe on the narrow edges of the conductors. Thisoccurs because: (1) The field equipotentialsmust terminate on current flow. (2) If the ACcurrent was distributed uniformly through thewindings, the field would be forced to cover awider extent. Greater volume of the fieldrequires more energy. The field pulls the currentin to the edges of the conductors in order tominimize the energy, even though the AC lossin the windings becomes astronomical. A dis-tributed gap in the center would be of little help.

Fig. 13. Distributed gap.

Fig. 14. Planar, single gap.

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Note that the DC component of the field inthe gap and its associated current does distributeuniformly through the windings, only the ACcomponent behaves badly. This problem may betolerable for a filter inductor if the AC current issmall compared to the DC component, also for aflyback transformer operating heavily into thecontinuous mode, as shown in Fig. 5(c).

But for a discontinuous flyback application,a planar core with centerleg gap may beunacceptable.

IV. CONCLUSIONS

In a true transformer, whether of helical orplanar construction, the field adjusts itself in amanner that minimizes leakage inductance andalso results in a uniform current distribution,which minimizes winding losses. In this case,the Law of Conservation of Energy is on theside of the magnetics designer. Furtherreduction of leakage inductance and ACwinding losses, discussed in the Unitrode/TIMagnetics Design Handbook include:

1. Maximize winding breadth by using a corewith a long, narrow window.

a. Interleaving the windings is the same as doubling the breadth and folding in half.

2. Minimize the number of layers (increasingthe breadth automatically does this.)

3. Minimize space between windings(insulation thickness) -- bifilar windingsare the ideal.

4. Excessive conductor thickness is deadly,especially with multiple layer windings.

In a flyback transformer, keep the windingsaway from the fringing field associated withdiscrete gaps, especially strip windings andwindings with multiple layers. Note that in thefield plots, (Figs. 9 through 12), the windingswere spaced considerably away from thecenterleg in order to provide visibility of thefield equipotentials. With the windings closer,the fringing field effects would be of evengreater concern.

1. If the windings do not occupy the entireavailable core window area, space thewindings away from the centerleg gap,rather than wasting the space on theoutside of the windings.

a. Substitute a spacer for the turns in the central portion of winding located in the fringing field.

2. Put strip windings on the outside of thewinding structure, away from the gap (mayalso have EMI benefit).

3. Divide single gap into multiple gaps, orbest, use a composite low permeabilitycenterleg.

4. Choose continuous mode operation toavoid high core losses and winding lossesassociated with AC fringing field.

5. Planar core winding orientation withrespect to centerleg gap makes it totallyunsuitable for discontinuous modeoperation.

REFERENCES

[1] FEMLAB, a product of COMSOL AB,Stockholm, Sweden, and COMSOL Inc.,Burlington, MA, USA.http://www.femlab.com

.

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APPENDIX A

The “k” Transformer Model — An Inappropriate Abstraction

The "k" transformer model is an abstractmodel with no connection to the reality ofunderlying magnetic theory. It assumes,superficially and erroneously, that the couplingcoefficient, k, between any two inductors is thesame in both directions, i.e. k12 = k21 = k. Thisassumption is valid only in the rare instanceswhen the transformer physical structure issymmetrical with respect to the windings. The“k” model results in a symmetrical pi or Tequivalent circuit, together with an idealtransformer. This model is reconciled and madeequivalent to the “true” equivalent circuit byadopting a turns ratio usually quite differentfrom the actual physical turns ratio. Thus, all ofthe values in the “k” model bear no directrelationship to the underlying physical/magneticproperties of the transformer, and provide littleinsight into improvement or optimization.

The “k” model, although it is a validequivalent circuit, should be discouraged. It hasled generations of engineers into believing thefalse notion that leakage inductance is alwaysdivided – half in series with the primary, half inthe secondary. This creates intellectual difficultywhen the engineer faces the realities ofmagnetics design. On the other hand, the “true”model uses the actual, physical turns ratio,usually resulting in an unsymmetrical T model.The parameters of the true model retain a directduality relationship with the underlying physicalparameters (reluctances, turns ratios) of thetransformer. Thus, the electrical performancecharacteristics can be easily translated back intothe physical structure, providing great insightinto improvement and optimization.

When the physical structure of themagnetically coupled device is examined, it canbe shown that k12 equals k21 only in that rareinstance when the two coils are geometricallyidentical (but not necessarily with equal turns).This is especially evident in the layeredconstruction of the typical transformer, whichhas concentric helical coils of differingdiameters.

