The method of electromotive force (emf) for calculation of antenna impedance is applicable to a very limited number of antenna configurations such as monopole and dipole antenna. In more general antenna structures, especially when complicated geometry such as fractals is involved numerical techniques are essential. There are Electromagnetic (EM) simulation softwares such as G-NEC, XFDTD for the modelling studies in which is essential an understanding of the fundamental theory behind these numerical techniques in creating the antenna models. In this paper, we compare the radiation from dipole antenna and the antenna with shape of the first iteration of Cantor set. The simulation of this problem has been done by COMSOL Multi-physics software

Azhini, Mehdi

Delfaan, Nasibeh

Pishkoo, Amir

English

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To Physics Journal Vol 4 (2019) ISSN: 2581-7396 http://purkh.com/index.php/tophy
Using COMSOL Multiphysics To Simulate Radiation from
Dipole Antenna and First Iteration Cantor Set Shape
Antenna
Nasibeh Delfan1, Amir Pishkoo2, Mahdi Azhini1
1Department of Mathematics, Science and Research branch, Islamic Azad University, Tehran, Iran
2Physics and Accelerators Research School, Nuclear Science and Technology Research Institute,P.O. Box 14395-836,
Tehran, Iran
apishkoo@gmail.com
Abstract
The method of electromotive force (emf) for calculation of antenna impedance is applicable to a very limited number
of antenna configurations such as monopole and dipole antenna. In more general antenna structures, especially when
complicated geometry such as fractals is involved numerical techniques are essential. There are Electromagnetic (EM)
simulation softwares such as G-NEC, XFDTD for the modelling studies in which an understanding of the fundamental
theory behind these numerical techniques in creating the antenna models is essential. In this paper, we compare the
radiation from dipole antenna and the antenna with shape of the first iteration of Cantor set. The simulation of this
problem has been done by COMSOL Multi-physics software.
Keywords: c.
1 Introduction
Antennas are essentially narrow-band devices. Their behaviour is highly dependent on the antenna size to the operat-
ing wavelength ratio. This means that for a fixed antenna size, the main antenna parameters (gain, input impedance,
pattern shape, secondary lobe level, and distribution) will suffer strong variations when changing the operating fre-
quency. The frequency dependence also implies that an antenna has to keep a minimum size relative to wavelength
to operate efficiently. That is, given a particular frequency, the antenna cannot be made arbitrarily small: it usually
has to keep a minimum size, typically on the order of a quarter wavelength [1]. The quest for smaller sized resonant
antennas has been on for decades. Several of antenna design principles may have to be modified while dealing with
small-sized antennas. By convention, a small antenna is defined as one occupying a fraction (typically < 16 ) of the
wavelength [2]. The primary concern in their design is in the impedance matching. This is better explained in the con-
text of dipole antennas. As the length of a dipole antenna is reduced, the real part of its input impedance approaches
zero, while the imaginary part tends to be an extremely large negative number. This causes major reflections at the
input terminal as the transmission line connected to it generally has a standard characteristic impedance (50). For a
thin cylindrical antenna of length 2l and radius a the current on the antenna is assumed to be [3]
I = Im sin(β(l − |z|), (1)
where β = 2piλ is wave number. It is also assumed that this current is distributed symmetrically around the axis of the
cylinder. It leads to a current distribution of
Jz =
I
2pia
. (2)
It can be shown that the electric field is given by [4].
Ez = −j ηI0
4pi
[2
e−jβr
r
cosβl − e
−jβR1
R1
− e
−jβR2
R2
], (3)
where r =
√
x2 + y2 + z2 , R1 =
√
x2 + y2 + (z − l)2 , R2 =
√
x2 + y2 + (z + l)2. It may be noted that when the
radius of the antenna approaches zero, the resonant length of the antenna may be obtained as 2l = λ2 .
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1.1 Fractal antenna
A fractal is a mathematical form showing self-repeating patterns [5, 6]. A fractal is a shape made of parts similar to
the whole in some way such as the following ”Cantor mass distribution” (see Fig. 1) and ”Cantor charge distribution”
(see Fig. 2) which represent the 1st, 2nd and 3rd iterations. The simplest way to define a fractal is as an object which
appears self-similar under varying degrees of magnification, and in effect, possessing symmetry across scale, with each
small part of the object replicating the structure of the whole. In fact, self-similarity can also be understood as the
property by which the fractal is found inside the fractal itself but at smaller scale [1]. The aim of this paper is to
compare the radiation from an ideal dipole antenna and the radiation from two dipole antennas which their size is 1/3
(the first iteration). Our simulation has been done in COMSOL Multi-physics software. Many electrical, mechanical,
