New electrical parameters
extraction method based on
simplified 3D model using finite
element analysis
Jorge Rafael González-Teodoro and
Enrique González Romero-Cadaval
Department of Electrical, Electronic and Control Engineering,
School of Industrial Engineering, University of Extremadura, Badajoz,
Spain
Rafael Asensi and Roberto Prieto
Polytechnic University of Madrid, Madrid, Spain, and
Vladimir Kindl
Faculty of Electrical Engineering, University of West Bohemia,
Pilsen, Czech Republic
Abstract
Purpose The purpose of this paper is the presentation of an electrical equivalent circuit for inductive
components as well as the methodology for electrical parameter extraction by using a 3 D finite element
analys is (FEA) tool.
Design/methodology/approach A parameter extraction based on energies has been modified for
three dimensions. Some s impli fications are needed in a real model to make the 3 D finite ele ment m ethod
(FEM) analysis operative for design engineers. Material properties for the components are modifi ed at
the pre modeling step and a corrector factor is used at the post modeling step to achieve the desired
accuracy.
Findings The current hardware computational limitations do not allow the 3 D FEA for every magnetic
component, and due to the component asymmetries, the 2 D analysis are not precise enough. The application
of the new methodology for three dimensions to several actual components has shown its usefulness and
accuracy. Details concerning model parameters extration are presented with simulation and measurement
results at different operation frequencies from 1 kHz to 1 GHz being the range of switching frequencies used
by power ele ctronic co nverters bas ed on Si, Si C or GaN semiconductors.
Practical implications This new model includes the high frequency effects (skin effect, proximity
effect, interleaving and core gap) and other effects can be only analyzed in 3 D analysis for non symmetric
components. The electrical parameters like resistance and inductance (self and mutual ones) are frequency
dependent; thus, the model represents the frequency behavior of windings in detail. These parameters
deter mine the ef fici ency fo r the in ductiv e com ponen t an d ope ration capab ili tie s fo r the pow er convert er s (as in
the volta ge boost fac tor), which de fine their suc cess on the market.
Originality/value The user can develop 3 D finite element method (FEM) based analyses with
geometrical simplifications, reducing the CPU time and extracting electrical parameters. The corrector factor
presented in this paper allows obtaining the electrical parameters when 3D FE simulation would have
This research has been supported by the Junta de Extremadura, under project IB18067 and by the
Ministry of Education, Youth and Sports of the Czech Republic under the project OP VVV Electrical
Engineering Technologies with High Level of Embedded Intelligence CZ.02.1.01/0.0/0. 0/18 069/
0009855 and by funding program of the University of West Bohemia number SGS2018 009.
developed without any geometry simplications. The contribution permits that the simulations do not need a
high computational resource, and the simulation times are reduced drastically. Also, the reduced CPU time
needed per simulation gives a potential tool to optimize the non symmetric components with 3 D FEM
analysis.
Keywords Electromagnetism, Finite element analysis, Power electronic simulation,
Computational electromagnetics, Magnetic equivalent circuit, Power electronic devices modeling,
Magnetics, 3D modeling, Transformers, Resistance, Inductance
Paper type Research paper
1. Introduction
Power electronics engineers need to use inductive components as an important part of their
converters. The electrical parameters for the inductor have a critical influence in the overall
efficiency and operation capabilities for the power converter.
The calculation of the winding resistance and inductance could be conducted by
application the Maxwell¨s equations. Nevertheless, the complete analytical solution for these
equations has only been available for components with 1 D magnetic field distribution.
Another alternative is to obtain the winding parameters by laboratory testing, but the
component design or optimization could be expensive and time consuming.
A finite element analysis (FEA) is an adequate tool to calculate the winding parameters
of any asymmetrical magnetic components (Iatcheva et
al.,201
8), but the current hardware
computational limitations (the Random Access Memory is usually the limiting factor) do not
natively allow to perform very complex 3 D simulation considering high-frequency effects,
interleaved winding or assembling air gap.
On the other hand, the calculation based on 1 D or 2 D FEA is not applicable due to the
hig
h degre
e of geometrical simplification. The only possible way for this type of calculation
is to significantly reduce the number of mesh elements and perform complex 3 D simulation.
Several various approaches for electrical parameters extraction have been developed
based on FEA optimization (Okamoto et
al.,201
6; Kiselev et al.,2016), original
methodologies (Ben Messaoud et al., 2016) or for particular components (Aljohani et al., 2016;
Phukan et al.,2016; Niyomsatian et al., 2018; Lu and Ngo, 2017) without present an actual
3 D model of the magnetic non-symmetrical component including all effects (skin effect,
proximity effect, interleaving, core air gap).
In this work, the attention is paid mainly to the components with EE and toroidal
mag
netic cor
es (non-symmetrical coil configurations) because they are common for
inductors, lack 3 D symmetry and the winding parameters (resistance and inductance)
cannot be calculated either using 1 D or 2 D FEA. The asymmetry of these inductive
components lies mainly in the square shape of the windings.
As shown in (Gonzalez-Teodoro et
al.,20
15; Coulomb, 2014), successful convergence of 3 D
simulation requires the adoption of certain simplifications that more or less affect the results.
The parameter extraction presented in this paper is based on an electrical equivalent
circuit dis
cussed in Asensi et al. (2007) and mod
ified for three dimensions (Figure 1.) This
equivalent circuit stands on the superposition theorem which generally introduces some
limitations.
