1. Introduction
Micro-strip patch antennas (MSPAs) have gained popularity due to their ease of production,
low production costs, planar nature, lightweight, and easy integration into integrated
circuits at microwave frequencies. Nevertheless, disadvantages exist, such as low
efficiency, poor gain, narrow bandwidth, and low return loss. Other significant challenges
that researchers face are the deviation in the desired design output when theoretical
values of design variables are obtained using complex design equations, simulated,
and produce deviations in subsequent experimental outputs.
Adjusting the theoretical values of the design variables is necessary to achieve the
desired design objective. Usually, this is done by trial and error, in which the designer
varies the design variables. The trial-and-error method is only effective when a limited
number of design variables can be modified to achieve limited design goals. The action
becomes cumbersome and may be difficult or even impossible when multiple variables
(such as the patch's length and width and the feed line's width) need to be optimized
to achieve various goals (such as minimal return loss, increased bandwidth, and accurate
resonance frequency).
Genetic Algorithm (GAs) was introduced by John Holland in 1975 and have been used
to optimize design requirements in electromagnetic engineering [3,6, ], especially in MSPA design optimization. In general, when applied to design optimization,
a GA involves formulating a fitness function that includes one or more desired design
outputs. The evaluation of the fitness function of every single value (chromosome)
within a set of values (generation) for its fitness or cost against of a preset value
is iterative. The process involves the selection of the best individual based on the
cost of forming a set of new values, which is called a generation. The evaluation
continues until the desired output or the specified number of generations is reached
(whichever occurs earlier).
This study surveyed the fitness functions formed using antenna parameters, such as
return loss, bandwidth, resonant frequency, and VSWR. These parameters were used individually
or in combination to optimize MSPAs for varying output requirements. A novel fitness
function formulation is proposed using graded penalties for deviations from the desired
operating frequency and required bandwidths of 5 GHz and 300 MHz, respectively, while
improving the return loss. The results compared with those of previous studies for
validation.
2. Design of MPSA
The "Transmission Line model" was used to model an MPSA. This model represents the
MPSA as equivalent to a conductor with width W, heighth, and length L. ${\lambda}$$_{0}$is
the wavelength in free space, and h is limited to the range of 0.333${\lambda}$$_{0}$
${\leq}$ h${\leq}$ 0.5${\lambda}$$_{0}$. The equations governing micro-strip patch
antenna design are summarized as follows:
Substrate selection
The power loss due to the dielectric [24] depends on the loss tangent. The dielectric constant usually lies in the range of
2.2 ${\leq}$ Ԑ$_{\mathrm{r}}$ ${\leq}$ 12. The substrate used here is RT-Duroid with
Ԑ$_{\mathrm{r}}$ =2.2. Its lowloss tangent reduces dielectric losses [8].
Designing dimensions of the patch
Standard equations calculate the dimensions of the MSPA, and the performance of the
patch antenna significantly depends on these dimensions. The following design equations
provide the theoretical values of the dimensions of the patch according to the requirements
[1,2,5,9].
(i) Width of the patch (W)
c = velocity of light in free space, f$_{\mathrm{r}}$= resonance frequency, and ${\varepsilon}$r
= dielectric constant of the substrate.
(ii) Effective dielectric constant
1. The effective dielectric constant is also a function of frequency \label{mark-1.}
Power loss due to the dielectric [24] is dependent on the loss tangent. Ԑ$_{\mathrm{r}}$ usually lies between 2.2 and 12.
The low loss tangent reduces the dielectric losses [8].
(iii) Length of the patch (L)
(iv) Extension of patch antenna length ${\Delta}$L
(v) Calculating dimensions of the ground
(vi) Dimensions of strip line
The length of the strip can be found from:
The width of the micro-strip line is given by:
Here, $\lambda _{0}=\frac{v_{0}}{f_{r}}$, where $\lambda _{0}$ is the wavelength in
free space, and $v_{0}$ is the velocity of light in free space. Also, $\lambda _{eff}=\frac{v_{0}}{f_{r}}\sqrt{\varepsilon
_{reff}}$, where $\lambda _{eff}$ is the effective wavelength in the substrate, and
$\varepsilon _{reff}$ is the effective dielectric constant of the substrate.
