Reference Design Section

8/3/2019 Reference Design Section

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Ocl94

Lloyd Dixon

A brief tutorial on magnetic fundamentals leadsdiscussion of magnetic core properties. A

version of Intusoft' s magnetic core modelpresented. Low1requency hysteresis is added to

model. making it suitable for magnetic amplifier

Fig 1. -Magnetic Core B-H Characteristicsurface 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 recoverablestored energy:

Units commonly used in magnetics design arewith conversion factors

the older cas system to the SI system (sys-nfernational- rationalized MKS). SI units arealmost universally throughout the world.

than their cas equivalents. Unfortunately,the published magnetics data is in the casespecially in the United States, requiring

Table I. CONVERSIONFACTORS,CGS to SIW/m3 = JHdB

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

SITesiaA-T/m41t.10.7

CGS CGS to SIGaUSS 10-4

Oersted 1000/41t1 41t.10.7

1cm2 10.4cm 10.2

Maxwell 10.8Gilbert 10/41t

109/41t41t.10.9

(Henry) 1Erg 10.7

DENSITY BINTENSITY H

space) ~orelative) J.Iy

(Core, Window) A(Core, Gap) .

FLUX = JBdA ~FIELD = JHdf F,MMF

= F/~ R= 1/R = L/N2 P= P-N2 L

m2m

WeberA-T

HenryJoule

Figure 1 is the B-H characteristic of a magneticmaterial -flux density (Tesla) vs. magneticintensity (A-Tlm). The slope of a line on thisof axes is permeability (p = B/H). Area on the Fig 2. -Core FlzlX vs. Magnetic Force

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

I = HIe~

is trnnsfonned into the total flux, ,hrough theentire cross-sectional area of the core, while fieldintensity is transfonned into total magnetic forcearound the entire magnetic path length of the core:

I!\ = B-A ; F = H.f't' e e

Area in Fig. 3 again represents energy. this timin electrical terms: W = JEIdt. The slope of a linin Fig 3 is inductance.

dt AL = E- = 11N2~di r- ,

e

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

metal alloys, amorphous metal alloys, and ferrite(ferrimagnetic oxides).[!]Figure 4 shows the low frequency electriccharacteristics of an inductor with an ungapptoroidal core of an idealized ferromagnetic mealloy. This homogeneous core becomes magnetizeat a specific field intensity H (corresponding specific current through the winding I=HI/N). Athis magnetizing current level, all of the magnedipoles (domains) within the core gradually aligcausing the flux to increase toward saturation. Tdomains cannot align and the flux cannot chaninstantaneously because energy is required. Thchanges occur at a rate governed by the voltagapplied to the winding, according to Faraday's Law

Thus, to magnetize this core, a specific magneting current, IM, is required, and the time to accomplish the flux change is a function of the voltagapplied to the winding. These factors combined

The slope of a line on these axes is permeance(P = ci>/F= pAJfe). Penneance is the inductance ofone 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. 2requires 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:~ = -~ ci>= 2- rEdtdt N N J'

Ampere's Law:Nl = fHdl ~ HI

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/dt). hus,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 trnnsfonned again into the equivalent

Fig 3. -Equivalent Electrical Characteristic Fig 4. -Ideal Magnetic Core

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The Effect or 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 currentlevel.A practical toroidal core has an O.D. substan-tially greater than Its I.D., causing magnetic pathlength to increase aDd field intensity to decreasewith increasing diameter. The electrical result isshown in Figure 5. As current increases, the criticalfield intensity, H, required to align the domains isachieved first at the inner diameter. According toAmpere:

NIT

NInD'"

H= =

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

voltage, and time -constitute energy putbetween the core characteristic and the vertical

this case, none of this energy is recoverableis all loss, incurred immediately while the

The vertical slope of the characteristic representsapparent infinite inductance. However, there isreal inductance -no recoverable stored energycharacteristic is actually resistive. (A resistorthe same vertical slope.)

