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IEEE TRANSACTIONS ON POWERELECTRONICS, VOL. 10, NO. 3, MAY 1995 349

Introduction to Modeling ofTransformers and Coupled Inductors

Arthur F. Witulski

Absb.act--A tutorial paper is presented on modeling and designof transformers and coupled inductors. Beginning with a briefreview of electromagnetic laws and magnetic circuit models,the magnetic and electric models of transformers and coupledinductors are developed, including both magnetizing and leakageeffects. It is shown that while the voltage waveforms on thewindings are primarily related by the turns ratio for both devices,the wiading currents of transformers and coupled inductorsare determined by very different mechanisms. An integratedstructure with both transformer and coupled inductor on tbesame core is also discussed, as well as the special case of thecoupled inductor used on a mdtiple-output transformer-isolatedconverter.

I. INTRODUCTION

LTHOUGH transformers and coupled-inductors eachA onsist of several windings on a magnetic core, and henceare superficially similar, thereare wide-ranging differences intheir principles of operation and in the role they play inswitching converter operation. The coupled-inductor has an

r gap, whereas the transformer does not, and they differ inpower flow, flux levels, current directions, current scaling, andin the effect that leakage inductances have on their terminalwaveforms.A transformeris primarily used for power transfer;the power entering the transformer ideally equals the powerexiting the transformer at any instant in time.A coupledinductor is a device primarily used for energy storage duringa power converter switching cycle; the power entering thecoupled-inductor is not the same as the power leaving it in agiven instant. Transformersare used for voltage and currentscaling, for dc isolation, and to obtain multiple outputs froma single converter. Coupled-inductorsare used to reduce

of transformer and coupled inductor operation, design, andfunctionality in switching converters. The paper is intendedto begin at an introductory level, and develop the necessaryconcepts that enable the practicing engineer to read an d profitfrom the research literature on transformers and coupled-inductors.The structure of the paper isas follows: First, a briefreview is presented of m agnetic and electric circuit modeling.Second, an extended discussion of the physical operation oftransformers and coupled inductors is presented, includingelectric and magnetic waveform s, voltage and current relation-ship between windings, the role of magnetizing inductance,and the direction of power flow. Third, the effects of leakageflux on each structure are examined, including the Occurrence

of the zero ripple condition in coupled inductors. Fourth,the possibility of winding both a transformer and a coupledinductor on the same core is exp lored. Fifth, the specific caseof a coupled inductor used in place of multiple single outputfilter inductors on m ultiple-output converters is examined. Tosummarize, a table is presented that highlights the differencesbetween transformers and coupled inductors.

n. VIEW OF MAGNETICMODELINGONCEITSBefore embarking on a comp arison of coupled inductors and

transformers, it is helpful to review the basic laws governingmagnetic structures, and to develop the m agnetic circuit modelfor analysis of these structures. Tothis end, the three lawsprimarily used in m agnetic analysis, namely, those of Faraday,

Ampe re, and Gauss,are briefly reviewed. The magnetic circuitmodel is seen to be a simplification of these general laws,which lends itself to analysis and design purposes.

converter volume by using one core instead of two (or more),to improve regulation of multi-output converters, andto steerthe ripple current [ l],[2] from one w inding to another.

Althwgh coupled inductorsare widely used, there is someconfusion as to just how they differ from a single windinginductor on the one hand, and from a transformer on theother. Since single-winding inductorsare well understood, itSeems most useful to e xm in e the distinction between coupledinductors and transformers, inthe hope that each can beunderstood better when examine d in the presence of the other.Consequently, this paper is presented as a tutorial review

A* Review Of Ezectmmagnetic Laws

integral form [ll asFaradays law is the first tobe discussed. It is given in its

E - d l = - -at L 2 . d ; (1)f - awhere E is the electric field andB is the magn etic flux density.A more familiar form of Faradays law is

2)d@

v = N -d t

which statesthe voltage induced in a conductor is proportionalto the change in the magnetic flux@ = BA passing throughthe surface enclosed by that conductor (Fig. l(a)). The quantityN is the number of turns of the conductor (only oneturn isshown in Fig. 1). Faradays law explains the Occurrence of

Manuscript received January 26, 1993; revised January 24, 1995.The author is with the Department of Electrical and Computer Engineering,

University of Arizona, Tucson, Az 85721 USA. Paper Was Presented atthe IEEE Applied Power Electronics Conference, San Diego CA, Mar. 8-11,1993.