(vii) Transition line dimensions
The transition line length is a quarter of the wavelength.
The width of the transition line W$_{\mathrm{T}}$ can be found from:
where $Z_{T}$ represents the characteristic impedance in the transition section.
Return loss
Here, the incident power is $P_{in}$, and the reflected power is $P_{ref}$ [24]. The higher the ratio $\frac{P_{in}}{P_{ref}}$ is, the better the power transfer
will be. The return loss equation $R_{L}$ is written using the reflection coefficient,
voltage-standing-wave-ratio $VSWR$, and impedance as follows:
where ${\rho}$ is the complex reflection coefficient, and $Z_{1}$ and $Z_{2}$ are
the impedances of the source and antenna.
3. The GA
A string of bits called a chromosome was used to represent the antenna’s design parameter
values, and a set of such strings formed one generation. Initially, one generation
is generated randomly. This generation is the first generation. Individual chromosomes
in a generation are tested for their fitness individually. The fitness value is then
compared with the desired fitness level of the cost function [8,9].
The chromosomes with the best fitness in the generation are identified and selected
to form the next generation. The bits of the best chromosomes undergo selection, crossover,
and mutation. This generation evaluation continues until the desired fitness level
is achieved for the cost function, as illustrated in Fig. 1 below.
The required performance as the optimization outcome is decided first and quantified
as the fitness value of a cost function. The cost function is formulated by including
the required antenna performance parameters to be optimized. The fitness value of
the cost function measures the antenna performance against the antenna design parameters
and is evaluated for its fitness during an optimization cycle for achieving the desired
output, thus making the formulation of the fitness function a crucial task in the
optimization procedure.
The design parameters that need to be optimized are coded suitably, and their range
is set for the optimization. The basic unit for optimization is a chromosome, which
is a code word containing a combination of codes for each parameter to be optimized.
Several chromosomes form a generation, and it the varies depending on the requirement.
Initially, a pre-determined number of chromosomes is randomly generated. Then, the
fitness of individual chromosomes in that generation is evaluated. The fitter chromosomes
are allowed to carry forward to the next generation, and this process continues until
the desired fitness value is attained or the number of generations defined in the
cycle is completed, whichever is earlier.
4. Literature Survey
An extensive survey of the formulation of fitness functions used in optimizing MSPA
using GA to improve the return loss and bandwidth was conducted. Articles on optimizing
the return loss (S$_{11}$) and bandwidth (BW) were studied in detail. The most frequently
used antenna parameter in the cost function formulation is the return loss(S$_{11}$)
or reflection coefficient ($\tau $) corresponding to the desired S$_{11}$. They were
used either alone or in combination with BW.
(i) Summation of Return loss/Reflection coefficient over a Single Frequency Band\label{mark-(i)}
The most widely used cost function formulation involves the evaluation of fitness
by summing S$_{11}$or $\tau $ for a sample of frequencies over the desired band. In
one study [23], the cost function evaluates fitness by summing up $\tau $ in the desired band for
27 sample frequencies.
where $L=\left\{\begin{array}{ll}
\rho , & \rho \geq -6dB\\
-6dB, & \rho <-6dB
\end{array}\right.$ Another fitness function [13] sums up $\tau $ for 59 samples in the cost function, as shown below:
and $L=\left\{\begin{array}{ll}
\rho , & \rho \geq -6dB\\
-6dB, & \rho <-6dB
\end{array}\right.$
In other studies [23,13], -6 dB is the value for which $\tau $ is calculated.In another study [12], the formulation of the cost function remains the same, but with the sample, the
frequencies increase to 136, and the decision level increases to -10 dB.
here $L\left(n\right)=\left\{\begin{array}{ll}
\rho , & \rho \geq -10dB\\
-10dB, & \rho <-10dB
\end{array}\right.$
$F=~ -\left(\sum _{i}L\left(fi\right)\right)$ is another fitness function [19], which is the same as in the above three cases but in a generalized form. Here, $\tau
$ was added to arrive at the fitness level. The fitness of each antenna was evaluated
using the following equation.
