Qci 94

An Electrical Circuit Model for Magnetic Cores

Lloyd Dixon

Summary:A brief tutorial on magnetic fundamentals leads

into a discussion of magnetic core properties. A

modified version of lntusoft' s magnetic core model

is presented. Low1requency hysteresis is added to

the model, making it suitable for magnetic amplifier

applications.

Fig I. -Magnetic Core B-H Characteristic

surface of Fig. 1 represents energy per unit volume.The area enclosed by the hysteresis loop is unre-coverable energy (loss). The area between thehysteresis loop and the vertical axis is recoverable

stored energy:W!m 3 = fBdH

In Figure 2, the shape is the same as Fig. I, but theaxis labels and values have been changed. Figure 2shows the characteristic of a specific core madefrom the material of Figure I. The flux density axis

Magnetic Fundamentals:Units commonly used in magnetics design are

given in Table I, along with conversion factorsfrom the older CGS system to the SI system (sys-feme infernational- rationalized MKS). SI units areused almost universally throughout the world.Equations used for magnetics design in the SIsystem are much simpler and therefore more intu-itive than their CGS equivalents. Unfortunately,much of the published magnetics data is in the CGSsystem, especially in the United States, requiringconversion to use the SI equations.

Table I. CONVERSION FACTORS, CGS to SI

CGS to SI

10.4

1000/41t

41t.10.7

1

10-4

10.2

10.8

10/41t

109/41t

41t.10.9

FLUX DENSITY 8FIELD INTENSITY H

PERMEABILrrv (space) J.l.PERMEABILrrv (relative) IIyAREA (Core, Window) ALENGTH (Core, Gap) .TOTAL FLUX = fBdA cIITOTAL FIELD = fHdI: F,MMF

RELUCTANCE = FlcII RPERMEANCE = 1/R = L/N2 PINDUCTANCE = P-N2 L

ENERGY W

m2

m

Weber

A.T

cm2

cm

Maxwell

Gilbert

HenIY

Joule

(Henry)Erg 10-7

Figure 1 is the B-H characteristic of a magneticcore material -flux density (Tesla) vs. magneticfield intensity (A-T/m). The slope of a line on thisset of axes is permeability (p = B/H). Area on the Fig 2. -Core Flux vs. Magnetic Force

6-1Electrical Circuit Model for Magnetic Cores

electrical characteristic of the magnetic core woundwith a specific number of turns, N, as shown inFigure 3.

I = HIeN

is tmnsfonned into the total flux, <II, through the

entire cross-sectional area of the core, while field

intensity is transfonned into total magnetic force

around the entire magnetic path length of the core:

"' = B-A .F = H.I'I' e ' e

Area in Fig. 3 again represents energy. this timein electrical terms: W = JEIdt. The slope of a line

in Fig 3 is inductance.

L -dt A-E- = nN2 e

di ,. T

e

Low-Frequency Core Characteristics:Ferromagnetic core materials include: Crystalline

metal alloys, amorphous metal alloys, and ferrites

(ferrimagnetic oxideS).[l]Figure 4 shows the low frequency electrical

characteristics of an inductor with an ungappedtoroidal core of an idealized ferromagnetic metalalloy. This homogeneous core becomes magnetizedat a specific field intensity H (corresponding tospecific current through the winding I=Hf/N). Atthis magnetizing current level, all of the magneticdipoles (domains) within the core gradually align,causing the flux to increase toward saturation. Thedomains cannot align and the flux cannot changeinstantaneously because energy is required. Thechanges occur at a rate governed by the voltageapplied to the winding, according to Faraday's Law.

Thus, to magnetize this core, a specific magnetiz-ing current, IM, is required, and the time to accom-plish the flux change is a function of the voltageapplied to the winding. These factors combined -

The slope of a line on these axes is permeance(P = 11>/F = JlAJ'e). Permeance is the inductance of

one turn wound on the core.Area in Fig. 2 represents total energy -hyster-

esis loss or recoverable energy.Changing the operating point in Fig. lor Fig. 2

requires a change of energy, therefore it can notchange instantaneously. When a winding is coupledto the magnetic core, the electrical to magneticrelationship is governed by Faraday's Law andAmpere's Law.