fluid, and chemical simulations have been done by using COMSOL Multiphysics software. It’s name is ”Multi-physics”
because in this software all relevant physical effects can be included and coupled. Fortunately, all industries benefit
from Multiphysics simulations. COMSOL helps student to be familiar with a complete simulation environment. In
fact, virtual testing is much faster than testing physical prototypes. ”Electromagnetic waves, frequency Domain
(emw) interface” is one of the many interfaces which can be seen and used in COMSOL. All problems are solved by
the numerical method called Finite Element Method (FEM) in branch of numerical analysis. FEM is widely used
as a numerical method for the solution of PDE problems. It divides the solution domain into simply shaped regions
called ”elements”. An approximate solution for the PDE can be developed for each of these elements. Then, the
total solution can be generated by linking together the individual solutions such that to be ensure continuity at the
inter-element boundaries. We deal with the system matrix which is merely a set of n simultaneous equations that can
be used to solve for the values of the dependent variable at the n nodes. The implementation of FEM usually follows a
standard step-by-step procedure that is started with ”discretization” and then ”element equations”. The finite element
method started with significant promise in the modelling of several mechanical applications related to aerospace and
civil engineering. This well-known software was chosen for our student because she could do mathematical modelling
and physical modelling together. One can setup his/her model, optimize design, and even share it as a simulation
application. This paper also helps one to be familiar with COMSOL’s environment.
Figure 1: Cantor mass distribution
The most simple and recognizable fractal is Cantor set which demonstrates the property of self-similarity. Its
fractal dimension is d = log 2log 3 = 0.63. It is used as the diffraction grating among notable applications in physics. The
mathematical language in fractal geometry is IFS and by combining geometric transformations such as translation,
rotation, and scaling, fractal structures can be formed [7, 8]. In the most comprehensive form, “fractal theory” has
been formed and used for fractured material, fluids, physics, chemistry, electronics and graphics. A fractal is a shape
made of parts similar to the whole in some way. In the past, antennas had simple form based on Euclidean geometry.
The use of fractal shaped geometries in antennas is one of many areas of science and engineering which has been
well-known. In fact, the use of fractal geometries has improved antenna features to varying extends. Depending on
antenna performance, fractal shaped geometries have been proved to result in multi-band characteristics and reduction
of antenna size. The space filling nature of some fractal shapes lead to significant reduction in antenna size. Recent
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Figure 2: Cantor charge distribution
efforts by several researchers around the world to combine fractal geometry with electromagnetic theory have led to a
plenty of new and innovative antenna designs. By virtue of its geometrical properties, a fractal structure can generate
multiple resonances in RF applications. Fractal antennas do not follow the Euclidean geometry design [1, 3, 9–15].
One can see the form of input impedance, its real part and the reactive component in [3]. It shows that by properly
choosing the antenna length, its reactance can be made zero. In this paper, the radiation from the dipole antenna and
the antenna with shape of the first iteration of the Cantor set are investigated.
2 Main Results
The dipole antenna is one of the most straightforward antenna configurations. It can be realized with two thin metallic
rods that have a sinusoidal voltage difference applied between them. The length of the rods is chosen such that they
are quarter wavelength elements at the operating frequency. Such an antenna has a well-known torus-like radiation
pattern.
In high-frequency systems, the wave length is comparable to the size of the system. Here λ0 = 4m and height of dipole
antenna is 1m. Using COMSOL Multiphysics software, in this paper one of the “Radio Frequency” interfaces is used
to compute the electric field in high-frequency system. This interface is called “Frequency Domain” which is used
to solve for time-harmonic electromagnetic field fractal distribution (Cantor set distribution). This type of physics
interface solves time-harmonic wave equation for electric field
−→
E . The governing equation is
1
µr
−→∇(×−→∇ ×−→E )− κ20(εr −
jσ
ω0
)
−→
E = 0, (4)
for the time-harmonic and eigen frequency problems. In the above equation µr, σ, and εr are relative permeability,
electrical conductivity and relative permittivity, respectively. The wave number of free space namely κ0 is defined as
κ0 =
ω
c , where c is the speed of light in vacuum.
Mathematical and physical modelling in COMSOL (Model builder) is performed in the following steps:
1. To define “parameters”
2. To draw “geometry”
3. To specify “materials”
