matlab代写-MARCH 1996 25
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996 25 1
A Generalized Dynamic Circuit Model of
Magnetic Cores for Low- and High-Frequency
Applications-Part 11: Circuit Model
Formulation and Implementation
S . Y. R. Hui, Senior Member, IEEE, Jian G.
Abstruct- This paper describes the formulation and imple-
mentation of a generalized dynamic magnetic core circuit model
suitable €or both low- and high-frequency applications. The
behavior of magnetic cores with any arbitrary flux waveforms
is modeled by a simple ladder network consisting of nonlinear
inductors and resistors. The nonlinear B-H loop and the hystere-
sis loss are incorporated in distributed nonideal inductors and
calculated by the Preisach scalar model of magnetic hysteresis.
The eddy current and anomalous losses are accounted for by
the generalized nonlinear equivalent resistors reported in Part I
of the paper. The transmission line modeling (TLM) method
is employed to solve the nonlinear state equations. Numerical
aspects and software implementation of the model are discussed.
The generalized model has been verified by simulations and
measurements at both low- and high-frequency operations.
I. INTRODUCTION
YNAMIC modeling of magnetic cores has long been a D difficult task owing to the nonlinear characteristics of
magnetic materials and the complicated mechanisms of core
losses. Accurate prediction of the instantaneous B-H operating
point is important because magnetic cores are often excited
with nonsinusoidal waveforms and at high frequency in most
power electronic circuits. Core loss assessment is another
important issue as the core loss increases with switching
frequency. It would be useful and practical to engineers if a
simple but accurate magnetic core model could be developed
for both low- and high-frequency applications. The model
must be simple to use, easy to formhate, based on data
supplied from manufacturers and/or obtained from simple
measurements, and able to predict all types of core losses to
a good degree of accuracy. In order to satisfy these criteria,
a circuit model of magnetic cores is preferred because of its
simplicity and time efficiency in simulation. However, a good
magnetic core circuit model, despite its inherent simplicity,
must be able to handle both the hysteresis behavior and total
eddy current losses including the anomalous loss.
Recently, the authors have proposed a dynamic discrete
circuit model for thin laminated magnetic cores for low-
Manuscript received February 14, 1995; revised October 20, 1995.
S . Y. R. Hui is with the Department of Electrical Engineering, University
J. G. Zhu and V. S . Ramsden are with the School of Electrical Engineering,
Publisher Item Identifier S 0885-8993(96)01916-3.
of Sydney, NSW 2006, Australia.
University of Technology, Sydney, NSW 2007, Australia.
I
1
Zhu, Member, IEEE, and V. S . Ramsden
frequency applications [ 11. This model includes all types
of core losses (i.e., hysteresis, classical eddy current, and
anomalous losses) and has been shown to provide significant
improvement in the accuracy of the core loss prediction
over traditional models without the anomalous loss. The
same concept was also applied successfully to a simple high-
frequency model for a solid magnetic core [2]. The results
reported indicate that the inclusion of anomalous loss into the
core loss model is achievable and practical. However, there
is so far no general approach that can be used to model both
laminated and solid magnetic cores for both low- and high-
frequency applications. Built on the two models previously
described, a generalized discrete dynamic magnetic core model
that can be readily implemented in digital computers for both
laminated and solid cores and for low- and high-frequency
applications is presented in this paper. The generalized model
is based on a simple ladder network consisting of nonlinear
inductors and resistors. The hysteresis effects are incorporated
into nonlinear inlductors and calculated by using the Preisach
theory [3]-[5]. Other core losses are included in nonlinear
resistors, the generalized expression of which is explained in
Part I of this paper. A recently introduced transmission-line
modeling (TLM) based discrete transform technique [6], [7] is
employed to develop the dynamic discrete circuit model. The
circuit model formulation and its implementation are described
in details. Criteria and factors influencing the size of the ladder
network for different core materials and different operating
frequencies are discussed. The model has been verified in
two examples, namely a laminated Lycore 140 magnetic core
at low-frequency operation and a solid ferrite core at high
frequency operaition. Both simulations and measurements are
included.