Here, $S_{11}$ was added for a predetermined number of sample frequencies $i$ in the
desired frequency band.
(ii)~Minimization of the Magnitude of $\boldsymbol{\tau }$ Corresponding to $~ \boldsymbol{S}_{11~
}$\label{mark-(ii)}
$fitness=\min \left(S_{11}\right)$ [11,14,20,15]
In some studies [11,15], $\tau $ was minimized, and fitness was evaluated for the desired frequency band
of 1.8GHz to 2.6 GHz and around 5.8 GHz, respectively. In another study [20], S$_{11}$ was minimized at three frequencies: 2.7 GHz, 3, and 3.3 GHz. Another study
[14] optimized the antenna for 5.8 GHz only
(iii) Minimization of $\boldsymbol{S}_{11}$ below the Desired Level
A cost function minimized S$_{11}$ below -15 dB for the desired frequency range of
402 to 405 MHz [10], and another cost function optimized an antenna that returns -10 dB at two frequencies
of 403.5 MHz and 2.45 GHz [21]. The minimization of S$_{11}$ for impedance matching in multiple frequency bands
was introduced in a cost function [18]. Here, S$_{11}$was minimized for <}= 10 dB over two specified bands of 2.5 to 2.69
GHz and 3.2 to 3.8 GHz.
Here, $d1<-10dB\,\,\mathrm{and}\,\,d2<-10dB$ are the constraints.
(iv) Maximization of the Magnitude of $~ \boldsymbol{S}_{11~ }$
A cost function maximized the magnitude of S$_{11}$ by optimizing the shape of a micro-strip
antenna [22]
(v) Maximization of the Sum of the Magnitude of $~ \boldsymbol{S}_{11~ }$
Another study [16] maximized the cost function, which includes the sum of two terms. The first term
is the summation of S$_{11}$over the desired band of frequencies n$_{1}$, and the
second term is also a summation of S$_{11}$, but with a penalty for a better S$_{11}$
value at the undesired band of frequencies n$_{\mathrm{2.}}$
5. Previous Work
5.1 Fitness Function Optimizing Antenna Design by Summing $S_{11}$
In other studies [8,9], the patch antenna was initially designed using theoretical equations, and the dimensions
obtained for 5 GHz are described below in Table 1 [9]. Fig. 2 shows the performance of the proposed antenna.
Other fitness functions used $\boldsymbol{S}_{11}$ in different formations to calculate
fitness of chromosomes [8, 9, 12, 13, 17, 19, 23]. Upon optimization using the GA,
the dimensions of the MSPA were corrected to the values shown in Table 2 below [8]. The antenna performance showed significant improvement for these dimensions (Fig. 3). The bandwidth was 280 MHz at an operating frequency of 5 GHz [8].
The fitness function used here is [8,9]:
Table 1. Dimensions calculated using standard equations.
|
Wy is the width (mm)
|
Lx is the length (mm)
|
|
23.71
|
19.01
|
Table 2. Dimensions obtained on optimization.
|
Wy is the Width (mm)
|
Lx is the Length (mm)
|
|
23.35
|
19.00
|
5.2 Fitness Function Minimizing $S_{11}$
In another study [14], the MSPA was expected to resonate at 5.8 GHz after optimization. The fitness function
used was $fitness=\min \left(S_{11}\right)$. Upon optimization, GA returned an optimized
antenna resonating at 5.7 GHz instead of 5.8 GHz. The optimized antenna deviated from
the desired design requirement of 5.8 GHz and had a minimum return loss of -37.41
dB at 5.7 GHz as per Fig. 4.
