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13Basic Magnetics Theory

Magnetics are an integral part of every switching converter. Often, the design of the magnetic devicescannot be isolated from the converter design. The power electronics engineer must not only model anddesign the converter, but must model and design the magnetics as well. Modeling and design of magnet-ics for switching converters is the topic of Part III of this book.

In this chapter, basic magnetics theory is reviewed, including magnetic circuits, inductor model-ing, and transformer modeling [1-5]. Loss mechanisms in magnetic devices are described. Winding eddycurrents and the proximity effect, a significant loss mechanism in high-current high-frequency windings,are explained in detail [6-11]. Inductor design is introduced in Chapter 14, and transformer design is cov-

ered in Chapter 15.

The basic magnetic quantities are illustrated in Fig. 13.1. Also illustrated are the analogous, and perhapsmore familiar, electrical quantities. The magnetomotive force or scalar potential, between two points

and is given by the integral of the magnetic field H along a path connecting the points:

where is a vector length element pointing in the direction of the path. The dot product yields the com-


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Basic Magnetics Theory

ponent of H in the direction of the path. If the magnetic field is of uniform strength H passing through anelement of length as il lustrated, then Eq. (13.1) reduces to

This is analogous to the electric field of uniform strength E, which induces a voltage between twopoints separated by distance

Figur e 13.1 also illustrates a total magnetic flux passing through a surface S having areaThe total flux is equal to the integral of the normal component of the flux density B over the surface

where dA is a vector area element having direction normal to the surface. For a uniform flux density of magnitude B as illustrated, the integral reduces to

Flux density B is analogous to the electrical current density J , and flux is analogous to the electric cur-rent I . If a uniform current density of magnitude J passes through a surface of area then the total cur-rent is

Faradays law relates the voltage induced in a winding to the total flux passing through the inte-rior of the winding. Figure 13.2 illustrates flux passing through the interior of a loop of wire. Theloop encloses cross-sectional area According to Faradays law, the flux induces a voltage v(t ) in the

wire, given by

where the polarities of v(t ) and are defined according to the r ight-hand rule, as in Fig. 13.2. For a


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13.1 Review of Basic Magnetics

uniform flux distribution, we can express v(t ) in terms of the flux density B(t ) by substitution of Eq.(13.4):

Thus, the voltage induced in a winding is related to the flux and flux density B passing through theinterior of the winding.

Lenzs law states that the voltage v(t ) induced by the changing flux in Fig. 13.2 is of thepolarity that tends to drive a current through the loop to counteract the flux change. For example, con-sider the shorted loop of Fig. 13.3. The changing flux passing through the interior of the loopinduces a voltage v(t ) around the loop. This voltage, divided by the impedance of the loop conductor,leads to a current i(t ) as illustrated. The current i(t ) induces a flux which tends to oppose thechanges in

a r o 9 9 5 ( t ) T J E T - 08 6 3 09 T c B T 8 5 5 0 0 1 4 6 2 7 40 1 8 3 0 1 7 5 8 T m c l o p p dt h e


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window in the center of the core, or i(t ). If the magnetic f ield is uniform and of magnitude H (t ), then theintegral is So for the example of Fig. 13.4, Eq. (13.7) reduces to

Thus, the magnetic field strength H (t ) is related to the winding current i(t ). We can view winding currentsas sources of MMF. Equation (13.8) states that the MMF around the core, is equal to thewinding current MMF i(t ). The total MMF around the closed loop, accounting for both MMFs, is zero.

The relationship between B and H , or equivalently between is determined by the corematerial characteristics. Figure 13.5(a) illustrates the characteristics of free space, or air:

The quantity is the permeability of free space, and is equal to Henries per meter in MKSunits. Figure 13.5(b) illustrates the BH characteristic of a typical iron alloy under high-level sinusoidalsteady-state excitation. The characteristic is highly nonlinear, and exhibits both hysteresis and saturation.The exact shape of the characteristic is dependent on the excitation, and is difficult to predict for arbitrarywaveforms.

For purposes of analysis, the core material characteristic of Fig. 13.5(b) is usually modeled bythe linear or piecewise-linear characteristics of Fig. 13.6. In Fig. 13.6(a), hysteresis and saturation areignored. The BH characteristic is then given by


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The core material permeability can be expressed as the product of the relative permeability and of Typical values of lie in the rangeThe piecewise-linear model of Fig. 13.6(b) accounts for saturation but not hysteresis. The core

material saturates when the magnitude of the flux density B exceeds the saturation flux density Forthe characteristic follows Eq. (13.10). When the model predicts that the core

reverts to free space, wi th a characteristic having a much smaller slope approximately equal toSquare-loop materials exhibit this type of abrupt-saturation characteristic, and additionally have a verylarge relative permeability Soft materials exhibit a less abrupt saturation characteristic, in whichgradually decreases as H is increased. Typical values of are 1 to 2 Tesla for iron laminations and sili-con steel, 0.5 to 1 Tesla for powdered iron and molypermalloy materials, and 0.25 to 0.5 Tesla for ferritematerials.

Unit systems for magnetic quantit ies are summarized in Table 13.1. The MKS system is usedthroughout this book. The unrationalized cgs system also continues to find some use. Conversionsbetween these systems are listed.

Figure 13.7 summarizes the relationships between the basic electrical and magnetic quantitiesof a magnetic device. The winding voltage v(t ) is related to the core flux and flux density via Faradays



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law. The winding current i(t ) is related to the magnetic field strength via Amperes law. The core materialcharacteristics relate B and H.

We can now determine the electrical terminal characteristics of the simple inductor of Fig.13.8(a). A winding of n turns is placed on a core having permeability Faradays law states that the flux

inside the core induces a voltage in each turn of the winding, given by

Since the same flux passes through each turn of the winding, the total winding voltage is

Equation (13.12) can be expressed in terms of the average flux density B(t ) by substitution of Eq. (13.4):



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When the core saturates, the magnetic device behavior approaches a short circuit. The device behaves asan inductor only when the winding current magnitude is less than Practical inductors exhibit somesmall residual inductance due to their nonzero saturated permeabilities; nonetheless, in saturation theinductor impedance is greatly reduced, and large inductor currents may result.

Figure 13.9(a) illustrates uniform flux and magnetic field inside a element having permeability lengthand cross-sectional area The MMF between the two ends of the clement is

Sincend and can express as

This equation is of the form

with

Equation (13.24) resembles Ohms law. This equation states that the magnetic flux through an element isproportional to the MMF across the elemen t. The constant of propor tional ity , or the reluc tance isanalogous to the resistance R of an electrical conductor. Indeed, we can construct a lumped-elementmagnetic circuit model that corresponds to Eq. (13.24), as in Fig. 13.9(b). In this magnetic circuit model,voltage and current are replaced by MMF and flux, while the element characteristic, Eq. (13.24), is rep-resented by the analog of a resistor, having reluctance

Complicated magnetic structures, composed of multiple windings and multiple heterogeneous


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where and are the MMFs across the core and air gap, respectively. The core and air gap character-istics can be modeled by reluctances as in Fig. 13.9 and Eq. (13.25); the core reluctance and air gap

reluctance are given by

elements such as cores and air gaps, can be represented using equivalent magnetic circuits. These mag-netic circuits can then be solved using conventional circuit analysis, to determine the various fluxes,MMFs, and terminal voltages and currents. Kirchoffs laws apply to magnetic circuits, and followdirectly from Maxwells equations. The analog of Kirchoffs current law holds because the divergence of B is zero, and hence magnetic flux lines are continuous and cannot end. Therefore, any flux line thatenters a node must leave the node. As illustrated in Fig. 13.10, the total flux entering a node must be zero.The analog of Kirchoffs voltage law follows from Ampere's law, Eq. (13.7). The left-hand-side integralin Eq. (13.7) is the sum of the MMFs across the reluctances around the closed path. The right-hand-sideof Eq. (13.7) states that currents in windings are sources of MMF. An n-turn winding carrying current i(t )

can be modeled as an MMF source, analogous to a voltage source, of value ni(t ). When these MMFsources are included, the total MMF around a closed path is zero.Consider the inductor with air gap of Fig. 13.11(a). A closed path following the magnetic field

lines is illustrated. This path passes through the core, of permeability and length andacross the airgap, of permeability and length The cross-sectional areas of the core and air gap are approximatelyequal. Application of Amperes law for this path leads to


