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246 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

Dynamic

art I: Theor

0 etic Cores for Low- and

of t ivalent Core Loss

Jian. G. Zhu, Member, IEEE, S. Y. R. Hui,

Abstruct- This paper describes the theoretical calculation of

the equivalent core loss resistance for a dynamic magnetic core

loss model. The equivalent core loss resistance incorporates the

effects of both the classical eddy current and anomalous losses.

Derivation of a generalized nonlinear core loss resistance expres-

sion is presented. This new equivalent core loss resistance can

be incorporated into a generalized dynamic magnetic core circuit

model suitable for low and high frequency applications (Part I1

of this paper).

I. INTRODUCTION

ITH the continuous improvements in switching speed

and power capability of modem power electronics, the

study of the magnetic core behavior becomes increasingly

important owing to the facts that magnetic cores are often

operated in the nonlinear characteristic regions and the core

loss increases with the operating frequency. A dynamic circuit

model for magnetic cores that can accurately predict such

nonlinear behavior would be very useful and practical to

engineers who want to assess the core loss and optimize the

switching frequency and efficiency of the overall system. The

model must be able to account for hysteresis, eddy current, and

also anomalous losses. It should be based on parameters that

can be obtained easily either from manufacturers' data sheets

or simple measurements, and it should be easy to implement.

Two major aspects of modeling magnetic cores by an equiv-

alent circuit are the representation of 1) the nonlinear dynamic

behavior of the hysteresis effects and 2) the equivalent core

loss resistance. Several papers have investigated various meth-

ods for modeling hysteresis effects [ 11-[4]. Among different

hysteresis modeling methods, the Preisach model appears

to be a practical and easy-to-use technique. It is strongly

related to the mechanisms of magnetic hysteresis and can

describe various macroscopic hysteretic phenomena. Although

the description of the classical Preisach theory requires compli-

cated mathematics, the resultant formula obtained by a simple

graphical approach is rather simple and easy to implement.

Details of such graphical approach can be found in [5]. The

Manuscript received February 14, 1995; revised October 20, 1995.

J. G. Zhu and V. S. Ramsden are with the School of Electrical Engineering,

S . Y. R. Hui is with the Department of Electrical Engineering, University

Publisher Item Identifier S 0885-8993(96)09195-1.

University of Technology, Sydney, NSW 2007, Australia.

of Sydney, NSW 2006, Australia.

Senior Member, IEEE, and V. S. Ramsden

results of Preisach modeling fit the experimental behavior of

different magnetic materials [ l]-[.5].

The modeling of core losses using an equivalent resistance,

however, needs more investigation. The total eddy current loss

includes the classical eddy current loss and anomalous loss. In

the past, most circuit models proposed included the hysteresis

and classical eddy current losses only. The anomalous loss,

which is due to the effects of magnetic domain wall movement

on the distribution of eddy currents [6]-[13], has not been

included.

Fig. l(a) shows a typical circuit model for magnetic core

together with the winding resistance R, of the coil wound in

the core. R, and L, represent the core loss and inductance

in the equivalent circuit, respectively. If anomalous loss is

included, it can be shown [14] that R, is composed of two

parallel resistances, namely the classical eddy current loss

resistance Re and the anomalous loss resistance R,, as shown

in a dynamic circuit model [Fig. l(b)], in which L( i ) represents

the differential inductance. At constant temperature, R, and

Re can be regarded as constants. R,, however, is a nonlinear

resistor. If hysteresis is involved, L( i ) is nonlinear and should

be determined by the magnetization history of the core using

a mathematical model of hysteresis.

It was pointed out in [14] that the circuit models shown in

Fig. l(a) and (b) were suitable only for low-frequency applica-

tions since the reaction of eddy currents on the distribution of

excitation magnetic field, which is known as the skin effect,

was not properly accounted for. In order to generalize the

model so that it can be used for high-frequency applications

as well, it is necessary to extend the circuit to a ladder network

as shown in Fig. 2. (Details of the ladder network will be given

in Part I1 of this paper.) In this approach, the cross section of

the magnetic core is divided into a number of assumed eddy

current paths. Within each path, the variation of eddy current

density with the depth can be approximately fit by a straight

line, and the distribution of flux density can then be considered

as uniform. Applying the circuit model shown in Fig. l(b) to

each assumed eddy current path results in the ladder network

model. In the ladder network, d@k/dzk ( k = 1 , 2 , . . . , n ,

where n is the number of stages of the ladder network) is a

differential inductance related to the kth eddy current loop, and

RI, ( k = 1 , 2 , . . . , n) is the equivalent core loss resistance of

0885-8993/96$05.00 0 1996 IEEE

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ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 241

t

(b)

Fig. 1.

model of magnetic core.

