## 3 Floquet-Bloch Analysis  
Floquet-Bloch analysis based on a unit cell is adopted to examine and estimate the stopband characteristics of the CMS-EBG structure. For a simple analysis, a new unit cell is defined, as shown in Figure 4a. The new definition of the unit cell is based on an irreducible Brillouin zone (IBZ) for the one-dimensional Floquet-Bloch analysis. The IBZ unit cell consists of half of the low-\(Z_{\text{o}}\) patch, the corresponding ground planes, the VCA branch, and ground vias. The ground vias are not shown in Figure 2 for simplicity. However, the ground vias must connect all ground planes in the CMS-EBG structure as shown in Figure 4a. Figure 4: **(a)** A unit cell with design parameters for Floquet-Bloch analysis and **(b)** an equivalent circuit model using a transmission line circuit to extract a dispersion equation. The equivalent circuit for one-dimensional propagation through the IBZ unit cell is extracted in Figure 4b. Half of the low-Z\({}_{\text{o}}\) patch is equivalently modeled as a transmission line circuit with a characteristic impedance of \(Z_{o,L}\), a propagation constant of \(\beta\), and the length of \(W_{P}/2\) while the equivalent circuit model of the VCA branch is a series of inductors. The inductances \(L_{b}\) and \(L_{v}\) are induced from the narrow patch in the high-Z\({}_{\text{o}}\) patch and via structure, respectively. The inductance of the ground vias is ignored to simplify the calculation. To derive the dispersion characteristics under the assumption of the Bloch wave, the ABCD parameters of the equivalent circuits are considered. The relationship between ABCD parameters are given by  
\[\left( \begin{array}{cc}A_{L}&B_{L}\\ C_{L}&D_{L}\end{array} \right)\left( \begin{array}{cc}A_{H}&B_{H}\\ C_{H}&D_{H}\end{array} \right)\left( \begin{array}{cc}A_{L}&B_{L}\\ C_{L}&D_{L}\end{array} \right)=\left( \begin{array}{cc}A_{\text{eq}}&B_{\text {eq}}\\ C_{\text{eq}}&D_{\text{eq}}\end{array} \right), \tag{1}\]  
where,  
\[\left( \begin{array}{cc}A_{L}&B_{L}\\ C_{L}&D_{L}\end{array} \right)=\left( \begin{array}{cc}cos(\beta W_{P}/2)&jZ _{\text{o}}Lin(\beta W_{P}/2)\\ j(Z_{\text{o}L})^{-1}sin(\beta W_{P}/2)&cos(\beta W_{P}/2)\end{array} \right), \tag{2}\]  
\[\left( \begin{array}{cc}A_{H}&B_{H}\\ C_{H}&D_{H}\end{array} \right)=\left( \begin{array}{cc}1&j\omega(2L_{b}+L_{v}) \\ 0&1\end{array} \right).\]  
The effective phase constant for the dispersion characteristics is obtained by calculating the A\({}_{\text{eq}}\) component of the equivalent transmission line  
\[\beta_{eff}=\frac{1}{W_{P}}cos^{-1}\bigg{\{}cos(\beta W_{P})-\frac{\pi fL_{ eff}}{Z_{oL}}sin(\beta W_{P})\bigg{\}}, \tag{3}\]  
where the effective inductance \(L_{\textit{eff}}\) of the VCA branch is \(2L_{b}+L_{V}\). \(Z_{oL}\) can be found using the equation in [14; 19] and a computer-aided simulation. \(L_{b}\) and \(L_{v}\) are given by [23; 24]  
\[L_{b}=\mu_{o}h\frac{d_{b}}{2W_{b}}, \tag{4}\]  
\[L_{v}=\frac{\mu_{o}h}{4}\left(ln\left(\frac{w_{p}^{2}}{\pi r_{v}^{2}}\right)+ \frac{\pi r_{v}^{2}}{w_{p}^{2}}-1\right), \tag{5}\]  
Using the effective phase constant \(\beta_{\textit{eff}}\), the electromagnetic bandgap characteristics of the CMS-EBG structure are explored. To present the effects of the design parameters on the bandgap characteristics, example values of the design parameters (\(w_{p}\), \(w_{b}\), \(d_{b}\), \(S_{b}\), \(S_{v}\), \(r_{v}\), \(t\), \(h\)) are selected. \(w_{p}\) is the length of the low-Z\({}_{\text{o}}\) patch. \(w_{b}\), \(d_{b}\), and \(S_{b}\) are the width, length, and location of the high-Z\({}_{\text{o}}\) patch, respectively. \(r_{v}\) is the radius of the via in the VCA branch and the ground vias. \(t\) is the thickness of the metal. \(h\) is the distance between metal layers. The nominal value of the design parameters \(w_{p}\), \(w_{b}\), \(d_{b}\), \(S_{b}\), \(S_{v}\), \(r_{v}\), \(t\) and \(h\) are 11, 1, 1, 5, 1, 0.2, 0.017, and 0.1 mm, respectively, which are practical and widely used in MPCBs.