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N A S A T E C H N I C A L N O T E

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HYSTERESIS A N DA TRANSFORMERA N A PP LIC A TIO N

by John BarrangerLewis Research CenterCleveland, Ohio

EDDY-CURRENT LOSSES OFLAMINATION VIEWED ASOF T H E P O Y N T I N G T H E O R E M

N A T I O N A L A E R O N AU T I C S A N D S PA CE A D M I N I S T R A T I O N W A S H I NG T O N , D . C. N O V E M B E R 1 9 6 5


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TECH LIBRARY KAFB. NMIl1111111111111I1llllI11I1MI

HYSTERESIS AND EDDY-CURRENT LOSSES O F A TRANSFORMERLAMINATION VIEWED AS AN APPLICATION O F

THE POYNTING THEOREMBy John Barranger

Lewis Resea rch CenterCleveland, Ohio

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONFor sale by the Clearinghouse for Federal Scientific and Technical Information

Springfield, Virginia 22151 - Price $ 1 . 0 0


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HYSTERESIS AN D EDDY-CURRENT LO SSES OF A TRANSFORM ER LA M I NA T I ONVIEW ED AS A N A PP L I CAT I ON OF THE POYN TI NG THEOREM

by J o h n B a r r a n g e rL e w i s R e s e a rc h C e n t e r

S U M M A R YAn energy-loss equation is developed based on the Poynting theo rem. The resultan t

integ rals provide a unified viewpoint that is shown to be the sa m e as the clas sical eddy-cu rr en t and hyste resis energy formulations. An expression is derived for the eddy-curre nt lo ss of a thin transformer lamination by assuming that the permeability is con-stant, and further, that the magnetic materia l is isotropic and homogeneous. The hys-te res i s loss is found by assuming that the introduction of the h ys te re si s loop does notalt er the fields. The re su lts indicate that, fo r a given total flux and at frequencies ap -proaching total shielding, both lo ss es a r e proportional to the 3/2 power of frequency andthe squa re root of th e conductivity and inversely p roportional to the squ are root ofpermeability.

I N T R O D U C T I O NAs part of a study of the distribution and us e of power on spacec raf t, the Lewis

Resea rch Center has been evaluating power sys tem components in unusual configurationsand at temp eratu res and frequencies outside the ordinary range s. Early in this study itbecame evident that neither classical nor modern approaches to magnetism could com-pletely predict trans form er prop ertie s and their variations with frequency.ventional engineering anal ysis which as su mes the magnetic ma ter ial to be isotropic andhomogeneous and defines the los se s in te rm s of the hys te re si s loop and classica l e lectro -dynamics. The modern theory of magnetism utiliz es cr ys ta l energies and magnetic do-main configurations to provide an understanding of lo ss es ( ref s. l to 3 ) . The classicallyderived loss es a r e characterized by measurements that are relatively easy to make whilethe modern theory relies on cry st al anisotropy energ ies and domain w a l l velocities tha t

The classical approach to the study of lo ss es in ferromagnetic ma terial s is the con-


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111 I1 I I Ill l I II I I1

a r e very difficult to evaluate and measu re in polycrystalline ma ter ial s.and eddy-current los s. Hysteresis lo ss is thought to be as sociat ed with the energy r e -quired to rotat e or move the walls of the magnetic domains over a full cycle. The hys-te res i s loss is proportional to the ar e a of the familiar hy ste re sis loop. The eddy-currentloss is due to the cu rr en ts generated within a re a l conductor subjected to a varying mag-netic field.

The purpose of th is rep or t is (1) o provide a bet ter understanding of c lass ica l theoryfo r the re ad er s already familiar with the basic engineering approach and (2) to derive ex-pressio ns useful in predicting the dependence of co re lo ss es on frequency and easilymeasured properties. Hysteresis and eddy -current los se s ar e usually treated separatelywith no common underlying basis. A mor e unified viewpoint will be provided startingwith Maxwell's equations and will show that considerations of the Poynting theorem yieldthe classical co re lo ss formulations. The separation of c or e los se s into their two t ra-ditional components results as a natura l consequence of t he ana lysis.