IEEE L o g Number 9410520.

08854i993/95 04.00 0 1995 IEEE


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350 IEEE TRANSACTIONS ON POWERELECTRONICS, VOL. 10, NO. , MAY 1995

(a) (b)Fig. 1 Faradays law and induced open circuit voltage. (a) side; b) top.

: -

Fig. 2 . . Application of Amperes law to a femte core (a) and the resultingmagnetic model, (b) where A c is area of the core.

an induced voltage in a conductor experiencing a changingmagnetic flux. As shown in Fig. l(b), an electric field isgenerated in a plane perpendicular tothe direction of themagnetic flux. Since the conductor isin this perpendicularplane, the electric field causes charge to move in the conductor.If the conductor is open-circuited, negative charges accumulateat one end and positive charges at the other, resulting in a netvoltage at the two ends of the wire. If the conductor forms aclosed path, current flows in the conductor.

The second law of interest is Amperes law, given in thisquasistatic integral form as

(3)

where H is the magnetic field intensity,dl is an incrementof path length, J is current density, and ZT is the totalcurrent passing through the surfaceS As shown in Fig. 2(a),Amperes law (3) states that the line integral of the magneticfield intensity H around a closed path is equal to the totalcurrent passing through the surfaceS enclosed by the pathC.Notice that the directions ofH and i obey the right-hand rule,i.e., if the thum b is placed in the direction of th e current, thefingers point in the direction of the magnetic field, or if thethumb is in the direction of the field, the fingers of the righthand point in the direction ofthe current. In Fig. 2(a), thewinding intersects the surfaceN times, so the total currentpassing through the surface is Ni While Faradays Lawgoverns the relation between a voltage and the corresponding

magnetic flux, Amperes Law governs the relationship betweena c m n t n a conductor and the corresponding magnetic field.

The third law of interest, Gauss law for magnetic fields,is given by

(4)

I I

(a) b)

Fig. 3. Gausss law: (a) physical model and b) magnetic circuit model.

which states that the net magnetic flux passing through anythree-dimensional body with surfaceSG is zero (Fig. 3(a)). Inother words, the magnetic flux entering the body is equal tothe magnetic flux leaving the body. Hence Gauss law statesthat there is no such thing as a magnetic point charge.

One further relationship is needed before the magneticcircuit model can be developed, and that is the relationshipbetween magnetic flux densityB and magnetic fieldH in agiven material. In free space, the relationship betweenB andH is given by

B = poi? (5 )where po is the perme ability of free space, equal to42~Wm. In a ferrom agnetic material, the relationship betweenBand H is given by the familiarB - H curve, which exhibits bothhysteresis and saturation.In the linear portion of the curve,the relation betweenB and H is

B = p r i = pTp (6)

where the permeabilityp is the product of the relative per-meability pr and the permeability of free spacepo. Forferromagnetic materials, the relative permeability is on theorder of lo3 or lo4, which results in very high flux densityBfor a given magnetic field intensity Hinside a ferromagneticcore.

B. The Magnetic Circuit Model

The magnetic circuit model is a means of simplifying theelectromagnetic lawsso that the line and surface integrals donot have tobe evaluated to find the m agnetic flux. The primaryassumption of the mag netic circuit model is thatthe magneticfield H is constant in a given region, andis aligned in thedirection of the path of integration.This is a good assumptionover a region of a uniform ferromagnetic material, or oversmall distances in free space. WhenH is constant and alignedwith the path, the integral in Amperes law 3) as shown inFig. 2(a) reduces to

H.e = Ni 7)

Application of 6) and the definition@ = B A , yields theexpression

(8)p @?R = Ni.

The quantity ?R is referred to as the reluctance of the pathC and the quantity i s known as the magnetomotive force

e

PAC


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35ITULSKI: INTROD UcIloN TO MODELING OF TRANSFORMERS AND COUPLED INDUCTORS

(mmf), also denoted by S. ince magnetic flux P is linearlyrelated to ia the reluctance, an analog toOhms law canbe established for magnetic circuits,as shown in Fig. 2 b), inwhich mmf is analogous to voltage, flux P is analogous tocurrent, and reluctance is analogous to resistance. Notice thatthe reluctance92 could be subdivided into smaller reluctances,each corresponding to a section of path lengthL 1 , L 2 , etc., inwhich case Amperes law couldbe stated as

