# Optimization of Gain, Impedance, and Bandwidth of

Authors: Munish Rattan; Manjeet Singh Patterh; B. S. Sohi
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

Particle swarm optimization (PSO) is a new, high-performance evolutionary technique, which has recently been used for optimization problems in antennas and electromagnetics. It is a global optimization technique-like genetic algorithm (GA) but has less computational cost compared to GA. In this paper, PSO has been used to optimize the gain, impedance, and bandwidth of Yagi-Uda array. To evaluate the performance of designs, a method of moments code NEC2 has been used. The results are comparable to those obtained using GA.


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## Body

## 1. Introduction

Yagi-Uda arrays are quite  common in  practice because they are
light weight, simple to build, low cost, and provide moderately desirable
characteristics for many applications. Yagi-Uda antenna was developed by Uda
and Yagi during 1930s. Since then, it has received much attention in the
literature. There are several gradient-based methods for optimization of Yagi
antenna [1–3]. The
shortcoming of gradient-based methods is that they are vulnerable to stuck in
local optima. Also, these heavily depend
on the choice of initial point, which depends on the experience of antenna
designer. The solution to the problems of gradient-based methods can be
overcome by using global optimization methods like genetic algorithm (GA) and particle
swarm optimization (PSO). GA optimizers are particularly effective when the
goal is to find an approximate global maximum in high dimension, multimodal
function domain in near optimal manner. Haupt has compared GA and gradient-based methods for
electromagnetic problems and found that the genetic algorithms are better than gradient-based
methods [4]. Many reserachers have used GA for optimization of Yagi-Uda array
for different design objectives [5–9]. In [7], the
Yagi-Uda array has been optimized for gain, impedance, and bandwidth using GA.
In this paper, we are also achieving the same design objective using PSO.PSO has been found to work better than GA in certain kind of
optimization problems. Compared to GA, it is easily implemented and has least
complexity [10]. Recently, it has been used for the synthesis of linear arrays
[10, 11]. It has also been used for the optimization of gain, impedance, and
relative side lobe level of Yagi-Uda array and has shown better performance
than GA [12]. Bandwidth is also an important design objective, which needs to be explored along with gain
and impedance. In this paper, PSO has been used for optimizing the gain,
impedance, and
bandwidth using length and spacing between elements as
variables. Section 2 describes the PSO
algorithm and the Yagi-Uda design optimization using PSO with simulation
example. The work has been concluded in Section 4.


## 2. Particle Swarm Optimization

PSO is an evolutionary algorithm based on the intelligence
and cooperation of group of birds or fish schooling. It maintains a swarm of
particles, where each particle represents a potential solution. In PSO,
algorithm particles are flown through a multidimensional search space, where
the position of each particle is adjusted according to its own experience and
that of its neighbors. Table1 shows some key terms used to describe PSO [13]. More details of algorithm can be found in [14, 15].Table 1
Key PSO vocabulary.
Some key terms used to describe PSOParticle/agentOne
single individual in the swarmLocation/positionAn
agent’s N-dimensional coordinates which represents a solution to the problemSwarmThe
entire collection of agentsFitness or costA
single number representing the goodness of a given solutionPersonal best (pbest)The
location in parameter space of the best fitness returned for a specific
agentGlobal best (gbest)The
location in parameter space of the best fitness returned for the entire swarmVmaxThe
maximum allowed velocity in a given direction


