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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.
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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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