When all of the domains have been aligned, thesaturated, at the flux level corresponding

little change in flux, and very little volt-can exist across the winding as the operatingmoves out on the saturation characteristic. The

rgy is being stored. With this idealis the same as if there were no

as shown by the dash line through theThe small amount of stored energy is repre-by a thin triangular area above the saturation

the vertical axis to the operatingIf the current is now interrupted, the flux will

to the residual flux level (point R) on theaxis. The small flux reduction requires

stored. (If the current isthe short voltage spike will be

amplitude.) As long as the currentzero (open circuit). the flux will remain

-at point R.If a negative magnetizing current is now applied

winding, the domains start to realign in theThe flux decreases at a rate

causing the operating point to move downcharacteristic at the left of the vertical axis. Asoperating point moves down. the cumulativebetween the characteristic and the vertical axis

energy lost in this process. When theaxis is reached, the net flux is zero -

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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, Em. 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: 1 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 characteistic. This slope arises from the inclusion of smanon-magnetic regions in series with the magnecore material. For example, such regions could bthe non-magnetic binder that holds the particletogether in a metal powder core, or tiny gaps at timperfect mating surfaces of two core halveAdditional magnetic force is required, proportionto the amount of flux, to push the flux across thesmall gaps. The resulting energy stored in thegaps is theoretically recoverable. To find out homuch energy is loss and how much is recoverablook at Figure 6. If the core is saturated, the enerwithin triangle S-V -R is recoverable because it between the operating point S and the vertical axand outside the hysteresis loop. That doesn't ensuthe energy will be recovered -it could end dumped into a dissipative snubber.

Another important aspect of the skewing resuing from the non-magnetic inclusions is that tresidual flux (point R) becomes much less than tsaturation flux level. To remain saturated, the comust now be driven by sufficient magnetizincurrent. When the circuit is opened, forcing tmagnetizing current to zero, the core will resitself to the lower residual flux level at R.

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

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

.Energy is stored, not dissipated. in non-magneregions..Magnetic materials do provide an easy path flux, thus they serve as "magnetic bus bars" link several coils to each other (in a transfonneor link a coil to a gap for storing .energy (inductor).

.High inductance does not equate to high enestorage. Flux swing is always limited by satution or by core losses. High inductance requirless magnetizing current to reach the flux limhence less energy is stored. Referring to Figu6, if the gap is made larger, further skewing characteristic and lowering the inductantriangle S-V -R gets bigger. indicating mostored energy.ig 6. -Non-Magnetic Inclusions

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Fig 7. -Non-HomogeneousEffects Fig 8. -Large Air Gapoutside 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 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 recoverablenergy. This is a desirable characteristic fo( Mag-

and conventional transformers. But filtertransformers require a great

stored energy, and the characteristics of4 -7 are unsuitable.Figure 8 is the same core as in Fig. 7 with much

gap(s) -a few millimeters total. This causes

Core Eddy Current Losses:Up to this point, the low frequency characteris-tics of magnetic cores have been considered. The

most 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

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u-. LIE

eddy cUlTent osses as requency dependent. Lossereally depend on rate of flux change, and thereforeaccording to Faraday s Law, upon the appliedvolts/turn. Frequency is relevant only in the case osinusoidal or symmetrical square wave voltagewaveforms.In a switching power supply operating at a fIXedfrequency, fs, core eddy CUlTent osses vary withpulse voltage amplitude squared. and inversely withpulse 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 maintained constant, eddy current loss varies with fs(but usually the flux swing is reduced at highefrequency to avoid excessive loss).

-0- --1.

Fig 9. -Core Eddy Current Modelhigh 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 lO, 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 hey arebased on frequency, assuming symmetrical drivewaveforms, which is not the case in switchingpower supplies.In fact, it is really not appropriate to think of

: rEtieI I, ,, ,, ,

!fl /f1o

-7"1" I

1 ,1112.f,

,,II

Forward Converter Illustration:Figure 11 provides an analysis of transfonne

operation in a typical forward converter. Accompanying wavefonns are in Fig. 12. The solid line inFig. 11 is the low-frequency characteristic of thferrite core. The dash lines show the actual path othe operating point, including core eddy currents reflected into the winding. Line X- y is the mid-poinof the low frequency hysteresis curve. Hysteresiloss will be incurred to the right of this line as thflux increases, to the left of this line when the fluxdecreases.Just before the power pulse is applied to thwinding, the operating point is at point R, thresidual flux level. When the positive (forwardpulse is applied, the current rises rapidly from R tD (there is no time constraint along this axis). Thcurrent at D includes a low-frequency magnetizingcomponent plus an eddy current component proportional to the applied forward voltage. The fluxincreases in the positive direction at a rate equal tthe applied volts/turn.As the flux progresses upward, some of thenergy taken from the source is stored, some loss. Point E is reached at the end of the positiv

