Nghiên cứu đặc tính điện từ trường đan xen trong hệ thống dây dẫn nhiều sợi bằng phương pháp phần tử hữu hạn

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(ISSN: 1859 - 4557) 
60 Số 13 tháng 11-2017 
STUDY OF ELECTROMAGNETIC BEHAVIOR IN MULTICONDUCTOR 
SYSTEM BY FINITE ELEMENT METHOD 
NGHIÊN CỨU ĐẶC TÍNH ĐIỆN TỪ TRƢỜNG 
ĐAN XEN TRONG HỆ THỐNG DÂY DẪN NHIỀU SỢI 
BẰNG PHƢƠNG PHÁP PHẦN TỬ HỮU HẠN 
Nguyen Duc Quang 
Electric Power University 
Abstract: 
This paper involves modeling and calculating the mutual electromagnetic characteristics in a 
multiconductor system using finite element method and equivalent energy equations. The approach 
is applied on a real three phase shielded cable. The finite element model of the cable is presented 
for calculating the mutual parameters which depend on the frequency. The high frequency 
phenomenas, the skin and proximity effect, are well studied. 
Keywords: 
Multiconductor, electromagnetic field, magnetodynamic, Maxwell’s equations, finite element method. 
Tóm tắt: 
Bài báo đề cập đến việc nghiên cứu các đặc tính điện từ trường đan xen trong một hệ thống đa dây 
dẫn bằng phương pháp phần tử hữu hạn kết hợp việc giải các phương trình năng lượng. Phương 
pháp nghiên cứu được trình bày chi tiết và áp dụng tính toán chi tiết một hệ thống đa dây dẫn cụ 
thể - cáp ba pha có đai bảo vệ. Các giá trị tương hỗ giữa các dây dẫn, cũng như các hiện tượng xuất 
hiện ở tần số cao như hiệu ứng bề mặt và hiệu ứng gần được xác định rõ nét. 
Từ khóa: 
Hệ thống đa dây dẫn, điện từ trường, điện động, hệ phương trình Maxwell, phương pháp phần tử 
hữu hạn. 
1. INTRODUCTION7 
Multiconductor systems are frequently 
used in energy transmission such as 
overhead lines and cables. The mutual 
electromagnetic effect is extremely 
7 Ngày nhận bài: 28/8/2017, ngày chấp nhận 
đăng: 20/9/2017, phản biện: TS. Trần Thanh 
Sơn. 
varied. The propagation of 
electromagnetic waves in transmission 
lines could be described by the Transverse 
Electromagnetic (TEM) mode. The terms 
of voltages and currents are calculated by 
using the circuit parameters of line. 
Moreover, the switching of 
semiconductor devices in power static 
converters can generate the 
TẠP CHÍ KHOA HỌC VÀ CÔNG NGHỆ NĂNG LƯỢNG - TRƯỜNG ĐẠI HỌC ĐIỆN LỰC 
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Số 13 tháng 11-2017 61 
Electromagnetic Interference (EMI). In 
power system, this high level of emission 
can produce the high frequency 
disturbance which propagate over the 
power cables [1,2]. In order to analyze the 
influence of transmission multiconductor 
system on the EMI level, it is necessary to 
precisely model the behavior of this 
system in the frequency domain. 
However, there are some difficulties in 
modeling of system due to several factors 
[3,5]. Firstly, the properties of materials, 
thicknesses of insulation and shielding are 
not fully known. Secondly, electrical 
wires and frame are twisted, sometimes in 
opposite sense. These physical parameters 
are insufficient to model a multiconductor 
system in the frequency domain; 
therefore, it is necessary to take into 
account the electromagnetic phenomena 
such as the skin effect and proximity 
effects [1,2,4]. To correct model, both of 
these effects are highly dependent on the 
characteristics of the materials and on the 
geometry; thus, the finite element method 
is proposed to use [4,7]. The number of 
simulations by finite element method will 
vary according to the number of 
conductors in the multiconductor system. 
Each simulation will provide an energy 
value that will allow us to determine the 
lumped parameter (resistance and 
inductance) matrices. Moreover, these 
simulations will be performed for several 
frequencies to capture the evolution of the 
skin and proximity effects. 