Faraday's Law:

.:!!. = ~ 11> = ..!.- rEdtdt N N J'

Ampere's Law:

Nl = JHdf z Hf

These laws operate bi-directionally. According toFaraday's Law, flux change is governed by thevoltage applied to the winding (or voltage inducedin the winding is proportional to d<1>/dt). Thus,electrical energy is transfonned into energy lost orstored in the magnetic system (or stored magneticenergy is transfonned into electrical energy).

Applying Faraday's and Ampere's Laws, theaxes can be transfonned again into the equivalent

Fig 4. [deal Magnetic CoreFig 3. -Equiva/ent Electrical Characteristic

UNITRODE CORPORATION6-2

The Effect of Core Thickness: Fig. 1 appliesonly to a very thin toroidal core with Inner Diame-ter almost equal to Outer Diameter resulting in asingle valued magnetic path length (7t'D). Thus thesame field intensity exists throughout the core, andthe entire core is magnetized at the same CUlTentlevel.

A practical toroidal core has an O.D, substan-tially greater than its I,D" causing magnetic pathlength to increase and field intensity to decreasewith increasing diameter. The electrical result isshown in Figure 5. As CUlTent increases, the criticalfield intensity, H, required to align the domains isachieved first at the inner diameter. According to

Ampere:H = NI -

T-

NI

""1tD

Hold on to your hats!! At fIrst, the domainsrealign and the flux changes in the new directiononly at the inner diameter. The entire outer portionof the core is as yet unaffected, because the fieldintensity has not reached the critical level except atthe inside diameter. The outer domains remain fullyaligned in the old direction and the outer fluxdensity remains saturated in the old direction. Infact, the core saturates completely in the newdirection at its inner diameter yet the remainder ofthe core remains saturated in the old direction.Thus, complete flux reversal always takes placestarting from the core inside diameter and progress-ing toward the outside.

In a switching power supply, magnetic devicesare usually driven at the switching frequency by a

cUlTent, voltage, and time -constitute energy putinto the core. The amount of energy put in is thearea between the core characteristic and the verticalaxis. In this case, none of this energy is recoverable-it is all loss, incUlTed immediately while thecurrent and voltage is being applied.

The vertical slope of the characteristic representsan apparent infinite inductance. However, there isno real inductance -no recoverable stored energy-the characteristic is actually resistive. (A resistordriven by a square wave, plotted on the same axes,has the same vertical slope.)

When all of the domains have been aligned, thematerial is saturated, at the flux level correspondingto complete alignment. A further increase in CUlTentproduces little change in flux, and very little volt-age can exist across the winding as the operatingpoint moves out on the saturation characteristic. Thesmall slope in this region is true inductance -

recoverable energy is being stored. With this idealcore, inductance Lo is the same as if there were nocore present, as shown by the dash line through theorigin. The small amount of stored energy is repre-sented by a thin triangular area above the saturationcharacteristic from the vertical axis to the operating

point (not shown).If the current is now interrupted, the flux will

decrease to the residual flux level (point R) on thevertical axis. The small flux reduction requiresreverse Volt-pseconds to remove the small amountof energy previously stored. (If the current isinterrupted rapidly, the short voltage spike will bequite large in amplitude.) As long as the currentremains at zero (open circuit), the flux will remain-forever -at point R.

If a negative magnetizing current is now appliedto the winding, the domains start to realign in theopposite direction. The flux decreases at a ratedetermined by the negative voltage across thewinding, causing the operating point to move downthe characteristic at the left of the vertical axis. Asthe operating point moves down, the cumulativearea between the characteristic and the vertical axisrepresents energy lost in this process. When thehorizontal axis is reached, the net flux is zero -

half the domains are oriented in the old direction,half in the new direction.

6-3Electrical Circuit Model for Magnetic Cores

voltage source. A voltage across the winding causesthe flux to change at a fixed rate. What actuallyhappens is the flux change starts at the core innerdiameter and progresses outward, at a rate equal tothe applied volts/turn, E/N .The entire core isalways saturated, but the inner portion is saturatedin the new direction, while the outer portionsremain saturated in the old direction. (This is thelowest energy state -lower than if the core werecompletely demagnetized.) There is in effect aboundary at the specific diameter where the fieldintensity is at the critical level required for domainrealignment. Flux does not change gradually anduniformly throughout the core!