A la rg e pa rt of the study is devoted to finding expression s fo r the hyste res is andeddy-current los se s fo r thin transfo rmer laminations. The magnetic materi al will beassum ed to be isotropic and homogeneous and to have a permeability that is constant.The permeability is obtained fr om the mean magnetization curve by taking the ra tio of theflux density to the corresponding magnetic field intensity.amined to ascert ain the behavior of the lo ss es with frequency.expressions , low- and high-frequency approximations will be derived. The re su lt s willalso be used to predict the important par am et ers of us e in high-frequency transfo rmerdesign. This should be of in te re st to design ers who are seeking methods of using lami-nated magnetic ma te ri al fo r high-frequency power app lications. It is possible that im -provements in weight and power efficiencies may be gained through the us e of high-frequency power systems.

Traditionally, tra ns fo rm er co re loss es have been divided into two groups, hy ste re sis

The formulas w i l l then be ex-To simplify the resultant

SYMBOLSa lamination thicknessB flux densityb length of laminationE potentialf frequencyH field intensityJ curren t density2


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1PStVW6ePP(T

height of laminationpower lossshape factortime variablevolumeenergyskin depthdummy variablepermeabilityfr ee charge densityconductivitytotal flux

POYNTING THEOREM AND CORE LOSSESEnergy transfer fr om a source to a magnetic material is accomplished through elec-

tromagnet ic induction, the amount of energy depending on the magnitude, dis tribution,and pha ses of the elec tri c and magnetic fields . Maxwell's equations, written in t e r m s ofthe total fields and cu rr en ts within the magnetic material, des crib e the electromagneticbehavior of the mate rial . The following conditions ar e appropriate to these equationswhen applied to good conductors: - -c(1)The free-charge te rm is zero, p = 0.(2 ) Conduction current density is given by Ohm's law, J(t)= aE(t).

(3 ) Displacement c ur ren t is negligible in comparison with conduction current.(4) The material is isotropic.

The two curl equations become - a i i ( t )V X E(t) = --t- -V x H(t) = J(t)-c 4If the divergence af E(t) X H(t) (the Poynting vec tor) is considered,

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v * [E(t) x .'(ti = G(t) * [v x ,-(t)] - G(t) . [v x .(ti = - E(t) . - E(t) T(t) (2)a tIntegrating over t he enclosed volume produces

- 1L -- / G(t) . + E(t) * J(t)dVa t/ v - [E(t) x G(t)ldV =V Vand using the divergence theorem r es ult s in

- 3-c-/ E(t) x H(t)] - d g =1 E&) -B(t) + E(t) J(t)dVa tS V(3 )

(4)

The total of all the te rm s to the right of the equal sign are identified as representing theenergy sto red and dissipated per unit time, the left side of equation (4 ) represe nts theenergy flow into the volume per unit time. The energy flow into the volume, then, be -tween two points of t ime t l and t2 is

W =lt2[.'(t) . ?!?@t + z(t) J(t)3 dV dt1

The time ra te of change of flux density fo r a stationary volume is-cdB(t) - 3% )dt a t

and

(5)

-c - 1W = jt2V . !?@ + E(t) - J(t)dV dtdtt l-b e -L +Since the material was assumed isotropic, B(t) and H(t), E(t) and J(t)a re in the same

direction and the energy expression becom es

(7)

4

.. .