2. METHODOLOGY 
In this section, the electromagnetic 
formulations used to calculate the lumped 
parameters are introduced. Based on 
energy method, the seft and mutual values 
are obtained from the finite element 
model [8]. 
2.1. Finite Element Method and 
Formulations 
Finite Element Method 
The finite element method (FEM) is a 
technique for the numerical resolution of 
partial differential equations. This method 
is powerful, general, robust and widely 
used in engineering. 
Figure 1. Decompostion of a studied object 
to finite elements 
In reality, the FEM solves the weak form 
of the partial differential equations by 
using a mesh which serves as support for 
the interpolation functions. 
The weak formulation is also called 
variational formulation. This formulation 
can be defined by considering a 
differential operator R and a function f 
such as finding u on Ω checking 

 fvvuR
for any adapted function v .
The distribution of electric field and 
magnetic field is described by Maxwell’s 
equations. The studied object can be 
discretized by the nodes, the edges, the 
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62 Số 13 tháng 11-2017 
facets and the volumes. 
Solving the final electromagnetic 
equations in a complex object, such as a 
multiconductor system, is extremely 
difficult. Therefore, the author used the 
finite element method and solved the 
problem by using its numerical tool as 
Salome software [6]. This is a software 
which provides a generic platform for 
numerical simulation. It is based on an 
open and flexible architecture made of 
reusable components. Salome can be used 
as standalone application for generation 
of Computer-aided design (CAD) model, 
its preparation for numerical calculations 
and post-processing of the calculation 
results. Salome can also be used as a 
platform for integration of the external 
third-party numerical codes to produce a 
new application for the full life-cycle 
management of CAD models. 
In this study, the value of the capacitance 
matrix is supposed not to be frequency 
dependent and not to be examined. 
However, for the resistance and 
inductance matrices which vary with the 
frequency, the two magnetodynamic 
potential formulations are used [9,10]. 
Magnetodynamic problem 
As mentioned above, the purpose is to 
determine the resistance and inductance 
matrices which depend on the skin and 
proximity effects. These resistance and 
inductance matrices are calculated in 
function of the frequency by solving the 
magnetodynamic formulations. 
The magnetic vector potential A and the 
electric scalar potential j are identified 
such that the magnetic field B and vector 
A are related by B=curlA and the electric 
field E is equal to E=jA-gradj. 
Combining the previous equations with 
the Ampere’s law (curlH = J, H as the 
magnetic field and J as the current 
density) and with the behavior laws 
(B=H and J=E with  as the 
permeability and  as the conductivity), 
the partial differential equation to be 
solved is: 
1
( )j  j

 Scurl curlA J A grad (1) 
The boundary conditions indicated on B 
(B.n=0) and E (E×n=0) are imposed on 
the application of A×n=0 on ΓB and 
A×n=0 and j=0 on ΓE respectively. 
There is another potential formulation. 
The electric vector potential formulation 
T and the magnetic scalar potential 
formulation Ω are introduced such that: 
= + = +S ind SJ J J curlT curlT (2) 
Where the source term JS=curlTS and the 
unknown term Jind=curlT. Consequently 
the equation to solve is given by a 
conductive part 
1
j

 S Scurl curlT curlT T T grad
(3)
The boundary conditions of type J and H 
on the boundary ΓH by imposing T×n=0 
and Ω =0 on ΓH. The main purpose when 
solving both formulations is to obtain two 
values of lumped parameters, one for each 
formulation. 
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2.2. Determination of impedance 
matrices 
Based on the calculation of the energy, 
Joule losses and magnetic energy, the 
values of R and L matrices can be found. 
In general, if the conductors are flown by 
an electric current, the Joule losses and 
the magnetic energy are expressed as 
follows : 
2
1 , 1;
2
1 , 1;
1
2
n n
Joules i ii i j ij
i i j i j
n n
mag i ii i j ij
i i j i j
P I R I I R
W I L I I L
 
 
 (4) 
where Rii, Lii are respectively the self 
resistance and inductance of conductor i; 
and Rij, Lij are the mutual reristance and 
inductance between conductor i and 
conductor j. 
To take into account the evolution of the 
resistance according to the skin and 
proximity effect, the simulations must be 
carried out at several frequency values. It 
should be noted that self resistance values 
corresponds to Joule losses in the three 
conductors when only one is supplied. 