(a) Traditional circuit model of magnetic core. (h) Dynamic circuit

a

d i'1

Fig. 2. A generalized equivalent circuit model (ladder network) for magnetic

cores.

the kth eddy current loop, which consists of both the classical

eddy current and anomalous losses represented by Re and R,

in Fig. l(b), respectively.

Part I of this paper deals with the calculation of the

equivalent core loss resistance in the generalized dynamic

circuit model for magnetic cores shown above. This work is

based on a low-frequency core loss model [14] and a high-

frequency model [ 151 previously published by the authors.

The new core loss resistance expression derived here includes

both the classical eddy current loss and the anomalous loss

and is developed in such a form that it not only suits both

low- and high-frequency applications, but it also simplifies

the formation of a generalized ladder network for the core

loss model. Details of the generalized core model and its

implementation are presented in Part I1 of this paper.

11. APPROXIMATE EDDY CURRENT

DISTRIBUTION I A MAGNETIC CORE

Fig. 3 illustrates the cross section of a magnetic core of

length I,, width a, and thickness b where 1, >> a >> b. When a

I

Fig. 3.

due to a changing magnetic field in the 2 direction.

Cross section of a magnetic core showing induced eddy current paths

time-varying magnetic field is applied in the 2-direction, eddy

currents are induced in the X and Y direction. For a sinusoidal

excitation, and since 1, >> a >> b, the phasor expressions of

flux density and eddy current density are found to be

cosh Q y

cosh

B, 1 B,-

and

QB, sinh ay

p cosh? (2)

where B, is the peak value of flux density at the core surface,

frequency, f the excitation frequency, and a and p are the

conductivity and permeability of the core material, respectively

[16]. The Y axis component of the eddy current density Jy,

and the variation of the eddy current density with x, can be

ignored since a >, b.

If 6 >> b, the magnetic core is regarded as a thin sheet

and approximate expressions of flux density and eddy current

density can be obtained from (1) and (2) as

J -__- x -

, 6 = ,E the skin depth, w = 2 ~ f the angular

B, N" B, (3)

and

J, s jwaB ,y . (4)

Therefore, in a thin sheet, the reaction of eddy currents on

the magnetic field distribution can be ignored. The magnetic

field can be considered as uniformly distributed over the cross

section of the core, and the amplitude of eddy current density

is proportional to the distance from the center of the core.

If b >> 6, however, the reaction of eddy currents on

the magnetic field distribution cannot be neglected. For an

approximate solution with an excitation of arbitrary waveform,

we divide the cross section of the core into a number of slice

pairs as shown in Fig. 4. Each slice pair forms an assumed

eddy current loop (J?, is ignored since a >> b). The thickness

of a slice Ay is chosen such that within the slice the variation

of eddy current density against y is almost linear. Inside the

slice, therefore, the reaction of eddy currents on the magnetic

field distribution can be ignored, and the flux density can be

considered as uniformly distributed, or independent of y.

In the kth slice pair, let the flux density be

B = B,k ( 5 )

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248 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

per unit volume, the instantaneous power loss in the kth slice

pair is

(12)

2a1, Y k

p e k = 7 Li, J,” dy .

From (9), we have

(13)

1

dy = ~ d J z .

a ( % ) ,

Hence

Fig. 4.

core.

An assumed eddy current path formed by a slice pair in a magnetic

where the bold face symbols are vectors, k is the unity vector

in the 2-direction in Fig. 4, and subscript z refers to the

2-direction component. Since a >> b, the eddy current density

in the kth slice pair is

J = J,i (4 )

where i is the unity vector in the X-direction in Fig. 4, and

subscript z denotes the X-direction component.

Substituting (5) and (6) into

J = aE

and

(7)

where E is the electric field strength in the kth slice pair, we

have

since B, is independent of y and Jz only varies in Y direction

in the slice pair.

The eddy current density in the kth slice pair can then be

obtained by performing integration on both sides of (9) as

where J k and J k - 1 are the eddy current densities at surfaces

y = yk and y = yk-1, respectively. Because B, is considered

as uniformly distributed inside the kth slice pair, as discussed

earlier, (%)k can also be treated as independent of y.