(9)i = @ ( 9 2 1 + 922 + e + Xin)or, for a general single-loop magnetic circuit,

n m

which is analogous to Kirchhoffs voltage law for electriccircuits. Finally, if mo re than on e flux path exists,as shown inthe physical model of Fig. 3(a), Gauss law states that the sumof the fluxes going into a magnetic node must be zero, whichis the m agnetic analog toKirchhoff s current law for electriccircuits, as shown in the magnetic circuit model of Fig.3(b).Thus for a given node inthe magnetic circuit model,

k

xCPi=i l

A . Ideal Transfonner Operation

The physical and magnetic model of a two-winding trans-former is shown in Fig. 4 Note that the primary is drivenby a voltage source, and the secondary is terminated by aresistance. The magnetic model consists of anmmf source foreach winding and a reluctance92 = L, /pA, that modelsthe properties of the core. When a voltagev s is applied tothe primary for a length of time, a magnetic flux9 1 mustflow in the core to satisfy Faradays law (Fig. 4(a)). Sincethe permeability p of the core is very high, it is a goodapproximation to assume that all of the flux generated by theprimary winding also links the secondary winding.This time-varying fluxCP1 induces a voltagev2 in the secondary windingaccording to Faradays law. Since the winding is terminatedby a resistance, a current22 flows, which by Amperes lawmust generate a magnetic fieldH 2 , and hence by (5) a fluxdensity B2 or a flux @ 2 in a direction opposite to the flux@ I .Consequently the mmf sourceN2i2 in Fig. 4(c) must have anopposite polarity to N1i l in the magnetic model. The totalflux @ in the core is the difference between91 and @ 2 , or

P = @ I - CP2. Now the voltage scaling relation canbe derivedfrom Faradays Law:

- -Since the same flux P links both sides, the volts perturn in(15) must be equal for both windings. Hence the relationshipbetween primary and secondary voltage is

where the positive direction of flux is defined into the nodeof the magnetic circuit.

It is possible to convert a magnetic circuit to an electric _ -(16)

The current scaling relation can be found from Amperes lawby summ ing themmfs around the loop in the magnetic circuit

circuit model so that well-known electrical circuit analysistechniques can be applied. For example, the magnetic circuit

- = -2 N 2 = nU1 N l

in Fig. 2 b) can be solved for the flux:

N i9 (12) model of Fig. 4(c):= - .

But Farad ays law tells us that the vo ltage induced- inthe N i i l @92 N2i2 = 0. (17)

For an ideal transform er, the core permeabilityp is consideredto be infinite, hence 92 is zero. Thus the current scalingrelation can be found from (17) as

multiple-turn winding shown in Fig.2(a) is

(13)

Hence the voltage canbe written from (12) and (13) as

where L = N 2 / R = p N 2 A ~ / l .s the electrical parameterof inductance in terms of the physical characteristics of thewinding and the magnetic core.

HI. OPERATION OF TRANSFORMER S AND COUPLED blDUcn>RS

In this section the operation of transformers and coupledinductors is described in terms of laws of Faraday, Ampere,and Gauss, and the magnetic circuit model. The physicalstructure of each is discussed, a magnetic model is derivedbased on the physical structure, and an electrical model isdeveloped from the magnetic model. Finally, a comparison ismade of the various magnetic and electric waveforms resultingwhen the same voltage waveform is applied to a transformerand a coupled inductor.

(18)i l N2 n

The electrical model of the ideal transformer is shown inFig. 5 . Thus, for an ideal transformer both current and voltageare scaled by the turns ratio n

The voltage scaling ratio is a result of Farad ays law, and thecurrent scalingratio is a result of Amperes law. Notice thatthere is no energy storage in anideal transformer, hence theinput power is equal to the output poweron an instantaneousbasis. Notice also that impedances canbe reflected to eitherside by use of the scaling laws of 16) and (18), e.g., animpedance 2 can be reflected to the primary side:

i 2 N I 1- = - = - .

Note that the dot convention indicates the polarity of thevoltages on the primary and secondary, i.e., if the voltage onthe primary is positive on the dot side, thenthe voltage of the


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352 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 10, NO. , MAY 1995

i l

ql -. .

a-

N l i l N2i 2

(C) (d)

Fig. 4.model of (c) transformer and (d) coupled inductor.

Structure of (a) transformer and b) coupled inductor, and magnetic

Fi g. 5. Ideal transformer circuit representation.

secondary is positive on the dot side of the secondary. Alsonotice that in the ideal transformer model, the current goesinto the doton the primary (energy-supplying) side and comesout of the doton the secondary (energy-ab sorbing) side. Hencethe direction of power flow in the ideal transformer is from

the primary to the secondary side.

