International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 708163, 12 pages

http://dx.doi.org/10.1155/2015/708163

## Efficiency Improvements of Antenna Optimization Using Orthogonal Fractional Experiments

Department of Electronic Engineering, National Taipei University of Technology, 1 Sec. 3, Zhongxiao E. Road, Taipei 10608, Taiwan

Received 22 June 2015; Revised 24 August 2015; Accepted 2 September 2015

Academic Editor: Toni Björninen

Copyright © 2015 Yen-Sheng Chen and Ting-Yu Ku. 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

This paper presents an extremely efficient method for antenna design and optimization. Traditionally, antenna optimization relies on nature-inspired heuristic algorithms, which are time-consuming due to their blind-search nature. In contrast, design of experiments (DOE) uses a completely different framework from heuristic algorithms, reducing the design cycle by formulating the surrogates of a design problem. However, the number of required simulations grows exponentially if a full factorial design is used. In this paper, a much more efficient technique is presented to achieve substantial time savings. By using orthogonal fractional experiments, only a small subset of the full factorial design is required, yet the resultant response surface models are still effective. The capability of orthogonal fractional experiments is demonstrated through three examples, including two tag antennas for radio-frequency identification (RFID) applications and one internal antenna for long-term-evolution (LTE) handheld devices. In these examples, orthogonal fractional experiments greatly improve the efficiency of DOE, thereby facilitating the antenna design with less simulation runs.

#### 1. Introduction

Following the rapid development of electromagnetic (EM) applications, it is of the utmost importance that the design cycle of antennas is reduced. Since antenna design problems usually involve a large number of design parameters, tuning the numerous unknowns by conventional trial-and-error approaches or one-factor-at-a-time parametric studies is very time-consuming. In order to cope with design problems more efficiently, nature-inspired heuristic algorithms have been successfully applied to many design instances. Among these optimizers, genetic algorithms (GAs) [1–4] and particle swarm optimization (PSO) [5–8] have drawn much more attention because of their simplicity and robustness. Also, other approaches including simulated annealing (SA) [9], ant colony optimization (ACO) [10], and invasive weed optimization (IWO) [11] have also been recognized as effective techniques for antenna design. However, heuristic algorithms usually need hundreds or even thousands of simulations due to their blind-search nature. No matter what the problem feature is, they just launch candidate solutions and search the solution domain by executing specific operators until a termination condition is met. Such a laborious process is so computationally intensive that the design cycle is inevitably expanded.

In opposition to the blind-search feature of these heuristic algorithms, a statistical optimization methodology called design of experiments (DOE) investigates the problem nature. DOE uses a predesigned simulation combination, analyzing the simulated results and building the response surface model of a design goal, which is the function of design parameters. In other words, the computational effort in DOE focuses on sequentially “building” the interested solution subdomain rather than iteratively “searching” the entire solution domain; therefore, it enables the design process to be more computationally efficient. DOE has been successfully applied to the EM design problems of flip-chip interconnects [12], microstrip vias [13], tag antennas for radio-frequency identification (RFID) [14], baluns [15], and annular slot ring antennas [16]. However, the above literature uses full factorial designs as the simulation combination. Although full factorial designs provide unbiased estimates on all the associated effects of design parameters, the number of required simulations grows exponentially as the number of design parameters increases. Therefore, it is often too costly to perform full factorial experiments. In these earlier reports, the number of design parameters is usually limited to not more than six to prevent hundreds of simulations; meanwhile, other parameters in the antenna structure are fixed at a certain level. If those fixed parameters are close to the true optimum value, these studies are still able to determine an optimum design having not more than six design parameters. However, real-world design problems usually involve a large number of unknowns. If prior knowledge of the fixed parameters is not included in the design process, the earlier approach may fail to find an optimum design, and this further lowers the usefulness of DOE.

In order to reduce the number of required simulations in DOE, this paper presents orthogonal fractional experiments as the simulation combination. The term “orthogonal” has two meanings. Firstly, in two-level experiments, all the column vectors in a simulation combination are perpendicular. That is, after normalizing the high level and low level of a parameter to +1 and −1, respectively, any two column vectors in a treatment are orthogonal since their inner product is zero. Secondly, the term “orthogonal” in statistics is interpreted in the combinatorial sense. For any pair of columns in a treatment, all combinations of parameter levels occur at equal times. On the other hand, the term “fractional” comes from the fact that only a small fraction, or subset, of a “full” factorial design of experiment is performed. The purpose of this paper is to lend validity to orthogonal fractional experiments, which run only a small fraction of full factorial designs, yet the optimized results are impressing. The capability of orthogonal fractional experiments will be illustrated via three examples. The first design concerning an inductive-feed meander dipole is successfully developed by conducting merely 8 simulations. The second design regarding a complex dual-antenna system within a compact area of 0.1 × 0.1*λ*^{2} is achieved by 16 simulations. Finally, the third antenna structure having 12 design parameters for long-term-evolution (LTE) handheld applications is determined by simply 32 simulations. In these examples, the efficacy of orthogonal fractional experiments will be compared with full factorial designs, and the optimized performances will also be verified by antenna measurements.

#### 2. Review of DOE

Before the achievement of orthogonal fractional experiments is detailed, this section reviews the procedure of DOE. In particular, the difference between full factorial designs and orthogonal fractional experiments will be thoroughly discussed.

DOE considers a design problem as a process, shown in Figure 1. The output responses are usually the design goals of an antenna, and the input factors are design parameters . To quantify how influential the design parameters are, antenna designers purposefully vary them at two levels. The degree of a parameter’s individual influence on a response variable is called the main effect of the parameter . Usually, the main effect of does not hold constant when other parameters alter their levels. The influence of () on is called the two-factor interaction between them. Likewise, two-factor interaction between and may change when the other parameter () is involved, and the resultant influence is called the three-factor between them. Similarly, other higher-order interactions between multiple factors can thus be defined. Estimating these effects as precisely as possible is the major computational effort in DOE.