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Ocl 94

Lloyd Dixon

provide a simple electrical equivalent circuit thatlends itself to circuit analysis.The magnetic structure shown in Figure 1 will bethe first demonstration of this technique. Thissimple inductor is built upon a ferrite core, with airgaps to store the required inductive energy createdby simply shimming the two core halves apart (notusually a good practice, but inexpensive).

Purpose:I. To define the electrical circuit equivalents of

magnetic device structures to enable improvedanalysis of circuit performance.

2. To define the magnitude and location ofrelevant parasitic magnetic elements to enableprediction of performance effects

3. To manipulate parasitic elements to obtainimproved or enhanced circuit performance4. To encourage the circuit designer to be more

involved with magnetic circuit design.Magnetic Definitions:

Systeme International (SI) Units and Equationsare used throughout this paper .Simplifying the Magnetic Structure:

The fIrst task is to take a magnetic devicestructure and reduce it to a few lumped elements -as few as possible for the sake of simplicity.because tllC equivalent electrical circuit will haveust as many elements. This is not an easy task -just about everything is distributed. not lumped: themagnetic force from current in the windings.distributed flux. fringing flux adjacent to gaps. andstray fields. Boiling this down to a few elementswith reasonable accuracy requires a little insightand intuition and experience -but it can be done.

Finite Element Analysis softwarc on the otherhand is extremely accurate because it does just tlleopposite -it chops the structure up unto a hugenumber of tiny elements.[!] It is a useful tool whichcan provide a great deal of knowlcdge about whatgoes on inside the magnetic device. but it does not

Fig ,. -Magnetic Structure -InductorThe first thing to do is to simplify -judiciously.

Looking at the inherent symmetry of the structurethe two outer legs can be combined into a singleleg with twice the area. Fringing fields around thegaps will be ignored, except the effective gap areamight be increased to take the fringing field intoaccount.

The number of flux paths will be minimized,eliminating those that are trivial. For example, fluxin the non-magnetic mateial adjacent to the corewill be ignored, because it is trivial compared tothe flux in the neighboring ferrite. On the otherhand, if there were two windings, the small amounof flux between the windings must not be ignored-it constitutes the small but important leakage

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combining the two centerpost halves into a singelement, likewise combining the outer leg portioon both sides of the gap. The magnetic field sourcthe ampere-turns of the winding, which is reallycirculatory field is assigned to any discrete poiwhere the nux is not divided. It would be incorreto locate this source in series with the outer lebecause t would drive the nux in the stray field the wrong direction.

The core will be divided into regions of similarand flux density. The distributed

force from the winding will be lumpedassigned a specific location in the physical

ASSUNE TYPICAL VALUES :A.=1cmz: te=10cmAw=2cmz (ETO3~)

NAX Ht=Nlm=JmAw=~OO.2=800

~u, i

~011,

z'r-- ~AX ENERGYFULL UTILIZ:

WmEHt.BmA.!Z=800..2S..0001!2-10mJGAP LENGTH FULL UTILIZ:

tg=NI!H-NII"!B=600...'..1 0-7j 25=.4cmFig 2. -Core Parameters and Utilization

For the purpose of illustration. the parameters ofcore (ETD34) are given in Figure 2. The

ampere-turns obtainable is calculatedthe window area times the max. current

of 400Ncm2. The maximum fluxthe saturation flux density

times the core area. From this. the maxi-. for a total of IOmJ. The gap

required to achieve this full utilization isthe modeling process. but they

the suitability of this core for .theThe Reluctance Diagram: Next, a reluctance

be created. modeling the physicalIt is a measure of the opposition to fluxany region of the magnetic device.

F H. .R=-=~--'~cI>

RELUCTANCE ~ MMfl--,

R -HliBA = 'I ~A

-. ! APPORTION ,. TO Re, Ro APPTN 19 TO Rcg, Rog.13 ~=~o~r, ~0=47T.10-7

~~ ~r= 1 (o;r), =3000 (f.rrlR Ro=(l.12)1(JJ.~rA.)ogI& =.051(JJ.oJJ.r..0001

Ro=.133.101 (omit 10Ra IS AN ESTIMATEFig 3. -The Simplified Reluctance ModelSo the reluctance diagram includes the reluctan

of the combined centerpost ferrite and also tcenterleg gap, the combined outer leg ferrite athe outer leg gap, and the reluctance of the strfield outside the core (the calculation is an educatguess). This "magnetic circuit" can be analyzed juas though it were an electrical circuit. Remembthat it is not an elecrical circuit -reluctance definitely not the same as resistance -it storenergy rather than dissipate it. But the reluctandiagram does follow the same rules as an electriccircuit, and the amount of flux in each path can calculated based on the magnetic force and treluctance.Much can be learned by examining the relutance model and playing .'what if'. games. Note thin each leg, the calculated gap reluctances are mothan 100 times greater than the adjacent ferrite colegs. This indicates that the core reluctances coube eliminated, impairing accuracy by less than I Note also that the flux in the suay field is actularge than the flux in the outer leg of the cobecause the suay field reluctance is less than tgap reluctance. This means that much noise propageted outside the core, and the inductan

---BA fL4The reluctance of each significant region of the

its area, length andand inserted with its specific value

as ShO\\'ll in Figure 3. Again, because ofthe model can be simplified by

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value obtained depends heavily on the stray field,which is difficult to calculate. If the centerleg gapis eliminated and the outer leg gap correspondinglyincrease, the amount of s.l:ray ield increases further.