When the operating point reaches the horizontalaxis, the net flux is zero, but this is achieved withthe inner half of the core saturated in the newdirection, while the outer half is simultaneouslysaturated in the old direction.

When voltage is applied to the winding, the netflux changes by moving the reversal boundaryoutward. The magnetizing current increases toprovide the required field intensity at the largerdiameter. If the O.D. of the core is twice the I.D.,the magnetizing current must vary by 2: I as the netflux traverses from minus to plus saturation. Thisaccounts for the finite slope or "inductance" in thecharacteristic of Fig. 5. The apparent inductance isan illusion. The energy involved is not stored -itis all loss, incurred while the operating point movesalong the characteristic -the energy involved isunrecoverable.

Non-Magnetic Inclusions: Figure 6 goesanother step further away from the ideal, with

considerable additional skewing of the character-istic. This slope arises from the inclusion of smallnon-magnetic regions in series with the magneticcore material. For example, such regions could bethe non-magnetic binder that holds the particlestogether in a metal powder core, or tiny gaps at theimperfect mating surfaces of two core halves.Additional magnetic force is required, proportionalto the amount of flux, to push the flux across thesesmall gaps. The resulting energy stored in thesegaps is theoretically recoverable. To find out howmuch energy is loss and how much is recoverablelook at Figure 6. If the core is saturated, the energywithin triangle S- V -R is recoverable because it isbetween the operating point S and the vertical axis,and outside the hysteresis loop. That doesn't ensurethe energy will be recovered -it could end updumped into a dissipative snubber.

Another important aspect of the skewing result-ing from the non-magnetic inclusions is that theresidual flux (point R) becomes much less than thesaturation flux level. To remain saturated, the coremust now be driven by sufficient magnetizingcurrent. When the circuit is opened, forcing themagnetizing current to zero, the core will resetitself to the lower residual flux level at R.

Reviewing some Principles:.Ideal magnetic materials do not store energy, but

they do dissipate the energy contained within thehysteresis loop. (Think of this loss as a result of"friction" in rotating the magnetic dipoles.)

.Energy is stored, not dissipated, in non-magnetic

regions..Magnetic materials do provide an easy path for

flux, thus they serve as "magnetic bus bars" tolink several coils to each other (in a transformer)or link a coil to a gap for storing energy (an

inductor)..High inductance does not equate to high energy

storage. Flux swing is always limited by satura-tion or by core losses. High inductance requiresless magnetizing current to reach the flux limit,hence less energy is stored. Referring to Figure6, if the gap is made larger, further skewing thecharacteristic and lowering the inductance,triangle S- V -R gets bigger, indicating nwrestored energy.Fig 6. -Non-Magnetic Inclusions

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Fig 8. -Large Air GapFig 7. -Non-Homogeneous Effects

outside of the hysteresis loop, is relatively huge.The recoverable energy is almost all stored in theadded gap. A little energy is stored in the non-magnetic inclusions within the core. Almost zeroenergy is stored in the magnetic core material itself.

With powdered metal cores, such as Mo-Permal-loy, the large gap is distributed between the metalparticles, in the non-magnetic binder which holdsthe core together. The amount of binder determinesthe effective total non-magnetic gap. This is usuallytranslated into an equivalent permeability value forthe composite core.

Non-Homogeneous Aspects: Figure 7 is thesame as Figure 6 with the sharp comers roundedoff, thereby approaching the observed shape ofactual magnetic cores. The rounding is due to non-homogeneous aspects of the core material and core

shape.Material anomalies that can skew and round the

characteristic include such things as variability inease of magnetizing the grains or particles thatmake up the material, contaminants, precipitation ofmetallic constituents, etc.

Core shapes which have sharp comers willparadoxically contribute to rounded comers in themagnetic characteristic. Field intensity and fluxdensity are considerably crowded around insidecomers. As a result, these areas will saturate beforethe rest of the core, causing the flux to shift to alonger path as saturation is approached. Toriodalcore shapes are relatively free of these effects.