B. Operation of the Coupled Inductor

The structure and magnetic model of a coupled inductor isshown in Figs. 4(b) and 4(d). Both windings are driven bya voltage source, and the direction of the currentsil and i2

C1 VL2

Fig. 6. a) Sepic dual, and b) Forward converter.

are both into the inductor structure. An air gap is present inthe coupled-inductor (CL) structure. Both mmf sources havethe same polarity in the magnetic model, and an additionalreluctance Rg = l g / p o A ,models the r gap. The presenceof the r gap imp lies substantial energy storage in the couple-inductor, which is contrasted to the transformer, in whichenergy storage is incidental.

For an ideal coupled inductor (no leakage) with a singleflux path, the voltage sources driving the windings must beproportional or identical to satisfy Faraday's law becau se eachwinding links the same d @ / d t .These voltage sourcesareprovided by the power converter in which the coupled-inductoris operating. Examples of converters with equal or proportionalvoltage waveformson the inductors are the Cuk, Sepic andSepic dual nonisolated converters. Also, multiple output con-verters with outpu t filter inductors fit this description, includingbuck-derived isolated converters such as the forward, push-pull, full- and half-bridge converters, and the isolated versionsof the Cuk, Sepic, or Sepic dual (zeta) converters[2], [3].The Sepic dual converter and the forward converter shown inFigs. 6(a) and (b) illustratethis property. It can be shown bydc analysis of the Sepic dual that the steady state voltageonthe capacitor C1 of the Sepic dual converter is equal to theoutput voltage V2, hence z 1 ~ 1 nd U L ~ re proportional duringboth D nd 1 D) switching intervals. Likewise,the voltages

on the secondaries of the forward converter transformer arescaled by the transformer turns ratios n = N1/NP andnz = N 2 / N p hence the voltages across the inductorL1 andL2 are proportional.

Now, let us resume the discussion of the physical operationof the coupled inductor. The voltage sourcez1,1 results in aflux @1 in the core as indicated in Fig. 4(b). Likewise, the


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WITULSKI: INTRODUCTION TO MODELING OF TRANSFORMERS A N D COUPLED NDUCTORS

voltage sourcewp results in a flux Q in the same directionas 9 1 . Hence the total flux in the core9 = 9 1 + is largerthan either individual flux. Since the same total flux9 inksboth the N I and N2 windings in Fig. 4(b), by Faradays lawthe voltage scaling relation for coupled inductors is the sameas for transformers ( 1 3 , (16). However, when the sum of themmf drops around the loop in the magnetic model is foundaccording to Amperes law, a different relation from that ofthe transformer is obtained.

N1il ( R c + R,) +Nzi2 = 0. 20)The presence ofR, implies that, in contrast to the transforme r,even if the core permeabilityp were infinite andRc ere zero,the relation betweenil and z would still depend on the fluxlevel in the core, i.e.,

Hence the current scaling law for transformers, (18), doesnot apply to coupled inductors. Rather, the relation betweeni l and i is determined by the extemal converter circuit,and the energy stored in the inductor adjusts appropriatelyto maintain the currents required by the extemal circuit. Forexample, the dc inductor currents of the Sepic dual converterof Fig. 6(a) are determined by the dc operating point of theconverter, i.e., the dc currentI2 must equal the load currentV 2 / Rand the current11 must equal [D/ 1- D)]I2whether ornot coupled inductorsare used, or regardless of the specificnumber of tums on the coupled inductor.This is so becausein steady-state the average current through capacitors mustbezero and the average voltage across inductors mustbe zeroregardless of the value of capacitance or inductance. Noticealso that the coupled inductor is an energ y storage element,soinstantaneous power in does not equal instantaneous powerout as in the transformer.

C. Transformer Operation with Finite Core PermeabilityEarlier, when discussing the ideal transformer, it was found

that if the core permeabilityp is infinite, then the relationbetween currentsil and z is govemed by the transformerturnsratio. However, when the permeabilityp of the core shown inFig. 4(a) is finite, the current relation for a transform er cannotbe described simply by theturns ratio because the reluctance)Ic is nonzero in (17) The fluxiD can be found from (17) for

this situation as

By Faradays law, the primary voltage01 is then given by

d im

. ) dtd 9 N: dd t R dt

211= N I

21 - 2 2= L , -. (23)

The proportionality factor is defined tobe the magnetizinginductor L,, and the resulting current is the magnetizingcurrent i of the transformer. Therefore an inductor Lappears in parallel with the primary in the electric model ofthe transformer as shown in Fig. 7(a). The voltage sca ling law,

353

I

b)

Fig. 7. Electrical models: a) transformer b) coupled inductor.

