A static hysteresis model for power ferrites

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002
453
A Static Hysteresis Model for Power Ferrites
Paiboon Nakmahachalasint, Student Member, IEEE, Khai D. T. Ngo, Senior Member, IEEE, and Loc Vu-Quoc
Abstract—Basso and Bertotti’s physics-based, yet simple, static
hysteresis model is brought to the power electronic community as
an alternative for simulation of magnetic components embedded
in a power electronic converter. The model is reviewed and its
equations cast in application/simulation-oriented forms. It is then
revised to better characterize the very soft saturation behavior
of commercial power manganese–zinc (MnZn) ferrites. The
procedures to extract the model parameters from voltage and
current measurements are described. The improved models have
been verified against experimental data for major and minor
hysteresis loops of three commercial power ferrites.
resnorm
Index Terms—Domain-wall, hysteresis modeling, power MnZn
ferrites, preisach model, soft magnetic materials.
NOMENCLATURE
flops
Magnetic flux density.
Magnetic flux density pertaining to a turning
point.
Maximum flux density of the major –
loop.
Remanent flux density.
Saturation flux density.
Magnetic flux density at which
is
zero.
Coefficient of the reversible process used in
(1).
Number of floating-point operations.
Magnetic field intensity.
Magnetic field intensity pertaining to a
turning point in Fig. 1.
Coercive force.
Magnetic field intensity at which
.
Half the width of the elemental hysteresis loop
in Fig. 9(b).
Bias field of the elemental hysteresis loop in
Fig. 9.
Magnetization;
.
Saturation magnetization.
Normalized magnetization.
Normalized magnetization pertaining to a
turning point.
Normalized magnetization at which
is zero.
Manuscript received February 19, 2002; revised March 5, 2002. This work
was supported by the National Science Foundation under Grant ECS-9906254,
the State of Florida Integrated Electronics Center, and the Royal Thai Government. Recommended by Associate Editor C. R. Sullivan.
The authors are with the Department of Electrical and Computer Engineering,
University of Florida, Gainesville, FL 32611-6200 USA.
Publisher Item Identifier 10.1109/TPEL.2002.801000.
Positive integer of the irreversible process
used in (3).
Parameter used in (7b) for an alternate form of
.
Domain-wall surface function given in (4),
(7a), and (7b).
Parameter used in (4) for
.
Squared 2-norm of the residual from
least-square-fitting.
Mean domain-wall position.
Domain-wall position pertaining to a turning
point in Fig. 1.
Mean domain-wall position of the initial
magnetization curve.
Mean domain-wall position of any return
branch.
Maximum susceptibility used in (1a) and (1b)
for .
Permeability,
.
I. INTRODUCTION
H
YSTERESIS models for power ferrites are an integral part
of a computer-aided design (CAD) system for power electronic converters. The models need to capture the dependence
of hysteresis on such physical phenomena as major loop, minor
loops, irreversible magnetization, reversible magnetization, dynamic effects, shape effects, and temperature effects. This paper
deals primarily with static hysteresis at room temperature. The
methods described in [1] and [2] could be used to incorporate
the dependence of the hysteresis phenomena on rate, temperature, and core shape.
The Stoner–Wolhfarth, Jiles–Atherton, Globus, and Preisach
models are four well-known physics-based macroscopic models
of static hysteresis. Their characteristics and applicability were
discussed and compared in [3]. These models generally do a
better job at describing the major loop than the minor loops
although power electronic transformers and inductors are normally designed to operate in minor loops. Static hysteresis is
better characterized than dynamic hysteresis although the later
is more pertinent in power converters.
Recently, Basso and Bertotti describes a model that is
physics-based and quite simple to use [4], and that could be
another choice for core models in circuit simulators for power
electronic engineers. This model was experimentally verified
for amorphous and nanocrystalline alloys [5], which exhibited
hard saturation. The applicability of the model to the soft
manganese–zinc (MnZn) power ferrites (e.g., MN8CX ferrite
from Ceramic Magnetics [6]) used in high-frequency power
transformers and inductors was demonstrated in [7].
0885-8993/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002
Fig. 1. Mean domain-wall position and B –H loop for H = 50 A/m, =
, c = 0.