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A magnetic circuit corresponding to Eqs. (13.26) and (13.27) is given in Fig. 13.1l(b). The winding is asourc e of MMF, of value ni. The core and air gap reluctances are effectively in series. The solution of themagnetic circuit is

The flux passes through the winding, and so we can use Faradays law to write

Use of Eq. (13.28) to eliminate yields

Therefore, the inductance L is

The air gap increases the total reluctance of the magnetic circuit, and decreases the inductance.Air gaps are employed in practical inductors for two reasons. With no air gap the

inductance is directly proportional to the core permeability This quanti ty is dependent on temperatureand operating point, and is difficult to control. Hence, it may be difficult to construct an inductor havinga well-controlled value of L . Addition of an air gap having a reluctance greater than causes thevalue of L in Eq. (13.31) to be insensitive to variations in

Addition of an air gap also allows the inductor to operate at higher values of winding current i(t )without saturation. The total flux is plotted vs. the winding MMF ni in Fig. 13.12. Since is propor-tional to B, and when the core is not saturated ni is proportional to the magnetic field strength H in the


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13.2 Transformer Modeling

core, F ig. 13.12 has the same shape as the core BH character istic. When the core is not saturated, isrelated to ni accord ing to the l inear relat ionsh ip ofEq . (13.28). When the coresaturates, is equal to

The w ind ing current at the onset of saturat ion is found by subst itut ion of Eq . (13.32) into (13 .28):

The character ist ics are plotted in Fig. 13.12 for two cases : (a) a ir gap present, and (b) no a ir gapIt can be seen that is increased by add ition of an a ir gap . Thus, the a ir gap allows increase of

the saturat ion current, at the expense of decreased inductance .

Cons ider next the two-w ind ing transformer of F ig. 13.13. The core has cross-sect ional area meanmagnet ic path length and permeab ility An equ ivalent magnet ic c ircuit is given in Fig. 13.14. Thecore reluctance is

Since there are two w ind ings in th is example, it is necessary to determ ine the relat ive polar ities of theMMF generators . Ampere s law states that


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This express ion could also be obta ined by subst itut ion of into Eq . (13.35).

In the ideal transformer, the core reluctance approaches zero . The causes the core MMF toalso approach zero . Equat ion (13 .35) then becomes

Also, by Faraday s law, we have

Note that is the same in both equat ions above : the same totalflux l inks both w ind ings . El iminat ion of leads to

Equat ions (13 .37) and (13 .39) are the equat ions of the ideal trans-former :

The ideal transformer symbol of F ig. 13.15 is def ined by Eq .(13 .40) .

For the actual case in wh ich the core reluctance is nonzero, we have

El iminat ion of y ields

The MMF generators are add itive, because the currents and pass in the same d irect ion through thecore w indow . Solut ion of F ig. 13.14 y ields


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This equat ion is of the form

where

are the magnetizing inductance and magnetizing current, referred to the pr imary w inding. An equ ivalentcircu it is illustrated in Fig. 13.16.

Figure 13 .16 co incides w ith the transformer model introduced in Chapter 6 . The magnet izinginductance models the magnet izat ion of the core mater ial . It is a real, phys ical inductor, wh ich exh ibitssaturat ion and hysteres is. All phys ical transformers must conta in a magnet izing inductance . For example,suppose that we d isconnect the secondary w ind ing . We are then left w ith a s ingle w ind ing on a magnet iccorean inductor . Indeed, the equ ivalent c ircu it of F ig. 13.16 pred icts th is behav ior, v ia the magnet izinginductance . The magnet izing current causes the rat io of the w ind ing currents to d iffer from the turnsrat io.

The transformer saturates when the core flux dens ity B(t ) exceeds the saturat ion flux dens ity

c l a rg e l s t (e )3 0 ( )T 0 Tc0 Tw BT / F8 1 T f8 3 7 0 6 T J E 5 4 6n d u ct 7 (s )23 (o ) rmer14 2 h T h m() T J E 1 0 0 2 45 Tc0 Tw BT 8 3 7 0 0 3 5 2 4 9t 6 0 0 9 c2 T m(mag n et )T J E T 0 TcBT /F8 1 T f8 3 7 0 9 5 7 1 9 t 6 0 0 9 c2 T m( )T J E T BT / F7 1 T f8 3 7 7 4 9 3 8 4 9 6 0 0 9 c2 T m(z) T J E T BT/ F8 1 T f8 3 7 7 0 9 3 8 2 t 6 0 0 9 c2 T m() T J E 1 0 0 2 2 3 TwBT / F7 1 T f8 3 7 0 0 9 1 8 4 9 t6 0 0 9 c2 T mn gn d u c 3 2 r eT m ( ) T J E 6 7 0 0 4 9 8 T c 0 2 8 7 3 T w B T 8 3 7 0 0 9 1 3 8 9 6 0 0 9 c 2 s m a l 2 7( r 1 8 ( n ) 17 ( a ) 6 ( r 1 8 ( t ) ( t ) 1 7 ( h )3 3 5 ( e ) 17 ( t r a n s f 3 3 5 ( T) 2 9 ) - 5 7 ( w ) T J E T 0 T c 0 T w B T / F 8 1 T f 8 3 7 0 6 0 9 6 0 6 7 t 6 0 0 9 c 2 T m ( ) T J 4 T 0 0 2 7 8 T c B T / F 7 1 T f 8 3 7 0 0 5 8 7 6 2 0t 6 0 0 9 c 2 T m ( n d ) T J E T 0 T c B T / F 8 1 T f 8 3 7 0 0 9 5 2 5 2 t 6 0 0 9 c 2 T m ( ) T J o Q 0 0 1 5 0 c 0 1 4 1 7 T w B T / F 7 1 T f 8 3 7 0 0 4 8 4 2 2 5 6 0 0 9 c 2 ) - 4 6 ( n ) -4 6 ( s ) 2 ( s ) c o m ( T ) 2 9 ( t u r n s) T J E 4 0 0 2 2 6 3 0 - 0 0 4 2 2 T w B T 8 3 7 3 3 0 9 1 4 2 t 6 0 0 9 c 2 h o r ) - 5 1 3 nt crcurcundsaturatT J E T 0 TcBT / F8 1 T f8 3 7 0 0 T f9 1 3 2 q 9 c2 T m( ) 2 8 Tc0 1 2 0 9 T 1 0 2 4 3 8 Tw BT / F7 1 T f8 3 7 5 6 6 5 6 0 8 4 1 3 2 q 9 c2 (T m(1 3 ) T J E T 0 Tc0 Tw BT / F8 1 T f8 3 7 0 9 3 4 5 5 1 3 2 q 9 c2 T m () 2 8 Tc0 1 2 T J E T BT / F7 1 T f8 3 7 7 2 7 2 7 2 1 1 3 2 q 9 c2 3 7 )x amp l eW e n d T m ( wh e re ) T J E T 0 T c0 T w B T/ F 8 1 T f8 3 7 0 6 40 9 9 1 3 2 q9 c 2 T m ( ) T J E T 0 0 17 1 T c 0 14 1 7 T wB T /F 7 1 T f 8 3 7 0 0 5 3 42 2 5 13 2 q 9 c2 sn a h m 19 s magnet B(smater

i

nd


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The l imits are chosen such that the integral is taken over the pos itive port ion of the appl ied per iodic volt-age waveform.

