Large signal inductor modeling for switch mode power supplies

Large signal inductor modeling for switch mode power supplies

Large signal inductor modeling for switch mode
power supplies
Shen Xiao, Silvio Misera, Sven Bönisch, Lutz Göhler
Fakultät Verkehrs- und Maschinensysteme
Technische Universität Berlin 10623 Berlin Germany
Fakultät für Ingenieurwissenschaften und Informatik
Hochschule Lausitz 01968 Senftenberg Germany
[email protected]
[email protected]
Kjellberg Finsterwalde GmbH
Leipziger Strasse 82, 03238 Finsterwalde Germany
[email protected]
Abstract— In this paper, a modified Jiles-Atherton model within
APLAC is introduced. This model can be used to to simulate
inductors and transformers in a switch-mode power supply at
large signal input. An optimization method is coupled to an
iterative algorithm to identify the Jiles-Atherton parameters for
a modified Jiles-Atherton model. The verification of the modified
model is done by comparing simulated hysteresis loops to
experimental results at different frequencies. The results state
that the modified model is valid.
Key Words— Hysteresis; Modified JA model; Parameter
identification; Genetic algorithm; Jiles-Atherton model
I. INTRODUCTION
Inductors and transformers which are used in switch-mode
power supplies usually work at large signal input. Large
signals may cause intensive inrush currents within
transformers and inductors and lead to saturation. Saturation,
as a phenomenon of lowering inductance with increasing
currents in inductors and transformers, is often lowering
system efficiency. It should be investigated by inspecting the
magnetic hysteresis loop. Therefore, hysteresis models are
needed for simulation of inductors and transformers.
Among all hysteresis models, the Jiles-Atherton (JA) model
is widely used and has been implemented in various system
simulation tools. However, if it is compared with the results
from experiment, its parameters should be further improved.
The purpose of this work is to use a modified JA model which
has already been implemented in APLAC, to propose an
appropriate identification technique for model parameters and
to let the hysteresis loop simulated by a modified JA model fit
the measurement results. The simulations are carried out at
different frequencies in order to investigate the validity of the
modified JA model. Finally, corresponding hysteresis loops
are compared to evaluate the determined parameters of this
modified JA model.
II.
MODIFIED JILES-ATHERTON HYSTERESIS
MODELING WITHIN APLAC
As a simple and accurate modeling technique for the
magnetic hysteresis loop, the Jiles-Atherton model has been
continuously developed and improved over time. In this
section, the principle of JA model is to be elucidated. Besides,
a modified JA model within APLAC is then introduced.
A. Original Jiles-Atherton Model
In the original JA model, total magnetization
can be
divided into two parts, an irreversible component
, which
is representing the magnetic domain wall displacement against
pinning site of defect in ferromagnetic material, and a
reversible component
, which accounts for the reversible
rotation of domain wall,
. The relation
between the magnetic effective field
and the magnetic field
is expressed as:
In Eq. (1), is the interdomain coupling coefficient. The
anhysteretic magnetization can be represented by using the
modified Langevin equation:
In Eq. (2),
is the saturated magnetization, characterizes
the shape of anhysteretic magnetization curve. Then, the
differential irreversible susceptibility can be obtained in the
simplified form [1]:
Finally, the total differential susceptibility which can describe
the dynamic behavior of magnetic hysteresis loop is derived
as follow:
B. Modified Jiles-Atherton Model in APLAC
In APLAC, a modified equation has been given, which
indicates that the magnetic effective field can be associated
with the magnetic field by using the anhysteresis
magnetization:
By adopting Eq. (5), the differential irreversible susceptibility
is obtained as:
In Eq. (6), the derivative of
depends on both the
derivative of the anhysteretic magnetization and the
displacement from anhysteretic curve. To fully describe
hysteresis, an additional equation of reversible magnetization
is needed:
After Eq. (7) is differentiated and added to Eq. (6), the
modified JA model can be given by：
Comparing Eq. (8) to Eq. (4), it can be found that the equation
of the modified JA model in APLAC is completely different
from that of the original JA model.
III.
Determination of modified Jiles-Atherton model
parameters
The procedure of parameter determination is split up into
two phases: iterative calculation for preliminary model
parameters and further optimization with a genetic algorithm.
A. Iterative Method For Parameter Calculation
The equations of the total differential susceptibility and its
complementary relations are given by:
Compared to the parameters of original JA model, those of the
modified JA model have different mathematical expressions
and are determined using experimental data.
