An Electrical Circuit Model for Magnetic Cores

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An Electrical Circuit Model for Magnetic CoresLloyd Dixon

Summary:A brief tutorial on magnetic fundamentals leadsinto a discussion of magnetic core properties. Amodified version of lntusoft' s magnetic core modelis presented. Low1requency hysteresis is added tothe model, making it suitable for magnetic amplifierapplications.

Fig I. -Magnetic Core B-H Characteristicsurface of Fig. 1 represents energy per unit voluThe area enclosed by the hysteresis loop is ucoverable energy (loss). The area between hysteresis loop and the vertical axis is recoverstored energy:

W!m 3 = fBdHIn Figure 2, the shape is the same as Fig. I, butaxis labels and values have been changed. Figurshows the characteristic of a specific core mfrom the material of Figure I. The flux density

Magnetic Fundamentals:Units commonly used in magnetics design aregiven in Table I, along with conversion factors

from the older CGS system to the SI system (sys-feme nfernational- 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. CONVERSIONFACTORS,CGS to SICGS to SI

10.41000/41t41t.10.7

110-410.210.8

10/41t109/41t

41t.10.9

FLUX DENSITY 8FIELD INTENSITY HPERMEABILrrv (space) J.l.PERMEABILrrv (relative) IIyAREA (Core, Window) ALENGTH (Core, Gap) .TOTAL FLUX = fBdA cIITOTAL FIELD = fHdI: F,MMFRELUCTANCE= FlcII RPERMEANCE= 1/R = L/N2 PINDUCTANCE= P-N2 LENERGY W

m2m

WeberA.T

cm2cm

MaxwellGilbert

HenIYJoule

(Henry)Erg 10-7Figure 1 is the B-H characteristic of a magnetic

core 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

lectrical Circuit Model for Magnetic Cores


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

I = HIeN

tmnsfonned into the total flux,


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The Effect of Core Thickness: Fig. 1 apponly to a very thin toroidal core with Inner Diamter almost equal to Outer Diameter resulting isingle valued magnetic path length (7t'D). Thussame field intensity exists throughout the core, the entire core is magnetized at the same CUlevel.A practical toroidal core has an O.D, substially greater than its I,D" causing magnetic plength to increase and field intensity to decrewith increasing diameter. The electrical resulshown in Figure 5. As CUlTent ncreases, the critfield intensity, H, required to align the domainachieved first at the inner diameter. AccordingAmpere:

H = NI -T-

NI

""1tD

Hold on to your hats!! At fIrst, the domrealign and the flux changes in the new direconly at the inner diameter. The entire outer porof the core is as yet unaffected, because the fintensity has not reached the critical level excethe inside diameter. The outer domains remain faligned in the old direction and the outer density remains saturated in the old direction.fact, the core saturates completely in the direction at its inner diameter yet the remaindethe core remains saturated in the old directThus, complete flux reversal always takes pstarting from the core inside diameter and progring toward the outside.In a switching power supply, magnetic devare usually driven at the switching frequency

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 operatingpoint (not shown).If the current is now interrupted, the flux willdecrease 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.



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A voltage across the winding causesflux to change at a fixed rate. What actuallyis the flux change starts at the core inner

outward, at a rate equal toapplied volts/turn, E/N .The entire core is

but the inner portion is saturated, while the outer portionsin the old direction. (This is the

rgy state -lower than if the core were

the critical level required for domaindoes not change gradually and

throughout the core!When the operating point reaches the horizontalflux is zero, but this is achieved with

of the core saturated in the new

When voltage is applied to the winding, the netchanges by moving the reversal boundary

magnetizing current increases tothe required field intensity at the larger

the O.D. of the core is twice the I.D.,must vary by 2: I as the net

traverses from minus to plus saturation. Thisfor the finite slope or "inductance" in the

Fig. 5. The apparent inductance isillusion. The energy involved is not stored -it

rred while the operating point movesthe characteristic -the energy involved is

Non-Magnetic Inclusions: Figure 6 goesstep 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-magneticregions..Magnetic materials do provide an easy path forflux, 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 (aninductor).

.High inductance does not equate to high energystorage. 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.ig 6. -Non-Magnetic Inclusions

UNITRODE CORPORATION


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Fig 8. -Large Air Gapig 7. -Non-HomogeneousEffectsoutside of the hysteresis loop, is relatively hThe recoverable energy is almost all stored inadded gap. A little energy is stored in the magnetic inclusions within the core. Almost energy is stored in the magnetic core material itWith powdered metal cores, such as Mo-Permloy, the large gap is distributed between the mparticles, in the non-magnetic binder which hthe core together. The amount of binder determthe effective total non-magnetic gap. This is ustranslated into an equivalent permeability valuethe 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 coreshape.Material anomalies that can skew and round thecharacteristic 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 charactics of magnetic cores have been considered.

most important distinction at high frequenciethat the core eddy currents become significant eventually become the dominant factor in losses. Eddy currents also exist in the windingmagnetic devices, causing increased copper loat high frequencies, but this is a separate topicdiscussed in this paper

Eddy currents arise because voltage is indwithin the magnetic core, just as it is induced inwindings overlaying the core. Since all magcore materials have finite resistivity, the indvoltage causes an eddy current to circulate wthe core. The resulting core loss is in additiothe low frequency hysteresis loss.