11. DYNAMIC CIRCUIT MODEL OF MAGNETIC ORES
Fig. 1 shows a simple circuit model for magnetic core. As
pointed out in Part I of this paper, this model is suitable
for low-frequency applications in which the magnetic flux
distribution in the core is close to uniform. At high-frequency
aperation, the eddy currents within the core are not identical,
md therefore, a ladder network incorporating the nonlinear
flux distribution within the core is needed. In fact, it is shown
in Section V that even for low-frequency operation, a ladder
0885-8993/96$05.00 0 1996 IEEE
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252 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996
Fig. 1. Traditional single-stage magnetic core circuit model.
d ?2
oil
Nis ' > / I
I
re
~
Fig. 3. Equivalent circuit of a generalized magnetic core model.
I
Fig. 2.
paths.
Cross section of a solid magnetic core with assumed eddy current
network can provide better accuracy than the single-stage
simple circuit in Fig. 1.
A. Generalized Ladder Network for Dynamic
Magnetic Core Circuit Model
Cross sections of laminated and solid magnetic cores can
be divided into a few segments for magnetic circuit analysis.
To calculate eddy currents, eddy current paths, and their
associated magnetic flux paths are assumed in each segment,
as shown in Fig. 2.
These eddy current paths are divided so that the reaction of
eddy currents on the distribution of flux density is negligible,
that is, the flux density in each path is assumed uniform. In
general, let's assume that there are n eddy current loops in the
core. The generalized circuit equations are
Coil:
(1)
d(@1 + @2 + . . . + an) - Vs
d t N
--
Core:
where R, is the winding ac resistance, N the number of
turns of the winding, is the current in the winding, @I, ( k =
1,2, . . . , n) the flux within each path, V, the terminal voltage
across the winding, and RI, ( k = 1,2, . . . , n) the equivalent
resistance of each assumed eddy current path representing
eddy current and anomalous losses.
The generalized set of (1)-(4) can be represented as an
equivalent circuit in the form of a ladder network, as shown
in Fig. 3.
B. Discrete Mathematical Description
of Dynamic Magnetic Core Model
be written in matrix form as
In terms of the mesh currents, these circuit equations can
The relationship between the assumed eddy currents i ~ ,
(k = 1,2, . . . ,n) in Fig. 2 and the mesh currents ii
( k = 1,2, . . . , n) in the ladder network (Fig. 3) is as follows:
Equations (5) and (6) give the mathematical description of the
generalized magnetic core model in a continuous form. In the
ladder network, while resistor RI, (k = 1 , 2 , . . . , n) represents
the core loss equivalent resistance in the kth eddy current path,
the hysteresis effects can be included in the terms in (5).
For simulations in digital computers, a iscrete circuit
model can be derived using the TLM-based discrete transform
".
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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11 253
technique [6], [7]. First, &% is replaced by the corresponding
differential inductance L(i,) in the kth loop, where di;
(7)
These differential inductances at any time instant can be
determined by the Preisach theory as described in Section III-
A. Second, express the 3 ( I C = 1,2 , e . . , n) terms in ( 5 )
as
using the TLM-based discrete transform technique, where
Zu = 2/Tp, Tp the propagation time (which equals the time
step used in the numerical solution), Vik’ is the incident
voltage pulse on inductor L(iL) used in the TLM method, and
V i is equal to the magnitude of d iL /d t or numerically the
voltage across an inductor of 1 Henry with the same &’,/at.
Hence, using (7) and (8), ( 5 ) can be rewritten as (9a) (see
equation at the bottom of the page), or
where L( i i ) ( k = 1,2 , . . . , n), are nonideal differential induc-
tors when magnetic hysteresis is considered and [V”] i s the
voltage vector containing the incident pulses. Equation (9) is
the generalized discrete-time model of magnetic cores and is
suitable for direct implementation on digital computers.
At the beginning of each time stepping procedure, incident
pulses V$’ and the instantaneous differential inductances
L( i i ) calculated from the ii ( I C = 1,2, . e . , n) in the previous
time step are substituted into (9) in order to determine the
new current vector [i’]. The differential inductances L(zL) can
then be updated according to the new iL and the history of
magnetization using the Preisach theory.
According to the TLM discrete transforms, it can be shown
that the incident pulses (which are required for each time step)
for the next time step can be obtained as follows:
where a‘, = ii, ih, . . . and ik, and the bracketed subscripts
(m) and (m + 1) denote the mth and (m + 1)th time steps,
respectively. From (9) and (lo), it can be seen that the discrete
algorithms of the dynamic core model are recursive and that
the discrete transform technique allows simple discrete circuit
formulation and easy handling of nonlinearities. Therefore,
a nonlinear magnetic core can be modeled by a tridiagonal
system of nonlinear algebraic equations which can be easily
solved.