I/ I--""IFig 10. -L.F. Hysteresis plus Eddy Current

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Fig 11. -Forward Converter Core F/ux,IM Fig 12. -Forward Converter WaveformsIn a forward converter operating at a fixed

switching 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 amountfrom D to E, but with higher VIN the flux changemore rapidly. Since the higher VIN s also across REthe equivalent eddy current resistance, the eddycurrent and associated loss will be proportional toVIN. Worst case for eddy current loss is at high line.[I] T.G. Wilson, Sr., Fundamentals ofMagneticMaterials, APEC Tutorial Seminar 1, 1987

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 thetrapewidal region A-C-R is therefore slightly lessthan on the way up through D and E.

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Eddy Current Losses in TransformerWindings and Circuit Wiring

Lloyd H. Dixon, Jr.energy is stored in air gaps, insulation betweenconductors, and within the conductors, whererelative permeability JLr is essentially 1.0 andconstant. The energy density then becomes:

w = \BH = \'tJ.(j/I2 J/m3where ILO s the absolute permeability of freespace ( =47r .10"7 in S.I. units). Total energy W(Joules) is obtained by integrating the energydensity over the entire volume, v, of the field:

W = \ILO f H2dv JoulesWithin typical transformers and inductors, themagnetic energy is almost always confined toregions where the field intensity H is relativelyconstant and quite predictable. This often oc-curs in circuit wiring, as well. In these cases:

W = \ILO H2A. e Joules (2)and from (1), He = NI. Substituting for H:

W = \ILO ~fA/e Joules (3)whereA is the cross-section area (m1 of theregion normal to the flux, and e is the length

of the region in meters (and the effectivelength of the field).4. Circuit inductance: Inductance is a meas-ure of an electrical circuit's ability to storemagnetic energy. Equating the energy stored inthe field from (3) with the same energy incircuit terms:

W = \Lf = \ILO N2fA/eL = ILON2A/e (4)

Skin EffectFigure 1 shows the magnetic field (flux lines)in and around a conductor carrying dc or lowfrequency current I. The field is radially sym-metrical, as shown, only if the return currentwith its associated field is at a great distance.At low frequency, the energy in the magneticfield is trivial compared to the energy loss inthe resistance of the wire. Hence the current

IntroductionAs switching power supply operating fre-quencies increase, eddy current losses andparasitic inductances can greatly impair circuitperformance. These high frequency effects arecaused by the magnetic field resulting fromcurrent flow in transformer windings and circuit

wiring.This paper is intended to provide insight intothese phenomena so that improved high fre-quency. performance can be achieved. Amongother things, it explains (I) why eddy currentlosses increase so dramatically with more wind-ing layers, (2) why parallelling thin strips does-n't work, (3) how passive conductors (Faradayshields and C. T .windings) have high losses,and (4) why increasing conductor surface areawill actually worsen losses and parasitic induc-tance if the configuration is not correct.Basic PrinciplesThe following principles are used in thedevelopment of this topic and are presentedhere as a review of basic magnetics.1. Ampere's Law: The total magneto-motiveforce along any closed path is equal to the totalcurrent enclosed by that path:

F = iHde = I, = NI Amps (1)where F is the total magneto-motive force(in Amperes) along a path of length e (m), His field intensity (Aim), and I, is the totalcurrent through all turns enclosed by the path.2. Conservation or energy: At any momentof time, the current within the conductors and

the magnetic field are distributed so as tominimize the energy taken from the source.3. Energy content or the field: The magneticfield is energy. The energy density at any pointin the field is:w = f HdB Jouleslm3

where B is the flux density (Tesla). Inswitching power supplies, almost all magneticR2-1


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Reference Design Section