The bandwidth is calculated as follows:
where $f_{max}$ is the maximum frequency, $S_{11}\leq -10dB$, $f_{min}$ is the minimum
frequency, and $f_{med}$ is the average between $f_{max}$ and $f_{min}$. This will
not ensure a symmetrical bandwidth spanning either side of the desired resonance frequency
of 5.8 GHz. Here, the minimum of $S_{11}$ was at 5.7 GHz. This will result in the
bandwidth shifting from the desired center frequency of 5.8 GHz.
5.3 Minimization of $S_{11}$ below the Desired Level
The fitness function $f\left(\overset{\rightarrow }{x}\right)=\begin{array}{l}
min\\
f\in F_{off}
\end{array}S_{11}+\begin{array}{l}
min\\
f\in F_{off}
\end{array}$$S_{11}=$ $d_{1}+d_{2}$ applied for GA optimization in another study
[18] minimizes $S_{11}$ at two different frequencies under constraints. The bandwidth
achieved in the first band is only 190 MHz with $S_{11}$ just below -16 dB as shown
in Fig. 5.
5.4 Fitness Function Maximizing the Magnitude of $S_{11}$
Maximization of the magnitude of $\boldsymbol{S}_{11}$ is illustrated in another study
[22] through GA optimization with the application of fitness given below:
The return loss shows improvement from -4.25 dB to -20 dB as in Figs. 6 and figure 7 below. No other parameter is optimized here.
Fig. 6. Before optimization.
Fig. 7. After optimization.
5.5 Fitness Function Maximizing the Sum of the Magnitude of $S_{11}$
Another study [16] maximized the cost function, which includes the sum of two terms. The first term
is the summation of S$_{11}$ over the desired band of frequencies n$_{1}$, and the
second term is also a summation of S$_{11}$, but with a penalty for better values
of S$_{11}$ at the undesired frequency band n$_{2}$. The optimized antenna using fitness
function returns an $S_{11}$ of nearly -13 dB, and the resonant frequency is shifted
from the desired value as shown in Fig. 8.
5.6 Fitness Function Averaging $S_{11}$ over Desired Frequency Bands
Other research [15] optimized path antenna dimensions for increasing bandwidth as shown above in Fig. 9. The fitness function used is:
6. Proposed Model
In this study, a novel fitness function formulation with a graded penalty is introduced
for deviations from the desired operating frequency and required bandwidth during
optimization. The proposed fitness function imposes separate penalties for deviation
from the targeted frequency of operation of 5 GHz and for a reduction in bandwidth
from the maximum bandwidth of 300 MHz. The higher the deviation, the higher will be
the penalty will be.
The new fitness function thus guides the optimization algorithm to arrive at dimensions
of MSPA, providing better results than earlier fitness functions used under similar
conditions. This is achieved by discouraging chromosomes that deviate from the required
resonant frequency and bandwidth from being selected for the next generation. The
proposed fitness function significantly improved $\boldsymbol{S}_{11}$ andbandwidth
while maintaining the frequency of operation at precisely 5 GHz.
$f_{p}$ is the fitness value after penalizing for an inaccurate operating frequency,
$b_{p}$ is the fitness value after penalizing for a reduction in bandwidth, and F
is the fitness of the chromosome. Here, the patch dimensions are subjected to optimization
to achieve a resonant frequency of 5 GHz and a bandwidth of 300 MHz while improving
the return loss.
Here, $Diff_{p}=\left(\frac{abs\left(f_{r}-5GHz\right)}{15}\right)\times 10$, and
$f_{r}$ is the actual resonant frequency.
$Diff_{p}$is the penalty for deviating from the desired resonant frequency of 5 GHz.