C. A Criterion for the Size of the Ladder Network for Low-
and High-Frequency Applications
For solid or laminated cores, the flux distribution within
the core becomes less uniform as the operating frequency
increases. The number of stages of the ladder network (which
represents the number of assumed eddy current paths) depends
on the operating frequency, conductivity and permeability of
the core material:;. Generally, the width of each eddy current
path (we) should be smaller than the skin depth 6 at the
fundamental excitation frequency f , i.e.,
we 5 6 = /= (1 1)
where CT is the conductivity, w = 2r f the angular frequency,
po = 47~ x the permeability of a vacuum in SI units,
and pr the relative permeability. Equation (11) shows that the
skin depth decreases with increasing frequency. Thus more
stages are required in the ladder network for a high-frequency
model. For low-frequency applications, the flux distribution
within the core is fairly uniform so that only one or two eddy
current paths are required. The generalized model reduces to
the low-frequency model developed in [ 13 when only one stage
of the ladder network is used. However, it should be noted
that (11) gives the minimum criterion in choosing the size
(i.e., the number of stages) of the ladder network. In general,
a larger ladder network would give more accurate results than
a smaller one.
UWPr Po
111. PARAMETERS AND LOSSES
All parameters of the model outlined in Section I1 can be
determined from the data sheet provided by the magnetic
material manufacturers.
A. Hysteresis Model
To model the nonlinear magnetization property and the hys-
teresis loss, the F’reisach model of hysteresis is adopted for its
easy parameter identification and considerable accuracy. Since
details of Preisach theory can be obtained in the literature
[3]-[5], this section describes its fundamental concepts only.
The Preisach imodel describes the hysteresis of a magnetic
material via an infinite set of magnetic dipoles, which have
rectangular hysteresis loops, as shown in Fig. 4(a). The flux
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254 E E E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996
density. Function T(a , P) , the area integration over the right
triangle of vertex (a , p) in the (a , p) plane, is calculated by
(17)
where B d and B, are the flux densities on the downward and
upward trajectories of the limiting B-H loop, and
F ( a ) = B d ( a ) - Bu(a) ( a 0) (18)
Bu(a) - Bd(P) + F ( a ) F ( - P )
2 T(a,P) =
2 d m
(a) (b)
Fig. 4.
diagram.
(a) Rectangular hysteresis loop of dipoles. @) Typical Preisach
density B due to the magnetic field H is expressed as
a = pL(a, P)%P(H) da dP
where 5’ is the triangular region Hsat 2 a 2 P 2 -Hsat
on the (a,P) plane shown in Fig. 4(b), Hsat the saturation
magnetic field strength, a and P are the magnetic field strength
in the increasing positive and negative directions respectively,
p(a, p ) the distribution function of the dipoles, ~ ~ a p ( H ) = 1
on enclosed area S+, and y,p(N) = -1 on enclosed area
S-. The interface between S+ and S- is determined by the
history and the present state of magnetization. Through certain
function transforms, the area integration can be related to the
limiting hysteresis loop, which is also the only information
required in the Preisach modeling.
When a magnetic material is magnetized from the initial
unmagnetized state, the magnetic flux density can be calculated
~31% [51 by
Bt = l+ P(Q, P ) d a dP - k- A@, P ) da @
= T ( H , - H ) (13)
where
T ( a , PI = la P ( X , Y) dx dY
P Y
is the area integration over the right triangle of vertex
T(a , a ) = 0, and subscript i denotes the initial B-H curve.
For general situation of Preisach diagram, as shown in
Fig. 4(b), where the magnetization curve has m local extrema,
namely, m reversal points, the flux density can be derived as
( % P I , T(a,P) = T(-W -P) since P(Q,P) = P(-& -a>,
B(H) = B(H,) - 2T(H,, H ) (15)
on a downward trajectory, and
B ( H ) = B ( H m ) + 2T(H, Hm) (16)
on an upward trajectory, where H , is the magnetic field of the
mth (last) reversal point, and B(H,) is the corresponding flux
or
F ( a ) = Jm ( a < 0). (19)
The new mathematical function F ( a ) in (18) and (19) now
provide information for the determination of T ( a , p) without
considering the distribution function p(a, p). It should be
noted that F ( a ) is expressed only in terms of flux density
values of the B-H limiting loop (i.e., B d and Bu). Thus, only
the B-H limiting loop is needed for the implementation of
the hysteresis model.
