## 3 Floquet-Bloch Analysis
From the geometrical values, \(L_{\textit{eff}}\) and \(Z_{oL}\) can be calculated as 0.25 nH and 0.8 \(\Omega\), respectively. The dispersion diagrams are obtained in Figure 5 using equation (3) and the finite-element method (FEM) simulation [25]. It is shown that passbands and stopbands alternately appeared in the dispersion diagram of the CMS-EBG structure. The first stopband is formed in the frequency range from low- and high-cutoff frequencies, which are denoted as \(f_{L}\) and \(f_{H}\). In the dispersion diagram of the example structure, the first stopband spans over the GHz frequency range. The extracted \(f_{L}\) and \(f_{H}\) values from Equation (3) are 1.98 GHz and 6.42 GHz, respectively. Those from the FEM simulation are 2.2 GHz and 6.1 GHz. The dispersion equation and the FEM simulation show good agreement. The bandwidth of the stopband is 4.2 GHz, which is sufficiently broad to suppress the electromagnetic noise in MPCBs. Hence, the CMS-EBG structure is expected to suppress the resonant modes in the frequency range and effectively mitigate the wideband GHz noise in MPCBs. Furthermore, a sensitivity analysis is performed with the equation of the effective phase constant and FEM simulation to examine the effects of the main design parameter \(w_{p}\). In the analysis, the nominal values of \(w_{p}\), \(w_{b}\), \(d_{b}\), \(S_{b}\), \(S_{w}\), \(r_{v}\), \(t\) and h are identical to those in Figure 5. \(f_{L}\) and \(f_{H}\) are calculated by sweeping the normalized coefficient k, which is defined as the ratio of a \(w_{p}\) value to the nominal \(w_{p}\) value (i.e., \(k=w_{p}/w_{po}\)). \(k\) varies from 0.7 to 1.3 and the corresponding \(w_{p}\) value varies from 7.7 mm to 14.3 mm with a nominal value of 11 mm. The extracted results of \(f_{L}\) and \(f_{H}\) are shown in Figure 6. For \(k=0.7\), \(f_{L}\) and \(f_{H}\) in Equation (3) are 2.82 GHz and 9.17 GHz, respectively; for \(k=1.3\), \(f_{L}\) is 1.52 GHz, and \(f_{H}\) is 4.94 GHz. The results derived from the FEM simulation show a good correlation with the results calculated using Equation (3). Hence, it is concluded that \(f_{L}\) and \(f_{H}\) are reduced as the \(w_{p}\) increases. The location and bandwidth characteristics of the stopband are additionally analyzed. The location of a stopband is characterized by the center frequency (\(f_{C}\)), which is defined as (\(f_{L}+f_{H}\))/2. The bandwidth characteristics are characterized by the fractional bandwidth (FBW), which is defined as (\(f_{H}-f_{L}\))/\(f_{C}\). As seen in Figure 7a, the center frequency \(f_{C}\) is shifted into the low-frequency range as \(w_{p}\) increases. Conversely, the FBW does not significantly change as \(w_{p}\) varies, as shown in Figure 7b. The FBW results obtained from the dispersion Equation (3) and FEM simulation are approximately 1.0 for various \(w_{p}\) values. In other words, the CMS-EBG structure maintains approximately 100% fractional bandwidth characteristics for the various \(w_{p}\) values. These characteristics enable a simple design of the CMS-EBG structure. The aforementioned dispersion characteristics of the  
Figure 5: Dispersion diagrams of the CMS-EBG structure from Floquet-Bloch analysis. Figure 6: Analysis of the \(w_{p}\) effects on the low- and high-cutoff frequencies. CMS-EBG structure are summarized in Table 1. The \(Z_{oL}\) values in Table 1 are obtained from the FEM simulation. The comparison results can be further summarized using the IEEE Standard for validation of the computational electromagnetics computer modeling and simulation with the feature selective validation (FSV) technique [26; 27; 28].