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where B1 and B2 a re the flux densities corresponding to t l and t2, respectively.The second te rm rep res ent s the energy lo ss due to conduction cu rre nt s or the eddy-curre nt loss. Over a complete cycle, the first te rm r epr ese nts the energy loss, whichis called hys ter esi s lo ss . Thus, according to Maxwell's equations the total lo ss or coreloss is

PC = P h + e (9)where Ph is the hysteresis loss and Pe is the eddy-current l oss .The two significant nonlinearities are

Consider now the nonlinearities commonly encountered in ferromagnetic subst ances.(1)The hystere sis loop(2) The mean magnetization curve

To simplify the analysis, the mean magnetization cur ve will be assumed to be a straightline, that is , the permeability is a constant. The ma ter ial will also be assumed to behomogeneous as well as isotropic.

CALCULATION OF CURRENT A N D FIELDBefore any calculation of co re lo ss can be made, th e cu rr en t and field distribution

within the volume of th e ma te ri al must be known. Th is is usually accomplished by solvinga differential equation in the quantity of intere_cst. By re turn ing once mo re to Maxwell'sequations, the cu rl of the magnetizing fo rc e H (t ) may be written

-c -cV X H(t) = cE(t)and taking the c ur l of both si de s yie lds

aB(t)a t

-c-V2G(t) + V [V * H(t)] = -u-Since it is desir able to solve the previous expression in ter m s of one variable, such asB(t), the cl as si c problem of re lat ing B(t) and H(t) for ferro magnetic substances must befaced. Even though the hys te re si s loop shows that this is a nonlinear functional relation,it will be assumed that the introduction of the loop al te rs only slightly the distributionand waveshape of the fields. Oriented silicon iro n approxi mate s thi s condition at low tomoderate f lux density levels. The functional relation then becomes

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G(t) = &t) (12)where the permeability p is the constant slope of the meanmagnetization curve. Substituting, the resultant equation1 becomes1

a& (13)''0

V B(t) = UP-tConsider the lamination of a transformer as a thin sheet

D of thickness a where the height of the lamination Z is muchgre ater than the thickness. Let the field be applied to the s u r -face of the lamination in such a way that the induced eddy cur-

Figure 1 - Lamination of transformer.

ren t s a r e as indicated in figure l. It is necessary to know how these current s a r e dis-tributed before an express ion for power lo ss can be found.of a standard differential equation sim ila r to the diffusion or hea t equation. By a similarderivation, the equation that gives the relation between space and time deriva tives ofcur ren t density at any point in a conductor is

The equation derived previously for the variation of g(t) n a conductor is in the form

V 2J(t)= UP-J(t)a t'etting J(t)be sinusoidal and introducing the complex exponential form

-LJ(t)= Re ( eJut)res ul ts in the previous expression becoming -v J = jwcryJ

Study of figure 1 will show that, since no net c ur rent can flow out of a conductor,the following symmetry and boundary conditions exist for thin sheets with an axiallyapplied field:

(1)Jx = 0(2)J = 0Y

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2(3 )-Jz --J c o

2aJz - a Jz - 0(4 )- -az az2(5) Jz( = Jox=(a/2)

= -Jox=-(a/2)By using the first four conditions, the cur re nt distribution equation becom es simply

d2Jz 2-- - w o p J z = T J,ndx"

where T is a constant and is equal to (taking the roo t with positive sign)

It is convenient to wr ite the solution of th e differential equation in te r m s of hyperbolicfunctions

(19)z = A sinh Tx + C cosh TxTo evaluate arb itr ary constants A and C, boundary conditions (5 ) and (6 ) (above) areutilized

Solving these re su lt s in

Jo = A sinh T a+ C cosh T 52 2a a-Jo = -A sinh T - + C cosh T -2 2

c = o7


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JOA =sinh T 52

andsinh Tx

asinh T -2J = J oZ

Defining the quantity 6 as 1 dw, the current denshy (If-s i & ( - ) .

0, = J

istribution may be written as

where 6 is usually called the skin depth or depth of penetr ation. In ter ms of a semi-infinite solid, it is the depth at which cu rren t density has de crease d to l / e (about 36.9percent) of its value at the surface.with the skin depth; they are the following:frequency in cre as es. Thus, m ate ria ls of high permeability have a sm all er depth ofpenetration, or conversely, skin effect takes place at a lower frequency.