To f ix a saturat ing transformer, the flux dens ity should be decreased by increas ing the number of turns, or by increas ing the core cross-sect ional area Add ing an a ir gap has no effect on saturat ion of convent ional transformers, s ince it does not mod ify Eq


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Figure 13 .18 illustrates a transformer electr ical equ ivalent c ircu it model, includ ing ser ies induc-tors wh ich model the leakage inductances . These leakage inductances cause the term inal volt-age rat io to d iffer from the ideal turns rat io In general, the term inal equat ions of a two-wind ing transformer can be wr itten

The quant ity is called the mutual inductance, and is given by

The quant ities and are called the pr imary and secondary self-inductances, given by

Note that Eq . (13.48) does not expl icitly ident ify the phys ical turns rat io Rather, Eq .

(13.48) expresses the transformer behav ior as a funct ion of electr ical quant ities alone . Equat ion (13 .48)can be used, however, to def ine the effective turns ratio

and the coupling coefficient

The coupl ing coeff icient l ies in the range and is a measure of the degree of magnet ic coupl ingbetween the pr imary and secondary w indings . In a transformer w ith perfect coupl ing, the leakage induc-tances and are zero. The coupl ing coeff icient k is then equal to 1 . Construct ion of low-voltage

cient is close to 1, then the effect ive turns rat io is approx imately equal to the phys ical turns rat io

and

transformers having coefficients in excess of 0.99 is quite feasible. When the coupling coeffi-


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Energy is required to effect a change in the magnet izat ion of a core material . Not all of this energy isrecoverable in electrical form; a fraction is lost as heat. This power loss can be observed electrically ashysteresis of the BH loop.

Consider an n turn inductor excited by periodic waveforms and i(t ) having frequency f. Thenet energy that flows into the inductor over one cycle is

We can relate this expression to the core BH characteristic: substitute B(t) for using Faradays law,Eq. (13.13), and substitute H (t ) for i(t ) using Amperes law, i.e. Eq. (13.14):

The term is the volume of the core, while the integral is the area of the BH loop:

The hysteresis power loss is equal to the energy lost per cycle, multiplied by the excitationfrequency f :

To the extent that the size of the hysteresis loop is independent of frequency, hysteresis loss increases

Magnetic core materials are iron alloys that, unfor-tunately, are also good electrical conductors. As a result, acmagnetic fields can cause electrical eddy currents to flowwithin the core material itself. An example is illustrated inFig. 13.19. The ac flux passes through the core. Thisinduces eddy currents i(t ) which, according to Lenzs law,flow in paths that oppose the time-varying flux Theseeddy currents cause losses in the resistance of the corematerial. The eddy current losses are especially significantin high-frequency applications.

According to Faradays law, the ac flux induces voltage in the core, which drives the cur-rent around the paths illustrated in Fig. 13.19. Since the induced voltage is proportional to the derivativeof the flux, the voltage magnitude increases directly with the excitation frequency f . If the impedance of

(energy lost per cycle) = (core volume)(area of BH loop ) (13.55)

directly with operating frequency.


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13.3 Loss Mechanisms in Magnetic Devices

the core mater ial is purely res istive and independent of frequency, then the magn itude of the inducededdy currents also increases d irectly w ith f . This impl ies that the eddy current losses should increaseas In power ferr ite mater ials, the core mater ial impedance magn itude actually decreases w ith increas-ing f . Over the useful frequency range, the eddy current losses typ ically increase faster than

There is a bas ic tradeoff between saturat ion flux dens ity and core loss . Use of a h igh operat ingflux dens ity leads to reduced s ize, we ight, and cost . Silicon steel and s imilar mater ials exh ibit saturat ionflux dens ities of 1 .5 to 2 T . Unfortunately, these core mater ials exh ibit h igh core loss . In part icular, thelow res istivity of these mater ials leads to h igh eddy current loss . Hence, these mater ials are su itable forf ilter inductor and low-frequency transformer appl icat ions . The core mater ial is produced in lam inat ionsor th in ribbons, to reduce the eddy current magn itude . Other ferrous alloys may conta in molybdenum,

cobalt, or other elements, and exh ibit somewhat lower core loss as well as somewhat lower saturat ionflux dens ities .

Iron alloys are also employed in powdered cores, conta ining ferromagnet ic part icles of suff i-ciently small d iameter such that eddy currents are small . These part icles are bound together us ing aninsulat ing med ium . Powdered iron and molybdenum permalloy powder cores exh ibit typ ical saturat ionflux dens ities of 0 .6 to 0 .8 T, w ith core losses s ign if icantly lower than lam inated ferrous alloy mater ials.The insulat ing med ium behaves effect ively as a d istr ibuted a ir gap, and hence these cores have relat ivelylow permeab ility . Powder cores f ind appl ication as transformers at frequenc ies of several kHz, and as f il-ter inductors in high frequency (100 kHz) sw itch ing converters .

Amorphous alloys exh ibit low hysteres is loss . Core conduct ivity and eddy current losses aresomewhat lower than ferrous alloys, but h igher than ferr ites . Saturat ion flux dens ities in the range 0 .6 to

Ferr ite cores are ceram ic mater ials hav-ing low saturat ion flux dens ity, 0 .25 to 0 .5 T.The ir res istivities are much h igher than othermater ials, and hence eddy current losses aremuch smaller . Manganese-z inc ferr ite cores f indwidespread use as inductors and transformers inconverters hav ing sw itch ing frequenc ies of 10kHz to 1 MHz . N ickel-z inc ferr ite mater ials canbe employed at yet h igher frequenc ies .

Figure 13 .20 conta ins typ ical total coreloss data, for a certa in ferr ite mater ial . Power lossdens ity, in Watts per cub ic cent imeter of coremater ial, is plotted as a funct ion of s inuso idalexcitation frequency f and peak ac flux dens ity

At a g iven frequency, the core loss can beapprox imated by an emp irical funct ion of theform

The parameters and are determ ined by f it-ting Eq . (13 .57) to the manufacturer s publ isheddata . Typ ical values of for ferr ite mater ialsoperat ing in the ir intended range of and f lie

1.5 T are obta ined .

in the range 2 .6 to 2 .8. The constant of proport ional ity increases rap idly w ith exc itation frequency f .The dependence of on f can also be approx imated by emp irical formulae that are f itted to the manu-


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facturer s publ ished data ; a fourth-order polynom ial or a funct ion of the form are somet imesemployed for th is purpose .

Sign if icant loss also occurs in the res istance of the copper w ind ings . Th is is alsoa ma jor determ inant of the s ize of a magnet ic dev ice: if copper loss and w ind ingresistance were irrelevant, then inductor and transformer elements could bemade arb itrar ily small by use of many small turns of small w ire .

Figure 13 .21 conta ins an equ ivalent c ircu it o f a w ind ing, in wh ich ele-ment R models the w inding res istance . The copper loss of the w ind ing is

where is the rms value of i(t ). The dc res istance of the w ind ing conductorcan be expressed as

1.724 for soft-annealed copper at room temperature . Th is res ist ivity increases to at 100C .

Eddy currents also cause power losses in w ind ing conductors . Th is can lead to copper losses s ign if icantlyin excess of the value pred icted by Eqs . (13.58) and (13 .59) . The spec if ic conductor eddy current mecha-nisms are called the skin effect and the proximity effect. These mechan isms are most pronounced in high-current conductors of mult i-layer w ind ings, part icularly in h igh-frequency converters .