$Diff_{p}$ was fixed at 10 units for a deviation of 15 MHz from the desired resonant
frequency. The actual center frequency $f_{r}$ is fixed at the frequency for which
maximum return loss is measured. With the desired 300-MHz bandwidth spanning either
side of the center frequency at 150 MHz each, a deviation of 150 MHz is penalized
by a value of 100. A shift in the actual center frequency beyond 150 MHz on either
side was treated with a maximum penalty of 100, thus discouraging those chromosomes
from being selected for the next generation.
The maximum fitness was assigned to the antenna operating at 5 GHz. For all the other
operating frequencies, a graded penalty was assigned based on the deviation of the
operating frequency from the desired frequency of 5 GHz. The higher the deviation,
the higher is the penalty is.
BW is the bandwidth around the center frequency, which is given by:
The desired bandwidth was set at 300 MHz, the frequencies on either side of the center
frequency were sampled at intervals of 15 MHz, and $S_{11}$was measured for the sampled
frequency and fitness values assigned as above. Sampling was performed on either side
up to 150 MHz from the center frequency. The fitness gained based on the achieved
bandwidth was calculated by adding the actual BW to the penalized value of $f_{p}$.
For BW less than 100 MHz or if $f_{p}=0$, the fitness value assigned is zero, and
for BW greater than 250 MHz, the fitness value is the sum of BW and the total value
of $f_{p}.$
For all the other BWs, the penalized value of $f_{p}$ is added to the actual BW. When
$f_{p}=0,b_{p}$ is also$0$. This ensures that the fitness gained from the bandwidth
achieved in the undesired range of frequencies does not contribute to overall fitness.
Thus, the fitness function selects only those antennas that operate near the desired
operating frequency and have the best bandwidth for the next generation.
Optimization was performed using HFSS and MATLAB. Antenna dimensions such as width
and length were coded into binary strings of 32 bits, with 16 MSBs representing the
width of antenna Wy and the remaining 16 bits representingthe length Lx. The first
generation was made up of 30 randomly generated chromosomes, each representing an
antenna dimension that was generated randomly using MATLAB from within the range set
by the values listed in Table 3 [9].
Each string of bits was passed on to the HFSS software from MATLAB. All 30 antenna
structures were simulated by HFSS one by one, measurements were taken, and their fitness
value was evaluated using fitness function F. Based on the fitness value, the best
chromosomes were passed on to the next generation after selection and then subjected
to crossover and mutation at a predetermined rate. The substrate used was RT Duroid
5880 with a height of 2 mm and all other dimensions are the same as in another study
[8].
This was repeated for 40 generations or until the desired result\textemdash{}that
is, a maximum fitness value of 400\textemdash{}was achieved. This process is illustrated
in Fig. 10. Dimensions of the antenna and the fitness value achieved after completion of optimization
are shown in Table 4. Antenna performance parameters such as $S_{11}$and bandwidth measured after simulation
for the optimized dimensions from Table 4 below are shown in Fig. 11, and the antenna is shown in Fig. 12.
Table 3. Range of dimension for optimization.
|
Wy is the Width (mm)
|
Lx is the Length (mm)
|
|
Min = 22.00
|
Min = 17.60
|
|
Max = 24.60
|
Mac = 21.00
|
Table 4. Dimensions obtained after optimization using graded penalty in the fitness function.
|
Wy is the Width (mm)
|
Lx is the Length (mm)
|
|
23.85
|
18.99
|
|
Fitness value = 400
|
7. Conclusion
GA is widely used for optimizing MSPAs. The core of the optimization is the formation
of a proper fitness function. The fitness function is also referred to as the cost
function or objective function and is formed using one or more of the antenna performance
parameters that need to be optimized for the desired result. Based on the desired
output, the fitness level to be achieved at the end of the optimization is set. The
formation of fitness function plays a key role in guiding optimization techniques
to achieve better results.
