International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 971646, 8 pages

http://dx.doi.org/10.1155/2015/971646

## Optimization and Design of Wideband Antenna Based on Factor

^{1}Hefei Electronic Engineering Institute, 460 Huangshan Road, Hefei, Anhui 230037, China^{2}Shengli Oilfield Company, SINOPEC, 483 Xisi Road, Dongying, Shandong 257051, China

Received 3 September 2015; Revised 5 November 2015; Accepted 9 November 2015

Academic Editor: Miguel Ferrando Bataller

Copyright © 2015 Han Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A wideband antenna is designed based on factor in this paper. Firstly, the volume-surface integral equations (VSIEs) and self-adaptive differential evolution algorithm (DEA) are introduced as the basic theories to optimize antennas. Secondly, we study the computation of of arbitrary shaped structures, aiming at designing an antenna with maximum bandwidth by minimizing the of the antenna. This method is much more efficient for only values at specific frequency points that are computed, which avoids optimizing bandwidth directly. Thirdly, an integrated method combining the above method with VSIEs and self-adaptive DEA is employed to optimize the wideband antenna, extending its bandwidth from 11.5~16.5 GHz to 7~20 GHz. Lastly, the optimized antenna is fabricated and measured. The measured results are consistent with the simulated results, demonstrating the feasibility and effectiveness of the proposed method.

#### 1. Introduction

Wideband communication systems have been drawing considerable attention among researchers and the industrial community due to the advantages provided by them such as high data rate, compact system size, and low power consumption. Antenna is an indispensable component of any wireless device. The optimal performance of a radio system depends greatly on efficient design of the antenna. So the research of wideband antennas has become a hot topic [1]. Many researchers have designed varieties of ultra-wideband (UWB) antennas with good performance in recent years. Samal et al. [2] proposed an UWB all-text antenna with full ground plane for off-body wireless body area networks (WBAN) applications. The key innovation of the proposed antenna is the use of the quite large and full ground plane, which avoids coupling to and consequently power absorption by the human body. As wideband antenna theory is not yet mature, its design is mostly based on experience with simulation software, which wastes much time. There are some researchers who have tried to design wideband antenna with numerical approaches and optimization algorithms. Ou et al. [3] proposed a method combining genetic algorithm (GA) and the method of moments (MoM) to optimize wideband antenna. The method operates in frequency domain and frequency range to be optimized is so large that there are too many that should be calculated, which costs too much time. Kim et al. [4] optimized an UWB antenna based on GA and finite-difference time-domain (FDTD). However, codes of FDTD are complicated and FDTD is based on difference operation, which has low accuracy compared with MoM. As a result, we should seek for more effective ways to design wideband antennas. We can find that factor is related to size of the antenna. The greater size the antenna occupies, the smaller the antenna has. Additionally, antenna bandwidth is also related to factor. The smaller the is, the wider the antenna bandwidth is. Therefore, we can infer the reduction of antenna size and the increase of bandwidth is contradictory in the process of antenna design. In this paper, we attempt to achieve the following goals. The size of fabricated antenna is as small as possible [5, 6]. The desired antenna bandwidth is as wide as possible. For the above purposes, we have two alternative resolutions: one is designing an antenna with maximum bandwidth in the case that antenna volume meets system requirements and the other is designing an antenna with minimum volume in the case that antenna performance meets system requirements. This paper will start with the first resolution to increase antenna bandwidth by optimizing factor.

The remainder of this paper is organized as follows. In Section 2, the basic theories, including the volume-surface integral equations (VSIEs) and self-adaptive differential evolution algorithm (DEA), are introduced. factor of arbitrary shaped structures is studied in Section 3. A wideband antenna is designed based on factor in Section 4. Finally, Section 5 is the conclusion.

#### 2. Basic Theory

##### 2.1. The Volume-Surface Integral Equations

Integral equation approach has been widely employed to solve electromagnetic (EM) problems due to its unique merits compared with other numerical approaches like FDTD and finite element method (FEM). Closed-form spatial domain Green’s function can be used to study microstrip antennas with infinite ground plane [7]. However, this method cannot work for antennas with finite ground plane. For microstrip antennas, the patch can be processed by surface integral equation and the substrate can be processed by volume integral equation. Therefore, the VSIEs, with high computational accuracy, can be used to analyze any antenna with arbitrary shape.

Assuming there is an antenna that contains perfectly conducting surfaces and dielectric volumes , there are RWG basis functions and SWG basis functions; the matrix equation based on the MoM can be expressed aswherewhere is the th SWG basis function and is the th RWG basis function. The subscripts (conductor) and (dielectric) in (3) are used to represent that the source or test basis function is on conductor and in dielectric, respectively. For example, represents interactions between conductor and dielectric.

With the establishment of the mother impedance matrix, antenna performance can be analyzed through the MATLAB programming based on the VSIEs [8].

