Post on 16-Aug-2020
Academy Journal of Science and Engineering 12 (1), 2018 Page 115 - 128
dsnyitamen@nda.edu.ng, simonagbendeh@gmail.com website: www.academyjsekad.edu.ng
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MODIFIED TRANSMISSION LINE MODEL EQUATIONS FOR
MICROSTRIP ANTENNA DESIGN
Nyitamen, Dominic S and Agbendeh, Simon T
Electrical Electronic Engineering Department, Nigerian Defence Academy, Kaduna
Abstract
Microstrip antennas are becoming more and more popular as the drive towards
miniaturization continues. Therefore, the need for accurate models is growing. Not
only accuracy is required, but also numerical efficiency, in order that the models are
suited for computer-aided-design (CAD) procedures involving optimization. The
transmission line model equations are used in the design of microstrip antennas.
However, the level of accuracy can be low and several iterations required in
optimization for better results. A modification of the dimensions obtained from TLM
method is modified before CAD application for higher accuracy. In this work the
transmission line model equations were modified by a factor and simulated.
Simulations were carried with the FDTD method of CST microwave Studio. Four
different substrates, namely FR-4 Rogers 5870 Roger RT 6010 Roger RT 5880 with
same thickness (1.6mm) but different dielectrics were tested experimentally with the
modified equations and compared with the conventional transmission line model at
1.75GHz. The three antenna parameters used for the studies were the return loss, S11,
the voltage standing wave ration VSWR and the radiation patterns. For the four
substrates used, the modification revealed improvements in the parameters of interest,
namely, the return loss, voltage standing wave ratio and radiation patterns. The
deviation in frequency (1.575GHz) was much less, hence better results.
Key words: antenna, microstrip patch, CAD, return loss, radiation pattern
1. Introduction
Various intricate techniques have been
proposed and used to analyze
microstrip antenna characteristics. The
analytical techniques include the
transmission line model, generalized
transmission line model, cavity model,
and multiport network model. The
Summerfield-type integral equations,
and the solutions of Maxwell’s
equations in the time domain form the
basis of microstrip antenna designs.
Numerical methods of analysis include
integral equation in the space domain,
or the finite-difference time-domain
(FDTD) approach. The methods based
on integral equation make one
important assumption: the dielectric
substrate and the ground plane are
infinite in extent. The solutions are
therefore more accurate when the
substrate and ground plane are several
wavelengths long [1,2]. The FDTD
technique is more efficient for finite-
sized antennas. The effect of the finite
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
This work is Licensed under a Creative Commons Attribution 4.0 International License (CC BY)
size is less severe on impedance
behavior because microstrip antennas
are inherently resonant structures and
their impedance characteristics are
primarily determined by the patch. A
more accurate and numerically
efficient Transmission Line method
proposed microstrip patch antenna.
The radiation behavior, on the other
hand, is considerably influenced by the
finite size of the substrate primarily
due to launching of the surface waves
and their diffraction at the edge of the
substrate [3]. Consequently, the theory
of diffraction is occasionally used in
conjunction with other methods to
improve the predication of the
radiation pattern. The analytical
models were the first to be developed
for microstrip antennas [4]. They use
simplifying assumptions, but generally
offer simple and analytical solutions,
well suited for an understanding of the
physical phenomena and for antenna
CAD, in the analytical methods or
models, the fields associated with the
antenna are divided into an interior
region and an exterior region [5]. The
interior region is formed by the patch
conductor, the portion of the ground
plane under the patch, and the walls
formed by the projection of the patch
periphery onto the ground plane [6].
The fields in this region can be
modeled as a transmission line section
or a cavity giving rise to the
designations transmission line model
and cavity model. The exterior region
is the rest of the space. This includes
the remainder of the ground plane, the
remainder of the dielectric, and the top
of the patch conducting surface.