Therefore

Resistance in the current paths along the Y axis is ignored

because the thickness b is very small when compared with

width a and length 1,. From (1 l), one obtains

Therefore, the classical eddy current loss in the kth slice pair

becomes

where

is a correction factor that takes into account the variation of

eddy current density J , with y. If the distribution of current

density is uniform, i.e., J k = J k - 1 , T J k becomes one.

w. CONSIDERATION OF ANOMALOUS LOSS

It has been pointed out in [14] that the eddy current loss

P, and the anomalous loss Pa expressions (in Wkg) in a thin

sheet are given by

2

P, = .($)

and

where where Ce and C, are the coefficients of the classical eddy

current loss and the anomalous loss, respectively, and B is

assumed uniform within the depth. The sum of the classical

eddy current loss and the anomalous loss is

AY = Yk - Yk-1.

111. CLASSICAL EDDY CURRENT Loss

The classical eddy current loss in the kth slice pair can be

expressed as the resistive loss of the eddy current path in the

kth slice pair. Since the classical eddy current loss P, = $ J 2

Pea = Pe + Pa

d B , 3’2 -ce __ + C __

- (d2r d t 1

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ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 249

where

is a correction factor if the anomalous loss (the effects of

domain wall movement) is included in the calculation of the

eddy current loss.

Therefore, the sum of the eddy current and anomalous losses

in the kth slice pair P e a k from (16) is

where

is the correction factor for the kth slice pair. For the time

instant at which I%Ik = 0, p e a k = 0, and Tck approaches

infinity. Numerically, a large number, for example 1020, is

assigned to Tck in this case.

The coefficients of the classical eddy current and anomalous

losses used in (21) for correction factor rc can be obtained by

the alternating core loss separation procedure described in [ 141.

V. EQUIVALENT CORE Loss RESISTANCE

Since a >> b, the induced emf in the kth slice pair at y = yk

is

VI, = 2 a E k . (23)

Since JI, = oEI,, then

for the kth slice pair.

From the ladder network shown in Fig. 2, it can be seen

that vk is also the voltage across the resistance R k ( k =

1 , 2 , . . . , n) since

where GJ = 2aAyB,, ( j = 1 , 2 , . . . , n) is the magnetic flux

through the cross section of the jth slice pair.

Therefore, the equivalent core loss resistance for the kth

slice pair can be calculated by

(26) vi? R k = -.

P e a k

Thus, from (22) and (24), one obtains

( F J k ) 2

( 2 a z / ) J ~ ~ J I , ~ ~ I , RI, =

where R D C k = 2% is the dc resistance of the kth slice pair.

if the variation

of eddy current dlensity in the kth path corresponding to the

skin effect in the magnetic core is considered. The resistance

expression changes to the form in (27) if the anomalous loss

is also included.

Equation (27) provides a simple general expression for the

equivalent eddy current and anomalous loss resistance for each

eddy current loop in the magnetic core. This equation can be

used in the generialized magnetic core model as explained in

Part I1 of this pape:r. It is important to note that RI, is expressed

in terms of the dc resistance, which can be easily determined.

This enables (27) to be applied to magnetic cores of different

cross-sectional shapes.

The dc resistance R D c k changes to

VI. CONCLUSION

This paper has presented the theoretical calculation of the

magnetic core loss in a generalized core model suitable for

both low and high-frequency applications. A new expression

for the effective core loss resistance, which includes both

the classical eddy current and anomalous loss, has been

derived and explained. This core loss resistor, together with

the hysteresis model, can be used to develop a generalized

dynamic magnetic core circuit model. The validity of this

new expression has been confirmed by measurements and

simulations. Implementation of and results obtained from such

models are presented in Part I1 of this paper.

REFERENCES

I. D. Mayergoyz, Mathematical Models of Hysteresis. New York

Springer Verlag,, 1991.

F. Ossart and G. Meunier, “Comparison between various hysteresis

models and experimental data,” ZEEE MAG-26, no. 5, pp. 2837-2839,

1990.

S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s

theory,” ZEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990

F. Preisach, “1Jber die magnetische Nachwirkung,” Zeifschhriji fur

Physik, vol. 94, pp. 277-302, 1935.