4. To choose “interface(s)” and type of study
5. To choose “mesh” size
6. To compute “numerical calculations”
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7. To choose “results” and postprocessing
In “Model Builder”, air and copper have been chosen for materials. Under a “Geometry node” one can define and
create the geometry sequence for the mode component.
In COMSOL Multiphysics Finite Element Method (FEM) is used to do numerical calculations. As it is shown in
Figure 3: Far field norm for the dipole antenna (right) and the antenna with shape of first iteration Cantor set (left)
Fig. 3, we have used “Far Field” plot for its electromagnetic field for which f = 7.4948× 107Hz. In Fig. 3 (right hand
side) three dimensional radiation pattern of an ideal dipole can be seen. If middle third of an antenna is removed at
the first iteration, then two smaller dipole antennas are made. Their radiation pattern in Fig. 3 (left hand side) has
different graphical representation. Far fields are computed in an effort to plot the radiation pattern and gain of the
antenna. One of the variables in the RF Module is the far field norm which is demonstrated in Fig. 4. “Far Field”
Figure 4: Far field plotted for a selected number of angles on a circle
plots are used to plot the value of a global variable which is here far field of an electromagnetic field. This variable
is plotted for a selected number of angles on a circle (in 2D). An ideal dipole as an isotropic radiator radiates power
with unit gain uniformly in all directions (right hand side in Fig. 4). However, when middle third of an antenna
is removed some lobes are formed (left hand side in Fig. 4). Convergence plots use graphics to show how an error
estimate or time step evolves during the solution process for nonlinear, time-dependent, and parametric solvers. These
plots show an error estimate against the iteration number. Fig. 5 gives convergence plot which shows how an error
estimate evolves during the solution process. The numbers appear in the Log window. For our configuration, solution
time was 68s while number of degrees of freedom solved for 712192. In the “parameters” section we enter parameters
manually. Furthermore, definitions which in someway refer to domains in a geometry are under the component branch
(see Fig. 6). Under a “Geometry node” we define and create the geometry sequence for the model component which
includes one sphere and two cylinders (see right side in Fig. 7). In Model Builder the “Materials column” lists our
materials which includes “air” and “copper” (see left side in Fig. 7). When adding a physics interface, we choose
the “Electromagnetic waves, frequency Domain (emw) interface” (see left side in Fig. 8). When a new component is
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Figure 5: Convergence plot shows how an error estimate evolves during the solution process
Figure 6: To introduce “parameters” and “definitions” in Model Builder
Figure 7: Used materials (air and copper) and defined geometry in “Model Builder”
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Figure 8: The RF module and used Mesh in Finite Element Method (FEM)
inserted to the Model Builder, a meshing sequence is added by default in the node “Mesh1” (see right side in Fig. 6).
Finite Element Method (FEM) as other numerical methods are used to provide engineering analysis. It divides the
body into smaller pieces. A solution can be computed depending on the sequence implemented for our model. The
Figure 9: Obtained “Results” for the second configuration antenna (the first iteration of Cantor set)
“Results branch” in the COMSOL Multiphysics model tree contain tools for postprocessing and analyzing the results
from our simulations (see Fig. 9).
3 Conclusion
In this paper, the rule of forming Cantor set (for just the first iteration) is used to compare the radiation of single
ideal dipole and two smaller ideal dipoles made in the first iteration. The physical and mathematical modelling and
simulation has been done in COMSOL Multiphysics software (Model builder) which is very convenient for solving
many kinds of problems in engineering, physics, and mathematics has been explained and demonstrated by figures.
Further research in this field can be done for Cantor set or other type of fractals such as Koch curve, complex tree
fractals, etc. and for second, third , and upper iterations in future by applying COMSOL.
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4 Acknowledgments
The authors are grateful to the referees and the editor for valuable remarks which contributed to the improvement
of the paper. Miss Nasibeh Delfan is Ph.D. student in mathematics whose supervisor is Dr. Amir Pishkoo and her
advisor is Dr. Mahdi Azhini.
5 Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
References
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Using COMSOL Multiphysics To Simulate Radiation from Dipole Antenna and First Iteration Cantor Set Shape Antenna

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Using COMSOL Multiphysics To Simulate Radiation from Dipole Antenna and First Iteration Cantor Set Shape Antenna

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