# Robust Design of 3D-Printed 6-18 GHz Double-Ridged TEM Horn Antenna
## 3 Procedure of the Robust Design Method  
In this paper, a novel robust design method is utilized to reduce the noise effects. The proposed method is realized by combining a genetical swarm optimization (GSO) algorithm and an orthogonal array (OA) matrix. The GSO algorithm, a hybrid optimization algorithm which combines a genetic algorithm (GA) and particle swarm optimization (PSO), is employed to find a new combination of robust parameters against the noise factors, while the OA is employed to evaluate the robustness of the parameters. Figure 8 shows a flowchart of the proposed robust design method. The detailed robust design procedure is described below. 1. Define the control and noise factors. The control factors are the design parameters for the TEM horn antenna and the noise factors are parameters that the designer cannot control. We assumed that the manufacturing process has tolerances along the \(x,y,\) and \(z\) axes and that the characteristic impedance of the SMA connector can vary. To realize the variation of the characteristic impedance in the simulation, the permittivity of the Teflon used in the SMA connector is changed. In the simulation, the tolerances are defined as \(nx\), \(ny\), and \(nz\), and the permittivity is represented by \(ni\). 2. Renew the dimensions of the control factors using the updating rule of the GSO. In each iteration, the design parameters are randomly divided into two parts by the hybridization coefficient (HC), and these are correspondingly updated by the GA and PSO [16]. For example, HC = 0 means that the parameters are evolved by only PSO. On the other hand, for HC = 1, the parameters are updated by only the GA. In this paper, HC is set to 0.8. 3. Evaluate the performance of the antenna with the dimensions of the parameters updated in step 2. The cost function for GSO must be defined to ensure that the performance of the optimized antenna is better than that of the initial antenna. To achieve this, the reflection coefficient and realized gain levels of the optimized antenna should be less than \(-13\) dB and higher than the realized gain of the initial antenna, respectively, as follows, \[Cost=\frac{1}{n}\sum_{i=1}^{n}\{S_{opt}(f_{i})+G_{opt}(f_{i})\}\] (4) \[S_{opt}(f_{i})=\left\{ \begin{array}{ll}1&\text{if }S_{11}(f_{i})>-13\text{ dB}\\ 0&\text{otherwise}\end{array} \right.\] (5) \[G_{opt}(f_{i})=\left\{ \begin{array}{ll}1&\text{if }gain_{opt}(f_{i})-gain_{ini}(f_{i})<0\\ 0&\text{otherwise}\end{array} \right.\] (6)  
Figure 7: Simulated results of the TEM horn antenna: (**a**) reflection coefficient and (**b**) realized gain. where \(n\) is the number of sampling frequencies and \(f_{i}\) is the \(i\)-th sampling frequency within the 6-18 GHz band. Additionally, \(S_{11}(f_{i})\) denotes the reflection coefficient level obtained at each sampling frequency. \(gain_{opt}\) and \(gain_{ini}\) are the gain levels obtained from step 2 and the initial antenna, respectively. 4. Steps 2 and 3 are repeated until the cost function is 0. 5. Build an OA matrix for noise factors and conduct the experiments in the OA matrix. In order to realize the orthogonality between the parameters in all possible combinations and to reduce the simulation time, an OA matrix with two strengths and three levels is employed. The OA matrix is described in Table 1.
6. Evaluate the robustness of the antenna as optimized by GSO. The robustness for the \(k\)-th experiment can be defined as the variation of the reflection coefficient, \[R_{e}^{k}=\frac{1}{n}\sum_{i=1}^{n}(|S_{opt}(f_{i})-S_{noi}^{k}(f_{i})|),(1\leq k \leq 9)\] (7) where \(S_{noi}^{k}(f_{i})\) is the reflection coefficient level at the sampling frequency. Next, the robustness \(R\) of the optimized antenna is expressed by averaging the robustness value for each experiment, as follows, \[R=\frac{1}{m}\sum_{k=1}^{m}R_{e}^{k},\] (8) where \(m\) is the number of experiments in the OA matrix. 7. Finally, steps 2 to 6 are repeated until the number of iteration reaches 100.