B. Differential Inductance and Hysteresis Loss
When the flux densities in two consecutive time steps are
obtained, the differential inductance in the kth stage of the
ladder network can be calculated as following
fork = 1 ,2 , . . . , n (20)
where A k is the cross-sectional area of the kth eddy current
path, and subscripts (m) and (m - I) refer to the mth and
(m - 1)th steps, respectively.
When the effect of hysteresis is considered, the differential
inductance L(iL) is nonideal. The average hysteresis loss
dissipated in the kth inductor can be calculated by
The total average hysteresis loss ph is therefore equal to
n di l
d t
iLL(iL)A dt . (21b)
k = l k=l
C. Equivalent Resistance and Eddy Current
and Anomalous Losses
It has been shown in Part I of this paper that the equivalent
resistance accounting for the classical eddy current loss P, and
anomalous loss Pa in the kth eddy current path is
(22) R k - v$ - RDck
P e a k T J k r c k
where v k is the voltage across the resistor R k , Peak =
( P e + P a ) k the sum of the classical eddy current and anomalous
losses, R D C ~ the dc resistance of the kth eddy current path,
r J k = 5 [I +- (I + +)+I the correction factor for the skin
effect, J k = U (g) kay + J k - 1 is the maximum eddy current
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HUI et al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 255
I F a jc = 1 2 , ... ,Nc
Hystcrrsis loso PN = 0, and eady C l t m n t
Fig. 5. Flow chart of dynamic circuit model.
density in the path, Jo = 0 Nm2, Ay the thickness of the
eddy current path, r,k = 1 + c, & the correction factor to
include the anomalous loss, and C, and C, are the eddy current
and anomalous loss coefficients, respectively, determined by
the standard core loss separation procedure as explained in [l]
and [2] .
In terms of the equivalent resistance, the sum of the classical
eddy Current and the anomalous losses in the kth eddy current
d t k
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256 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996
r - - - - - - - - -
1 Scalar Preisach Model
I
I
I
I
Stackempty? 7
H going up? +
I
I
I
I
Calculate B on an upward
trajec%oq by Eqns. (16) aod (17)
Calculae B on the initial curve
Fig. 6. Flow chart of the Preisach model subroutine.
path is equal to where V, = K / N - R,(Ni,) is the voltage across resistor
&, and Pea and Ph are the sum of the eddy current and
anomalous losses and the hysteresis loss, respectively. The
term on the left hand side of (24) is the total power fed into the
T
P e a k 1 1 R k i z d t . ~ 3 ~ )
" 1 sum of the classical eddy current and anomalous losses, and the
second term on the right hand side is the total hysteresis loss
dissipated in the nonideal inductors L(iL), ( I C = 1 , 2 , . . . , n),
for an n-stage ladder network.
'ea = 2 peak = T 1 R k i E dt. (23b)
k=l k=l
D. Total Core Loss Calculation
The general core loss expression of the magnetic core model Iv. SOFTWARE IMPLEMENTATION
can be described, based on the law of energy conservation, as
Fig. 5 illustrates the flow chart of the operation of the
magnetic core model. The program can be used to run the
core model routine N, times. User-chosen N, points can
be obtained within each switching period. Depending on the
nature of the excitation, a variable time step TLM technique [8]
has been implemented in the routine so that any nonlinear ex-
citation waveforms with transients can be incorporated without
$ LT VRn(Nis) d t
T T
R k i i d t + 2 $1 iLL(ii) (5) dt
k=l k=l
= pea + ph (24)
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257 HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART 11
I Generator H Amplifier f - I ' I I
I I I IQlannel2 I I
I I 1 1
Data Precision
6OOO Signal
voltage (v)
Current (A)
1.5
1
0.5
0
-0.5
-1
2
Fig. 7. Experimental setup of annular ring example.
TABLE I
PARAMETERS AND DIMENSIONS OF THE LYCORE-140 CORE EXPERIMENTAL SETUP
losing transient information. This is particularly important in
the study of magnetic core behavior in many modern switched
mode power supplies because the excitation waveforms are
normally highly nonlinear with rapidly changing edges.