Adding a Large Air Gap: The cores depictedin Figures 4 -7 have little or no stored recoverableenergy. This is a desirable characteristic for Mag-amps and conventional transformers. But filterinductors and flyback transformers require a greatdeal of stored energy, and the characteristics ofFigures 4 -7 are unsuitable.

Figure 8 is the same core as in Fig. 7 with muchlarger gap(s) -a few millimeters total. This causesa much more radical skewing of the characteristicThe horizontal axis scale (magnetizing current) isperhaps 50 times greater than in Figure 7. Thus thestored, recoverable energy in triangle S- V -R,

Core Eddy Current Losses:Up to this point, the low frequency characteris-

tics of magnetic cores have been considered. Themost important distinction at high frequencies isthat the core eddy currents become significant andeventually become the dominant factor in corelosses. Eddy currents also exist in the windings ofmagnetic devices, causing increased copper lossesat high frequencies, but this is a separate topic, notdiscussed in this paper .

Eddy currents arise because voltage is inducedwithin the magnetic core, just as it is induced in thewindings overlaying the core. Since all magneticcore materials have finite resistivity, the inducedvoltage causes an eddy current to circulate withinthe core. The resulting core loss is in addition tothe low frequency hysteresis loss.

Ferrite cores have relatively high resistivity. Thisreduces loss, making them well suited for high fre-quency power applications. Further improvements in

6-5Electrical Circuit Model for Magnetic Cores

0--L

RE

eddy culTent losses as Irequency dependent. Lossesreally depend on rate of flux change, and thereforeaccording to Faraday , s Law, upon the applied

volts/turn. Frequency is relevant only in the case ofsinusoidal or symmetrical square wave voltagewaveforms.

In a switching power supply operating at a fixedfrequency, Is, core eddy CUlTent losses vary withpulse voltage amplitude squared. and inversely ~ithpulse width -exactly the same as for a discreteresistor connected across the winding:

2Vp tpLoss = --RE T

If the pulse voltage is doubled and pulse widthhalved, the same flux swing occurs, but at twice therate. V; is quadrupled, tp is halved -losses double.

If the flux swing and the duty cycle is main-tained constant, eddy CUlTent loss varies with 1s2(but usually the flux swing is reduced at higherfrequency to avoid excessive loss).

o-UJFig 9. -Core Eddy Current Model

high frequency power ferrite materials focus onachieving higher resistivity. Amorphous metal coresand especially crystalline metal cores have muchlower resistivity and therefore higher losses. Thesecores are built-up with very thin laminations. Thisdrastically reduces the voltages induced within thecore because of the small cross section area of eachlamination.

The core can be considered to be a single-turnwinding which couples the eddy current loss resis-tance into any actual winding. Thus, as shown inFigure 9, the high frequency eddy current lossresistance can be modeled as a resistor RE inparallel with a winding which represents all of thelow frequency properties of the device.

In Figure 10, the solid line shows the low fre-quency characteristic of a magnetic core, with dashlines labeled /1 and /2 showing how the hysteresisloop effectively widens at successively higher fre-quencies. Curves like this frequently appear onmanufacturer's data sheets. They are not very usefulfor switching power supply design, because they arebased on frequency, assuming symmetrical drivewaveforms, which is not the case in switching

power supplies.In fact, it is really not appropriate to think of

I rEele

1-7"'I I

I f, ,, ,

I'

:f1 !f1-

I I

,/fz. If,

I, ,

I ,I I

Forward Converter Illustration:Figure 11 provides an analysis of transformer

operation in a typical forward converter. Accompa-nying waveforms are in Fig. 12. The solid line inFig. 11 is the low-frequency characteristic of theferrite core. The dash lines show the actual path ofthe operating point, including core eddy currents re-flected into the winding. Line x- y is the mid-pointof the low frequency hysteresis curve. Hysteresisloss will be incurred to the right of this line as theflux increases, to the left of this line when the fluxdecreases.

Just before the power pulse is applied to thewinding, the operating point is at point R, theresidual flux level. When the positive (forward)pulse is applied, the current rises rapidly from R toD (there is no time constraint along this axis). Thecurrent at D includes a low-frequency magnetizingcomponent plus an eddy current component propor-tional to the applied forward voltage. The fluxincreases in the positive direction at a rate equal tothe applied volts/turn.