S. Y. R. Hui and J. G . Zhu, “Numerical modeling and simulation of

hysteresis effects in magnetic cores using the transmission line modeling

and Preisach theory,” ZEE Proc.-Sci. Meas. Technol., vol. 142, no. 1,

R. H. Pry and IC. P. Bean, “Calculation of the energy loss in magnetic

sheet materials using a domain model,” J. Appl. Phys., vol. 29, no. 3,

F. Brailsford arnd R. Fogg, “Anomalous iron losses in cold reduced

grain-oriented transformer steel at very low frequencies,” in Proc. IEE,

vol. 113, pp. 1562-1564, 1966.

J. A. Robey, “Domain wall motion and eddy current loss in grain

oriented silicon iron sheet,” J. Sci. & Tech., vol. 36, no. 1, pp. 3-10,

1969.

C. D. Graham, “Physical origin of losses in conducting ferro-magnetic

materials,” J. Appl. Phys., vol. 53, no. 11, pp. 8276-8280, 1982.

pp. 57-62, 1995.

pp. 532-533, 1958.

Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply.

250 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11. NO. 2, MARCH 1996

[lo] K. Foster, F. E. Werner, and R. M. Del Vecchio, “Loss separation

measurements for several electrical steels,’’ J. Appl. Phys., vol. 53,

no. 11, pp. 8308-8310, 1982.

[ I l l T. Sat0 and Y. Sakaki, “Physical meaning of equivalent loss resistance

of magnetic cores,” IEEE MAG-26, no. 5, pp. 2894-2897, 1990.

[12] F. Fiorillo and A. Novikov, “An improved approach to power losses in

magnetic laminations under nonsinusoidal induction waveform,” IEEE

MAG-26, no. 5, pp. 29042910, 1990.

[I31 H. Pfutzner, B. Erbil, 6. Harasko, and T. Klinger, “Problems of loss

separation for crystalline and consolidated amorphous soft magnetic

materials,” IEEE MAG-27, no. 3, pp. 34263432, 1991.

[14] J. G. Zhu, S. Y. R. Hui, and V. S . Ramsden, “Discrete modeling

of magnetic cores including hysteresis, eddy current, and anomalous

losses,” IEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993.

[15] J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “A dynamic circuit model

for solid magnetic cores,” in IEEE Power Electronics Specialists Cons

(PESC’93) Rec., Seattle, WA, June 20-24, 1993, pp. 11 11-1 115.

[16] R. L. Stoll, The Analysis of Eddy Currents. Oxford, U.K.: Clarendon,

ch. 1, sec. 2.7, pp. 21-23, 1974.

Jian 6. Zhu (M’92) was bom in Shanghai, China,

in 1958 He received the B.Eng. degree from Jiangsu

Institute of Technology (JIT), China, in 1982, the

M Eng degree from Shanghai University of Tech-

nology, China, in 1987, and the Ph.D. degree from

the University of Technology, Sydney (UTS), Aus-

tralia, in 1995.

From 1987-1990, he worked as a Lecturer at JIT,

China. He joined the School of Electrical Engineer-

ing, UTS, as a Lecturer in 1994. His research areas

include numerical analysis of electromagnetic fields,

measurement and modeling of magnetic properties of materials, and electrical

machines and drive systems.

S. Y . R. (Ron) Hui (SM’94) was bom in Hong

Kong in 1961. He obtained the B Sc. (Hons.) at the

University of Birmingham, U K., in 1984, and the

D.I.C. and Ph.D. degrees at the Impenal College of

Science and Technology, London, U K , in 1987.

He was appointed Lecturer in power electronics

at the University of Nottmgbam, U.K., in 1987.

In 1990, he went to the University of Technology,

Australia, where he became a Senior Lecturer in

1991. In January 1993, he joined the University

of Sydney, Australia, where he is now a Reader

of Electrical Engineering and Coordinator of Power Electronics Research.

He has pubhshed over 30 refereed joumal papers and over 40 conference

papers on various aspects of power electronics and simulation techniques His

current research interests include soft-switching techniques for converters and

inverters, power electronic clrcuit and magnetic core simulation, power factor

correction circuits, PWM methods, new topologies for power conversion, and

electronic ballast.

Dr. Hui is a Fellow of the IEAust.