(16) still holds, but now in the presenc e of finitep the terminalcurrents are no longer related by (18). The magnetizing currenti models the energy needed to magnetize the core. Itis usual to design for low magnetizing current, i.e., usinghigh pr and no intentional air gap. (Foran introductionto design of transformers and inductors,see Chapter 17 of[2].) The magnetizing inductor L, has the properties ofany ferromagnetic core inductor, including saturation andhysteresis.

D . Electrical M odel of the Coupled Inductor

The electrical model of the coup led inductor with finitep issimilar, but not identical to that of the transformer.To obtainthe electrical model. we follow the same derivationas for thetransformer but start with the magnetic model of Fig. 4(d).The equation obtained by summing the mmf drops around theloop 20) can be solved for the flux flowing in the core:

The voltage on the primary side is then given by Faradayslaw:

211 dt R c + R , dtd imd t

L , -.

Once again, a magnetizing inductance is present in the model,as shown in Fig. 7(b). Both the m agnetizing inductance and

current are different from those of the transformer model.Inmost cases, p is much larger thanpo, hence )Ig is much largerthan Rc, o the magnetizing inductance for a coupled inductoris usually approximatedas

26):L , M -.


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354 IEEE TRANSACTIONS ON POWERELECTRONICS, VOL. 10, NO. 3, MAY 1995

The magn etizing current is now the sum of the primary currentil and the current Np lN1 ) 2 .Hence in the electrical modelof Fig . 7(b), the current ip flows in the dot of the idealtransformer model, and N 2 / N 1 ) i 2 lows out of the dot onthe N I side. In the transformer the magnetizing inductanceL , models the energy stored inthe core, but in the coupledinductor model L , models the energy stored in the air gap.

E. Comparison of Waveforms

In Fig. 8 the magnetic and electric waveforms of a trans-former and coupled inductor driven by the same voltage sourcev1 are shown. The quantities are labeled as inFig. 4. Thetransformer primary is driven byw = V I and the coupledinductor is driven by v1 on the N I side and by up = V Ion the N2 side. Since the transformer secondary voltage isa square wave and the load on the secondary is resistive, thereflected secondary current on the primary side is a squarewave, which adds to the magnetizing current to give thewinding currentil. The coupled inductor (CL) primary currenthas a dc component with a triangular ripple component. Theflux @ I of the CL has a dc component. The total flux9 nthe transformer is less than either of the fluxes9 1 r @ 2 , butthe total flux@ of the C L is greater than its compo nent fluxes.The transformer magnetizing current is an ac w aveform, butthe inductor magnetizing amount is a dc waveform with atriangular ac ripple component.

I

ti *Iv. EFFECTS F LEAKAGELUX (a) b)In a physical transformer or coupled inductor, notall of the

flux generated by a wind ing flows in the core, a portion of theflux flow s instead into the r around the windingas shown inFig. 9(a). In the magnetic model these leakage paths canberepresented as additional reluctances in parallel with the mmfsources N1il and N2i2, as shown in Fig. 9(b).

These leakage reluctances give rise to leakage inductances

Lex and L,p in the electrical models of both the transformersand the coupled inductor (Fig.lo), which affect the volt-age and current relations of the transformers and coupledinductors differently. (Becauseof the property of dualitybetween magnetic circuits and electric circuits, a parallel

fig. 8. Comp arison of (a) transformer and (b) coupled inductor ~ av ~ fo r m s.

branch in the magnetic circuit mustbe represented by aseries branch in the electrical circuit[4]). Notice, first ofall, that the transformer is terminated by a resistor, and theCL is terminated on the secondary by a voltage source.In the general case, the transformer secondary may be acombination of active and passive elements, but cannotbea simple voltage source. On the other hand, the CL must beterminated by a voltage source in all cases. Consequently, thesecondary voltage is a depend ent quantity for transformers, butan independent quantity for coupled inductors. Hence, to firstorder, the leakage inductances affect the voltage wav eforms oftransformers, and the current waveform s of coupled indu ctors.The voltage on the secondary of the transformer is given by

m

b)

Fig. 9. (a) Physical and (b) magnetic models of the nonideal transformer.

