Electromagnetic Optimization Using Taguchi Method: A Case Study of Linear Antenna Array Design. Wei-Chung Weng*, Fan Yang, Veysel Demir, and Atef ...
Electromagnetic Optimization Using Taguchi Method: A Case Study of Linear Antenna Array Design Wei-Chung Weng*, Fan Yang, Veysel Demir, and Atef Elsherbeni Department of Electrical Engineering, The University of Mississippi University, MS 38677, USA 1. Introduction Taguchi method is a methodology based on orthogonal arrays (OAs) concept, which effectively reduces the number of test iterations required in an optimization process [1]. Although Taguchi method has been successfully applied in many fields such as chemical engineering, mechanical engineering, IC manufacture, etc., it is rarely used in electromagnetic field. This paper presents our pioneer work to introduce Taguchi method to the electromagnetic community. As a robust design technique, Taguchi method is used in this paper for electromagnetic optimizations. The implementation procedure is presented, and a linear antenna array design is used as an example to illustrate this procedure step by step. Instead of genetic algorithm (GA) and particle swarm optimization (PSO) methods [2-4], Taguchi method is used to design linear antenna arrays that produce a null control radiation pattern or a sector beam radiation pattern. This study shows that the method can quickly converge to the desired design since it takes less computational resource. Most important of all, Taguchi method is easy and straightforward to implement. Numerical results show that employing Taguchi method successfully reaches the desired radiation pattern. 2. Implementation Procedure of the Taguchi Method 2.1 Problem Initialization The flow chart of Taguchi method is shown in Fig 1. The optimization procedure starts with the problem initialization, which includes the selection of a proper orthogonal array (OA) and an appropriate design of the fitness function. The orthogonal array, which has a profound background from statistics [5], plays an essential role in the method. The selection of an orthogonal array depends on the input parameters of an optimization problem. For the symmetrical 20 elements linear array shown in Fig. 2, there are ten excitation magnitudes and ten excitation phases that should be optimized in sector beam pattern problem, and ten parameters for excitation magnitudes should be optimized since all excitation phases are set at zero in null control pattern problem. In order to present the non-linear characteristics, three (3) levels are used for each input variables. After searching references [6], an L27310 OA, which offers ten columns for ten parameters is used for null control pattern problem, as shown in Table 1, and an L81320 OA is used for sector beam pattern problem. The fitness function is devised according to the optimization goal. For a linear array problem, the following fitness function is used. π (1) Fitness = ∫ f d (θ ) − f (θ ) dθ , 0
where, the f d (θ ) is the desired pattern, and f (θ ) is the pattern obtained from the array factor function. Basically, the smaller the value of the fitness function is, a better performance is obtained.
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2.2 Match OA with Level Values of Input Parameters Next, one needs to decide the three level values of each parameter. For the first iteration, the level 2 is selected at the center of the optimization range. Values of levels 1 and 3 are calculated by adding/subtracting the value of level 2 with a variable called level difference. The level difference ( LD1 ) in the first iteration can be obtained by the following equation: 0.999− 0.001 up − low (2) LD1 = = = 0.2495 , number of level + 1
3 +1
where, up is upper-bound of computational range, and low is lower-bound of computational range. Therefore, each entry of the OA in Table 1 can be converted into a proper value of the input parameter a(n)1k , where n indicates the nth antenna element, the subscript 1 indicates the first iteration, and superscript k indicates the level 1, 2, or 3. 2.3 Conduct Experiments After converting OA entries to proper input values, one can calculate the fitness function for each experiment in the OA. For example, in Table 1 the fitness value for experiment 1 with all variables’ level being 1 is calculated using (1) and the result is 49.971. Next, the S/N ratio η (dB) in the Taguchi method [1] is calculated below: (3) η = −20log ( Fitness ) (dB) . After taking all experiments in the first iteration, the fitness and corresponding S/N ratio are obtained in Table 1. These results are used to build a response table by averaging the S/N ratios of the same level for each parameter at each level, as shown in Table 2. 