On the other hand, if the outer leg gap is closedand the centerleg gap is widened, the stray field isalmost eliminated, because the reluctance of thecenterleg without gap is much less than the strayfield reluctance. In this manner, the reluctancemodel is useful without even converting it to theequivalent electrical circuit.

to the dual.A planar circuit is defined as one that can b

drawn on a plane surface with no crossovers. Thduality process fails if there are crossovers, whiccan occur with complex core structures.ISJActuallywith only three windings on a simple core, if threluctance model includes every theoreticallpossible flux linkage combination between windings, there will be crossovers. But most of thetheoretical linkages are trivial, and should bignored, for the sake of simplicity if nothing elsThis is where common sense comes in.

Creating the Dual: The process for creating thelectrical dual from the reluctance model is actualquite simple, as illustrated in Figure 4.

1~ ,-51-'+.~111/, \: ! I 0, , Ir ' I" ' ,, , l ' "" 1 \ U' '": I -c ';: -0 \ RS" I \ ~-- -I " \\ .,J fl .., ".-,..' -RCG OG,

Magnetic-Electrical Duality:Fifty years ago, E. Colin Cherry published apaper showing the duality between magnetic circuitsand electrical circuits.[3] It is well know that twoelectrical circuits can be duals of each other -theCuk converter is the dual of the flyback (buck-boost), for example. Electrical circuits that are dualsare not therefore equivalent. Magnetic circuits andelectrical circuits are in a different realm, and yetin this case the duals are truly equivalent.

A dual is created by essentially turning thecircuit inside out and upside down. Some of themagnetic-electrical duality relationships and rulesare: ', ~

Fig 4. -The Magnetic / Electrical DualFirst, identify each mesh, or loop in the reluc

tance model. In this case there are 3 loops. (Thoutside is always considered a loop. Topologicallya simple circle has two loops -the inside and thoutside.) Put a dot in tile center of each loop (anconvenient location on the outside). These dots wibe the nodes of the electrical circuit. Draw a dasline from electrical node to node though everintervening element. The dash lines are branches ithe electrical circuit. The intervening elemenbecome elements of the new circuit, but they atransformed: Reluctances become their reciprocal-permeances, the magnetic winding becomes thelectrical terminals, with d(j)l/dt translating intVl/Nl (Faraday's Law), and magnetic force tranlating into NlIl in series with the terminals. Notthat the 5 nodes in the original reluctance modare automatically converted into 5 loops in thelectrical dual. (Don't forget the outside is a loop

Nodes Meshes (loops)Open ShortSeries Elements Parallel ElementsMagn. Force Ampere- Turnsdcl>/dt Volts/turnReluctance Permeance

Polarity Orientation: Rotate in same directionCircuits must be planar

Permeance is the reciprocal of reluctance:p=2.=~R l

Pemleance is actually the inductance for 1 turn.Multiply pemleance by N2 to obtain the inductancevalue referred to all N-turn winding.

Polarity Orientation means that for elementssuch as the windings that have polarity, to assignthe proper polarity in the dual, rotate all polarityindications in the same direction from the original

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AIl of the values in the equivalent electrical circuitat this stage pertain to a one-turn winding. Theelectrical equivalent circuit is redrawn ~ shown inFigure 5:

UNITS: Henry./Turn 2

inductance. But if the outer leg gap was closed upits inductance would become infinite, making thstray field inductance irrelevant. The overall inductance would then equal the 14~ of the centerlegap.A simple transformer: A transformer with twwindings is shown in Figure 7. The transformer bano gap -energy storage is undesirable. The flubetween the two windings, although small, is verimportant because the energy contained between thwindings constitutes leakage inductance.

p = 1/R = ~A/1 = L/N2

Po = 1/Ro = 7.5~H/N2

V1/N1 = d-l/dl

N111 = Hi

Fig 5. -The Equivalent Electrical CircuitIf the winding has fifteen turns, penneances are

converted to inductance values by multiplying by152. Likewise, tenninal V/N is multiplied by 15 tobecome tenninal voltage, and NI is divided by N tobecome tenninal current. The final electrical equiv-alent circuit is shown below:WITH N = 15 TURNS

Figure 8 is the reluctance model with specifivalues calculated for the same Em34 core used ithe previous example, but ungapped. The amperturns in the two windings cancel except through thregion between the windings (RI2). Thus, when thtransformer is loaded, this is the only place thfields don't cancel, storing considerable leakaginductance energy as a function of load current.