The flow chart of the Preisach hysteresis model routine is
shown in Fig. 6. The input data are the limiting hysteresis
loop, and the specified magnetic field for simulation. The stack
keeps the flux density B and magnetic field H values of each
reversal point (local extremum) and is initialized empty (start
from the initial curve) before start. In checking the stack, a
pair of reversal points is popped out of the stack whenever
the specified H exceeds the value kept in the stack. This is to
wipe out a completed minor loop since it will not have any
effect on the future state of magnetization.
V . EXPERIMENTAL VERIFICATION
A. Annular Ring of Silicon Iron with Square
Wave Voltage Excitation
Fig. 7 is an experimental setup for measuring the response
of an annular ring of silicon iron Lycore-140 0.35 mm lam-
ination, under square wave voltage excitation. The primary
and the secondary coils are identical. The primary voltage,
primary current and the secondary voltage are recorded via
-lS -2 f
Hyst.+Eddy+Anom. I
Hyst.+Eddy+Anom. I
-1.51
(b)
Fig. 8. Results of discrete modeling under 100 Hz square wave voltage
excitation. (a) Comparison of voltage and current waveforms; - - - measured,
- calculated. The ideal value of V, used in the calculation is slightly
different from the actual signal because of the nonideal characteristcs of the
power amplifier. (b), Comparison of equivalent hysteresis loops.
three sepatate channels in the signal processing system. The
parameters and the dimensions of the setup are listed in Table I.
simulated
and measured current and equivalent hysteresis loops at 100
Hz are shown in Fig. 8. Since the skin depth (0.5 mm) is
greater than the steel sheet thickness (0.35 mm), only one
layer of eddy current path is assumed, which is identical to
the model used in [l]. The time step used to obtain the
result is 10 ,us. The total calculated core loss is 0.0691
W, and the measured 0.0726 W. The percentage error is
4.82%. Comparing the various B-H loops in Fig. 8(b), the
accuracy improvement in incorporating the anomalous loss in
the magnetic core model is obvious.
2 ) Lycore-140 Core-I kHz Operation with Width of Eddy
Current Path Larger Than Skin Depth: Fig. 9 shows the
simulated and measured results of Lycore-140 core at 1 kHz
I ) Lycore 140 Core-I00 Hz Operation: The
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258
Voltage (VI
IEI
Fig. 9.
network model; - - - measured, __ calculated.
Simulated and measured results at 1 kHz with a one-stage ladder
Voltage gr) Cur" {A)
4 T T 0'4
41 I -0.4
Fig. 10.
network model; - - - measured, ~ calculated.
Simulated and measured results at 1 kHz with a two-stage ladder
excitation, assuming only one eddy current loop. At this
frequency, the skin depth (6) for the core is 0.134 mm. The
width of each lamination in the core is 0.35 mm, giving the
width of each eddy current path (we) to be 0.175 mm. Thls
means that the condition set in (11) is not satisfied. It can
be seen that the calculated current is lower and flatter than
measured. The calculated specific core loss per Hertz is 0.036
W/kg/Hz, and the measured core loss is 0.0336 WkgMz. The
percentage error is 7.2%. This inaccuracy can be attributed to
the reaction of eddy current on the distribution of flux density
in the core [I], 121.
-with Width of Eddy Current Path Smaller Than Skin
Depth: Applying the condition set in (1 l), the cross section
of the core is divided into two layers of eddy current paths,
resulting in the width of each layer being 0.0875 mm. Width of
the eddy current path (we) is now smaller than the skin depth
(6). The core model with two assumed eddy current paths
is represented by a two-stage ladder network (i.e., n = 2).
Fig. 10 shows the simulated results using a two-stage ladder
network. The simulated current has better agreement with the
measured current than that in Fig. 9. The calculated specific
core loss per Hz is 0.0346 W/kg/Hz, and the percentage error
is reduced to 2.98%.
B. A Ferrite Toroid in a Switching Circuit-100-kHz Operation
A SiemensMatsushita N47 soft ferrite R16 toroid core in
a high-frequency switching circuit, as shown in Fig. 11 was
3E TRANSACTIONS ON POWER ELECTRONICS, VOL. 11, NO. 2, MARCH 1996
4-0
1 I
I I
2
Fig. 11. Schematic of a high-frequency switching circuit.