Figure 13 .22(a) illustrates a current i(t ) flow ing through a sol itary conductor . This current induces mag-net ic flux whose flux l ines follow c ircular paths around the current as shown . Accord ing to Lenz slaw, the ac flux in the conductor induces eddy currents, wh ich flow in a manner that tends to oppose theac flux F igure 13 .22(b) illustrates the paths of the eddy currents . It can be seen that the eddy cur-rents tend to reduce the net current dens ity in the center of the conductor, and increase the net currentdens ity near the surface of the conductor .

The current d istribution w ithin the conductor can be found by solut ion of Maxwell s equat ions .For a s inuso idal current i(t ) of frequency f , the result is that the current dens ity is greatest at the surface of the conductor . The current dens ity is an exponent ially decay ing funct ion of d istance into the conductor,with character istic length known as the penetration depth or skin depth. The penetrat ion depth is g ivenby

where is the w ire bare cross-sect ional area, and is the length of the w ire . The res ist ivity is equal to


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13.4 Eddy Currents in Winding Conductors

For a copper conductor , the permeab ility is equal to and the res istivity is g iven in Sect ion 13 .3.2.At 100C, the penetrat ion depth of a copper conductor is

w ith f expressed in Hz . The penetrat ion depth of copper conductors is plotted in F ig. 13 .23, as a funct ionof frequency f . For compar ison, the w ire d iameters d of standard Amer ican W ire Gauge (AWG) conduc-tors are also l isted . It can be seen that = 1 for AWG #40 at approx imately 500 kHz, wh ile = 1 for


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AWG #22 at approx imately 10 kHz .The sk in effect causes the res istance and copper loss of sol itary large-d iameter w ires to increase

at h igh frequency . High-frequency currents do not penetrate to the center of the conductor . The currentcrowds at the surface of the w ire, the ins ide of the w ire is not ut ilized, and the effect ive w ire cross-sec-tional area is reduced . However, the sk in effect alone is not suff icient to expla in the increased h igh-fre-quency copper losses observed in mult iple-layer transformer w ind ings .

A conductor that carr ies a h igh-frequency current i(t )induces copper loss in a ad jacent conductor by a phenomenonknown as the proximity effect. Figure 13 .24 illustrates two copperfo il conductors that are placed in close prox imity to each other .

Conductor 1 carr ies a h igh-frequency s inuso idal current i(t ) , whosepenetrat ion depth is much smaller than the th ickness h of conduc-tors 1 or 2 . Conductor 2 is open-c ircu ited, so that it carr ies a netcurrent of zero . However, it is poss ible for eddy currents to beinduced in conductor 2 by the current i(t ) flow ing in conductor 1 .

The current i(t ) flow ing in conductor 1 generates a fluxin the space between conductors 1 and 2 ; th is flux attempts to

penetrate conductor 2 . By Lenz s law, a current is induced on thead jacent (left) s ide of conductor 2, wh ich tends to oppose the flux

If the conductors are closely spaced, and if then theinduced current w ill be equal and oppos ite to the current i(t ), as

illustrated in Fig. 13.24.Since conductor 2 is open-c ircu ited, the net current in con-

ductor 2 must be zero . Therefore, a current + i(t ) flows on the r ight-side surface of conductor 2 . So the current flow ing in conductor 1induces a current that c irculates on the surfaces of conductor 2 .

Figure 13 .25 illustrates the prox imity effect in a s imple transformer w ind ing . The pr imarywind ing cons ists of three ser ies-connected turns of copper fo il, hav ing th ickness and carry ing netcurrent i(t ). The secondary w ind ing is ident ical ; to the extent that the magnet izing current is small, thesecondary turns carry net current i(t ). The w ind ings are surrounded by a magnet ic core mater ial thatencloses the mutual flux of the transformer .

The h igh-frequency s inuso idal current i(t ) flows on the r ight surface of pr imary layer 1, ad jacentto layer 2 . Th is induces a copper loss in layer 1, wh ich can be calculated as follows . Let be the dcres istance of layer 1, g iven by Eq . (13 .59), and let I be the rms value of i(t ). The sk in effect causes thecopper loss in layer 1 to be equal to the loss in a conductor of th ickness w ith un ifor m current dens ity .Th is reduct ion of the conductor th ickness from to effect ively increases the res istance by the same fac-tor. Hence, layer 1 can be v iewed as hav ing an ac res istance given by

The copper loss in layer 1 is

The prox imity effect causes a current to be induced in the ad jacent (left-s ide) surface of pr imarylayer 2, wh ich tends to oppose the flux generated by the current of layer 1 . If the conductors are closelyspaced, and if then the induced current w ill be equal and oppos ite to the current i(t ), as illustrated


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in Fig. 13.25. Hence, current i(t ) flows on the left surface of the second layer . Since layers 1 and 2 areconnected in ser ies, they must both conduct the same net current i(t ). As a result, a current + 2i(t ) mustflow on the r ight-s ide surface of layer 2 .

The current flow ing on the lef t surface of layer 2 has the same magn itude as the current of layer1,and hence the copper loss is the same : . The current flow ing on the r ight surface of layer 2 has rmsmagn itude 2 I ; hence, it induces copper loss The total copper loss in pr imary layer 2 istherefore

The copper loss in the second layer is f ive t imes as large as the copper loss in the f irst layer!The current 2i(t ) flow ing on the r ight surface of layer 2 induces a flux as illustrated in Fig.

13.25. This causes an oppos ing current 2 i(t ) to flow on the ad jacent (left) surface of pr imary layer 3 .Since layer 3 must also conduct net current i(t ), a current + 3i (t ) flows on the r ight surface of layer 3 . Thetotal copper loss in layer 3 is


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Likew ise, the copper loss in layer m of a mult iple-layer w ind ing can be wr itten

It can be seen that the copper loss compounds very qu ickly in a mult iple-layer w ind ing .The total copper loss in the three-layer pr imary w ind ing is More gen-

erally, if the w inding conta ins a total of M layers, then the total copper loss is

If a dc or low-frequency ac current of rms ampl itude I were appl ied to the M -layer w inding, its copperloss would be Hence, the prox imity effect increases the copper loss by the factor

This express ion is val id for a fo il w ind inghav ingAs illustrated in Fig. 13.25, the currents in the secondary w ind ing are symmetr ical, and hence

the secondary w ind ing has the same conduct ion loss .The example above, and the assoc iated equat ions, are l imited to and to the w ind ing

geometry shown . The equat ions do not quant ify the behav ior for nor for round conductors, nor arethe equat ions suff iciently general to cover the more compl icated w ind ing geometr ies often encounteredin the magnet ic dev ices of sw itch ing converters . Opt imum des igns may, in fact, occur w ith conductorthicknesses in the v icinity of one penetrat ion depth . The d iscuss ions of the follow ing sect ions allow com-putat ion of prox imity losses in more general c ircumstances .

As descr ibed above, an externally-appl ied magnet ic f ield w ill induce eddy currents to flow in a conduc-tor, and thereby induce copper loss . To understand how magnet ic f ields are or iented in w ind ings, let uscons ider the s imple two-w ind ing transformer illustrated in Fig. 13.26. In th is example, the core has largepermeab ility The pr imary w ind ing cons ists of e ight turns of w ire arranged in two layers, andeach turn carr ies current i(t ) in the d irect ion ind icated . The secondary w ind ing is ident ical to the pr imarywind ing, except that the current polar ity is reversed .

Flux l ines for typ ical operat ion of th is transformer are sketched in Fig. 13.26(b) . As descr ibed inSect ion 13 .2, a relat ively large mutual flux is present, wh ich magnet izes the core . In add ition, leakage

flux is present, wh ich does not completely l in k both w ind ings . Because of the symmetry of the w ind inggeometry in Fig. 13.26, the leakage flux runs approx imately vert ically through the w ind ings .