2. Conventional Transmission
Line Model
The transmission line model was the
first technique used to analyze a
rectangular microstrip antenna by
Munson in 1974 [7]. In this model, the
interior region of the patch antenna is
modeled as the section of transmission
line. It was indicated earlier that the
transmission-line model is the easiest
of all but it yields the least accurate
results and it lacks the versatility.
However, it does shed some physical
insight. Basically the transmission-line
model represents the microstrip
antenna by two slots, separated by a
low-impedance Zc transmission line of
length L.
a. Effective Dielectric
The dimensions of the patch are finite
along the length and width, the fields
at the edges of the patch undergo
fringing.The fringing and height are
calculated using the same equations
used for a matched transmission line.
This model assumes that some electric
field lines will pass out of the dielectric
and into empty space [8]. Because of
this, the permittivity will not strictly be
the relative permittivity of the material
it will instead have an effective
permittivity (𝜀𝑟𝑒𝑓𝑓). The effective
permittivity or reference permittivity is
affected by the width of the patch and
the height of the substrate. If it can be
assumed that the width is greater than
the height of the patch, then the
reference permittivity can be described
by equation (1) [8].
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𝜀𝑟𝑒𝑓𝑓 =𝜀𝑟+1
2+
𝜀𝑟−1
2[1 + 12
ℎ
𝑊]
−1
2 1
b. Effective Length
The length of the patch looks
electrically slightly larger than the
usual length of the design, because of
the fringing field along the patch
width, the dimensions of the patch
along its length have been extended on
each end by a distance Δ𝐿, which is a
function of the effective dielectric
constant 𝜀𝑟𝑒𝑓𝑓and the width-to-height
ratio (W/h). A very popular and
practical approximate relation for the
normalized extension of the length is
[4]. this parameter can be calculated by
using Equation 2:
∆𝐿 = 0.412ℎ(𝜀𝑟𝑒𝑓𝑓+0.3)(
𝑊
ℎ+0.264)
(𝜀𝑟𝑒𝑓𝑓−0.258)(𝑊
ℎ+0.8)
2
After the calculation of each of
effective and extended lengths of the
patch, the actual value of the patch
length (L) is calculated by using
Equation 3:
𝐿 = 𝐿𝑒𝑓𝑓 − 2∆𝐿 3
c. Effective Width
For an efficient radiator, a practical
width that leads to good radiation
efficiencies is given by the following
formula equation 4:
𝑊 =𝑐
2𝑓𝑟√
2
𝜀𝑟+1 4
d. Resonant Frequency
For the dominant TM010 mode, the
resonant frequency of the microstrip
antenna is a function of its length.
Usually it is given by equation 5:
𝑓 =𝑐
2𝐿√𝜀𝑟 5
3. Materials and Methodology
The microstrip antenna is made up of a
dielectric substrate and conducting
patches. The choice of substrates and
their dimensions are critical factors at
the desired frequency of operation.
Through numerical computations and
simulation, certain factor was obtained
to modify the conventional equations.
The FDTD method of CST microwave
Studio is used for simulations. Four
different substrates were chosen with
same thickness for experimentation at
a fixed frequency of 1.575GHz used by
GPS. The four variants of the
microstrip antenna designs were
carried out using the conventional
TLM and tested for results. The same
designs were modified by introducing
a numerical factor, k and the tested for
results. These results were then
compared for the four cases
(substrates) against return loss, voltage
standing wave ratio and antenna
patterns. The factor k is obtained
numerically through series of
simulations.
Modified Equations for the microstrip
antenna dimensions
Page 117
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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A half wavelength variation in surface
current is observed during simulation
along the patch boundary (L + W).
[𝐿+𝑊]
𝐾=
𝜆𝑔
2 6
where k is an empirically derived
parameter that includes the effect of
the substrate, and 𝜆𝑔 is the wavelength
in the dielectric that is computed from
the free space wavelength 𝜆0 as:
𝜆𝑔 =𝜆0
√𝜀𝑒𝑓𝑓 7
and 𝜀𝑒𝑓𝑓 is the effective permittivity of
the substrate.