The average voltage of 212 is the same as before, namelyzero, but the shape of the voltage waveform is changed. Forcoupled inductors, as mentioned earlier, the average value ofthe currents Il nd I2 is determined by the external circuit,


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WITULSKI: INTRODUCTION TO MODELING OF TRANSFORMERSAND COUPLED INDUCIORS 355

hence the addition of leakag e inductances does not change theaverage valueof the currents. Instead, the leakage inductanceschange the waveshape, or ripple, ofthe current. Consider theelectrical model of the C L in Fig. lO(b).As shown in [3], theeffect of the leakage inductances on the ripple current canbeexplained in the following way: Suppose thatv1 = v2. Thenthe voltage vi that appears acrossLm is

The voltagev on the secondary of the ideal transformer modelin the CL electrical circuit is given by

Consequently, in most cases a voltage is present acrossLe2and an ac ripple current flows inLe2 . However, consider thespecial case when theturns ratio is equal to the inverse of theinductive divider:

Then v i = v l = 212, so there is no voltage drop acrossLe2, hence no ac ripple current can flow.In other words,a null condition has been established acrossL e 2 , so bothcurrent and voltage are zero.This is the origin of the wellknown zero ripple condition[2], [3]. Notice that zero-ripplecan be achieved on either the input orthe output, but noton both at the same time. On a multiple-output converterwith coupled inductors, suchas the forward converter withcoupled output inductors, the ripple can be steered fromone output into another.Also, note that in practical casesthe condition 30) cannot be achieved exactly, because ofproduction tolerances, temperature changes, etc.,so a smallvoltage still appears acrossthe leakage inductance. Thus, it isadvantageous to have a large leakage inductance to minimizethe amount of ripple current that flows. Therefore in practice

it is desirable to have a large leakage inductance present ina coupled inductor design to minimize current ripple, but asmall leakage inductance in a transformer design, to minimizevoltage distortion. Consequently, a coupled inductor shouldbewound with the w indingsas physically far apartas is practical,but a transformer shouldbe wound with the windingsas closetogether as possible, preferably in a bi-filar or CO-filar ashion.

v. INTEGRATION OF TRANSFORMER ANDCOUPLED INDUCTOR ON A SINGLE CORE

In many converter applications it is desirable to have bothtransformer isolation and a coupled inductor structure. Forexample in Fig. 11, the inductor L1 of the nonisolated Sepicdual is used as the magnetizing inductance ofan isolation

transformer. The voltage on the primary and the secondaryof the transformer, and on the inductorL2 are proportionalto each other, so both the transformer and the inductor canbe wound on a common core,as shown in Fig. ll(b). Asmentioned earlier, the primary winding and the transformersecondary winding shouldbe closely coupled, but the primaryand the coupled inductor secondary shouldbe loosely coupled.

+v 2-

b)

Fig. 10. Electrical models of (a) transformer and b) coupled inductor withleakage.

L 1

1 I Core II

b)

Fig. 11. Transformer isolated Sepic dual. (a) topology; b) construction oftransformer/coupled inductor integrated structure.

In such an arrangement, the ripple current from the inductorL2 can be steered intothe magnetizing inductance L 1 ) of theprimary w inding.This is also possible with multiple outputs.Thus, more than one transformer secondary could be woundtightly coupled tothe L 1 winding, and more than one coupledinductor winding couldbe wound loosely coupled as shownfor L2, all on a single core. Winding both the transformer andthe CL on the same core changes the conditions needed toobtain zero ripple in the coupled inductor[3], 151.

VI. COUPLED-INDUCTORSN MULTI-OUTPUTCONVERTERS

As mentioned previously, an important use for coupledinductors is as the filter element on multi-output converters,


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WITULSKI. INTRODUCTIONTO MODELING OF TRANSFORMERS A N D COUPLED INDUCTORS

Arthur F. WitulsLi obtained the B.S., M.S., ndPh.D. degrees from the University of Colorado,Boulder, in 1981, 1986, and 1988, respectively.

He has worked as a design engineer and productengineer at a major manufacturer in the area ofpower supply design for large magnetic data storagesub-systems. He is currently an Assistant Professorin the Department of Hectrical and Computer En-gineering at the University of Arizona, Tucson. isresearch interests a ~ .n resonant power conversion,distributed electronic power systems, high power

factor rectification, modeling of magnetic components, and integration ofpower converters for on chip or multichip-module power processing.

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