To determ ine the magn itude of the leakage flux, we can apply Ampere s Law . Cons ider theclosed path taken by one of the leakage flux l ines, as illustrated in Fig. 13.27. Since the core has largepermeab ility, we can assume that the MMF induced in the core by th is flux is negl igible, and that the


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total MMF around the path is dom inated by the MMF across the core w indow . Hence, Ampere sLaw states that the net current enclosed by the path is equal to the MMF across the a ir gap :

where is the he ight of the w indow as shown in Fig. 13.27. The net current enclosed by the pathdepends on the number of pr imary and secondary conductors enclosed by the path, and is therefore afunct ion of the hor izontal pos ition x. The f irst layer of the pr imary w ind ing cons ists of 4 turns, each car-rying current i(t ). So when the path encloses only the f irst layer of the pr imary w inding, then the enclosedcurrent is 4 i(t ) as shown in Fig. 13.28. L ikew ise, when the path encloses both layers of the pr imary w ind-ing, then the enclosed current is 8i(t ). When the path encloses the ent ire pr imary, plus layer 2 of the sec-ondary w ind ing, then the net enclosed current is 8i(t ) 4 i(t ) = 4 i(t ). The MMF across the corewindow is zero outs ide the w ind ing, and r ises to a max imum of 8 i(t ) at the interface between the pr imaryand secondary w ind ings . Since the magnet ic f ield intens ity H ( x ) is proport ional to the

sketch of F ig. 13.28.


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It should be noted that the shape of the curve in the v ic in i ty of the w ind ing conductorsdepend s on the d istribution of the current w ithin the conductors . Since th is distribution is not yet known,the curve of F ig. 13.28 is arb itrar ily drawn as stra ight l ine segments .

In general, the magnet ic f ields that surround conductors and lead to eddy currents must be

determ ined us ing f inite element analys is or other s imilar methods . However, in a large class of coax ialsoleno idal w ind ing geometr ies, the magnet ic f ield l ines are nearly parallel to the w ind ing layers . Asshow n below, we can then obta in an analyt ical solut ion for the prox imity losses .

The w ind ing symmetry descr ibed in the prev ious sect ion allows s impl if icat ion of the analys is. For thepurposes of determ ining leakage inductance and w ind ing eddy currents, a layer cons ist ing of turns of round w ire carry ing current i(t ) can be approx imately modeled as an effect ive s ingle turn of fo il, wh ichcarr ies current The steps in the transformat ion of a layer of round conductors into a fo il conductorare formal ized in Fig. 13.29 [6, 811] . The round conductors are replaced by square conductors hav ingthe same copper cross-sect ional area, F ig. 13.29(b) . The th ickness h of the square conductors is therefore


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equal to the bare copper w ire d iameter, mult iplied by the factor

These square conductors are then joined together, into a fo il layer [F ig. 13.29(c)] . Finally, the w idth of the fo il is increased, such that it spans the w idth of the core w indow [F ig. 13.29(d)] . Since th is stretch ingproces s increases the conductor cross-sect ional area, a compensat ing factor must be introduced suchthat the correct dc conductor res istance is pred icted . Th is factor, somet imes called the conductor spacing

factor or the w inding porosity, is def ined as the rat io of the actual layer copper area [F ig. 13.29(a)] to thearea of the effect ive fo il conductor of F ig. 13.29(d). The poros ity effect ively increases the res ist ivity of

the conductor, and thereby increases its sk in depth :

If a layer of w idth conta ins turns of round w ire hav ing d iameter d, then the w ind ing poros ity isgiven by

A typ ical value of for round conductors that span the w idth of the w ind ing bobb in is 0.8. In the follow-

ing analys is, the factor is given by for fo il conductors, and by the rat io of the effect ive fo il conduc-tor th ickness h to the effect ive sk in depth for round conductors as follows :

In th is sect ion, the average power loss P in a un iform layer of thickness h is determ ined . As illustrated in F ig. 13.30, the mag-net ic f ield strengths on the left and r ight s ides of the conductor aredenoted H (0) and H (d ) , respect ively . It is assumed that the compo-nent of magnet ic f ield normal to the conductor surface is zero .These magnet ic f ields are dr iven by the magnetomot ive forces

and respect ively . Sinuso idal waveforms are assumed,and rms magn itudes are employed . It is further assumed here that

H (0) and H (h) are in phase ; the effect of a phase sh ift is treated in[10].

With these assumpt ions, Maxwell s equat ions are solvedto f ind the current dens ity d istr ibut ion in the layer . The power loss

dens ity is then computed, and is integrated over the volume of thelayer to f ind the total copper loss in the layer [10] . The result is


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where is the number of turns in the layer, and is the dc res istance of the layer . The funct ionsand are

If the w ind ing carr ies current of rms magn itude I , then we can wr ite

Let us further express in terms of the w inding current I , as

The quant ity m is therefore the rat io of the MMF to the layer ampere-turns Then,

The power d issipated in the layer, Eq . (13.74), can then be wr itten

where

We can conclude that the prox imity effect increases the copper loss in the layer by the factor

Equat ion (13.81), in con junct ion w ith the def initions (13 .80), (13 .77), (13 .75), and (13 .73), can be plot-ted us ing a computer spreadsheet or small computer program . The result is illustrated in Fig. 13.31, forseveral values of m.

It is illum inat ing to express the layer copper loss P in terms of the dc power loss thatwould be obta ined in a fo il conductor hav ing a th ickness Th is loss is found by d ividing Eq . (13.81)by the effect ive th ickness rat io

Equat ion (13 .82) is plotted in Fig. 13.32. Large copper loss is obta ined for small s imply because thelayer is th in and hence the dc res istance of the layer is large . For large m and large the p rox imity effectleads to large power loss ; Eq. (13.66) pred icts that is asymptot ic to for largeBetween these extremes, there is a value of wh ich m inimizes the layer copper loss .


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Let us aga in cons ider the prox imity loss in a convent ional transformer, in which the pr imary and second-

ary w ind ings each cons ist of M layers . The normal ized MMF d iagram is illustrated in F ig. 13.33. Asgiven by Eq . (13 .81), the prox imity effect increases the copper loss in each layer by the factorThe total increase in pr imary w ind ing copper loss is found by summat ion over al l of the pr imary lay-ers:

Ow ing to the symmetry of the w ind ings in th is example, the secondary w ind ing copper loss is increasedby the same factor . Upon subst i tu t ing Eq . (13 .80) and collect ing terms, we obta in

The summat ion can be expressed in closed form w ith the help of the ident ities

Use of these ident i t ies to s impl ify Eq . (13 .84) leads to

This express ion is plotted in Fig. 13.34, for several values of M. For large tends to 1, wh i le


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tends to 0 . It can be ver if ied that then tends to the value pred icted by Eq . (13.68).We can aga in express the total pr imary power loss in terms of the dc power loss that would be

obta ined us ing a conductor in wh ich Th is loss is found by d ividing Eq . (13.86) by

This express ion is plotted in Fig. 13.35, for several values of M . Depend ing on the number of layers, theminimum copper loss for s inuso idal exc itat ion is obta ined for near to, or somewhat less than, un ity .

One way to reduce the copper losses due to the prox imity effect is to interleave the w ind ings . Figure13.36 illustrates the MMF d iagram for a s imple transformer in wh ich the pr imary and secondary layersare alternated, w ith net layer current of magn itude i. It can be seen that each layer operates w ith onone s ide, and on the other . Hence, each layer operates effect ively w ith m = 1. Note that Eq . (13.74)is symmetr ic w ith respect to and hence, the copper losses of the interleaved secondary andprimary layers are ident ical . The prox imity losses of the ent ire w ind ing can therefore be determ ineddirectly from F ig. 13.34 and 13 .35, w ith M = 1. It can be shown that the m inimum copper loss for th is

case (w ith s inuso idal currents) occurs w ith although the copper loss is nearly constant for anyand is approx imately equal to the dc copper loss obta ined when It should be apparent thatinterleav ing can lead to s ign if icant improvements in copper loss when the w ind ing conta ins several lay-ers.