V. S. Ramden,

publication.

photograph and biography not railable at the time of

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学霸联盟

Dynamic

art I: Theor

0 etic Cores for Low- and

of t ivalent Core Loss

Jian. G. Zhu, Member, IEEE, S. Y. R. Hui,

Abstruct- This paper describes the theoretical calculation of

the equivalent core loss resistance for a dynamic magnetic core

loss model. The equivalent core loss resistance incorporates the

effects of both the classical eddy current and anomalous losses.

Derivation of a generalized nonlinear core loss resistance expres-

sion is presented. This new equivalent core loss resistance can

be incorporated into a generalized dynamic magnetic core circuit

model suitable for low and high frequency applications (Part I1

of this paper).

I. INTRODUCTION

ITH the continuous improvements in switching speed

and power capability of modem power electronics, the

study of the magnetic core behavior becomes increasingly

important owing to the facts that magnetic cores are often

operated in the nonlinear characteristic regions and the core

loss increases with the operating frequency. A dynamic circuit

model for magnetic cores that can accurately predict such

nonlinear behavior would be very useful and practical to

engineers who want to assess the core loss and optimize the

switching frequency and efficiency of the overall system. The

model must be able to account for hysteresis, eddy current, and

also anomalous losses. It should be based on parameters that

can be obtained easily either from manufacturers' data sheets

or simple measurements, and it should be easy to implement.

Two major aspects of modeling magnetic cores by an equiv-

alent circuit are the representation of 1) the nonlinear dynamic

behavior of the hysteresis effects and 2) the equivalent core

loss resistance. Several papers have investigated various meth-

ods for modeling hysteresis effects [ 11-[4]. Among different

hysteresis modeling methods, the Preisach model appears

to be a practical and easy-to-use technique. It is strongly

related to the mechanisms of magnetic hysteresis and can

describe various macroscopic hysteretic phenomena. Although

the description of the classical Preisach theory requires compli-

cated mathematics, the resultant formula obtained by a simple

graphical approach is rather simple and easy to implement.

Details of such graphical approach can be found in [5]. The

Manuscript received February 14, 1995; revised October 20, 1995.

J. G. Zhu and V. S. Ramsden are with the School of Electrical Engineering,

S . Y. R. Hui is with the Department of Electrical Engineering, University

Publisher Item Identifier S 0885-8993(96)09195-1.

University of Technology, Sydney, NSW 2007, Australia.

of Sydney, NSW 2006, Australia.

Senior Member, IEEE, and V. S. Ramsden

results of Preisach modeling fit the experimental behavior of

different magnetic materials [ l]-[.5].

The modeling of core losses using an equivalent resistance,

however, needs more investigation. The total eddy current loss

includes the classical eddy current loss and anomalous loss. In

the past, most circuit models proposed included the hysteresis

and classical eddy current losses only. The anomalous loss,

which is due to the effects of magnetic domain wall movement

on the distribution of eddy currents [6]-[13], has not been

included.

Fig. l(a) shows a typical circuit model for magnetic core

together with the winding resistance R, of the coil wound in

the core. R, and L, represent the core loss and inductance

in the equivalent circuit, respectively. If anomalous loss is

included, it can be shown [14] that R, is composed of two

parallel resistances, namely the classical eddy current loss

resistance Re and the anomalous loss resistance R,, as shown

in a dynamic circuit model [Fig. l(b)], in which L( i ) represents

the differential inductance. At constant temperature, R, and

Re can be regarded as constants. R,, however, is a nonlinear

resistor. If hysteresis is involved, L( i ) is nonlinear and should

be determined by the magnetization history of the core using

a mathematical model of hysteresis.

It was pointed out in [14] that the circuit models shown in

Fig. l(a) and (b) were suitable only for low-frequency applica-

tions since the reaction of eddy currents on the distribution of

excitation magnetic field, which is known as the skin effect,

was not properly accounted for. In order to generalize the

model so that it can be used for high-frequency applications

as well, it is necessary to extend the circuit to a ladder network

as shown in Fig. 2. (Details of the ladder network will be given

in Part I1 of this paper.) In this approach, the cross section of

the magnetic core is divided into a number of assumed eddy

current paths. Within each path, the variation of eddy current

density with the depth can be approximately fit by a straight

line, and the distribution of flux density can then be considered

as uniform. Applying the circuit model shown in Fig. l(b) to

each assumed eddy current path results in the ladder network

model. In the ladder network, d@k/dzk ( k = 1 , 2 , . . . , n ,

where n is the number of stages of the ladder network) is a

differential inductance related to the kth eddy current loop, and

RI, ( k = 1 , 2 , . . . , n) is the equivalent core loss resistance of

0885-8993/96$05.00 0 1996 IEEE

Authorized licensed use limited to: University of Sydney. Downloaded on September 02,2020 at 08:32:26 UTC from IEEE Xplore. Restrictions apply.

ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 241

t

(b)

Fig. 1.

model of magnetic core.

(a) Traditional circuit model of magnetic core. (h) Dynamic circuit

a

d i'1

Fig. 2. A generalized equivalent circuit model (ladder network) for magnetic

cores.

the kth eddy current loop, which consists of both the classical

eddy current and anomalous losses represented by Re and R,

in Fig. l(b), respectively.

Part I of this paper deals with the calculation of the

equivalent core loss resistance in the generalized dynamic

circuit model for magnetic cores shown above. This work is

based on a low-frequency core loss model [14] and a high-

frequency model [ 151 previously published by the authors.

The new core loss resistance expression derived here includes

both the classical eddy current loss and the anomalous loss

and is developed in such a form that it not only suits both

low- and high-frequency applications, but it also simplifies

the formation of a generalized ladder network for the core

loss model. Details of the generalized core model and its

implementation are presented in Part I1 of this paper.

11. APPROXIMATE EDDY CURRENT

DISTRIBUTION I A MAGNETIC CORE

Fig. 3 illustrates the cross section of a magnetic core of

length I,, width a, and thickness b where 1, >> a >> b. When a

I

Fig. 3.

due to a changing magnetic field in the 2 direction.

Cross section of a magnetic core showing induced eddy current paths

time-varying magnetic field is applied in the 2-direction, eddy

currents are induced in the X and Y direction. For a sinusoidal

excitation, and since 1, >> a >> b, the phasor expressions of

flux density and eddy current density are found to be

cosh Q y

cosh

B, 1 B,-

and

QB, sinh ay

p cosh? (2)

where B, is the peak value of flux density at the core surface,

frequency, f the excitation frequency, and a and p are the

conductivity and permeability of the core material, respectively

[16]. The Y axis component of the eddy current density Jy,

and the variation of the eddy current density with x, can be

ignored since a >, b.

If 6 >> b, the magnetic core is regarded as a thin sheet

and approximate expressions of flux density and eddy current

density can be obtained from (1) and (2) as

J -__- x -

, 6 = ,E the skin depth, w = 2 ~ f the angular

B, N" B, (3)

and

J, s jwaB ,y . (4)

Therefore, in a thin sheet, the reaction of eddy currents on

the magnetic field distribution can be ignored. The magnetic

field can be considered as uniformly distributed over the cross

section of the core, and the amplitude of eddy current density

is proportional to the distance from the center of the core.

If b >> 6, however, the reaction of eddy currents on

the magnetic field distribution cannot be neglected. For an

approximate solution with an excitation of arbitrary waveform,

we divide the cross section of the core into a number of slice

pairs as shown in Fig. 4. Each slice pair forms an assumed

eddy current loop (J?, is ignored since a >> b). The thickness

of a slice Ay is chosen such that within the slice the variation

of eddy current density against y is almost linear. Inside the

slice, therefore, the reaction of eddy currents on the magnetic

field distribution can be ignored, and the flux density can be

considered as uniformly distributed, or independent of y.

In the kth slice pair, let the flux density be

B = B,k ( 5 )

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248 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996

per unit volume, the instantaneous power loss in the kth slice

pair is

(12)

2a1, Y k

p e k = 7 Li, J,” dy .

From (9), we have

(13)

1

dy = ~ d J z .

a ( % ) ,

Hence

Fig. 4.

core.

An assumed eddy current path formed by a slice pair in a magnetic

where the bold face symbols are vectors, k is the unity vector

in the 2-direction in Fig. 4, and subscript z refers to the

2-direction component. Since a >> b, the eddy current density

in the kth slice pair is

J = J,i (4 )

where i is the unity vector in the X-direction in Fig. 4, and

subscript z denotes the X-direction component.

Substituting (5) and (6) into

J = aE

and

(7)

where E is the electric field strength in the kth slice pair, we

have

since B, is independent of y and Jz only varies in Y direction

in the slice pair.