:
0 01 (A/m)
One objective of the paper is to introduce the Basso–Bertotti
static hysteresis model to the power electronic readers. This is
accomplished in Section II-A and the Appendix, which review
the model described in [4] and [5] from the simulation/application standpoint. The second objective of the paper is to describe
a refined Basso-Bertotti model for soft ferrites. This is accomplished in Section II-B, which identifies the need to improve the
original model and suggests the corresponding modifications.
While rigorous implementation of the refined model in a commercial circuit simulator remains a future topic, a numerical algorithm is suggested in Section II-C for the simulation of a static
hysteresis loop using the model. Parameter extraction and experimental verification based on voltage and current measurements
of sample toroids are discussed in Section III. The main results
are summarized in Section IV.
Fig. 2. Comparison of theoretical and measured R(m).
netization), (maximum susceptibility,
), (associated
(coercive force), and the adwith reversible magnetization),
ditional parameters in
(1a)
(1b)
(2a)
or
(2b)
where
II. MODEL DESCRIPTION AND IMPLEMENTATION
A. Review of Basso–Bertotti Model for Static Hysteresis
Basso–Bertotti’s model for static hysteresis accepts as an
input the applied magnetic field intensity , and outputs the
(related to the magnetic flux density
by
magnetization
) via an intermediate parameter (the mean
and , the
domain-wall position). The model also requires
and of the most recent turning point, which is the tip of a
-loop/branch, to generate the minor and major hysteresis
loops.
From the application standpoint, Basso-Bertotti static hysteresis model can be summarized by (1)–(3). (These equations
are derived in the Appendix for those readers interested in the
physics of magnetism.) Let the magnetization process start at
. The “initial magthe de-magnetized state with
netization curve” is generated by (1a), which computes from
, and (2b), which computes
from . For any
-branch
,
is found from
via (1b) and (2b) in a simwith
ilar fashion. Equation (1) is illustrated in Fig. 1. As explained
is associated with the irrein the Appendix, the function
(saturation magversible process. The model parameters are
(3)
is the inverse function of
, i.e.,
.
and
(called “domain-wall surface function” in
The function
[4]) is critical in shaping the saturation regions of the hysteresis
is given in [4]
loop. One choice of
(4)
by the curve labeled
which is illustrated in Fig. 2 for
. When
is employed in (2b),
as plotted in Fig. 3. The corresponding
loop generated by (1)–(4) saturates rather abruptly as shown in Fig. 1.
Although (4) has been found to be satisfactory for the two
sample materials in [5], a good fit between modeling and measurement has been found to be difficult for the very soft saturation characteristic of power MnZn ferrites, as is evident in Fig. 4.
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NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL
455
Fig. 5.
Fig. 3.
(2b).
Comparison of theoretical and measured dR(m)=dm.
Normalized magnetization vs mean domain-wall position obtained by
Thus, the suitability of
for a given magnetic material can
with the measured
be assessed by comparing
. This is done in Fig. 5 to evaluate the appropriateness of (4) for MN8CX soft ferrite. The need for alternate
is evident.
forms of
B. Static Hysteresis Model for Power Ferrites
can be obtained from measureAs suggested by (5),
from the major loop data.
ments by computing
These data points are plotted in Fig. 2 along the
suggested in [5], which is (4) with
. Much better fit,
however, is achieved by the other two curves in Fig. 2 that correspond to
and
(7a)
(7b)
and
for MN8CX ferrite. Note that the
in (7a) was suggested in [7]. The
given by (7b) may
be considered to be the generalized from of the original BassoBertotti’s function in (4). The two parameters ( and ) that
(7b) has more than (7a) are expected to provide more flexibility
in fitting a variety of hysteresis shapes. On the other hand, the
in (7a) is more
mathematics below will show that the
efficient numerically.
where
Fig. 4. Measured and fit major loops at room temperature and 10 kHz for
MN8CX ferrite. The fit major loop is generated by the original Basso & Bertotti
model [4], using the parameters in Table I for R(m) = 1 m .
