## 3 Floquet Analysis
### Analysis of the IEP Effect on Stopband
The effective propagation constant \(\beta_{\textit{eff}}\) is given by  
\[\beta_{\textit{eff}}=\frac{1}{d_{c}}cos^{-1}\{cos(\beta_{c}d_{c})-k\textit{ \cdot}sin(\beta_{c}d_{c})\}, \tag{5}\]  
where  
\[k=(\omega C_{\textit{P}}Z_{\textit{oc}})/\left(2-2\omega^{2}L_{\textit{eff}}C _{\textit{P}}\right). \tag{6}\]  
Using Equation (5), we obtained the example of the dispersion diagram of the IEP-EBG structure in Figure 3. The design parameters and corresponding values of the example of the IEP-EBG structure are shown in Table 1. The values were reasonably determined considering the conventional manufacturing processes used for high-speed packages and PCBs. Figure 3: Dispersion diagram of the IEP-EBG structure showing the stopband characteristics. In the dispersion diagram shown in Figure 3, the passbands and stopbands are shown alternately. In the first stopband, the slow-wave effect is observed and the stopband starts from the low cut-off frequency \(f_{L}\). It is anticipated that the PPW modes are efficiently suppressed in the stopband. As a consequence, the SSN is significantly mitigated over the frequency range from \(f_{L}\) to a high cut-off frequency (\(f_{L}\)). In this example of the dispersion results, the stopband is obtained in the frequency range from 1.6 GHz to 2.86 GHz. To thoroughly examine the IEP effects, a closed-form solution of \(f_{L}\) was derived by realizing that a real part of \(\beta_{\mathit{eff}}d_{c}\) was equal to \(\pi\) (i.e., Re\(\{\beta_{\mathit{eff}}d_{c}\}=\pi\)). In addition, we assumed that the value of \(\beta_{\mathit{eff}}d_{c}\) was too small for the small angle approximation of trigonometric functions in Equation (5) because it was far from the Brillouin zone boundary at the low cut-off frequency. Then, a closed-form expression for \(f_{L}\) can be extracted as  
\[f_{L}=\left(Z_{oc}C_{p}d_{c}/\left(4v_{p}\right)+L_{\mathit{eff}}C_{p}\right)^ {\left(-\frac{1}{2}\right)} \tag{7}\]  
where  
\[v_{p}=c/\left(\varepsilon_{\mathit{eff}}\right)^{\frac{1}{2}} \tag{8}\]  
\(v_{p}\) is the phase velocity. \(c\) is the speed of light and \(\varepsilon_{\mathit{eff}}\) is the effective relative permittivity of the dielectric material. Using Equation (7), the IEP effects on\(f_{L}\) were characterized by sweeping the \(L_{\mathit{eff}}\) values from 0.6 nH to 1.4 nH as shown in Figure 4. The \(L_{\mathit{eff}}\) values used herein were chosen considering a conventional design of high-speed packages and PCBs. The \(f_{L}\) analysis with Equation (7) was not limited to these values. As can be seen in Figure 4, it was found that the \(f_{L}\) value decreased as the \(L_{\mathit{eff}}\) value increased. Applying the IEP technique to the EBG structure increased the effective inductance of the EBG UC. As a result, the \(f_{L}\) was shifted down in the low frequency range. The advantage of \(f_{L}\) reduction with the IEP technique is the efficient miniaturization of the UC. When we have a design objective for the \(f_{L}\) in high-speed packages and PCBs, the value of the \(f_{L}\) can be controlled by changing the \(d_{c}\), \(L_{\mathit{eff}}\), \(Z_{\alpha c}\), and \(C_{p}\) values. To shift the \(f_{L}\) down into the low frequency range and satisfy the design target, first, \(d_{c}\) can be increased, thus enlarging the UC size. Alternatively, increasing \(Z_{\alpha c}\) enables a low \(f_{L}\) value. For a large \(Z_{\alpha c}\), the thickness of the packages and PCBs needs to be significantly increased, which is prohibited in most applications of high-speed packages and PCBs. Additionally, increasing \(C_{p}\) reduces the \(f_{L}\) value. However, increasing \(C_{p}\) can be accomplished by using a high dielectric constant material, which is a costly and unreliable method. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{d_{c}}\) & \(\mathbf{Z_{oc}}\) & \(\mathbf{C_{p}}\) & \(\varepsilon_{\mathbf{r}}\) & \(\mathbf{L_{\mathit{eff}}}\) \\ \hline Values & 5.2 mm & 9.2 \(\Omega\) & 10.3 pF & 4.4 & 0.9 nH \\ \hline \hline \end{tabular}
\end{table}