COIL 1

COIL 2

Fig. 2. Coupling coefficients.Consider two cylindrical, helical air core

coils, both 40 turns, both 4 cm in length. Coil 1is one cm in diameter, Coil 2 is 2 cm indiameter. Although not necessary for thisdemonstration, the two coils are inserted snuglyinto a pillbox of high permeability magneticmaterial (a core with centerleg removed). Theopen ends of both coils are thus bounded by theflat surfaces of the pillbox. This eliminatesfringing fields, providing greater accuracy incalculating inductance. The independentinductance of each coil can then be calculatedwith an error less than 1 or 2 percent,confirmable by measurement. Coil 2 has exactlyfour times the inductance of Coil 1, because ithas four times the cross-section area. Using theinductance formula in the SI system

�AµNL 2= ; then L1 = 4µH, L2 = 16µH.

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Assume Coil 1 is inside of Coil 2. (Theydon’t have to be concentric.) With Coil 1shorted, the inductance measured across Coil 2equals 12µH. With Coil 2 shorted, theinductance measured across Coil 1 is 3µH.

Apply 40V (1V per turn) to Coil 1 (thesmaller coil), with Coil 2 open circuited .According to Faraday's law, the voltage appliedto Coil 1 produces a precise rate of change offlux within that coil. Because of the cylindricalgeometry, bounded at each end, the flux densityis uniform within Coil 1. At the ends of Coil 1,the flux lines proceed through the highpermeability pillbox, completing their closedloop paths outside of Coil 2. Thus, virtually allof the flux generated within Coil 1 also linksCoil 2. In the region between coils 1 and 2, thereis essentially zero flux. (The magnetic force inthis region is zero, because the flux takes the“short-circuit” path through the outside of thehigh permeability pillbox. Thus, the entire fluxchange generated within Coil 1 is also linked toCoil 2. The coupling coefficient in this directionequals 1.0. Since Faraday's law works in bothdirections, the same 1V per turn is induced inCoil 2, resulting in 40V open circuit voltageacross Coil 2..

Then, with Coil 1 open circuit, apply 40 Vto Coil 2. According to Faraday’s Law, theexact same rate of change of flux is produced asbefore, this time uniformly distributed withinCoil 2. However, the flux density is only one-fourth as much as before, because the cross-section area is four times greater. Since Coil 1has only one-fourth the area of Coil 2, Coil 1contains only one-fourth of the total flux changeproduced in Coil 2. In this direction, thecoupling coefficient is 0.25. The open circuitvoltage induced in Coil 1 is only 10V. Thus, allof the flux generated within Coil 1 links to Coil2, but only ¼ of the flux generated within Coil 2links to Coil 1.

The electrical 2-port model which directlycorrelates with this structure has two inductiveelements: a mutual inductance (L1 = 4 µH) inshunt, directly across Port 1, and a leakageinductance (L2 minus L1 = 12 µH) in serieswith Port 2. (Note that although k12 is unequalto k21, M12 = M21 = 4 µH, maintaining

reciprocity.) An additional ideal transformerwould have a turns ratio equal to the physicalturns ratio, in this case 1:1.

Using the “k” model, the resultingsymmetrical T network must be "corrected" tomake it equivalent, by using a transformer turnsratio unequal to the actual physical turns ratio,in order to create the necessary asymmetry seenat the two ports. For example, the valuesmeasured above would result in a symmetrical Tmodel, referred to Port 1, consisting of twoseries inductors of 2 µH each, with a central2 µH shunt inductor. An ideal transformer with1:2 turns ratio is then required to couple the Tnetwork to Port 2. (The actual turns ratio is 1:1,remember.) The coupling coefficient, k equals0.5 (which is the geometric mean between theactual k21 = 1.0 and k12 = 0.25.)

It is incorrect to refer to the series inductorvalues in the “k” model as “leakage inductance”or the shunt inductance as “mutual inductance”.Each of these inductor values is an abstraction,each derived from a combination of the trueleakage and mutual inductances evident only inthe “true” model (whose turns ratio equals thephysical turns ratio).

It is possible to derive the “true” model frommeasured values, if the actual turns ratio isknown and applied to the solution, and allowingunequal values for k12 and k21.

Externally, the “k” model exhibits the samemeasurements as the “true” model, and willprovide the same simulation results. It is indeedequivalent, but totally abstract, and as such itprovides little insight into the structure of thetransformer and the path for its improvement.

The SPICE Coupled Inductor model uses the"k" model which does produces the correctsimulation result, but with abstract transformervalues. If the real turns ratio is known, it isbetter to use the SPICE Inductor model tosimulate the real leakage inductances, and usethe Coupled Inductor model with k = 1 tosimulate the mutual inductance and real turnsratio.

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