0
Two arguments are now given to explain the difficulty in fitting
is shown in Appendix B and in [5]
near saturation. First,
of the saturation loop. Thus,
to be proportional to
ought to approach zero very slowly as approaches unity for
of power ferrites approaches zero
power ferrites, for the
very slowly in the saturation region. This is not seen in Fig. 2.
increases monoIn fact, the (magnitude of the) slope of
decreases from
tonically as increases from 0 to 1, or as
1 to 0.
Secondly, the softness of the static hysteresis loop can be
with respect to
quantified to be the rate of change of
, which can be obtained from the upward branch of the major
loop using (A.11)
as
or
C. Model Implementation
(5)
To recap, the improved domain-wall model for static hysteresis of soft ferrites consists of the following equations:
(1)–(3), and (7a) or (7b).
The pseudo-code for the implementation of the preceding
equations is outlined in Algorithm 1, which calls Algorithm 2
for a given
according to (2a)
to numerically solve for
if a closed-form solution or a lookup table for in term of is
given in (7a), Algorithm 2
unavailable via (2b). For the
is not needed since a closed-form solution for is available
(6)
(8)
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456
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002
For the
given in (7b), Algorithm 2 is needed during parameter extraction since a closed-form solution for in term of
is generally unavailable. Once all model parameters have been
extracted, a lookup table can be generated and used to determine
for a given , instead of Algorithm 2, to reduce computation
time.
In steps 3 and 13 of Algorithm 1, is assumed to be negli,
gible relative to , permitting one to write
etc. The derivation of step 10 in Algorithm 2 is shown in Appendix C. The Matlab [8] codes for Algorithms 1 and 2 are listed
should not be allowed to equal 1
in Appendix D. Note that
unless the solution algorithm for (2) can handle the singularity
.
at
Algorithm 1. Generation of any hysteresis
branch
.
1) Data: model parameters
and
.
2) Input:
.
3) Calculate
4) Goal: compute , , and
5) if
, (initial magnetization
curve)
.
6) set
via (1a);
7) Calculate
8) else (any return branch)
via (1a).
9) Calculate
via (1b).
10) Calculate
11) endif
12) Calculate
via, e.g., (8), or via
is unavailable.
Algorithm 2 if
.
13) Find
Algorithm 2. Numerical solution of (2) for
by Newton’s method
1) Data: Tolerance (tol), Iterative-limit
(iter), and -limit (mlim)
,
, ,
2) Input: ,
3) Let
. (initial guess).
and
.
4) Set
and
,
5) while
6) Set
7) if
,
.
8) Set
9) endif
10) Calculate
.
.
11) Update
12) endwhile
III. PARAMETER EXTRACTION AND EXPERIMENTAL
VERIFICATION
The model using (7a) for
has one integer parameter
and four real parameters ( , , , and
) to be extracted. In
addition to these parameters, (integer) and (real) need to be
.
extracted if (7b) is used for
Fig. 6. Experimental setup for B –H loop measurement.
Fig. 7. Measured and fit major loops at room temperature and 10 kHz for
MN8CX ferrite. The fit major loop is generated by the improved model, using
the extracted parameters in Table I for R(m) = (1 m )
.
0
The first step in parameter extraction is the acquisition of the
quasistatic hysteresis loops via voltage and current measurements [1], [9]. In this paper, the hysteresis loops were measured
with a 10 kHz sinusoidal excitation on an MN8CX test toroid at
room temperature. The test core has the following dimensions:
outer diameter 12.70 mm, inner diameter 6.35 mm, height
3.18 mm, mean path length 29.92 mm, and cross-sectional
area 10.1 mm . A primary winding and a secondary winding
were wound bifilar on the core, both having 25 turns of gauge
AWG #28.
LabVIEW [10] was used to automatically acquire hysteresis
loop data by controlling the measuring instruments via IEEE
488 interface as shown in Fig. 6. The test core was demagnetized before obtaining each set of the measured data by slowly
reducing the magnitude of the ac excitation voltage from its saturation value to zero [9]. To prevent self-heating, the test core
was excited for only few seconds before each set of data was
measured and stored for post-processing.
The first quadrant of the measured major static hysteresis
loop is shown in Fig. 7. The measured major hysteresis loop
is shown together with several measured minor loops in Fig. 8.
describes the reduction of the magnetization near
Since
the saturation region, its parameters are extracted from the major
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NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL
457
TABLE I
EXTRACTED PARAMETERS FOR THE HYSTERESIS MODEL FOR MN8CX FERRITE
loop data. Nevertheless, the minor loop data are also included to
check the accuracy of the improved model with respect to minor
loop behavior.