Part ial interleav ing can lead to a part ial improvement in prox imity loss . Figure 13 .37 illustratesa transformer hav ing three pr imary layers and four secondary layers . If the total current carr ied by eachprimary layer is i(t ) , then each secondary layer should carry current 0 .75i(t ). The max imum MMF aga inoccurs in the spaces between the pr imary and secondary w ind ings, but has value 1 .5i(t ).

To determ ine the value for m in a g iven layer, we can solve Eq . (13.78) for m:


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The above express ion is val id in general, and Eq . (13 .74) is symmetr ical in and However,when F (0) is greater in magn itude than it is conven ient to interchange the roles of andso that the plots of F igs . 13 .31 and 13 .32 can be employed .

In the leftmost secondary layer of F ig. 13.37, the layer carr ies current 0 .75i. The MMFchanges from 0 to 0 .75i. The value of m for th is layer is found by evaluat ion of Eq . (13.88):

The loss in th is layer, relat ive to the dc loss of th is secondary layer, can be determ ined us ing the plots of Figs. 13 .31 and 13 .32 w ith m = 1 . For the next secondary layer, we obta in

Hence the loss in th is layer can be determ ined us ing the plots of F igs. 13.31 and 13 .32 w ith m = 2.Thenext layer is a pr imary-w ind ing layer . Its value of m can be expressed as

The loss in th is layer, relat ive to the dc loss of th is pr imary layer, can be determ ined us ing the plots of Figs. 13.31 and 13 .32 w ith m = 1 .5. The center layer has an m of

The loss in th is layer, relat ive to the dc loss of th is pr imary layer, can be determ ined us ing the plots of Figs. 13.31 and 13 .32 w ith m = 0.5. The rema ining layers are symmetr ical to the correspond ing layersdescr ibed above, and have ident ical copper losses . The total loss in the w inding is found by summ ing thelosses descr ibed above for each layer .


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Interleav ing windings can s ignif icantly reduce the prox imity loss when the pr imary and second-ary currents are in phase . However, in some cases such as the transformers of the flyback and SEPIC con-verters, the w ind ing currents are out of phase . Interleav ing then does l ittle to reduce the MMFs andmagnet ic f ields in the v icinity of the w indings, and hence th e prox imity loss is essent ially unchanged . Itshould also be noted that Eqs . (13.74) to (13 .82) assume that the w ind ing currents are in phase . Generalexpress ions for out-of-phase currents, as well as analys is of a flyback example, are g iven in [10] .

The above procedure can be used to determ ine the h igh-frequency copper losses of more com-plicated mult iple-w ind ing magnet ic dev ices . The MMF d iagrams are constructed, and then the powerloss in each layer is evaluated us ing Eq . (13 .81) . These losses are summed, to f ind the total copper loss .The losses induced in electrostat ic sh ields can also be determ ined. Several add itional examples are g iven

in [10]. It can be concluded that, for s inuso idal currents, there is an opt imal conductor th ickness in thevicinity of that leads to m inimum copper loss . It is h ighly advantageous to m inimize the number of layers, and to interleave the w ind ings . The amount of copper in t h e v icinity ofthe h igh-MMF port ions of wind ings should be kept to a m inimum . Core geometr ies that max imize the w idth of the layers, wh ileminimizing the overall number of layers, lead to reduced prox imity loss .

Use of Litz wire is another means of increas ing the conductor area wh ile ma intaining low prox-imity losses . Tens, hundreds, or more strands of small-gauge insulated copper w ire are bundled together,and are externally connected in parallel . These strands are tw isted, or transposed, such that each strandpasses equally through each pos ition ins ide and on the surface of the bundle . Th is prevents the c ircula-tion of h igh-frequency currents between strands . To be effect ive, the d iameter of the strands should be

suff iciently less than one sk in depth . Also, it should be po inted out that the L itz w ire bundle itself is com-pose d of mult iple layers . The d isadvantages of L itz w ire are its increased cost, and its reduced f ill factor .

The pulse-w idth-modulated waveforms encountered in sw itching converters conta in s ignif icant harmon-ics, wh ich can lead to increased prox imity losses . The effect of harmon ics on the losses in a layer can bedeterm ined v ia f ield harmon ic analys is [10], in wh ich the MMF waveforms and of Eq .(13.74) are expressed in Four ier ser ies. The power loss of each ind ividual harmon ic is computed as inSect ion 13.4.4, and the losses are summed to f ind the total loss in a layer . For example, the PWM wave-form of F ig. 13.38 can be represented by the follow ing Four ier ser ies:

with Th is waveform conta ins a dc component plus harmon ics of rms magn itude

proport ional to 1/ j. The transformer w ind ing current waveforms of most sw itch ing converters follow th isFour ier ser ies, or a s imilar ser ies.

Effects of waveforms harmon ics on prox imity losses are d iscussed in [810] . The dc componentof the w ind ing currents does not lead to prox imity loss, and should not be included in prox imity loss cal-culat ions . Fa ilure to remove the dc component can lead to s ign if icantly pess imist ic est imates of copper

where


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loss . The sk in depth is smaller for h igh frequency harmon ics than for the fundamental, and hence the

waveform harmon ics exh ibit an increased effect ive Let be g iven by Eq . (13.73), in wh ich is foundby evaluat ion of Eq . (13 .60) at the fundamental frequency . Since the penetrat ion depth var ies as theinverse square-root of frequency, the effect ive value of for harmon ic, j is

In a mult iple-layer w ind ing exc ited by a current waveform whose fundamental component hasclos e to 1, harmon ics can s ignif icantly increase the total copper loss . This occurs because, for m > 1,

is a rap idly increas ing funct ion of in the v icinity of 1 . When is suff icient ly greater than 1,then is nearly constant, and harmon ics have less infl uence on the total copper loss .

For example, suppose that the two-w inding transformer of F ig. 13.33 is employed in a converter

such as the forward converter, in wh ich a w ind ing current waveform i(t ) can be well approx imated by theFour ier ser ies of Eq . (13.93). The w ind ing conta ins M layers, and has dc res istance The copper lossinduced by the dc component is

The copper loss ascr ibable to harmon ic j is found by evaluat ion of Eq . (13 .86) w ith

The total copper loss in the w inding is the sum of losses ar ising from all components of the harmon icseries:

In Eq . (13 .97), the copper loss is expressed relat ive to the loss pred icted by a low-frequencyanalys is. This express ion can be evaluated by use of a computer program or computer spreadsheet .

To expl icitly quant ify the effects of harmon ics, we can def ine the harmon ic loss factor as

with given by Eq . (13.96). The total w ind ing copper loss is then g iven by


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13.5 Several Types of Magnetic Devices, Their BH Loops, and Core vs. Copper Loss

with given by Eq . (13 .86). The harmon ic factor is a funct ion not only of the w inding geometry, butalso of the harmon ic spectrum of the w ind ing current waveform . The harmon ic factor is plotted inFig. 13.39 for several values of D, for the s imple transformer example . The total harmon ic d istort ion(THD) of the example current waveforms are : 48% for D = 0.5, 76% for D = 0.3, and 191% for D = 0 .1.The waveform THD is def ined as

It can be seen that harmon ics s ign if icantly increase the prox imity loss of a mult ilayer w ind ing when isclose to 1 . For suff iciently small the prox imity effect can be neglected, and tends to the value

For large the harmon ics also increase the prox im ity loss ; however, the increase is lessdramat ic than for near 1 because the fundamental component prox imity loss is large . It can be con-cluded that, when the current waveform conta ins h igh THD and when the w ind ing conta ins several lay-ers or more, then prox imity losses can be kept low only by choos ing much less than 1 . Interleav ing thewindings allows a larger value of to be employed .

