## 3 Floquet Analysis
### Demonstration of \(f_{\text{L}}\) Reduction and Miniaturization  
In this section, the \(f_{\text{L}}\) reduction and miniaturization of the IEP-EBG structure were demonstrated by applying Floquet theory to the example designs. The stopband characteristics of the example designs of the proposed IEP-EBG structures were compared to those of the conventional mushroom-type EBG (MT-EBG) structure. Their dispersion diagrams were derived using Floquet theory based on a full-wave simulation. The ABCD parameters of the proposed IEP-EBG and conventional MT-EBG structures were extracted using finite element method (FEM) simulations performed by HFSS. In this analysis, the FEM simulation technique was employed for a practical design approach based on the geometric design. The \(f_{\text{L}}\) reduction was examined for the proposed IEP-EBG and previous MT-EBG structures having the same UC size. The design parameters and corresponding dimensions are summarized in Table 2. The dimensions were selected considering a conventional design guide for high-speed packages and PCBs. The copper and FR-4 were adopted as the metal in the conduction layers and the dielectric material for the example design. The copper thickness and relative permittivity of the dielectric material were assumed to be 18 \(\upmu\)m and 4.4, respectively. The dispersion characteristics were extracted as shown in Figure 6. The \(f_{\text{L}}\) value of the previous MT-EBG structure was 4.1 GHz, while that of the proposed IEP-EBG structure was 1.6 GHz. With the same UC size, the proposed  
Figure 5: IEP effects on the high cut-off frequency \(f_{H}\) with various \(L_{\textit{eff}}\) values. IEP-EBG structure significantly reduced the \(f_{L}\) value, up to 61%, when compared to the conventional MT-EBG structure. To demonstrate the miniaturization, we compared the dispersion characteristics of the proposed IEP-EBG and previous MT-EBG structures, which had the same \(f_{L}\) values of 1.6 GHz, as shown in Figure 7. The dimensions of the IEP-EBG structure were the same as the example structure shown in the \(f_{L}\) reduction analysis. The MT-EBG structure was enlarged from 5.2 mm to 11.7 mm to have the same \(f_{L}\) as the IEP-EBG structure. As shown in Figure 7, the IEP structure achieved a significant reduction in UC size when compared to the MT-EBG structure, while the stopband of the IEP-EBG structure was not significantly reduced. Figure 6: Dispersion analysis of the \(f_{L}\) reduction of the IEP-EBG structure. Figure 7: Miniaturization analysis of the UC of the IEP-EBG structure using a dispersion diagram. An 80% size reduction was accomplished using the IEP technique without stopband deterioration, additional materials, or cost increase. In the analysis, the distinguished characteristics of the \(f_{L}\) reduction and miniaturization of the IEP-EBG structure were proven for the specific design. However, it should be noted that the design is not limited to just this specific case. To further investigate the effects of design parameters such as the slot length \(d_{s}\) and the slot rotation angle \(\theta_{s}\), the stopband characteristics were obtained using Floquet analysis for various values of design parameters. Figure 8 depicts the effects of slot length on the \(f_{L}\) and \(f_{H}\) with respect to various slot length values, varying from 0.2 mm to 1.8 mm. As shown in Figure 8, the \(f_{L}\) and \(f_{H}\) were approximately inversely proportional to the slot length \(d_{s}\). This can be explained by the \(d_{s}\) increase, resulting in the large effective inductance \(L_{\textit{eff}}\) of the UC and lowering of the stopband. The dispersion characteristics were compared for three cases of the slot rotation angle 0, \(\pi/4\), and \(\pi/2\) radians in Figure 9. The Floquet analysis was performed for those cases. As seen in the results, \(f_{L}\) and \(f_{H}\) were not significantly different for the various slot rotation angles.