For example, a simple two-conductor 
system can be seen as below: 
R12
L12
u1
R11
L11
R22
L22
C10
C20
u2
i1
i2
C12
Figure 2. The mutual relationship 
in the two conductor system 
In the magnetodynamic problem, the 
relaitonship of resistance and inductance 
between the condutors can be defined as 
follows : 
11 12 11 12
21 22 21 22
;
R R L L
R L
R R L L
 (5) 
where R11, L11, R22, L22 are respectively 
the self resistance and inductance of 
conductor 1 and conductor 2; and R12, L12 
(or R21, L21) are the mutual resistance and 
inductance between conductor 1 and 
conductor 2. R12 represents the effect of 
proximity of conductor 1 to conductor 2 
and L12 is the mutual inductance between 
these two conductors. 
The energy equations (4) in this case 
become: 
2 2
1 11 2 22 1 2 12
2 2
1 11 2 22 1 2 12
2
1 1
2 2
Joules
mag
P I R I R I I R
W I L I L I I L
 (6) 
The approach principle is the variation of 
input currents in FEM model to calculate 
the energy equations. The findings of 
PJoules and Wmag values are based on this 
FEM model. 
Thus, in order to calculate the resistance 
and inductance of conductor 1 (R11 and 
L11), the established FEM model is 
applied to the currents on two conductors 
(I1, I2) as (1,0) A. Based on PJoules and 
Wmag of FEM model, the energy equations 
(6) are calculated to obtain the resistance 
and inductance of conductor 1. In order to 
get the mutual values (R12, L12), the two 
applied currents have to be different and 
non-zero. 
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3. CASE STUDY 
3.1. Geometry and parameters 
This cable has three cores, and each 
conductive core consists of 61 non-
insulated copper wires. Each core is also 
surrounded by a semi-conductive tape and 
a XLPE insulation, and then there is the 
jam, the sealing sleeve, the armature as 
well as the outer sheath as being shown in 
Figure 3. 
Figure 3. Configuration of the cable 
Table 1. Parameters of the cable 
As a part of the study, all of the copper 
strands are assimilated to a uniform 
section. This assumption is valid as far as 
the strands are not insulated from each 
other and are wrapped by an insulating 
sheath which contributes to increasing the 
contact areas. Each conductor is 
surrounded by semiconductor layers. As a 
part of this work, these semiconductor 
layers are consided playing a role of 
insulation. Therefore, they are modeled in 
the case of magnetodynamic model by a 
non-conductive and non-magnetic 
material in the case of electrostatic model 
by a dielectric material r = 2,4 
corresponding to the XLPE insulation 
surrounding the conductor. The studied 
system is simulated for 1 m of length. 
3.2. Mesh 
Since there are three conductors and 
conductive armature (Figure 4), the linear 
parameters to be determined will be 
expressed in a matrix form of around 4*4. 
Figure 4. Representation of conductive parts 
in modeling cable 
Figure 5. Index of matrix [R] and [L] 
The numbering of the various conductors 
is given in Figure 5, and in Figure 6, the 
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Số 13 tháng 11-2017 65 
equivalent circuit of this cable is 
represented by the coefficients of the 
matrices [R] and [L]. 
Instead, the magnetodynamic model will 
present well the distribution of induced 
currents of cable. The calculations are 
carried out with the amor connection 
condition as in Figure 6. 
This figure shows the equivalent circuit of 
the multiconductor system. The values 
R11, R22, R33, Ra and L11, L22, L33, La are 
respectively the resistance and inductance 
of conductor 1, 2, 3 and of the amor. The 
values R12, R13, R23, R1a, R2a, R3a are 
respectively the mutual resistance 
between the conductors as well as 
between a conductor with the amor of 
cable. The rule of inductance is similar. 
Figure 6. Equivalent circuit of studied cable 
The Figure 6 (bottom) represents the 
equivalent circuit in case of forward 
current in conductor 1 and back current in 
two conductors 2 and 3. In this case, the 
amor of cable is open circuit (ia = 0). 
3.3. Solution and Results 
The distribution of the induced current in 
cable at f = 1 kHz is well presented in the 
Figure 7. 