In Eq. (10),
and
are flux density and magnetic field at
saturation of the hysteresis loop. Both of them are extracted
from experimental results. According to the measured
magnetic properties and Eq. (9a), the initial susceptibility is
calculated at the initial point (
) of the
initial magnetization curve (
):
Eq. (11b) is derived from reference [3], once is initially
known, the parameter can be determined by using Eq. (11a),
which is written as:
At coercivity (
,
previously determined, parameter
the following equation:
), once , and are
can be calculated using
At remanence
,
parameter can be obtained by the following equation:
the
Finally, regarding the use of the coordinates of the loop tip
(
), considering the previous
determination of parameters , and , the parameter can
be obtained by solving:
Since there are four equations needed for determining the
parameters which are expressed implicitly in terms of these
and other parameters, by using Eq. (12), Eq. (13), Eq. (14) and
Eq. (15) successively in an iterative algorithm, the value
of , , and can be obtained. First, the initial value of
and are given. After the first four parameters are calculated
as discussed above, the procedure for the calculation is then
repeated for several times and the preliminary group of model
parameters are finally given.
B. The Relation Between Parameters And Hysteresis Loop
Because the calculation result of the iterative algorithm is
very sensitive to its initial values, it is necessary to elucidate
the effect of each parameter for the hysteresis loop. The
parameter reflects the coupling between magnetic domains,
when increases, the slope of hysteresis loop increases. The
parameter a represents the shape of the anhysteretic
magnetization curve, the slope of the hysteresis loop decreases
with increasing . According to the change trend of the
hysteresis loop, two suitable estimated values of and can
be given for the iterative calculation. The parameter and
relate to the hysteresis loss, the hysteresis loss increases with
decreasing parameter or increasing parameter .
C. Parameters Optimization with a genetic algorithm
After the preliminary parameters are calculated with an
iterative algorithm, a further improvement for matching of
experimental data and simulation results can be applied to
above relationships. First, the hysteresis loop with those five
calculated parameters is simulated in APLAC. After, the result
of the model simulation is compared to the result of the
measurement, five proper variable domains can be generated
by introducing another five parameters. The change of each
parameter is limited in between the values of the calculated
preliminary parameter and simulated parameter. The fitness
function f(s) is implemented as follows:
While
In Eq. (17), is the sampling number of a half period
hysteresis loop from experiment. The magnetization
is evaluated by Eq. (8) and compared with the
jth point of the experimental results under analysis for the
same H. In Eq. (16), in order to minimize the differences
between measurement and simulation, an optimum group of
parameters is determined in terms of evaluation of the highest
fitness function. A genetic algorithm (GA) is an ideal method
to retrieve the maximum value of the fitness function as well
as all optimum parameters. The five preliminary model
parameters are input to the GA which gives rise to 50
chromosomes. Each chromosome is codified with a binary
code of 50bits, and each parameter of the model corresponds
to 10bits. On the basis of the above discussion, the selection
of the GA is to find the highest fitness function. Then, the
crossover and mutation of GA creates a new optimum of
parameters on each generation. This GA proceeds until its
30th generation. Thus, improved parameters are obtained, and
the simulated hysteresis loops become more close to
experimental results.
B-H characteristics at different frequencies
0.6
0.4
B [T]
0.2
0
500Hz
1kHz
2kHz
5kHz
10kHz
20kHz
40kHz
PARAM ETERS EXTRA CTED FROM EXPERIM ENT
Parameter
Measured value
Parameter
Measured value
1995
56.87
3966
4167
259
0.393
9.53
0.063
TABLE II
IDENTIFIED PARAM ETERS OF THE M ODIFIED JA M ODEL
Identified parameter
Value
342901
22
5e-7
0.75
18
Once the identified parameters are obtained, all hysteresis
loops can be calculated and plotted using the Gear-Shichman
method for numerical integration in APLAC. The hysteresis
loops of measurement as well as the simulation results at
2kHz and 20kHz are respectively shown in Fig. 2.
Comparison of B-H characteristics of TR
0.6
0.4
simulation at 2kHz
simulation at 20kHz
measurement at 2kHz
measurement at 20kHz
0.2
0
-0.2
-0.4
-400
-0.2
-200
0
H [A/m]
200
400
Fig. 2 Co mparison between measurement and simulat ion of the transformer at
different frequencies
-0.4
-400
TABLE I
B [T]
IV. Results and discussion
To verify the applicability of the modified JA model in the
simulation of an inductor at large signal input, a transformer
which is used in the intermediate circuit of a power supply is
selected as test object. To obtain various hysteresis loops, the
primary side of the transformer is saturated with an
appropriate input current at different frequencies. At the mean
time, the secondary side of the transformer is an open circuit
during measurements.