(2) The cu rre nt does not fail to penetr ate below the depth 6; t his is merely the pointat which curre nt densities and fields have decreased to l / e of their value at the surface.

(3 ) The skin depth may be consider ed a constant fo r a given material at a frequencyThe curr ent distribution at low frequenci es may be obtained by making the ra ti o

a/26 much less than one, that is , the skin depth is much grea ter than the thickness.Taking the absolu te value of the c ur re nt density gives

There are a number of impo rtant fac tor s associa ted(1)The depth of penetr ation decreases as the conductivity, t l e permeability and the

f and is a useful para me ter in the analys is of thin shee ts.

I JZI = IJoIX X X 2 x6 6 6 6sinh -co s - + j cosh - sin? I /sinh - + sin - (23)a a asinh -cos - j cosh - in-26 26 26 26

and i f a/26


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This distribution is linear with x. The re rem ains only I JoI to beUsing Faraday's line integr al and integrating around the path

4 (t)o dlc = --@(t)dt evaluated.ABCD in figure 1yield(25)where @(t)s the total flux within the line ABCD. Multiplying both si de s by conductivityQ and using the rela tion J(t)o= oE(t), make the following obtainable:-L -L

If the shee t has height 1 and the height is much greater than the thickness, the curr en tdensity Jo in complex notation form is

If the absolute value is taken191IJOI =y

then [ inha f + sin2UQ14lJzl =- 2 a 2 a2' sinh -+ sin -The previous exp ression rep re se nt s the curren t density distribution, which will be

used to find the eddy-current lo ss. In the analysis of hys ter es is loss, the flux densitydistribution must alsr, be known. This may be found directly fro m the c ur ren t densityexpression. Consider the cu rl equation

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I I I I I I I I 1111 II I I l l I I 1111111111111111 11 1 I I I 11111 1 111 ,1 1 1 , I

+V X E(t) = --tUsing the complex exponential for m

+and noting tha t J, is the only component of J result in

Y- jwoBJzax- - _ -

or

If the formulas for Jz and Jo found previously are now substituted,

The absolu te value of the flux density becomes

EDDY- CU R RENT LO S SThe first section showed that the eddy-current loss may be expressed as

J /t" % t dV10

(36)


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It can be shown that the avera ge power lo ss in complex notation is

If the element of volume is b2 dx (fig. 2), then the total power may be ex press ed as

The previous expression repre sen ts the power los s in a trans form er lamination dueto eddy cur ren ts. Latour (ref. 4)considered the sa me problem fr om the standpoint of aconstant lag of phase angle between the magnetic induction B( t) and magnetizing fieldE(t). His los s formula is the same as the one derived (eq. 3 8 ) . For a given laminationmate rial, thickness, and frequency, the lo ss va ri es as @ .

-c

2 Simpler expressions may

11 a - /bIFigure 2 - Lamination of transformershowing element of volume.

be obtained by examing the behavior of the equation as a func-tion of frequency. A t low frequencies o r when a/6


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when a/6 >> 1, the power l os s isob6* 'max

81

and the ave rage power pe r unit volume is2 1 J 2

3/2f3/2 2 1/2a 'maxu2 z 2ap1/2Pe(H) =

Thus, i f the sinusoidal field is regulated so that there is constant amax,he eddy-current lo ss varies as the sq ua re of the frequency at power f requen cies and as the 3/2power of frequency at frequencies approaching total eddy-current shielding.

H Y S T E R E S I S L O S SIt w a s stated previously that the integral

B

/ ireprese nts the hy steresis energy loss when itis taken over a full cycle, that is ,

I / I /

Figure 3. - Hysteresis loop.