Ferrite cores have relatively high resistivity. reduces loss, making them well suited for highquency power applications. Further improvemen



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0-- LRE

eddy culTent losses as requency dependent. Lossesreally depend on rate of flux change, and thereforeaccording to Faraday s Law, upon the appliedvolts/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 osses vary withpulse voltage amplitude squared. and inversely ~ithpulse width -exactly the same as for a discreteresistor connected across the winding:

2Vp tpLoss = --RE TIf 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 oss varies with 1s2(but usually the flux swing is reduced at higherfrequency to avoid excessive loss).

o-UJFig 9. -Core Eddy Current Model

resistivity. Amorphous metal coresespecially crystalline metal cores have muchresistivity and therefore higher losses. Theseare built-up with very thin laminations. This

reduces the voltages induced within theof the small cross section area of each

he core can be considered to be a single-turnwhich couples the eddy current loss resis-

into any actual winding. Thus, as shown in

a winding which represents all of then Figure 10, the solid line shows the low fre-

characteristic of a magnetic core, with dashlabeled /1 and /2 showing how the hysteresis

widens at successively higher fre-Curves like this frequently appear on

data sheets.They are not very usefulon frequency, assuming symmetrical drive

which is not the case in switching

I rEele1-7"'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,, fI II II

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



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Fig 12. -Forward Converter Waveformsig 11. -Forward Converter Core Flux, IMIn a forward converter operating at a fix

switching frequency, a specific V -ps forward puis required to obtain the desired VOUT.When changes, pulse width changes inversely. The traformer flux will always change the same amoufrom D to E, but with higher VIN the flux chanmore rapidly. Since the higher VIN s also across Rthe equivalent eddy current resistance, the edcurrent and associated loss will be proportional VIN. Worst case for eddy current loss is at high li

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 c

model is shown in Fig- 13- This model is a twterminal device (plus a third terminal for monitoriflux level) that can be used in a wide rangemagnetic core applications- It can directly simulan inductor or saturable reactor (magamp). Addto an ideal transformer, it can simulate a flybackconventional transformer -A Spice subcircuit net list is given in Fig. 1Electrical parameters must be calculated from magnetic design and inserted into the model eitdirectly, replacing the expressions within cubrackets with numerical values, or by parampassing with the subcircuit call-

Definable parameters include (SI Units):SVSEC Volt-sec at saturation = BsAT-AE-NIVSEC Volt-sec Initial condition = B-AENLMAG Unsaturated Inductance = )lo)l;N2AE/LSA T Saturated Inductance = )loN2AE/'E

6lectrical Circuit Model for Magnetic Cores


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LSAT: Use core dimensions but with Pr = I(saturated core is non-magnetic)REDDY One approach is to determine frequency

where 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

Magnetizing I at O Flux = Hl.pjNEddy current loss resistance

Notes on parameter calculation:For ungapped core, I. = I.E, (total patharound core). For gapped core, Pr = I,I. = gap length. AE = core area, m2.

Fig ]4 -Core Model Net ListCOREH 1 2101 9 DHYST

2 {SVSEC/250} IC={IVSEC/SVSEC*250}

56 {LSAT*250/SVSEC}67 DCLAMP86 DCLAMP

2 VM 1



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the integration capacitor CI value is calculated reach a nonnalized value of SOOV when the intgrated input V -ps reaches the value SVSEC corrsponding to saturation. Thus the voltage on tcapacitor is not equal to the actual V -ps, but tV -ps times SOO/SVSEC. Also, in the originmodel, the voltage viewed at tenninal 3 indicatthe integrated V-ps across Cl only when tennin2 is at ground potential. In applications whetenninal 2 is not grounded, tenninal 3 does nshow the voltage across C I, as intended. Theproblems are eliminated by adding E2, whicprovides a corrected, ground referenced V -ps valviewed at tenninal 10.Finally, the original model requires the use peak-to-peak V -ps values for SVSEC and IVSEcorresponding to p-p flux swing from minus to plsaturation. In the modified model, SOOV (P-p) changed to 2S0V (peak) so that SVSEC and IVSEvalues based on peak flux levels can be used, ascustomary .References:[1] T. G. Wilson, Sr., "Fundamentals of Magne

Materials," APEC Tutorial Seminar 1, 198[2] "Saturable Reactor Model," lsSpice Use

Guide 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


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An Electrical Circuit Model for Magnetic Cores

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