The measurement results of the primary side are summarized
in Fig. 1. All traces of all hysteresis loops overlap once
magnetic field reaches saturation. According to these results,
it is possible to use one model to simulate the hysteresis loop
at different frequencies.
To identify the modified JA model, some needed
parameters which are extracted from the experiment are
presented in TABLE I. These are then utilized for the
calculation of the preliminary key parameters of the hysteresis
loop. After that, the five obtained parameters are optimized by
using GA in Matlab. The identified parameters are finally
determined and listed in TABLE II.
-200
0
H [A/m]
200
400
Fig. 1 Hysteresis loop of the transformer at different frequencies
Two hysteresis loops saturate at 2kHz, however, another two
are not saturated at 20kHz. The former shape is called major
loop, and the latter shape is referred as minor loop. To verify
the validity of the model parameters, all of the simulated
results should match with the corresponding measured
hysteresis loops. The similarities of these loops can be
inspected by using some specific parameters of the hysteresis
loop. TABLE III shows the errors between measurement and
simulation.
TABLE III
MODELING ERROR OF HYSTERESIS LOOP
Verified
Parameter
Experiment
2kHz 20kHz
259.7 43.17
0.393 0.192
9.53
8.28
0.063 0.048
4167 4305
3966 4038
17.16 4.92
Simulation
2kHz 20kHz
251.5 41.34
0.401 0.178
9.26
8.63
0.058 0.051
3994
4236
3799
4236
16.17 5.41
Error
2kHz 20kHz
3.2% 4.2%
2%
7.1%
2.8% 4.1%
7.9% 5.9%
4.2% 1.2%
4.2% 4.7%
5.8% 9.1%
All parameters shown here extracted from the corresponding
hysteresis loops as given by Fig. 2. The upper six parameters
describe the size and slope of each loop. They show good
match for all loops. As a crucial parameter of criteria among
others, the last parameter represents the relative area of the
hysteresis loop, which reflects the summation of the dynamic
change of the loop during magnetizing. All errors are below
10%, pointing to the validity of the simulated hysteresis loop.
To sum up, a good agreement between measurement and
simulation is obtained for both frequencies. This result
indicates the effectiveness of this technique used to identify
model parameters. In this case, the modified JA model can be
used for the simulation of the magnetic hysteresis of a
transformer at different frequencies.
V. Conclusion
The modified JA model as well as an identification
technique based on an iterative method coupled with a genetic
algorithm for parameter optimization has been described. The
simulated hysteresis loops are compared to experimental
results at different frequencies. The results of comparison
indicate that the optimized identified parameters of the
modified JA model are valid and accurate. They can be
adopted to simulate the hysteresis loop of a transformer which
is used in a switch mode power supply. Moreover, as
hysteresis loops show in Fig. 2, this model can not only
manage saturated loops at sinusoidal excitation, but also can
describe unsaturated minor loops. This is beneficial to
completely reflect the behavior of magnetic components
within system at normal working conditions.
ACKNOWLEDEGMENT
This paper has been produced in collaboration with
Kjellberg Finsterwalde GmbH, the most professional
manufacturer of plasma cutting devices in Germany. I would
therefore sincerely thank the Kjellberg colleagues who
participated in the project “Störarme Stromquellen“ for
support in every aspect.
I especially thank Dr. Silvio Misera who provide the
specific inductors and transformer for testing and carefully
answer every question regarding these products.
I am heartily thank my supervisor, Prof. Sven Bönisch, who
gave me the opportunity to live such experience and guided
me during the development phase as well as writing this
publication.
Last, I offer my regards to all of those who supported and
encouraged me during the completion of the project especially
Prof. Lutz Göhler and all the co-workers in EMC laboratory
of Hochschule Lausitz.
REFERENCES
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Journal of Magnetism and Magnetic Materials 61 (1986), 48-60
[2] Jarmo Virtanen, Toroid Model in APLAC, November 11, 1997
[3] David C. Jiles, Sen ior Member IEEE, J.B. Thoelke, and M.K. Dev ine,
Nu merical Determination of Hysteresis parameters for the modeling of
magnetic properties using the theory of ferromagnetic Hysteresis, IEEE
Transactions on Magnetics, January 1992
[4] M. Hamimid , M. Feliachi,S. M.Mimoune, Modified Jiles -Atherton model
and parameters identification using false position method , Physica B, January,
2010
[5] X. Wang, D.W.P. Tho mas, M. Su mmer, J. Paul, and S.H.L. Cabral,
Characteristics of Jiles-Atherton model parameters and their application to
transformer inrush current simu lation, IEEE, Trans actions on Magnetics VOL.
44 No. 3 March, 2008