BH

A var iety of magnet ic elements are commonly used in power appl icat ions, wh ich employ the propert iesof magnet ic core mater ials and w indings in d ifferent ways . As a result, qu ite a few factors constra in thedesign of a magnet ic dev ice. The max imum flux dens ity must not saturate the core . The peak ac flux den-sity should also be suff iciently small, such that core losses are acceptably low . The w ire size should besuff iciently small, to f it the requ ired number of turns in the core w indow . Sub ject to th is constra int, thewire cross-sect ional area should be as large as poss ible, to m inimize the w inding dc res istance and cop-per loss . But if the w ire is too th ick, then unacceptable copper losses occur ow ing to the prox imity effect .An a ir gap is needed when the dev ice stores s ign if icant energy . But an a ir gap is undes irable in trans-former appl icat ions . It should be apparent that, for a g iven magnet ic dev ice, some of these constra ints areactive wh ile others are not s ignif icant .

Thus, des ign of a magnet ic element involves not only obta ining the des ired inductance or turnsratio, but also ensur ing that the core mater ial does not saturate and that the total power loss is not toolarge . Several common power appl icat ions of magnet ics are d iscussed in th is sect ion, wh ich illustrate thefactors govern ing the cho ice of core mater ial, max imum flux dens ity, and des ign approach .

A f ilter inductor employed in a CCM buck converter is illustrated in Fig. 13.40(a) . In th is appl icat ion, thevalue of inductance L is usually chosen such that the inductor current r ipple peak magn itude is a smallfract ion of the full-load inductor current dc component I , as illustrated in Fig. 13.40(b) . As illustrated inFig. 13.41, an a ir gap is employed that is suff iciently large to prevent saturat ion of the core by the peak current


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The core magnet ic f ield strength is related to the w ind ing current i(t ) accord ing to

where is the magnet ic path length of the core . S ince is proport ional to i(t ), can be expressedas a large dc component and a small super imposed ac r ipple where

A sketch of B(t ) vs. for th is appl ication is given in Fig. 13.42. This dev ice operates w ith the m inor BH loop illustrated . The s ize of the m inor loop, and hence the core loss, depends on the magn itude of the inductor current r ipple The copper loss depends on the rms inductor current r ipple, essent ially


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13.5 Several Types of Magnetic Devices, Their BH Loops, and Core vs. Copper Loss

In the transformer appl icat ion, core loss and prox imity losses are usually s ign if icant . Typ icallythe max imum flux dens ity is l imited by core loss rather than by saturat ion. A h igh-frequency mater ialhav ing low core loss is employed . Both core and copper losses must be accounted for in the des ign of thetransformer . The des ign must also incorporate mult iple w ind ings . Transformer des ign w ith flux dens ityoptimized for m inimum total loss is descr ibed in Chapter 15 .

A coupled inductor is a f ilter inductor hav ing mult iple w indings . Figure 13 .47(a) illustrates coupledinductors in a two-output forward converter . The inductors can be wound on the same core, because thewind ing voltage waveforms are proport ional . The inductors of the SEPIC and converters, as well asof mult iple-output buck-der ived converters and some other converters, can be coupled . The inductor cur-ren t ripples can be controlled by control of the w inding leakage inductances [12,13] . DC currents flow ineach w ind ing as illustrated in F ig. 13.47(b), and the net magnet izat ion of the core is proport ional to thesum of the w ind ing ampere-turns :

As in the case of the s ingle-w ind ing f ilter inductor, the s ize of the m inor BH loop is proport ional to thetotal current r ipple (F ig. 13.48). Small r ipple impl ies small core loss, as well as small prox imity loss . Anair gap is employed, and the max imum flux dens ity is typ ically l imited by saturat ion.


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As d iscussed in Chapter 6, the flyback transformer funct ions as an inductor w ith two w ind ings . The pr i-mary w ind ing is used dur ing the trans istor conduct ion interval, and the secondary is used dur ing thediode conduct ion interval . A flyback converter is illustrated in Fig. 13.49(a), w ith the flyback transformermodeled as a magnet izing inductance in parallel w ith an ideal transformer . The magnet izing currentis proport ional to the core magnet ic f ield strength Typ ical DCM waveforms are g iven in F ig.13 .49(b) .

Since the flyback transformer stores energy, an a ir gap is needed . Core loss depends on the mag-nitude of the ac component of the magnet izing current . The BH loop for d iscont inuous conduct ionmode operat ion is illustrated in Fig. 13.50. When the converter is des igned to operate in DCM, the coreloss is s ign if icant . The peak ac flux dens ity is then chosen to ma inta in an acceptably low core loss .For CCM operat ion, core loss is less s ignif icant, and the max imum flux dens ity may be l imited only bysaturat ion of the core . In e ither case, w ind ing prox imity losses are typ ically qu ite s ign if icant . Unfortu-nately, interleav ing the w ind ings has l ittle impact on the prox imity loss because the pr imary and second-ary w ind ing currents are out of phase .


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13.6 Summary of Key Points

9.

Magnet ic dev ices can be modeled us ing lumped-element magnet ic c ircu its, in a manner s imilar to thatcommonly used to model electr ical c ircu its. The magnet ic analogs of electr ical voltage V , current I , and

and h igh core loss Lam inated iron alloy cores exh ib it the h ighest but also the h ighestwh ile ferr ite cores exh ibit the lowest but also the lowest Between these two extremes are pow-dered iron alloy and amorphous alloy mater ials .

The sk in and prox im ity effects lead to eddy currents in w ind ing conductors, wh ich increase the copper lossin h igh-current h igh-frequency magnet ic dev ices . When a conductor has th ickness approach ing or

1.

2 .

3 .

4 .

5 .

6 .

7 .

8 .

res istance R, are magnetomot ive force (MMF) and reluctance respect ively .

Faraday s law relates the voltage induced in a loop of w ire to the der ivat ive of flux pass ing through theinter ior of the loop .

Ampere s law relates the total MMF around a loop to the total current pass ing through the center of theloop . Ampere s law impl ies that w ind ing currents are sources of MMF, and that when these sources areincluded, then the net MMF around a closed path is equal to zero .

Magnet ic core mater ials exh ib i t hysteres is and saturat ion . A core mater ial saturates when the flux dens ity B reaches the saturat ion flux dens ity

Air gaps are employed in inductors to prevent saturat ion when a g iven max imum current flows in thew ind ing, and to stab ilize the value of inductance . The inductor w ith a ir gap can be analyzed us ing a s implemagnet ic equ ivalent c ircu i t, conta in ing core and a ir gap reluctances and a source represent ing the w ind ingMMF .

Convent ional transformers can be modeled us ing sources represent ing the MMFs of each w ind ing, and thecore MMF . The core reluctance approaches zero in an ideal transformer . Nonzero core reluctance leads toan electr ical transformer model conta ining a magnet izing inductance, effect ively in parallel w ith the idealtransformer . Flux that does not l ink both w indings, or leakage flux, can be modeled us ing ser ies induc-tors .

The convent ional transformer saturates when the appl ied w ind ing volt-seconds are too large . Add ition of an a ir gap has no effect on saturat ion . Saturat ion can be prevented by increas ing the core cross-sect ionalarea, or by increas ing the number of pr imary turns .

Magnet ic mater ials exh ibit core loss, due to hysteres is of the BH loop and to induced eddy currents flow-ing in the core mater ial. In ava ilable core mater ials, there is a tradeoff between h igh saturat ion flux dens ity


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10.

11.

12.

larger than the penetrat ion depth magnet ic f ields in the v ic in ity of the conductor induce eddy currents inthe conductor . Accord ing to Lenz s law, these eddy currents flow in paths that tend to oppose the appl iedmagnet ic f ields .