Fig 6. -Final Electrical Equivalent CircuitThis simple example obviously produces a trivial

result. There are five inductive elements whichrepresent the five reluctances in the reluctancemodel. The five inductances combine to form asingle inductance value of IO~. But each of thefive elements is clearly identifiable back to thereluctance it represents. Note that the series reluc-tances of the centerleg ferrite and centerleg gapshow up as parallel inductances in Fig. 6. and thestray field reluctance which was in parallel with theouter leg are now series inductances. The circuitshows that the high inductance values contributedby the centerleg ferrite and outer leg ferrite areirrelevant in parallel with the much lower gapinductances. It shows that the stray field inductancemakes a very significant contribution to the overall

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transfonner is added to one tenninal pair to allowfor a turns ratio other than 1:1, and to providegalvanic isolation.

Coupled Inductor: Topic M7 in the DesignReference Section of the Seminar Manual describethe benefits of coupled filter inductors in multi-output buck regulators. Figure 12 shows the physi-cal structure and resulting equivalent circuit whichcan provide insight into the design of this device.

L12--r-rv;'"'-I, N2 = 15 TURNSL IN J.LH Log,

14.(~~~~ ~~

R.f.Topic...7. PJ

4 LOOPS, 4 NODES IN BOTHFig 9. -The Transformer Electrical Dual

WITH Np .20 TURNS, N. -2 TURNS -:Leg L12

Lo +L. -~

Fig 12. -Coupled InductorFractional Tur~: Transfonners with fractiona

turns have been featured in previous SeminarsFigure 13 provides insight into the design andbehavior of this device.

Fig 10. -Transformer Equivalent CircuitA Three-Winding Transformer: Figure 11 is

a three-winding transfonner showing the physicalstructure and the equivalent electrical circuit. Thereluctance model is not shown. The equivalentcircuit model shows how the leakage inductancesare distributed between windings and their magni-tudes.

~ Vln ,.. ;Hpl2 L1 IV."I~] IN."""L2

ErrECTIVE LEAKAGE L --0HOW TO STIrrEN UP Vp V. l3

y -1

Fig 13. -Transformer with Fractional Turns

'L'

The Gyrator-Capacitor Approach:Another method for defining an equivalen

electrical circuit of a magnetic device structurecompletely avoids the Reluctance/Dualityapproachthat has been discussed up to this point.l4.5J Th

PER~EANCE IN ~H/turn2

Fig 11. -Three-Winding Transformer

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equivalent circuit layout achieved with the Gyrator-Capacitor follows exactly the pattern of the magnet-ic structure. This makes it somewhat easier to relateelectrical performance back to the magnetic ele-ments. A major advantage of this method is that itdoes not require that the magnetic circuit be "pla-nar", as the Reluctance/Dualitymethod does. Propo-nents of the Gyrator-Capacitor method cite theseadvantages.However, the equivalent electrical circuit thatresults from this method looks nothing like aclassical inductor or transformer. Inductive energystorage elements are replaced by capacitors (theelectrical dual of inductance), and transformerwindings are replaced by gyrators. (A gyrator is anideal two-port element which one port reflects thereciprocal of the impedance at the other port, andscales the impedance according to a factor ~ 0) Inother words, the gyrators translate the perfonnanceof the equivalent circuit, employing capacitors, intoits electrical dual, employing inductances, but onenever actually sees he dualized equivalent circuit -the gyrators take care of it transparently. Figure 14shows the equivalent circuit of a simple flybackconverter modeled using the Gyrator-Capacitormethod.

but it does no1 resemble the magnetic devicephysical structure, diminishing insight into thephysical-electrical relationship, or(2) An equivalent circuit with capacitos andgyrators replacing inductive elements whose layoutclosely resembles the structure of the magneticdevice. This facilitates insight into the physical-electrical relationship, but severely diminishesinsight into circuit analysis.

The author prefers to remain with the Reluc-tanceIDuality method, but admits that it's probablya matter of what one is used to and more comfort-able with.References:[1] R. Sevems, "Finite Element Analysis in Power

Converter Design," IEEE APEC Proc. pp. 3-9,Feb. 1994

[2] Dauhajre and Middlebrook, "Modelling andEstimation of Leakage Phenomena in MagneticCircuits,"IEEE PESC Record, pp. 213-226,1986

[3] E. C. Cherry, "The Duality Between Electric andMagnetic Circuits and the Formation of Trans-former Equivalent Circuits," Proc. Physical Soc.London, VoI62Bpp.101-111, Feb. 1949

[4] D.C. Hamill, "Lumped Equivalent Circuits ofMagnetic Components: The Gyrator-CapacitorApproach," IEEE Trans. on Power Electronics,vol. 8, no.2, pp. 97-103, April 1993

[5] D.C. Hamill, "Gyrator-Capacitor Modeling: ABetter Way of Understanding Magnetic Compo-nents,"IEEE APEC Proc. pp. 326-332, Feb. 1994

+Yo

Fig 14. -Gyrator-Capacitor Modelof a Forward ConverterSo the choice is -do you prefer:

(I) A conventional electrical circuit whose mag-netic elements include leakage and magnetizinginductances and transformer windings, etc. Thisfacilitates intuitive and insightful circuit analysis,

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