TABLE I1
DIMENSIONS AND PARAMETERS OF THE N47 FERRITE CORE AND WINDINGS
Outer diameter of the CQE 16.33 mm
also simulated. Table I1 lists the dimensions and parameters
of the core and windings.
Fig. 12 shows the simulated and measured current wave-
forms and the 23-H loops of the core at 100 kHz switching
frequency. The cross section of the core was divided into five
paths. The time step is 50 ns. The calculated specific core loss
per Hz, corresponding to the area enclosed by the calculated
B-H loop in Fig. 12(b), is 2.592 x W/kg/Hz, and the
measured specific core loss, corresponding to the area enclosed
by the measured B-H loop in Fig. 12(b), is 2.566 X
WkgEIz. The percentage error is about 1%.
VI. CONCLUSION
A generalized dynamic discrete circuit model of magnetic
core for both low- and high-frequency applications has been
presented. The dynamic circuit core model takes all sources of
nonlinearity and core losses into consideration. The nonlinear
B-H loop and the hysteresis loss are incorporated in the
distributed nonideal inductors and calculated by the Preisach
scalar model of hysteresis. The classical eddy current and
anomalous losses are included in generalized nonlinear resis-
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HUI ef al.: A GENERALIZED DYNAMIC CIRCUIT MODEL OF MAGNETIC CORES-PART I1 259
Flux Density 0
0.2
0.15
0.1
0.05
0
-0.05
0 . 1
-0.15
-0.2
-150 -100 -50 0 50 100 150
Field Strength (Nm)
(b)
Fig. 12. Comparison of simulated and measured. (a) Current waveforms. (b)
B-H loops.
tors. All model parameters, such as limiting hysteresis loop
and specific core loss coefficients, can be obtained from the
manufacturers’ data sheets.
The discrete transform technique together with the Preisach
hysteresis model provides a useful means to handle hysteresis
nonlinearity. The resultant tridiagonal nonlinear system model
equation features easy programming and fast computation. A
criterion for choosing the number of stages of the ladder net-
work is discussed and demonstrated with numerical examples.
Comparisons of the simulations for two types of cores and
different frequencies confirm the flexibility and accuracy of
the generalized model.
REFERENCES
J. G. Zhu, S. Y. R. Hui, and V. S. Ramsden, “Discrete modeling
of magnetic coires including hysteresis, eddy current, and anomalous
losses,” ZEE Proc.-A, vol. 140, no. 4, pp. 317-322, 1993.
-, “A dynamic circuit model for solid magnetic cores,” in ZEEE
Power Electron. Spec. Con$ (PESC ’93) Rec., Seattle, WA, June 20-24,
1993, pp. 1111-1115.
I. D. Mayergoyz, “Mathematical models of hysteresis,” IEEE MAG-22,
no. 5 , pp. 603408, 1986.
S. R. Naidu, “Simulation of the hysteresis phenomenon using Preisach’s
theory,” IEE Proc.-A, vol. 137, no. 2, pp. 73-79, 1990.
S. Y. R. Hui and J. G. Zhu, “Numerical modeling and simulation of
hysteresis effecls in magnetic cores using the transmission line modeling
and F’reisach thieory,” in ZEE Proc.-Sei. Meas. Technol., 1995, vol. 142,
no. 1, pp. 57-62.
S. Y. R. Hui and C. Christopolous, “Discrete transform technique for
solving coupled integro-differential equations in digital computers,” ZEE
Proc.-A, vol. 138, no. 5, pp. 273-280. 1991.
__ , “Discrete transform technique for solving nonlinear circuits and
equations,” ZEE Proc.-A, vol. 139, no. 6, pp. 321-328, 1992.
S. Y. R. Hui, K. K. Fnng, M. Q. Zhang, and C. Christopolous, “Variable
time step technique for transmission line modeling,” ZEE Proc. -A, vol.
140, no. 4, pp. 299-302, 1993.
S. Y. R. (Ron) Hui (SM’94) for a photograph and biography, see this issue,
p. 250.
Jian G. Zhu (S’93--M’93) for a photograph and biography, see this issue,
p. 250.
V. S. Ramsden photograph and biography not available at the time of
publication.
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