As the flux progresses upward, some of theenergy taken from the source is stored, some isloss. Point E is reached at the end of the positive

I ff f,, f

I II I

I

Fig 10. -L.F. Hysteresis plus Eddy Current

6-6 UNITRODE CORPORATION

Fig 12. -Forward Converter WaveformsFig 11. -Forward Converter Core Flux, IM

In a forward converter operating at a fixedswitching frequency, a specific V -ps forward pulseis required to obtain the desired VOUT. When VINchanges, pulse width changes inversely. The trans-former flux will always change the same amount,from D to E, but with higher VIN the flux changesmore rapidly. Since the higher VIN is also across RE,the equivalent eddy current resistance, the eddycurrent and associated loss will be proportional toVIN. Worst case for eddy current loss is at high line.

pulse. The energy enclosed by X-D-E- Y -X has beendissipated in the core, about half as hysteresis loss,half eddy current loss as shown. The energy en-closed by R-X- Y -B-R is stored (temporarily).

When the power switch turns off, removing theforward voltage, the stored energy causes thevoltage to rapidly swing negative to reset the core,and the operating point moves rapidly from E to A.Assuming the reverse voltage is clamped at thesame level as the forward voltage, the eddy currentmagnitude is the same in both directions, and theflux will decrease at the same rate that it increasedduring the forward interval.

As the operating point moves from A to C, thecurrent delivered into the clamp is small. Duringthis interval, a little energy is delivered to thesource, none is received from the source. Most ofthe energy that had been temporarily stored at pointE is turned into hysteresis and eddy current loss asthe flux moves from A to C to R. The only energyrecovered is the area of the small triangle A-B-C.

Note that as the flux diminishes, the current intothe clamp reaches zero at point C. The clamp diodeprevents the current from going negative, so thewinding disconnects from the clamp. The voltagetails off toward zero, while previously stored energycontinues to supply the remaining hysteresis andeddy current losses. Because the voltage is dimin-ishing, the flux slows down as it moves from C toD. Therefore the eddy current also diminishes. Thetotal eddy current loss on the way down through thetrapezoidal region A-C-R is therefore slightly lessthan on the way up through D and E.

Magnetic Core Circuit Model:A modified version of Intusoft's magnetic core

model is shown in Fig- 13- This model is a two-terminal device (plus a third terminal for monitoringflux level) that can be used in a wide range ofmagnetic core applications- It can directly simulatean inductor or saturable reactor (magamp). Addedto an ideal transformer, it can simulate a flyback orconventional transformer -

A Spice subcircuit net list is given in Fig. 14 .Electrical parameters must be calculated from themagnetic design and inserted into the model eitherdirectly, replacing the expressions within curlybrackets with numerical values, or by parameterpassing with the subcircuit call-

Definable parameters include (SI Units):

SVSEC Volt-sec at saturation = BsAT-AE-N

IVSEC Volt-sec Initial condition = B-AEN

LMAG Unsaturated Inductance = )lo)l;N2 AE/'

LSA T Saturated Inductance = )loN2 AE/'E

6-7Electrical Circuit Model for Magnetic Cores

LSAT: Use core dimensions but with Pr = I

(saturated core is non-magnetic)

REDDY One approach is to determine frequencywhere permeability vs frequency is 3dBdown. REDDY equals LMAG reactanceat this frequency.

Description of the model: Magnetizing currentassociated with low frequency hysteresis is providedby current sinks IHI and IH2. With no voltageacross the terminals I and 2, these currents circulatethrough their respective diodes, and the net terminalcurrent is zero. When voltage is applied, the appro-priate diode starts to block, and its current sinkbecomes active.

Terminal voltage is applied to source G I whosegain is INI V. G I output current drives integratorcapacitor CI, whose voltage V(3,2) is proportionalto the integrated V-ps across the terminals. CIvalue is calculated so the V -ps at saturation for thecore being simulated translates into 250V acrossC I. Source E I, with gain I V II V, prevents C I frombeing loaded by downstream circuit impedances.

V(4,2), same as V(3,2) drives resistor RB whichsimulates the normal inductance (below saturation).Resistor RS simulates the saturated inductance ofthe core, but RS cannot conduct until V(4,2) ex-ceeds the 250V sources VS lor VS2.