The eddy current density in the kth slice pair can then be

obtained by performing integration on both sides of (9) as

where J k and J k - 1 are the eddy current densities at surfaces

y = yk and y = yk-1, respectively. Because B, is considered

as uniformly distributed inside the kth slice pair, as discussed

earlier, (%)k can also be treated as independent of y.

Therefore

Resistance in the current paths along the Y axis is ignored

because the thickness b is very small when compared with

width a and length 1,. From (1 l), one obtains

Therefore, the classical eddy current loss in the kth slice pair

becomes

where

is a correction factor that takes into account the variation of

eddy current density J , with y. If the distribution of current

density is uniform, i.e., J k = J k - 1 , T J k becomes one.

w. CONSIDERATION OF ANOMALOUS LOSS

It has been pointed out in [14] that the eddy current loss

P, and the anomalous loss Pa expressions (in Wkg) in a thin

sheet are given by

2

P, = .($)

and

where where Ce and C, are the coefficients of the classical eddy

current loss and the anomalous loss, respectively, and B is

assumed uniform within the depth. The sum of the classical

eddy current loss and the anomalous loss is

AY = Yk - Yk-1.

111. CLASSICAL EDDY CURRENT Loss

The classical eddy current loss in the kth slice pair can be

expressed as the resistive loss of the eddy current path in the

kth slice pair. Since the classical eddy current loss P, = $ J 2

Pea = Pe + Pa

d B , 3’2 -ce __ + C __

- (d2r d t 1

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ZHU et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I 249

where

is a correction factor if the anomalous loss (the effects of

domain wall movement) is included in the calculation of the

eddy current loss.

Therefore, the sum of the eddy current and anomalous losses

in the kth slice pair P e a k from (16) is

where

is the correction factor for the kth slice pair. For the time

instant at which I%Ik = 0, p e a k = 0, and Tck approaches

infinity. Numerically, a large number, for example 1020, is

assigned to Tck in this case.

The coefficients of the classical eddy current and anomalous

losses used in (21) for correction factor rc can be obtained by

the alternating core loss separation procedure described in [ 141.

V. EQUIVALENT CORE Loss RESISTANCE

Since a >> b, the induced emf in the kth slice pair at y = yk

is

VI, = 2 a E k . (23)

Since JI, = oEI,, then

for the kth slice pair.

From the ladder network shown in Fig. 2, it can be seen

that vk is also the voltage across the resistance R k ( k =

1 , 2 , . . . , n) since

where GJ = 2aAyB,, ( j = 1 , 2 , . . . , n) is the magnetic flux

through the cross section of the jth slice pair.

Therefore, the equivalent core loss resistance for the kth

slice pair can be calculated by

(26) vi? R k = -.

P e a k

Thus, from (22) and (24), one obtains

( F J k ) 2

( 2 a z / ) J ~ ~ J I , ~ ~ I , RI, =

where R D C k = 2% is the dc resistance of the kth slice pair.

if the variation

of eddy current dlensity in the kth path corresponding to the

skin effect in the magnetic core is considered. The resistance

expression changes to the form in (27) if the anomalous loss

is also included.

Equation (27) provides a simple general expression for the

equivalent eddy current and anomalous loss resistance for each

eddy current loop in the magnetic core. This equation can be

used in the generialized magnetic core model as explained in

Part I1 of this pape:r. It is important to note that RI, is expressed

in terms of the dc resistance, which can be easily determined.

This enables (27) to be applied to magnetic cores of different

cross-sectional shapes.

The dc resistance R D c k changes to

VI. CONCLUSION

This paper has presented the theoretical calculation of the

magnetic core loss in a generalized core model suitable for

both low and high-frequency applications. A new expression

for the effective core loss resistance, which includes both

the classical eddy current and anomalous loss, has been

derived and explained. This core loss resistor, together with

the hysteresis model, can be used to develop a generalized

dynamic magnetic core circuit model. The validity of this

new expression has been confirmed by measurements and

simulations. Implementation of and results obtained from such

models are presented in Part I1 of this paper.

REFERENCES

I. D. Mayergoyz, Mathematical Models of Hysteresis. New York

Springer Verlag,, 1991.

F. Ossart and G. Meunier, “Comparison between various hysteresis

models and experimental data,” ZEEE MAG-26, no. 5, pp. 2837-2839,

1990.

S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s

theory,” ZEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990

F. Preisach, “1Jber die magnetische Nachwirkung,” Zeifschhriji fur

Physik, vol. 94, pp. 277-302, 1935.