The bracket re pre sen ts the are a of the hyst er-es is loop for the element dV under cons idera-tion, If th e area is expressed as a function ofthe variable V, then the formula may be inte -gra ted over the ent ire volume to yield the totalhyster esis loss per cycle. The H and Bwithin the bracket are obtained from the statichys tere sis loop of the mat eria l. The end point

1 2


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of th e hysteres is loop correspon ds to the maximum value of B. Thus , the solution of thedifferential equation in B may also be used to describ e thi s point. The a re a of thehys tere sis loop will be found first for the su rfac e of the sheet, and then the resul ts willbe extended to provide an expression for the a re a at any point.loop, a dummy variable 8 defined by the following equation must b e introduced:

-The hyste resis loop for the surfac e is given in figure 3 . To find the a re a of this

B = -B, C O S e (44)where Bm is the maximum value of B. If 8 is limited to a half period, the magnetiz-ing forc e H, may be described by the half-range Fourier expansion

H = aO m + alHm cos 8 + a2Hm cos 20 + - - - + anHm cos ne (45)where Hm is the maximum value of H . Examination of figure 3 shows that, as the fluxdensity tr av er se s the loop from -Bm to +Bm and back to -Bm, the ar ea over thecomplete cycle is twice the ar ea between -Bm to +B,. Thus,

f H dB = 2 liBmdBm

and i f it is noted that 8 = 0 and 8 = 1~ corre spond to -Bm and +Bm respectively, thear ea becomes

= 2 1'H B sin 8 + al cos 8 + a2 cos 28 + . . . + an cos n8)de0 m m2 - - a 4 - ' * + - 2an ) for n even21 - n5

Let the summation within the parentheses be called the shape factor S. The value of theshape fact or may be determined by solving the par ametri c equations fo r B and H forthe hyste res is loop being investigated. The number of t er ms in the summation depends

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on the number of data points and the clo sen ess of f i t desired. Since the permeability wasassu me d constant, B and H are related by

Bm = PHm

and the a re a of the hy ste resi s loop isThe hy ste res is energy integral c an now be solved

2IJ.

W(per cycle) = [$(H dB)] dV = dV

According to figure 2 (p. ll), c2+(per cycle = 4 t band since S and p a r e assumed constant, they will not depend on the coordinate x.Thus,

4lbSp2Bm dx(per cycle) =- O

(49)

It is now nec essary to find Bm in t e rm s of x to integrate the expression. It was foundpreviously that

(cos::+ sinh: r .1 '2sin2a+inh -

and by noting that

2Bm may now be solved14


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Substituting into the energy equation yields

W(per cycle) = 1 (COS' ?+ sinha E)&I-1 6 6

26 26

-

The hystere sis loss is f tim es the energy lost per cycle. Thus,

P - f W =h -

(55)

The previous expression rep re sen ts the power lo ss in a transformer lamination dueto hysteresis effects. Latour (ref. 4) ound the s am e formula except that this form ulafor h yst ere sis lo ss ha s the added advantage of being defined in te rm s of data taken fromthe static hy st ere si s loop. At power frequencies or when a/S


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which again is frequency tim es the a re a of the hyst eres is loop.

isA t freque ncies where skin effect predominates (i.e . , when a/S >> 1) the power l os s

and the average power pe r unit volume is

Thus, i f the sinusoidal field is regulated so that ther e is constant omax, the loss variesas f 3/ 2 .

TOTAL CORE LOSSA t high freque ncies, the hyste resi s and eddy-current los se s can be combined to

form the total core los ses

2 2-2 -2a f Bmaxoa6 fSBm,a+2 P6c(H) = Pe(H)+ ph(H) =where the averag e flux density Bmax is defined as

After substitution, the previous equation yieldsa

pc(H) =

-For a given Bm, and freq uency, then, the core loss at high frequenc ies is proportional16


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to the sq ua re root of conductivity and inversely proportiona l to the square roo t of thepermeability. Clearly, as the frequency is increased, the eddy-current lo ss var iesfro m the square to the 3/2 power of frequency, while the hys te re si s lo ss var ie s fr om thefirst to the 3/2 power of frequency.