From the versus data for the major hysteresis loop,
and, then,
are computed. Appendix B
suggests the following algorithm to estimate the initial values
, and :
for the five real parameters , , ,
(9)
(10)
(11)
Fig. 8. Measured and fit major and minor loops at room temperature and 10
kHz for MN8CX ferrite. The fit B –H loops are generated by the improved
model, using the extracted parameters in Table I for R(m) = cos (m=2).
the model can be attributed to the physics base of the model.
The model has also been verified for 3D3 ferrite [11] (resnorm
0.002 016) [7] and for ferrite [12] (resnorm 0.006 720).
(12)
IV. CONCLUSION
is the magnetic flux density at which
is
where
zero. Using the preceding equations and unity as the initial guess
for , the initial guesses for MN8CX ferrite are found to be
T,
(A/m) ,
,
A/m,
.
and
Matlab [8] was then used to extract the model parameters.
Since and are integers, they could not be extracted using
the least-square-fitting function, lsqcurvefit, in a straightforward
manner. Thus, was swept between 1 and 5, and between 1
pair, the five real parameters , ,
,
and 4. For each
, and were extracted using lsqcurvefit, and the squared
2-norm error (called resnorm in Matlab) recorded. The sets of
parameters with the minimum squared 2-norm errors are listed
in Table I. The numbers of floating-point operations (flops) of
is about the
the improved model using
same as the flops of the original model, whereas the flops of
is about
the improved model using
two hundred times of the flops of the original model (without
retable-lookup). The model using
quires more computation because of the integration of (2) via
Algorithm 2. The resnorms of the improved models are at least
twenty times less than the resnorm of the original model, indicating that the improved models provide a better fit to the measured data.
Fig. 7 compares the measured major loop with the fit major
. The close agreeloop calculated with
ment between theory and experiment justifies the use of the improved models for power MnZn ferrites.
Fig. 8 compares the measured and predicted minor loops
. Interestingly, the fit between
using
theory and experiment is still good although the minor loop
data were not used to extract the parameters. The robustness of
To better characterize the very soft saturation characteristic
functions with a particular shape have
of power ferrites,
been proposed for Basso-Bertotti model of static hysteresis so
approaches zero asymptotically as the
that the slope of
functions have resulted
core saturates. These improved
in good agreement between measured and fit data for the major
and minor loops of three commercial power ferrites.
The model parameters for the other commercial power ferrites will be extracted to make this work complete. In addition, other physical phenomena that are important to power electronics, such as temperature, frequency, and core shape, ought to
be incorporated into the model. The complete model should be
implemented in a commercial circuit simulator, offering power
electronic designers another choice for magnetic core models.
APPENDIX A
DERIVATION OF THE MEAN DOMAIN-WALL POSITION
Similar to the classical Preisach model [13], the simplified
Preisach model in [4] employs a collection of noninteracting,
statistically distributed elemental hysteresis loops like those
shown in Fig. 9 to describe the macroscopic hysteresis phenomena. Each elemental hysteresis loop is associated with an
“idealized” one-dimensional domain-wall that is allowed to
is
move freely when an external magnetic field intensity
applied. Domain-wall motion could be reversible (no loss) or
irreversible (with loss). The reversible domain-wall motion is
modeled by the zero-width elemental hysteresis loop shown
in Fig. 9(a), and the irreversible domain-wall motion by the
elemental hysteresis loop in Fig. 9(b).
The elemental hysteresis loops are distributed statistically ac. In [5],
cording to the Preisach distribution function
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 17, NO. 4, JULY 2002
Fig. 9. Elemental hysteresis loops for (a) reversible and (b) irreversible
domain-wall motions.
Fig. 11. Preisach diagrams for the derivation of the mean domain-wall motion:
(a) for the initial magnetization, (b) for the downward branch, and (c) for the
upward branch.