##### 2.2. Self-Adaptive Differential Evolution Algorithm

In this paper, the MoM is employed to analyze the antenna, so an optimization algorithm is needed to design the antenna combined with the MoM. DEA, one of the fastest optimization algorithms, has many advantages such as simple process, good robustness, and fast convergence. Parameters of DEA only need to be set once and these parameters can be reused in other cases [9, 10]. The optimization of antenna structures is generally focused on the retention or removal of antenna grids that binary coding is corresponding to. As standard DEA is real coding, we need to convert it to binary coding. The process of binary coding self-adaptive DEA, which is used to optimize antenna structures, is explained as follows.

###### 2.2.1. Initialization

A population should be initialized randomly at the beginning of adaptive DEA, which consists of vectors. Each vector represents a possible solution corresponding to a kind of antenna structure. It can be expressed aswhere is the vector number and is the generation number.

###### 2.2.2. Mutation Operation

The mutation vector in standard DEA can be obtained byIn (6), , , and are randomly chosen indices from the population and is the mutation control parameter. In binary coding DEA, the mutation vector can be expressed aswhere “+” is “OR” operator, “” is “AND” operator, and “” is “XOR” operator. Each element in is 1 or 0 with certain probability.

###### 2.2.3. Crossover Operation

In order to increase the diversity of population, crossover operation is carried out. Trial vectors becomeIn (9), is length of code, is a number from a uniform random distribution within , and CR is the crossover index within .

In order to avoid the premature convergence and enhance local search ability, CR can be set associated with generation [11], which can be defined aswhere and are the upper and lower limits of set according to the actual question. is the current generation, and is the maximum generation.

###### 2.2.4. Selection Operation

The offspring vector replaces the father vector only when it produces a higher value than the father vector in the next generation, which can be expressed aswhere , are the fitness values of the offspring and father vector, respectively.

#### 3. Factor of Arbitrary Shaped Structures

Kenneth S. Johnson, the first person to propose the concept of factor, defined as the ratio of inductance value to resistance value. Chu [12] used spherical waves to express the stored and radiated energies outside the smallest circumscribing sphere of an antenna structure, and the approach has dominated the research on small antennas and offers many results on factor. This theory was later extended by Harrington [13] to include circularly polarized antennas and is expressed asAfter that, McLean [14] reexamined the case of small antennas [15] and expressed as

Earlier works [12–14] do not consider actual current distribution, so they have to deal only with bounds related to dimensions of the enclosing sphere. What is more, the results obtained by Chu use a lot of approximations. Therefore, earlier work have limitation in guiding the design of antennas with theoretical minimum or minimum size.

In recent years, Thal Jr. [16] and Jelinek et al. [17] have considered the energy in the circumscribing sphere, who figured out new radiation limits. As the Jelinek bound is only correct when , it may not be effective in the case of high frequency or antennas with big size.

Many other authors have followed the same paths and given of any radiator under different circumstances [18]. But in general, computation of is focused on narrow band antennas, which has limitation in guiding the study of wideband antennas.

factor is applied to the computation of a wideband antenna in this paper. In order to introduce factor to the case of wideband antennas [19, 20], we need to start with the most primitive expression of factor. According to related literature [21], for arbitrary radiator, the expression of can be written asIn order to compute antenna , it is required to obtain the radiated power , time-average stored electric energy , and time-average stored magnetic energy . By using the method in [22], we can obtainThe strict expression of , obtained by the above computation, can be used in any radiator. Antenna is inversely proportional to bandwidth only for small, high , single-resonant antennas [23], and the ratio of the stored energy to the lost energy per one cycle is not uniquely proportional to the fractional bandwidth [24]. For single at resonant frequency, the expression above is correct, but in this paper we focus on wideband antenna, and the optimization is on the sum of . The relation between the sum of and antenna bandwidth is complex. In a word, with the increase of the sum of , antenna bandwidth becomes narrow.

Generally speaking, we can optimize directly to obtain an antenna with specific bandwidth. But for wideband antennas, the optimization range of bandwidth is so large that at too many frequency points should be computed. If the maximum bandwidth is required, the optimization range of the frequency band may not be enough and the optimization range needs to be increasingly expanded. Therefore, the maximum bandwidth cannot be obtained by direct optimization on bandwidth, which can only meet the requirements of specific frequency band. If factor is used as optimization function, the problem above will be easily solved.

We find that expressions of the radiated power, time-average stored electric energy, and time-average stored magnetic energy are similar to the mother impedance matrix of the VSIEs, where is the multiplication of current basis functions and is the multiplication of electric scalar potentials.

When optimizing antenna bandwidth with the MoM, we usually determine the removal or retention of rows and columns in mother impedance matrix to achieve optimization goal. Similar “mother” matrices are computed at the same time for the stored electric and magnetic energies and radiated power. As shown in Figure 1, by optimizing antenna patch’s shape, the rows and columns of removal are obtained, which will be removed in the radiated power matrix. After optimization, the matrix is multiplied by the coefficient of the surface current or electric displacement vector. Then the modules of all elements in matrix are in sum for computing the stored electric and magnetic energies and radiated power, which can easily obtain the value at a certain frequency point.