It is only applied on the magnetic fields and the
current densities (not on energies or
losses) when analyzing a linear system. The FEM simulation therefore considers a constant
permeability and coercivity set for the magnetic core material (Eddy current solver used). In
other words, the model is not applicable to the systems with non-linear saturation (BH
curve).
In Section 2, the different analyses, the parameter extraction procedure and research
developed to calculate the correction factor will be explained. Section 3 will introduce the
original methodology to be used by the power engineers. Section 4 is dedicated to the
experimental validation f or eight different inductive components and showing the CPU
reduction time. Conclusions and advantages for the proposed procedure are given in
Section 5.
2.
Analysis for non-symmetrical inductive components
The procedure of obtaining the correction factor is based on elementary geometry FEM
analyses (referring to cases I, II, III, IV) and proper mathematical regressions. The cases
have been modeled with the geometry assistant in the FEA (Figure 2). The modeling has
be
en done
following next process:
Selected solver has been the Eddy Current Type with a frequency in the range from
1 Hz to 1.2 GHz.
Material properties of the core and windings have been adjusted using the model
descriptor function
Figure 1.
Electrical equivalent
circuit for non
symmetric
components
Figure 2.
Ansys Maxwell
environment for
modeling and
simulation of
magnetic components
the mesh used for FEM analysis is automatic generated with these settings: 1 per
cent error,1 25 maximum number of steps and step refinement of 20 per cent.
Finally, the excitations (both, input and output signals) has been introduced using
the sheets created into windings.
The material properties have been adjusted in the software database, the current density has
been int
roduced according to the Transient solver and the mesh has been set up
automatically. More detail about FEA methodology is given in Section 3.
2.1 Polygonal model
The conductor cross-sections for analyzed case studies are modified from a circular shape
(original geometry) to a polygonal shape to reduce the FE number and achieve easier
computational convergence. These modification change both the cross-section area and the
distance between conductors which consequently affects the winding resistance and
inductance. The material properties (1-2) must be accordingly tuned in the model to achieve
the same resistance and inductance values that were measured for original geometry
(Gonzalez-Teodoro et
al., 2015):
s
Pol
¼
S
Circular
S
Polygon
s
Circular
(1)
m
Pol
¼
S
Circular
S
Polygon
m
Circular
(2)
In (1) and (2), the parameters
s
Pol
and
m
Pol
represent the conductivity and the permeability
for the polygonal conductor,
s
Circular
and
m
Circular
represent the same but for the circular
conductor and S
Pircular
and S
Polygonal
stand for the cross-section areas of both geometrical
variants respectively.
To minimize any distortion of the results, the zero vector potential boundary condition is
set on the surface of enclose “free-space” region (having a size 5 times higher that of the
component being analyzed) surrounding the conductor.
The analyzed inductive component is named “Polygonal Model” in the paper. The
simulation is performed using Eddy Current Solver (Ansys Maxwell) with an energy error of
2 per cent and using standard defined mesh. The mesh is imported from a transformer
(multiwinding component) to compare the results from different inductive components
(single winding) having the same mesh.
At the post-processing stage, the parameter extraction is accomplished according to the
magnetic-electrical equivalent circuit shown in Figure 1. The resistance and the inductance
fo
r the
windings are calculated using a JAVA script based on equations (3) and (4)
introduced in Asensi et al. (2007) and modified for three dimensions:
L
ij
¼
1
I
2
0
∰
V
Re B
!
i0
H *
!
j0
dv (3)
R
ij
¼
1
I
2
0
∰
V
Re J
!
i0
J *
!
j0
dv (4)
where I
o
is the winding current (rms value), B
!
is the magnetic flux density, H*
!
is the
complex conjugate value of the magnetic field strength, and J
!
is the current density, J*
!
is
the complex conjugate of the current density. The indexes correspond to the mutual
coefficients for the electrical parameters when there are more than two windings in the
magnetic component.
For a wide supply frequency range, the current density distribution within the wire is
affected by alternating internal and external electromagnetic fields. These phenomena are
usually presented as a skin and proximity effect and have different behavior for circular and
polygonal cross-section. Therefore, a correction factor k must be applied to the extracted
resistance (R
FEM
ij
) to get a proper resistance value. In this case the inductance is practically
not influenced by the polygonal model and therefore the correction factor is not necessary
(Zhang et al., 2017; Di
as et al., 2018; Igarashi, 2017; Christian Bednarz et al.,2018).
2.2 Correction factor depending on the frequency
The corre
ction factor [equation (6)] depen
ding on the frequency is applied to the extracted
resistance [equation (4)] introduced for the Polygonal Model:
R
real
ij
¼ k R
FEM
ij
(5)
where R
real
is the resistance for the original model (circular cross-section) and R
FEM
is the
resistance obtained from the Polygonal Model using equation (4). The cor
rection factor is
divided in two coefficients corresponding to the skin and proximity effects that depend on
the frequency. According to Ferreira (1994), the skin and the proximity effect can be
calculated separately due to the orthogonal relation existing between them:
k ¼ k
skin
þ k
proximity
(6)
A large number of analysis have been performed to calculate the correction factor. It is
based on the ratio of the resistance calculated for the real component and the resistance
calculated for the Polygonal Model (7):
EQ
R
¼
R
cirle
R
polygon
(7)
The correction factor also depends on the polygonal cross-section used in the simulation. A
new term is introduced in the study to involve the polygonal cross-section dependency
labeled A
pol
and given in equation (8) (Figure 3):
Figure 3.
Modified
conductivity
