## 3 Floquet Analysis
### Analysis of the IEP Effect on Stopband  
To obtain a dispersion diagram and examine the IEP effects, an equivalent circuit model of the UC of the proposed IEP-EBG structure was derived using transmission lines and a LC resonator, as shown in Figure 2. In the equivalent circuit model, only one-dimensional (1-D) propagation of the SSN is considered to simplify the dispersion analysis. However, it can be extended for a 2-D propagation analysis. The difference in the cut-off frequencies of 1-D and 2-D propagation analysis can be further investigated [17]. As described previously, the IEP technique significantly increases the inductance value of the EBG structure. The inductor enhanced by the IEP technique is represented as \(L_{\textit{eff}}\) in the equivalent circuit model. The transmission line depicts the power and ground planes in the UC, which are vertically separated by the distance \(h_{c}\). Its characteristic impedance \(Z_{\textit{oc}}\) is mainly determined by the UC width \(d_{c}\), dielectric thickness \(h_{c}\), and relative dielectric permittivity \(\varepsilon_{r}\). The \(Z_{\textit{oc}}\) value can be calculated by the closed-form equation in [12], or numerical field solvers such as HFSS, MWS, and Momentum when physical dimensions are given. The patch and corresponding ground plane forms a capacitor with  
Figure 2: An equivalent circuit model of the UC of the proposed structure using an inductance-enhanced patch (IEP-EBG) for Floquet analysis. a capacitance value of \(C_{p}\). The via and IEP induces an inductance \(L_{\textit{eff}}\), which is the sum of the via inductance and the patch inductance, which is significantly increased by the IEP. The LC resonator formed by the capacitor and inductor are connected in shunt with transmission lines. The \(C_{p}\) is simply extracted from a rectangular-shape parallel plate capacitor. The via inductance and patch inductance enhanced by the IEP are difficult to extract separately using an analytical equation or numerical solvers. In this paper, the sum of the via and IEP inductances were used as the \(L_{\textit{eff}}\) for the simplicity of analysis. To obtain the dispersion characteristics, we applied [PERSON]'s theory to the IEP-EBG structure. [PERSON]'s theory for periodical structures is based on a UC analysis. It assumes that UCs are arranged periodically and infinitely. [PERSON]'s theory shows that the voltages and currents at all ports of a UC can be defined, and their relationships are expressed as network parameters. In particular, the use of ABCD parameters results in the simplification of the dispersion analysis for a two-port UC. For the IEP-EBG structure, the ABCD parameter of the UC is expressed as a product of the ABCD parameters of the transmission lines and the LC resonator circuit as follows:  
\[\left( \begin{array}{cc}A_{\textit{eq}}&B_{\textit{eq}}\\ C_{\textit{eq}}&D_{\textit{eq}}\end{array} \right)=\left( \begin{array}{cc}A _{\textit{T}}&B_{\textit{T}}\\ C_{\textit{T}}&D_{\textit{T}}\end{array} \right)\left( \begin{array}{cc}A_{ \textit{R}}&B_{\textit{R}}\\ C_{\textit{R}}&D_{\textit{R}}\end{array} \right)\left( \begin{array}{cc}A_{ \textit{T}}&B_{\textit{T}}\\ C_{\textit{T}}&D_{\textit{T}}\end{array} \right) \tag{1}\]  
where  
\[\left( \begin{array}{cc}A_{\textit{T}}&B_{\textit{T}}\\ C_{\textit{T}}&D_{\textit{T}}\end{array} \right)=\left( \begin{array}{cc}cos( \beta_{c}d_{c}/2)&jZ_{\textit{oc}}sin(\beta_{c}d_{c}/2)\\ j(Z_{\textit{oc}})^{-1}sin(\beta_{c}d_{c}/2)&cos(\beta_{c}d_{c}/2)\end{array} \right) \tag{2}\]  
\[\left( \begin{array}{cc}A_{\textit{R}}&B_{\textit{R}}\\ C_{\textit{R}}&D_{\textit{R}}\end{array} \right)=\left( \begin{array}{cc}1&0\\ Y_{\textit{R}}&1\end{array} \right) \tag{3}\]  
\[Y_{\textit{R}}=(j\omega C_{\textit{P}})/\left(1-\omega^{2}L_{\textit{eff}}C_{ \textit{P}}\right). \tag{4}\]  
Evaluating the \(A_{\textit{eq}}\) from the equations above, the dispersion equation is derived in terms of the characteristics of the transmission lines and the resonator circuit.