Based on this, design equations are
derived relating to the geometry and
operating frequency band of the
proposed antenna. The design
procedure can be framed as follows:
For a design involving a 50Ω
transmission line on a substrate with
permittivity 𝜀𝑟 and thickness h. The
calculation of wavelength in the
dielectric, 𝜆𝑔 is carried out using
Equation 7.
The design dimensions of the patch are
found as
𝐿 = 𝑊 = 0.47𝜆𝑔 and
The design the dimensions of the
ground using
𝐿𝑔 = 𝑊𝑔 = 0.64𝜆𝑔
For any given specification of the
patch antenna, the patch and the
ground plane dimensions are modified
by k, which is given in equation 6.
This modification is verified
experimentally.
4. Experimental Results and
Discussions
The numerical value of was found and
used to modify the design before
simulation to compare with the direct
TLM design. The validity of the
modified equations proposed is
highlighted by comparing the results of
the reflection coefficient, the VSWR
and radiation pattern. Simulations were
performed using conventional
transmission line model (TLM) and
modified equations for four substrate
materials at the GPS resonate
frequency of 1.575 GHz. The results
were compared. Figure 1 shows a
rectangular microstrip antenna fed by a
50Ωcoaxial probe. The following
parameters in Table 1 were used in
evaluating the proposed factor
introduced.
Page 118
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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Figure 1. Rectangular Microstrip antenna fed by 50 ohms coaxial probe.
Table 1: Antenna Description at frequency of 1.575 GHz
Antenna 1 Antenna 2 Antenna 3 Antenna 4
Substrate FR-4 Rogers 5870 Roger RT 6010 Roger RT 5880
h (mm) 1.6 1.6 1.6 1.6
𝜀𝑟 4.4 2.32 10.2 2.2
𝜀𝑒𝑓𝑓 4.08 2.25 9.52 2.13
𝑥𝑓 6.85 9.28 4.47 9.52
k 4.08 2.25 9.52 2.13
Antenna 1 using FR4 at the Resonate
Frequency of 1.575 GHz
Figure 2 is the return loss plot of
Antenna 1 whose parameters are
shown in Table 1, using FR4 substrate
at the resonate frequency of 1.575
GHz. The plot with marker 1
represents the simulation of the
antenna using the conventional TLM,
while marker 2 represents the
simulation of the antenna using the
modified equations. The modified
equations gave better results as
compared with the conventional TLM.
Page 119
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Academy Journal of Science and Engineering 12 (1), 2018
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Figure 2 The Return Loss of Antenna 1 of the conventional TLM and the Modified
Equations
The simulated VSWR of Antenna 1
using the conventional TLM and
modified equations is as shown in
Figure 3. The modified equations
“marker 2” gave better results close to
the desired frequency as compared
with the conventional TLM “marker
1”.
Figure 3 The VSWR of Antenna 1 of the conventional TLM and the Modified
Equations
Figure 4 is the plot of the simulated
radiation pattern of Antenna 1. As seen
from the plot, the modified equations
gave wider angle of the main lobe as
compared with the conventional TLM.
Page 120
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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Figure 4 The Radiation Pattern of Antenna 1 of the conventional TLM and the
Modified Equations
Antenna 2 using Rogers 5870 at the
Resonate Frequency of 1.575 GHz
Figure 5 is a return loss plot of
Antenna 2 using Rogers RT 5870
substrate (Table 1) at the resonate
frequency of 1.575GHz. The plot with
marker 1 represents the simulation of
the antenna using the conventional
TLM, while marker 2 represents the
simulation of the antenna using the
modified equations. It is clear that the
modified equations gave better results
as compared with the conventional
TLM
Figure 5 The Return Loss of Antenna 2 of the conventional TLM and the Modified
Equations
Page 121
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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The simulated VSWR of Antenna 2
using the conventional TLM and
modified equations is as shown in
Figure 6. The modified equations
“marker 2” gave a VSWR of 1.03 at a
frequency of 1.582 GHz as compared
with the conventional TLM “marker 1”
with VSWR of 1.05 at 1.526 GHz.