In the classical Preisach model, the magnetization under a
certain field history is found by proper integration of the distri. In the Basso-Bertotti domain-wall
bution function
model, however, magnetization is calculated from the domain-wall position that, in turn, is found by proper integration
.
of
described in
With the Preisach distribution function
(A.1)–(A.3) and the Preisach diagram in Fig. 11, the mean domain-wall position for the initial magnetization curve, the downward branch, and the upward branch are derived, respectively, as
shown in (A.4)–(A.6). The generalized equation (A.7) for any
return branches can then be obtained by combining (A.5) and
(A.6)
Fig. 10. Irreversible Preisach distribution function
50 A/m, = 0.
c
is uniform with respect to
cording to
p (h ) in (A.2) for H
and depends on
=
(A.4)
ac-
(A.1)
(A.5)
(A.2)
(A.3)
consists of two terms,
and
The function
describing irreversible and reversible processes,
,
, and
are defined
respectively. Since
,
, and
such that
, the coefficient is used to weigh the
contribution of the reversible process; the coefficient
then represents the contribution of the irreversible process.
in (A.2) is plotted in Fig. 10 for various
The function
is the Dirac delta function
values of . The function
enables the initial susin (A.3). As explained in [5],
ceptibility and the susceptibility at a turning point (e.g., point
in Fig. 1) to be non-zero.
(A.6)
(A.7)
versus is exemplified in Fig. 1, where it is
A plot of
could reach infinity as approaches infinity.
noted that
versus
are exemplified in Fig. 1 for a
The plots of
downward branch originating from a large positive value of ,
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NAKMAHACHALASINT et al.: STATIC HYSTERESIS MODEL
459
and for an upward branch originating from a large negative value
of .
Thus
APPENDIX B
, AND
DERIVATIONS OF , ,
or
By chain rule
If the test core is driven back from saturation to
as
(A.17)
at
(A.8)
as defined by (A.6)
With
(A.18)
as
(A.19)
Thus
If
FIND
(A.9)
(A.20)
APPENDIX C
NEWTON’S METHOD
BY
Using (2b), define
Substitution of (A.9) in (A.8) yields
(A.10)
Thus
as
or
(A.11)
(A.21)
as
(A.12)
If the test core is driven to saturation from the demagnetized
state
(A.13)
By definition
(A.14)
Thus
as
If the test core is driven back from saturation to
(A.15)
at
(A.16)
APPENDIX D
MATLAB CODES FOR ALGORITHMS 1 AND 2
The Matlab codes for Algorithms 1 and 2 are listed below
along with comments, signified by the % sign.
function B =
findB(H,H0,B0,chi,c,Hc,Bs,n,r,q)
%Data: model parameters {chi, c, Hc, Bs,
n, r, q}
%Input: H and {H0, B0}
m0 = B0/Bs;
%Goal: compute delx = x-x0, m, B
if (H0 == 0) & (B0 == 0), %initial magnetization curve
x0 = 0;
x = chi*sign(H)*((1c)*Pirr(abs(H),Hc,c,n)+c*abs(H));
else %any return branch
x0 = chi*sign(H0)*((1c)*Pirr(abs(H0),Hc,c,n)+c*abs(H0));
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x = x0+2*chi*sign(H-H0)*…
((1-c)*Pirr(abs(H-H0)/2,Hc,c,n)+c*abs(HH0)/2);
end
m = G(x,x0,m0,r,q); % or m =
2/pi*atan(pi/2*x);
B = m*Bs;
% subfunction Pirr(delH)
function output = Pirr(delH,Hc,c,n)
sum = 0;
for k = 1:n,
sum = sum+k/n/prod (1:n-k)*(n*delH/Hc*(1c))^(n-k);
end
output =(delH-Hc/(1-c))+Hc/(1-c)*exp(n*delH/Hc*(1-c))*sum;
function m = G(x,x0,m0,r,q)
%Data: Tolerance (tol), Iterative-limit
(iter), and m-limit (mlim).
%Input: x, x0, m0, r, q.
iRm = inline(’1./(1m.^(2*r)).^q’,’m’,’r’,’q’); %
1/R(m)
tol = 0.0001; iter = 500; mlim = 0.99;
m = 0; %initial guess
delm = tol; i = 0;
while (abs(delm) >= tol) & (i < iter),
i = i+1;
if abs(m) > mlim, m = sign(m)*mlim; end
delm = (quad8(iRm,m0,m,[ ],[ ],r,q)-x+x0)
/iRm(m,r,q);
m = m-delm;
end
[6] Engineered Ferrites Catalog, Ceramic Magnetics, 1999.