2.4 Identify Optimal Level Values and Conduct Confirmation Experiment From Table 2, one can find the optimal level for each parameter by finding the largest S/N ratio in each column. For example, the optimum value for the first iteration is Next, a confirmation a(1)11 , a (2)12 , a (3)12 , a (4)12 , a(5)11, a(6)12 , a(7)12 , a (8)11, a(9)11 , and a(10)11 . experiment is performed using the optimal level combination. The fitness value obtained will be compared with the design objectives and the fitness of previous iteration. If the design goals are achieved or the result converges, the optimization process will end. 2.5 Reduce the Optimization Range One should proceed to the next iteration for optimization if the previous step results do not converge. The level values of the optimal combination are used as values of level 2 for the next iteration. The LDi will be reduced by a reduced rate ( RR ) for level difference LDi +1 of the (i + 1) th iteration: (4) LDi +1 = RR ⋅ LDi . The RR can be set between 0.5 and 0.95 depending on the problems. The larger RR is, the slower convergence is. The same procedure will be repeated until the fitness value is converged or the optimization objectives are met. 3. Numerical Results Using Taguchi method, the optimizations of a linear array quickly converge to the design goals, as shown in Fig. 3. The optimized null control pattern and sector beam pattern are shown in Fig. 4 and Fig. 5, respectively. Figure 4 shows the beam width at -40 dB sidelobe level is 20.90 , HPBW is 7.40 , and nulls are below to –55dB in the angle ranges
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of [50º, 60º] and [120º, 130º]. Figure 5 shows a sector beam pattern with 0.25 dB ripples between 800 and 1000 . Its beam width at -25 dB SLL is 40.20 , and HPBW is 26.60 . 4. Conclusion This paper introduces a novel electromagnetic optimization technique using Taguchi method. To illustrate this technique, a linear antenna array is optimized to realize a null control pattern and a sector beam pattern. It is found that Taguchi method is easy to implement and it converges to the desired patterns quickly. This method is a good candidate for optimizing EM applications. References [1] Genichi Taguchi, Introduction to quality Engineering, White Plain, NY: Uni Pub, 1986. [2] M. M. Khodier; C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas and Propagat., Vol. 53, pp. 2674 - 2679, Aug. 2005. [3] D. W. Boeringer, D. H. Werner, and D. W. Machuga,” A simultaneous parameter adaptation scheme for genetic algorithms with application to phased array synthesis,” IEEE Trans. Antennas and Propagat., Vol. 53, pp. 356 - 371, Jan. 2005. [4] D. Gies and Y. Rahmat-Samii, “Particle swarm optimization for reconfigurable phased-differentiated array design,” Microw. Opt. Tech. Lett., vol. 38, no. 3, pp. 168 - 175, Aug. 2003. [5] A. S. Hedayat, N. J. A. Sloane, and John Stufken, Orthogonal Arrays:Theory and Applications, SpringerVerlag New York Inc., New York, 1999. [6] http://www.research.att.com/~njas/oadir/
Table 1. The L27 310 Orthogonal Array, Fitness values, and S/N Ratio of First Iteration Element Experiment 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
1 2 3 4 5 6 7 8 9 10
Fitness
S/N Ratio (dB)
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
49.971 89.021 94.098 96.733 66.527 140.52 127.19 97.299 89.165 90.632 145.05 116.0 75.165 70.484 112.76 119.93 75.456 73.755 103.12 108.8 119.56 100.14 114.67 117.59 115.34 115.15 92.35
-33.974 -38.99 -39.472 -39.711 -36.46 -42.955 -42.089 -39.762 -39.004 -39.146 -43.231 -41.289 -37.52 -36.962 -41.043 -41.578 -37.554 -37.356 -40.267 -40.733 -41.552 -40.012 -41.189 -41.407 -41.24 -41.225 -39.309
1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3
1 2 3 1 2 3 1 2 3 2 3 1 2 3 1 2 3 1 3 1 2 3 1 2 3 1 2
1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2 1 2 3 2 3 1 3 1 2
1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 1 2 3 3 1 2 1 2 3 2 3 1
1 3 2 2 1 3 3 2 1 2 1 3 3 2 1 1 3 2 3 2 1 1 3 2 2 1 3
1 3 2 3 2 1 2 1 3 2 1 3 1 3 2 3 2 1 3 2 1 2 1 3 1 3 2
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1 1 1 3 3 3 2 2 2 3 3 3 2 2 2 1 1 1 2 2 2 1 1 1 3 3 3
1 2 3 1 2 3 1 2 3 3 1 2 3 1 2 3 1 2 2 3 1 2 3 1 2 3 1
1 3 2 2 1 3 3 2 1 1 3 2 2 1 3 3 2 1 1 3 2 2 1 3 3 2 1
Problem Initialization Select orthogonal array Design fitness function
Match OA with level values of input parameters
Fig. 2. Geometry of a 20 elements equally spaced linear array.
Conduct experiments and build response table
Reduce the optimization range
Identify optimal level values and conduct confirmation experiment No Termination criteria met? Yes
Fig. 3. Convergence curves of fitness function.
End
Fig. 1. Flow chart of Taguchi method. Table 2. Response Table of First Iteration Element Level
1
2
3
4
5
6
7
8
9
10
1 2 3
-39.50 -39.57 -40.38
-39.85 -39.70 -39.90
-39.85 -39.53 -40.08
-39.70 -39.54 -40.20
-39.62 -39.94 -39.89
-39.79 -39.53 -40.13
-39.86 -39.54 -40.05
-39.06 -39.88 -40.51
-39.53 -39.60 -40.31
-38.19 -39.79 -41.47
Fig. 5. The optimal normalized sector beam pattern.
Fig. 4. The prescribed nulls located between 500 and 600 and between 1200 and 1300 are below to -55dB.
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