# Performance-Driven Yield Optimization of High-Frequency Structures by Kriging Surrogates
## 4 Demonstration Case Studies
### Case IV: Compact Microstrip Rat-Race Coupler  
The second verification structure is a miniaturized microstrip rat-race coupler (RRC) presented in Figure 10 and described in Figure 11 [68]. The compact size of the circuit is a result of folding of the transmission lines that constitute ts interior.  
Figure 10: Layout of the miniaturized folded rat-race coupler [68]; the numbered circles (1 through 4) mark the structure ports.  
Figure 9: Monte Carlo analysis of antenna of Figure 7 using EM simulations (gray plots): (**a**) nominal design; (**b**) yield-optimized design obtained using the algorithm of Section 3.1. Black plots show the antenna response at the nominal and optimized designs, respectively.  
Parameter deviations are assumed to be independently and uniformly distributed with the maximum deviation of \(\delta_{\text{max}}=0.05\) mm. For the discussed algorithm, the metamodel of the relative RMS error of 2.3% was set up within a domain of size \(\delta_{c,k}=2\delta_{\text{max}}\), and the training data set comprised 72 training samples. In the case of Algorithm 1, the surrogate was constructed with 400 samples allocated over the domain of size \(10\delta_{\text{max}}\), and the relative RMS error was equal to 3.4%. For Algorithm 2, the first metamodel set up with 50 samples within the domain encompassing \(\mathbf{x}^{(0)}\) featured a relative RMS error of 2.2% (the size-defining parameter has been set to \(3\delta_{\text{max}}\)). Table 3 provides the relevant results for the discussed and benchmark algorithms. The following optimal design was yielded: \(\mathbf{x}^{*}=[4.65\ 11.10\ 21.87\ 0.71\ 0.95\ 0.81]^{T}\).  
The results of Monte Carlo analysis at the initial and optimal designs are shown in Figure 12 (500 uniformly distributed samples were used with \(\delta_{\text{max}}=0.05\) mm). In this case, the discussed approach and Algorithm 2 were able to estimate the yield in a reliable manner. As for Algorithm 1, the values of the yield predicted using the surrogate and EM simulations differed considerably. This is because the surrogate rendered across a domain of a larger size featured worse predictive power.  
As in the previous case, it has been verified that the yield optimization within a confined domain enables the obtainment of the benefits of both benchmark procedures. Thus, the entire process may be concluded in one stage due to spanning the domain along the most consequential directions (as opposed to the iterative Algorithm 2). Moreover, as the domain is of reduced size, it was possible to construct the surrogate at a remarkably low cost. The computational savings reached up to 82 and 64 percent (the discussed procedure versus Algorithms 1 and 2, respectively). At the same time, high design quality was maintained in all cases, and the value of the optimized yield is similar for all compared algorithms. Furthermore, the obtained results agree with those of EM-based Monte Carlo analysis.  
Figure 11: Description of the rat-race coupler of Figure 10[68].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optimization Algorithm** & \multicolumn{2}{c}{**Initial Yield**} & \multicolumn{2}{c}{**Optimized Yield**} & \multicolumn{2}{c}{**CPU Cost\({}^{\lx@sectionsign}\)**} \\
**Estimated by Metamodel** & **EM-Based** & **Estimated by Metamodel** & **EM-Based** & **CPU Cost\({}^{\lx@sectionsign}\)** \\ \hline Algorithm 1 & 67\% & 62\% & 90\% & 83\% & 400 \\ Algorithm 2 & 66\% & 62\% & 86\% & 83\% & 200 * \\ Algorithm of Section 3.1 & 63\% & 62\% & 84\% & 82\% & 72 \\ \hline \hline \end{tabular} \({}^{\lx@sectionsign}\) Optimization cost in number of EM analyses of the antenna structure. \({}^{\lx@sectionsign}\) Algorithm convergence after four iterations (surrogate setup cost 100 training samples per iteration).  
\end{table}
Table 3: Yield optimization of the compact coupler of Figure 10.