CONCLUDING REMARKSIt ha s been demonstrated that the us e of Maxwell's equations and the Poynting

theorem yield the class ical eddy-current and hy st er es is lo ss formulations. The resultantintegrals w ere evaluated to find expression s for each los s for a thin transformer lami-nation by assuming that the permeability was constant, and fur the r, that the magneticmate rial was isotropic and homogeneous. The expressions w ere then simplified to yieldlow-frequency formulas. The hy st er es is lo ss was found to be proportional to the area ofthe hys teres is loop, while the eddy-current lo ss became th e conventional engineering ex-pression .

Bmaxand inversely proportional to the permeab ility. Thus, to minimize los ses , high-frequency trans for mer laminations should be made of a mate rial that has high resistivityand high permeability.

Examination of the co re lo ss exp ressi ons a lso showed that for a given laminationmaterial, thickness, and 9 as the frequency is increased, the eddy-current lo ssvar ies from the squ are to the 3/2 power of frequency while the hyste res is lo s s va ri esfrom the first to the 3/2 power of frequency. Th is means that the eddy-current lo ss con-tribution to the total lo ss is l e s s than would be expected fro m the power frequency ex-press ion and that the hyster esis- loss contribution to the total los s is gr ea te r than wouldbe expected fro m the power -frequency considerations alone. Thus, hyst eresi s l os splays an important role in transformer cor e los s at high as well as low frequencies.

Since the expressio ns for the thin trans form er lamination assum ed that the per me -ability w a s constant, application of the lo ss formu las without r eg ar d to variati ons ofpermeability with the applied field and saturation effects will res ul t in large e r ro rs . Itis felt that by maintaining the s am e bas ic approach, an approximation scheme could bedevised that would take the tr ue permeability into account (refs. 5 to 7 ) . The assumptionthat the material is isotropic and homogeneous would still have to be made.

By making a high-frequency approximation, it ha s been found that for a given averageand frequency, the co re los s at high frequenc ies is proportional to the conductivity

Lewis Research Center,National Aeronautics and Space Administration,

Cleveland, Ohio, September 1, 1965.

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REFERENCES1. Williams, H. . ; Shockley, W . ; and Kittel, C. : Studies of the Propagation Velocity of

a Ferromagnetic Domain Boundary. Phys. Rev., vol. 80, no. 6, Dec. 15, 1950,pp. 1090-1094.

2. Pry, R. . and Bean, C. P. : Calculation of th e Energy L os s in Magnetic SheetMaterials Using a Domain Model. J . Appl. Phys., vol. 29, no. 3, Mar. 1958,pp. 532-533.

3. Mulhall, B. . : Eddy-Current L os se s in Thin Magnetic Sheet. Pr oc . Inst. Elec.Engrs., vol. 111, no. 1, Jan. 1964, pp. 183-192.

4. Latour, Marius: Note on Lo ss es in Sheet Iron at Radio Frequencies. Proc. Inst .Radio En gr s. , Feb . 1919, pp. 61-71.

5 . Poritsky, H . ; and Butler, J . M. : A-C Flux Pene tra tion into Magnetic Materials withSaturation. AIEE Trans. on Communications and Electronics, no. 70, Jan. 1964,pp. 99-111.

6. Kesavamurthy, N. Rajagopalan, P. K. ; Subba Rao, V. : Effect s of Saturation on theCore Loss of Thin Fer rom agnetic Plates Subjected to Alternating Flux. Pr oc .Inst. Elec. Engrs., vol. 110, no. 2, Feb. 1963, pp. 459-473.

7. Brailsford, F. : Investigation of the Eddy-Current Anomaly in Electrica l Sheet Steels .J . Inst. Elec. Engrs., London, England, vol. 95, no. 43, pt. 11, 1948, pp. 38-48.

18 NASA-Langley, 1965 E-2992


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