## 3. The Design Process

The goal of the design process is to develop an antenna that
meets or exceeds some desired performance characteristics. A few of the
characteristics that  define the antenna performance are sidelobe level,
beamwidth, bandwidth, front-to-back ratio, size, gain, and  input impedance. The
quality of a design is expressed mathematically by an objective function. This
paper uses the function which tries to maximize gain and to obtain optimum
impedance, that is,Z=50 Ω for the 10% bandwidth. The objective function used is
given by
(1)F=aG(x)−b|50−ReZ(x)|−c|ImZ(x)|,
where G is the Gain of antenna x in the endfire direction; Z
is the impedance; and |⋅| denotes the absolute value. The value of a was taken as 40 and b and c equal to 2. The values of a, b, and c have been taken by hit and trail. It is clear that
if the real part of impedance is far from 50 or the imaginary part is large,
the fitness will be worse. To optimize the pattern for bandwidth, fitness
operation is done in three frequencies: up, central, and down as in [7]. For
example, if it is desired 10% of bandwidth, the operation is made in three
frequencies: fup,  fc, and  fdown,
where, fup=1.05∗fc andfdown=.95∗fc.Here,fc, is the central frequency. Figure 1 shows the
geometry of a K element Yagi-Uda array. The six-element Yagi-Uda array
has been optimized for gain, impedance, and bandwidth using PSO algorithm. The
distances between consecutive elements (five different distances) and the
lengths of each element are the parameters to be optimized. The problem with
the antenna design is that if we increase gain, the impedance gets bad or
bandwidth decreases. So, in order to have good antenna it should have high gain,
low VSWR, and good bandwidth. The cross section
radius is the same for all elements and is set equal to 0.003369 λ wavelengths
at 299.8 MHz. The NEC2 simulation program [6] has been used to evaluate all
antenna designs. The source element for excitation was specified to be the
middle segment of the driven element. The z location of the reflector
element was always set to 0. The antenna was analyzed in free space. The
spacing between elements was allowed to vary between 0.10 λ and 0.45 λ, and
the length of each element is allowed to vary between 0.15 λ and 0.35 λ. Figure 2
shows the maximum directive gain versus frequency response for GA and PSO. As
can be seen, PSO shows better gain characteristics compared to GA. Figure 3
shows the VSWR versus frequency for GA and PSO. It can be observed that GA has
better impedance characteristics. But it has to be noted that in [7] the
position of feedpoint has been varied for getting better impedance
characteristics, whereas in this work the feedpoint position is fixed. Table 2
shows the relative performance comparisons. The convergence rate of the PSO has
been shown in Figure 4. In general, PSO converges in a few iterations compared
to GA since it has less complexity and can be easily implemented. In this very
particular problem, PSO converges after 2500 iterations, whereas GA took 10 000
iterations. This results in considerable time saving. The  memory complexity is
also less in PSO as PSO is inherently less complex compared to GA.Table 2
Length, spacing, and performance comparisons. The central frequency is 300 mHz. The radius of
elements is 0.003369λ. Length and spacing are also in terms of
wavelength.
GA optimized gain, impedance,
and bandwidth [7]PSO optimized for gain, impedance, and
bandwidthElementLengthSpacingLengthSpacing1.52—.4939—2.47.182.4665.22693.42.152.4213.16624.41.229.4149.28835.39.435.4120.32136.39.272.4199.2731Feedpoint0.0250Gain
(dBi)(fdown/fup)9.3911.2910.2212.12Z (Ω)52.4−j6.647.1+j2.649.08−j2049.99+j0.50Figure 1
Geometry of a K element Yagi-Uda array.Figure 2
Maximum directive gain in dB versus normalized
frequency for GA and PSO.Figure 3
Voltage standing wave ratio versus
normalized frequency  for GA and PSO.Figure 4
Convergence rate of the PSO algorithm, showing
fitness value versus number of iterations.


## 4. Conclusion

Particle swarm optimization is currently being used for
antenna optimization due to its low complexity and global nature. Earlier
designs using PSO did not consider bandwidth as optimization variable. The
bandwidth is an important parameter as antennas work in the specific frequency
bands. In the present work, PSO has been successfully used for the optimization
of gain, impedance, and bandwidth of Yagi-Uda array. Results show that PSO is
well suited for these kinds of multimodal problems. The results are comparable
to those obtained using GA taking length and spacing as variable. More control
over the pattern can be obtained by taking length, spacing, and radii, that is,
all three as variables. Also, other design objectives like sidelobe level and
beamwidth can be included in the optimization process.



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*Source: 193242-2008-05-18.xml*