Figure 6 The VSWR of Antenna 2 of the conventional TLM and the Modified
Equations
Figure 7 below is the simulated
radiation pattern plot of Antenna 2. As
seen from the plot, the modified
equations wider angle of the main lobe
and smaller back lobe as compared
with the conventional TLM
Figure 7 The Radiation Pattern of Antenna 2 of the conventional TLM and the
Modified Equations
Page 122
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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Antenna 3 using Rogers RT 6010 at
the Resonate Frequency of 1.575 GHz
Figure 8 presents the return loss plot of
Antenna 3 using Rogers RT 6010
substrate at the resonate frequency of
1.575 GHz. The plot with marker 1
represents the simulation of the
antenna using the conventional TLM,
while marker 2 represents the
simulation of the antenna using the
modified equations. It is clear that the
modified equations gave better results
as compared with the conventional
TLM
Figure 8 The Return Loss of Antenna 3 of the conventional TLM and the Modified
Equations
The simulated VSWR of Antenna 3
using the conventional TLM and
modified equations is as presented in
Figure 9. The modified equations
“marker 2” gave better results close to
the desired frequency as compared
with the conventional TLM “marker
1”.
Figure 9 The VSWR of Antenna 3 of the conventional TLM and the Modified
Equations
Page 123
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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Figure 10 shows the simulated
radiation pattern plot of Antenna 2 and
from the plot, the modified equations
gave a slight narrow angle of the main
lobe and back lobe as compared with
the conventional TLM, this is as a
result of the effect of a high dielectric
constant of the substrate material.
Figure 10 The Radiation Pattern of Antenna 3 of the conventional TLM and the
Modified Equations
Antenna 4 using Rogers RT 5880 at
the Resonate Frequency of 1.575 GHz
Figure11 presents the return loss plot
of Antenna 4 using Rogers RT 5880
substrate at the resonate frequency of
1.575 GHz. The plot with marker 1
represents the simulation of the
antenna using the conventional TLM,
while marker 2 represents the
simulation of the antenna using the
modified equations. It is clear that the
modified equations gave better results
as compared with the conventional
TLM
Figure 11 The Return Loss of Antenna 4 of the conventional TLM and the Modified
Equations
Page 124
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
This work is Licensed under a Creative Commons Attribution 4.0 International License (CC BY)
The simulated VSWR of Antenna 4
using the conventional TLM and
modified equations is as presented in
Figure 12. The modified equations
“marker 2” gave better results close to
the desired frequency as compared
with the conventional TLM “marker
1”.
Figure 12 The VSWR of Antenna 4 of the conventional TLM and the Modified
Equations
Figure 13 is the simulated radiation
pattern plot of Antenna 2, and from the
plot, the modified equations provide
wider angle of the main lobe and
smaller back lobe as compared with
the conventional TLM
.
Figure 13 The Radiation Pattern of Antenna 4 of the conventional TLM and the
Modified Equations
Page 125
Modified Transmission Line Model……… Dominic S,Agbendeh, Simon T
Academy Journal of Science and Engineering 12 (1), 2018
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5. Conclusion
In this paper, a modified and
computation-efficient transmission line
model is developed to analyze the
microstrip antenna. The results so far
show that the modified equations can
be successfully used to design the
microstrip antenna and even though the
model is conceptually simple, it still
produces better results in a relatively
short period of computing time. The
results obtained highlight an excellent
agreement between the modified
equations and the FDTD method of
CST microwave Studio. A comparison
of the results produced by the final
model with the FDTD data showed the
validity of the proposed model. This
allows the analysis of different
substrate materials and frequencies for
linear polarization. Based on these
characteristics, the modified equations
can be useful for microstrip antenna
designs.
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