# A Phase Angle-Modulated Bat Algorithm with Application to Antenna Topology Optimization
## 4 Experimental Results and Discussions
### Compact Dual-Band Planar Monopole Antenna Design
#### 4.2.2 Antenna Topology Optimization Design  
Multi-band antennas meet the needs of different communication standards with only one device. The advantage of a multi-band device is that it has adaptability and scalability, and the number of frequency bands can differ. Therefore, multi-band antennas are very popular in the field of antenna optimization designs. To verify the effectiveness of the P-AMBA, herein, we present a design example of a dual-band planar monopole antenna and compare it with the BPSO [23], BBA [22], AMBA [27], and A-AMBA [28] optimization algorithms. The initial antenna geometry is given in Figure 7, where the left and right pictures are the top views and side views, respectively. It consists of a 14.6 \(\times\) 12.8 mm rectangular patch and a 16.8 \(\times\) 4 mm rectangular ground plane. The antenna is fed by a 50\(\Omega\) microstrip line and printed on a 16.8 \(\times\) 24.6 mm FR4 substrate with a 1.6 mm thickness, a 4.4 permittivity, and a 0.02 loss tangent. The antenna was designed to achieve the reflection coefficients lower than \(-\)10 dB (i.e., S\({}_{11}<-\)10 dB) in the two frequency bands of 2.3\(-\)2.5 GHz and 5.1\(-\)5.9 GHz, covering the entire 2.4/5.2/5.8 GHz WLAN bands. The fitness function can be expressed as:  
\[Min\ F=\frac{1}{m}\sum\limits_{i=1}^{m}\ (Q(f_{i})) \tag{17}\] \[s.t.\ Q(f_{i})=\left\{ \begin{array}{l}-10,S_{11}(f_{i})\leq-10 \\ S_{11}(f_{i}),S_{11}(f_{i})>-10\end{array} \right.\]  
Figure 6: The relationship between the topology structure of the antenna and its corresponding binary code. where \(f_{i}\) denotes the \(i\) th sampling frequencies within the given operating bands; \(\text{S}_{11}(f_{i})\) is the reflection coefficient of sample \(f_{i}\); \(m\) is the total number of samples; and \(F\) is the fitness value. The initial antenna setting discretized the rectangular patch into \(8\times 7\) pixels, that is, 56 pixels. The size of each pixel was \(2\times 2\) mm. To ensure that the two sub patches that were only in contact at the opposite corners could pass the current, each pixel was overlapped at the edge with an overlapping width of 0.2 mm [43]. In the algorithm parameter settings, the population size was set as 10, and the maximum iteration number was set as 30. To ensure that the antenna has enough metal sheets to achieve radiation performance, the curve of the angle modulation function was controlled above the y-axis as much as possible, so the initial range of the parameters \(d\) and \(g\) was set within \([-0.5,1]\). The setting range of the other parameters of the algorithm is the same as the test data of the above knapsack problem. After testing, these parameter ranges can be well applied to antenna designs. A schematic diagram of the application of phase angle-modulated technology in antenna optimization is shown in Figure 8. By optimizing the six-dimensional vector, the binary data of the \(8\times 7\) matrix corresponding to the antenna model could be mapped. Then, the HFSS was called for electromagnetic simulation, and the obtained return loss values were calculated as the fitness values. Finally, an iterative update of the optimization algorithm was performed. A comparison of the convergence performance of the five optimization algorithms is shown in Figure 9, where the abscissa and the ordinate represent the number of iterations and the fitness value, respectively. It can be seen that the P-AMBA proposed in this paper achieved a good balance between exploitation and exploration, showing a strong optimization ability in the early stage of iteration and not falling into the local optimum prematurely. The final antenna topology and \(\text{S}_{11}\) results optimized by the P-AMBA are shown in Figure 10. The obtained antenna structure met the design objective requirements of the dual-band operation. Figure 7: The top view (**a**) and the side view (**b**) of the initial antenna geometry (unit: mm). The antenna topology design is different from the traditional antenna structure design.