S. Y. R. Hui and J. G . Zhu, “Numerical modeling and simulation of

hysteresis effects in magnetic cores using the transmission line modeling

and Preisach theory,” ZEE Proc.-Sci. Meas. Technol., vol. 142, no. 1,

R. H. Pry and IC. P. Bean, “Calculation of the energy loss in magnetic

sheet materials using a domain model,” J. Appl. Phys., vol. 29, no. 3,

F. Brailsford arnd R. Fogg, “Anomalous iron losses in cold reduced

grain-oriented transformer steel at very low frequencies,” in Proc. IEE,

vol. 113, pp. 1562-1564, 1966.

J. A. Robey, “Domain wall motion and eddy current loss in grain

oriented silicon iron sheet,” J. Sci. & Tech., vol. 36, no. 1, pp. 3-10,

1969.

C. D. Graham, “Physical origin of losses in conducting ferro-magnetic

materials,” J. Appl. Phys., vol. 53, no. 11, pp. 8276-8280, 1982.

pp. 57-62, 1995.

pp. 532-533, 1958.

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250 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11. NO. 2, MARCH 1996

[lo] K. Foster, F. E. Werner, and R. M. Del Vecchio, “Loss separation

measurements for several electrical steels,’’ J. Appl. Phys., vol. 53,

no. 11, pp. 8308-8310, 1982.

[ I l l T. Sat0 and Y. Sakaki, “Physical meaning of equivalent loss resistance

of magnetic cores,” IEEE MAG-26, no. 5, pp. 2894-2897, 1990.

[12] F. Fiorillo and A. Novikov, “An improved approach to power losses in

magnetic laminations under nonsinusoidal induction waveform,” IEEE

MAG-26, no. 5, pp. 29042910, 1990.

[I31 H. Pfutzner, B. Erbil, 6. Harasko, and T. Klinger, “Problems of loss

separation for crystalline and consolidated amorphous soft magnetic

materials,” IEEE MAG-27, no. 3, pp. 34263432, 1991.

[14] J. G. Zhu, S. Y. R. Hui, and V. S . Ramsden, “Discrete modeling

of magnetic cores including hysteresis, eddy current, and anomalous

losses,” IEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993.

[15] J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “A dynamic circuit model

for solid magnetic cores,” in IEEE Power Electronics Specialists Cons

(PESC’93) Rec., Seattle, WA, June 20-24, 1993, pp. 11 11-1 115.

[16] R. L. Stoll, The Analysis of Eddy Currents. Oxford, U.K.: Clarendon,

ch. 1, sec. 2.7, pp. 21-23, 1974.

Jian 6. Zhu (M’92) was bom in Shanghai, China,

in 1958 He received the B.Eng. degree from Jiangsu

Institute of Technology (JIT), China, in 1982, the

M Eng degree from Shanghai University of Tech-

nology, China, in 1987, and the Ph.D. degree from

the University of Technology, Sydney (UTS), Aus-

tralia, in 1995.

From 1987-1990, he worked as a Lecturer at JIT,

China. He joined the School of Electrical Engineer-

ing, UTS, as a Lecturer in 1994. His research areas

include numerical analysis of electromagnetic fields,

measurement and modeling of magnetic properties of materials, and electrical

machines and drive systems.

S. Y . R. (Ron) Hui (SM’94) was bom in Hong

Kong in 1961. He obtained the B Sc. (Hons.) at the

University of Birmingham, U K., in 1984, and the

D.I.C. and Ph.D. degrees at the Impenal College of

Science and Technology, London, U K , in 1987.

He was appointed Lecturer in power electronics

at the University of Nottmgbam, U.K., in 1987.

In 1990, he went to the University of Technology,

Australia, where he became a Senior Lecturer in

1991. In January 1993, he joined the University

of Sydney, Australia, where he is now a Reader

of Electrical Engineering and Coordinator of Power Electronics Research.

He has pubhshed over 30 refereed joumal papers and over 40 conference

papers on various aspects of power electronics and simulation techniques His

current research interests include soft-switching techniques for converters and

inverters, power electronic clrcuit and magnetic core simulation, power factor

correction circuits, PWM methods, new topologies for power conversion, and

electronic ballast.

Dr. Hui is a Fellow of the IEAust.

V. S. Ramden,

publication.

photograph and biography not railable at the time of

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