The novel fitness function proposed here optimized the antenna dimensions to achieve
better results compared to the fitness function used in another study [8]. Using that fitness function [8], the antenna dimensions were optimized to bring the resonant frequency to 5 GHz with
a bandwidth of 280 MHz at a minimum $\boldsymbol{S}_{11}$ of -24.16 dB compared to
theoretically calculated antenna dimensions [8]. When the proposed fitness function was applied for optimization under similar conditions,
antenna parameters such as bandwidth showed significant improvement while maintaining
the resonant frequency precisely at 5 GHz. $\boldsymbol{S}_{11}$ was improved from
-24.16 Db [9] to -34.79 dB, an increase of 43 percent, while resonating at the precise resonant
frequency of 5 GHz. The bandwidth also improved from 280 MHz to 300 MHZ, an increase
of 7.14 percent.
The optimization was aimed at improving the two performance parameters of the MSPA
using the proposed novel fitness function, and the same has been achieved. When compared
to another study [14], the proposed fitness function addresses the deviation from the desired resonant
frequency and the subsequent shifting of bandwidth effectively. In another study [18], the fitness function returns a nominal bandwidth of 190 MHz and minimum return loss
at -16 dB compared to 300 MHz and -34.79 dB in the proposed work. The fitness function
used in another study [22] improves only the return loss from -4.25 dB to -20 dB compared to the improvement
achieved in three parameters\textemdash{}bandwidth, return loss, and resonant frequency\textemdash{}using
the proposed novel fitness function.
Compared to the fitness function used in another study [16], the proposed fitness function provides exact resonant frequency on optimization,
in addition to a decent return loss of -34.79 dB compared to the -13 dB level in the
other study [16]. The performance of the proposed fitness function was compared with other results
[15]. The bandwidth alone was considered for optimization in that study [15], where as in the proposed work, the fitness function optimizes the patch antenna
for return loss, bandwidth, and an exact resonant frequency.
The optimization process was iterated over 40 generations of 30 chromosomes each,
and it took 12 h for completion on a laptop with an I3 processor and 4 GB of RAM.
The maximum fitness value was set to 400 for the desired bandwidth of 300 MHz. However,
the optimization exceeded this expectation by returning a bandwidth greater than 300
MHz. By setting higher fitness values and increasing the number of generations, the
antenna design can be further improved. Other antenna design parameters, such as feed
position and width, can also be included to improve the optimization of antenna designs
using GA.
ACKNOWLEDGMENTS
The authors thank the authorities of the Department of Instrumentation under Cochin
University of Science and Technology for allowing us to carry out the research and
for the valuable support and guidance.
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Author
Manoj B. is an assistant professor of Electronics and Communication at the College
of Engineering Munnar in the state of Kerala, India. He received his B.E. degree in
Electronics and Communication engineering from Madurai Kamaraj University and an M.E.
degree in Computer and Communication engineering from Anna university Chennai. Presently,
he is pursuing his PhD at Cochin University of Science and Technology(CUSAT). He has
vast experience in teaching at undergraduate and graduate levels in engineering, along
with industrial experience in handling mega information technology projects for the
Government of Kerala such as State Wide area Network (KSWAN), Kerala Fibre Optic network(K-FON),
Public Wi-Fi(K-Fi) and State Data Center. He was also the technical committee member
for the Thiruvananthapuram smart city project. He has many publications to his credit
on Micro-strip antenna research in several peer-reviewed journals and international
conference articles. His research interests are antenna and wave propagation, AI,
optimization techniques in engineering, and wireless communication
Stephen Rodriguez was former head of the Department of Instrumentation at Cochin
University of Science and Technology (CUSAT) and is a professor. He received his Bachelor
of Science and Master of Science in physics with specialization in electronics and
PhD from Cochin University of Science and Technology in the state of Kerala, India.
He has vast experience in handling undergraduate and graduate courses at CUSAT and
has served as a member in various boards of studies of the university and also as
member of an academic council. His current research interests include planar antennas,
microwaves, and antenna technologies.