[7] P. Nakmahachalasint and K. D. T. Ngo, “An improved domain-wall
model of static hysteresis for power ferrites,” Electron. Lett., vol. 36,
no. 24, pp. 2020–2022, Nov. 2000.
[8] Optimization Toolbox User’s Guide, Version 2, MathWorks, Natick,
MA, 1999.
[9] IEEE Standard for Test Procedures for Magnetic Cores, IEEE Std 3931991.
[10] LabVIEW User Manual for Windows, Version 5.1, National Instruments,
Austin, TX, 1999.
[11] Soft Ferrites, Data Handbook MA01, Phillips Components, 2000.
[12] Ferrite Cores Design Manual, Magnetics, Inc., 1999.
[13] G. Bertotti, Hysteresis in Magnetism. Boston, MA: Academic, 1998.
Paiboon Nakmahachalasint (S’99) received the
B.Eng. degree in industrial instrumentation from
King Mongkut’s Institute of Technology Ladkrabang, Thailand, in 1991, and the M.S. degree in
electrical engineering from the University of Florida,
Gainesville, in 1994, where he is currently pursuing
the Ph.D. degree.
Since 1992, he has been with Thammasat University, Thailand, where he is currently an Assistant
Professor. His current research interests include
magnetic materials and components.
Khai D. T. Ngo (S’82–M’84–SM’02) received the
B.S. degree from California State Polytechnic University, Pomona, in 1979, and the M.S. and Ph.D.
degrees from the California Institute of Technology,
Pasadena, in 1980 and 1984, respectively, all in electrical and electronics engineering.
He was a Member of Technical Staff, General
Electric Corporate Research and Development
Center, Schenectady, NY, from 1984 to 1988. He has
been an Associate Professor in the Department of
Electrical and Computer Engineering, University of
Florida, since 1988. His current research interests are magnetic materials and
components, power converters, and power integrated circuits.
ACKNOWLEDGMENT
The authors wish to thank Dr. V. Basso for his valuable comments during the course of this work.
REFERENCES
[1] J. T. Hsu and K. D. T. Ngo, “A Hammerstein-based dynamic model
for hysteresis phenomenon,” IEEE Trans. Power Electron., vol. 2, pp.
406–413, May 1997.
[2]
, “Application of field-based circuits to the modeling of magnetic
components with hysteresis,” IEEE Trans. Power Electron., vol. 2, pp.
422–428, May 1997.
[3] F. Liorzou, B. Phelps, and D. L. Atherton, “Macroscopic models of magnetization,” IEEE Trans. Magn., vol. 36, pp. 418–428, Mar. 2000.
[4] V. Basso and G. Bertotti, “Hysteresis models for the description of domain wall motion,” IEEE Trans. Magn., vol. 32, pp. 4210–4212, Sept.
1996.
[5] V. Basso, “Hysteresis models for magnetization by domain wall motion,” IEEE Trans. Magn., vol. 34, pp. 2207–2212, July 1998.
Loc Vu-Quoc received the Dip.Ing. degree in structural engineering (with highest honors) from the Institut National des Science Appliquées, Lyon, France,
in 1979, the M.S. degree in structural mechanics from
the Illinois Institute of Technology, Chicago, in 1981,
and the the M.S. degree in electrical engineering and
computer science and the Ph.D. degree in structural
engineering and structural mechanics from the University of California, Berkeley, in 1985 and 1986, respectively.
He worked for two years (1979–1981) developing
finite element codes at the Centre Technique des Industries Mécaniques, Senlis,
France. He joined the University of Florida in 1988, after two years of postDoctoral work at Stanford University, Stanford, CA, and Berkeley, CA, he is
currently Professor of aerospace engineering, mechanics and engineering science. His current research interests are in applied/computational electromagnetics/mechanics, and in power electronics simulation.
Dr. Vu-Quoc received the NSF Presidential Young Investigator award in 1990.
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