Figure 7. Density of current in cable (A/m
2
) 
This result is corresponding to the case of 
the current flowing through the conductor 
1. The distribution of current in conductor 
1 is according to the skin effect. At the 
same time, the induced currents are 
produced in the conductors 2 and 3. This 
is the proximity effect that occurs at high 
frequency in the multiconductor system. 
Figure 8. Density of current in amor (A/m
2
) 
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66 Số 13 tháng 11-2017 
This proximity effect also appears on the 
amor of cable. The Figure 8 shows that 
the induced current is maximum at 
position near conductor 1 and minimum 
corresponding to the farthest distance 
from the conductors. 
Therefore, the high frequency 
phenomenas like the skin effect and the 
proximity effect in conductors and also in 
amor of cable are clearly demonstrated. 
Solving the problem magnetodynamic by 
finite element method, value of Joules 
losses and magnetic energy are obtained 
according to frequency. 
The equations (4) in this case become as 
follow: 
4 4
2
1 , 1;
4 4
2
1 , 1;
1
2
Joules i ii i j ij
i i j i j
mag i ii i j ij
i i j i j
P I R I I R
W I L I I L
 
 
 (7) 
Solving equations (7) by obtained energy, 
values of resistance and inductance 
depend on frequency. The parameters 
variation of conductors (R11, L11) and 
shield (R44, L44) as well as mutual values 
between two conductors (R12, L12) and 
between a conductor and shield (R14, L14) 
are calculated and shown in Figure 9 and 
in Figure 10. 
The value of resistance increases and of 
inductance decreases with the frequency 
of source. In the frequency range [0; 
100kHz], the calculated resistance is 
relatively small. It is explained by a good 
conductivity material of this study cable. 
As the frequency increases, due to skin 
and proximity effect, the resistance value 
increases and the inductance decreases. 
The difference of result obtained by A-j 
and T-Ω formulations is also evident at 
high frequency. 
Figure 9. Evolution of resistances depends on frequency 
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Figure 10. Evolution of inductances depends on frequency 
The self values of resistance (Rii) and the 
inductance (Lii) are always greater than 
the mutual value between the conductors 
(Rij, Lij). However, this mutual value 
cannot be ignored and that is the 
electromagnetic interference effect in 
multiconductor system. It is perfectly 
consistent with the theory when the skin 
effect and the proximity effect appear to 
produce the induced current in the 
conductors of system. 
4. CONCLUSION 
This paper presents a method which is 
applied to determine the impedances of a 
multiconductor system according to the 
frequency. The results can assert greatly 
the phenomena HF of the cable: the skin 
and proximity effect. There are two major 
advantages when conducting this method. 
Firstly, the modeling of proximity effect 
in high frequency is carried out 
successfully. Secondly, another benefit is 
the introduction of the connection matrix. 
The impedance of other configuration of 
system according to frequency can be 
determined by changing this matrix. This 
method can be applied in calculation, 
planning and operation of cable and 
distribution network. Furthermore, this 
approach will be help for the next study to 
determine the resonant frequency of the 
transmission system. 
REFERENCES 
[1] Y. Weens, N. Idir, R. Bausiere and J. J. Franchaud, “Modeling and simulation of unshielded 
and shielded energy cables in frequency and time domains”, IEEE Transactions on Magnetics, 
Volume: 42, Issue: 7, p. 1876 - 1882, 2006 
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[2] H. De Gersem, A. Muetze, “Finite-Element supported transmission line models for calculating 
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htttp://www.salome-platform.org 
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Multiconductor Transmission Lines Excited by an Electromagnetic Field”, IEEE Transactions on 
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Biography: 
 Nguyen Duc Quang received his Engineer diploma degree from the Hanoi 
University of Science and Technology, Vietnam in 2007; M.S degree from the 
Lille 1 University, France, in 2009 and Ph.D. degree from the Ecole Nationale 
Superieure d’Arts et Metiers Paristech, France, in 2013. All were in electrical 
engineering. He is currently Lecturer of the department of Electrical 
Engineering, at the Electric Power University, Vietnam. His research interests 
are in the fields: numerical modeling methods, electromagnetic field, 
electrical machines and renewable energy.