The magnet ic f ield strengths in the v icinity of the w ind ing conductors can be determ ined by use of MMFdiagrams . These d iagrams are constructed by appl icat ion of Ampere s law, follow ing the closed paths of the magnet ic f ield l ines wh ich pass near the w ind ing conductors . Mult iple-layer non interleaved w ind ingscan exh ibit h igh max imum MMFs, w ith result ing h igh eddy currents and h igh copper loss .

An express ion for the copper loss in a layer, as a funct ion of the magnet ic f ield strengths or MMFs sur-round ing the layer, is g iven in Sect ion 13 .4.4. Th is express ion can be used in con junct ion w ith the MMFdiagram, to compute the copper loss in each layer of a w ind ing . The results can then be summed, y ield ing

the total w ind ing copper loss . When the effect ive layer th ickness is near to or greater than one sk in depth,the copper losses of mult iple-layer non interleaved w ind ings are greatly increased .

Pulse-w idth-modulated w ind ing currents conta in s ign if icant total harmon ic d istort ion, wh ich can lead to afurther increase of copper loss . The increase in prox imity loss caused by current harmon ics is most pro-nounced in mult iple-layer non- interleaved w ind ings, w i th an effect ive layer th ickness near one sk in depth .

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

MIT S TAFF , Magnetic Circuits and Transformers, Cambr idge : The MIT Press, 1943 .

J. K. W ATSON , Applications of Magnetism, New York : John W iley & Sons, 1980 .

R. P. SEVERNS and G . E. BLOOM , Modern Dc-to-Dc Switchmode Power Converter Circuits, New York :Va n Nostrand Re inhold, 1985 .

A. D AUHAJRE and R . D . M IDDLEBROOK , Model ing and Est imat ion of Leakage Phenomena in Magnet icCircu its, IEEE Power Electronics Specialists Conference, 1986 Record, pp . 213-226 .

S. E L-HAMAMSY and E . C HANG , Magnet ics Model ing for Computer-A ided Des ign of Power Electron icsC ircu its, IEEE Power Electronics Specialists Conference, 1989 Record, pp . 635-645 .

P. L. DOWELL , Effects of Eddy Currents in Transformer W indings, Proceedings of the IEE, Vol . 113, No .

8, August 1966, pp . 1387-1394 .

M. P . PERRY , Mult iple Layer Ser ies Connected W inding Des ign for M inimum Loss, IEEE Transactionson Power Apparatus and Systems, Vol . PAS-98, No . 1, Jan . /Feb . 1979, pp . 116-123 .

P. S . V ENKATRAMAN , Winding Eddy Current Losses in Sw itch Mode Power Transformers Due to Rect-angula r Wave Currents, Proceedings of Powercon 11, 1984, pp . A 1 .1 - A 1 .11 .

B. CARSTEN , High Frequency Conductor Losses in Sw itchmode Magnet ics, High Frequency Power Converter Conference, 1986 Record, pp . 155-176 .

J. P . V ANDELAC and P . Z IOGAS , A Novel Approach for M inimizing H igh Frequency Transformer Copper

Losses, IEEE Power Electronics Specialists Conference, 1987 Record, pp . 355-367 .

A. M . URLING , V . A . N IEMELA , G . R. SKUTT , and T . G . W ILSON , Character izing H igh-Frequency Effectsin Transformer W indingsA Gu ide to Several S ignif icant Art icles, IEEE Applied Power ElectronicsConference, 1989 Record, pp . 373-385 .


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Problems

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[13]

S. and R . D. M IDDLEBROOK , Coupled-Inductor and Other Extens ions of a New Opt imum TopologySwitch ing DctoDc Converter, IEEE Industry Applications Society Annual Meeting, 1977 Proceed ings,pp. 1110-1122 .

S. and Z . ZHANG , Coupled-Inductor Analys is and Des ign, IEEE Power Electronics SpecialistsConference, 1986 Record, pp . 655-665 .

The core illustrated in Fig. 13.51(a) is 1 cm th ick . All legs are 1 cm w ide, except for the r ight-hand s ide

vert ical leg, wh ich is 0.5 cm w ide. You may neglect nonun iform ities in the flux d istribution caused byturn ing corners .

Two w ind ings are placed as illustrated in Fig. 13.52(a) on a core of un iform cross-sect ional areaEach w ind ing has 50 turns . The relat ive permeab ility of the core is

Sketch an equ ivalent magnet ic circu it, and determ ine numer ical values for each reluctance .

Determ ine the self- inductance ofeach w ind ing .

Determ ine the inductance obta ined when the w indings are connected in ser ies as in Fig.13.52(b) .

Determ ine the inductance obta ined when the w ind ings are connected in ant i-ser ies as in Fig.13.52(c) .

Determ ine the magnet ic c ircu it model of th is dev ice, and label the values of all reluctances inyour model .

Determ ine the inductance of the w inding.

A second w ind ing is added to the same core, as shown in Fig. 13.5 l(b) .

Mod ify your model ofpart (a) to include th is wind ing .

The electr ical equat ions for th is circu it may be wr itten in the form

Use superpos ition to determ ine analyt ical express ions and numer ical values for and


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All three legs of the magnet ic dev ice illustrated in Fig. 13.53 are of un iform cross-sect ional areaLegs 1 and 2 each have magnet ic path length wh ile leg 3 has magnet ic path length Both w ind ingshave n turns . The core has permeab ility

Sketch a magnet ic equ ivalent c ircu it, and g ive analyt ical express ions for all element values .

A voltage source is connected to w ind ing 1, such that is a square wave of peak value andper iod W ind ing 2 is open-c ircu ited .

Sketch and label its peak value .

Find the flux in leg 2 . Sketch and label its peak value .

Sketch and label its peak value .

The magnet ic dev ice illustrated in Fig. 13.54(a) cons ists of two w ind ings, wh ich can replace the twoinductors in a SEPIC, or other s imilar converter . For th is problem, all three legs have the same un i-form cross-sect ional area The legs have gaps of lengths and respect ively . The core perme-ability is very large . Yo u may neglect fr ing ing flux . Legs 1 and 2 have w ind ings conta ining andturns, respect ively .

Der ive a magnet ic c ircu it model for th is dev ice, and g ive analyt ical express ions for each reluc-tance in your model . Label the polar ities of the MMF generators .

Wr ite the electr ical term inal equat ions of th is dev ice in the matr ix form


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Problems

and der ive analyt ical express ions for and

Der ive an electr ical c ircu i t model for th is dev ice, and g ive analyt ical express ions for the turnsrat io and each inductance in your model, in terms of the tu rns and reluctances of part (a) .

Th is s ingle magnet ic dev ice is to be used to real ize the two inductors of the converter, as in F ig.

13.54(b) .Sketch the voltage waveforms and mak ing the l inear r ipple approx imat ion as appro-pr iate . You may assume that the converter operates in the cont inuous conduct ion mode .

The voltage waveforms of part (d) are appl ied to your model of parts (b) and (c) . Solve yourmodel to determ ine the slopes of the inductor current r ipples dur ing intervals andSketch the steady-state inductor current waveforms and and label all slopes .

By sk illful cho ice of and the a ir gap lengths, it is poss ible to make the inductor current r ip-ple in e ither or go to zero . Determ ine the cond itions on and thatcause the current r ipple in to become zero . Sketch the result ing and and label allslopes .

It is poss ible to couple the inductors in th is manner, and cause one of the inductor current r ipples to go tozero, in any converter in wh ich the inductor voltage waveforms are proport ional .

Over its usable operat ing range, a certa in permanent magnet mater ial has the BH character ist ics illus-trated by the sol id l ine in F ig. 13.55. The magnet has length and cross-sect ional area

T. Der ive an equ ivalent magnet ic c ircu it model for the magnet, and label the numer ical values of the elements .


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