The capacitance of diodes DS I and DS2 is

IHYST Magnetizing I at O Flux = Hl.pjN

REDDY Eddy current loss resistance

Notes on parameter calculation:

LMAG: For ungapped core, I. = I.E, (total path

around core). For gapped core, Pr = I,I. = gap length. AE = core area, m2.

Fig ]4 -Core Model Net List

.SUBCKT COREH 1 210DH1 1 9 DHYSTIH1 9 1 {IHYST}DH2 2 9 DHYSTIH2 9 2 {IHYST}G1 2 3 1 2 1C1 3 2 {SVSEC/250} IC={IVSEC/SVSEC*250}E1 4 2 3 2 1VM 45RB 5 2 {LMAG*250/SVSEC}RS 56 {LSAT*250/SVSEC}D1 67 DCLAMPVP 7 2 250D2 86 DCLAMPVN 2 8 250F1 1 2 VM 1E2 10 0 3 {SVSEC/250}.MODEL DHYST D.MODEL DCLAMP D(CJO={3*SVSEC

+/(250*REDDY)} VJ=25).ENDS

6-8 UNITRODE CORPORATION

the integration capacitor CI value is calculated toreach a nonnalized value of SOOV when the inte-grated input V -ps reaches the value SVSEC corre-sponding to saturation. Thus the voltage on thecapacitor is not equal to the actual V -ps, but theV -ps times SOO/SVSEC. Also, in the originalmodel, the voltage viewed at tenninal 3 indicatesthe integrated V-ps across Cl only when tenninal2 is at ground potential. In applications wheretenninal 2 is not grounded, tenninal 3 does notshow the voltage across C I, as intended. Theseproblems are eliminated by adding E2, whichprovides a corrected, ground referenced V -ps valueviewed at tenninal 10.

Finally, the original model requires the use ofpeak-to-peak V -ps values for SVSEC and IVSEC,corresponding to p-p flux swing from minus to plussaturation. In the modified model, SOOV (P-p) ischanged to 2S0V (peak) so that SVSEC and IVSECvalues based on peak flux levels can be used, as iscustomary .

References:

[1] T. G. Wilson, Sr., "Fundamentals of MagneticMaterials," APEC Tutorial Seminar 1, 1987

[2] "Saturable Reactor Model," lsSpice User'sGuide pp 329-337, Intusoft, 1994

calculated to provide a current simulating the eddycurrent loss resistance.

The total current flowing from the output of Elis sensed by the measuring source VM (0 Volts),and injected into the terminals I and 2 by sourceFI, whose gain is INIA.

It may seem strange to have no inductors in amodel that basically simulates inductance. Insteadof using RB to simulate LMAG, why not putLMAG directly across the terminals??

In a simulation run it is usually necessary toestablish initial conditions for each inductor andcapacitor. In this application it is almost impossibleto define a set of initial conditions that would notconflict, causing the run to fail. It is always best tominimize the number of elements that store energy,to reduce the number of initial conditions that mustbe specified.

In this model, it is fundamentally necessary tointegrate the terminal voltage to know when satura-tion is reached. The initial condition for the inte-grating capacitor is Volt-seconds, corresponding toflux in the core. In switching power supply applica-tions, magnetic devices are usually driven byvoltage sources, and the volt-second operating pointis much easier to predict than inductor magnetizingcurrents, which depend on many uncertain vari-abies.

So it is much easier to go with the integratingcapacitor as the only energy storage device requir-ing an initial condition, and derive all the othervalues indirectly from the integrated terminalvoltage across CI.

Since the voltage across CI is not the terminalvoltage, but the integral of the terminal voltage,resistance (RB, Rs) is required to simulate induc-tance values, voltages (V SI, VS2) are required tosimulate saturation flux <fv -ps), and diode capaci-tance (DSI, DS2) simulates eddy current loss resis-tance (REDDY)

The original Intusoft version has one seriousshortcoming -it does not include low frequencyhysteresis. This precludes its use in magamp appli-cations, because without dc hysteresis, residual fluxis zero, and flux collapses when open-circuited. Themodified model includes dc hysteresis.

Another problem with the original model is that

Electrical Circuit Model for Magnetic Cores 6-9

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