## 2 Design of Compact and Multi-Stack Electromagnetic Bandgap (CMS-EBG) Structure
### Characteristic-Impedance Analysis  
The principle of noise suppression of the CMS-EBG structure is based on the stepped impedance resonator. To explain the noise suppression characteristics, a lattice diagram for the characteristic impedance (Z\({}_{\rm o}\)-lattice diagram) is extracted from the CMS-EBG structure as shown in Figure 3. The characteristic impedances and the associated physical geometries of the unit cells are shown in the Z\({}_{\rm o}\)-lattice diagram. The aforementioned low-Z\({}_{\rm o}\) part is represented as Z\({}_{\rm L}\), and the high-Z\({}_{\rm o}\) part of the VCA branch is represented as Z\({}_{\rm H}\). The Z\({}_{\rm o}\)-lattice diagram describes the relationship between the unit cells considering the electromagnetic wave theory. The Z\({}_{\rm o}\)-lattice diagram shows that the stepped impedance characteristics are formed in the CMS-EBG structure, and the low- and high-Z\({}_{\rm o}\) parts are two-dimensionally and quasi-periodically arranged. The multi-stack technique achieves the 3-D physical geometry of the EBG structure which contains a 2-D lattice structure of the characteristic impedances.  
In the Z\({}_{\rm o}\)-lattice diagram, adjacent low-Z\({}_{\rm o}\) parts (Z\({}_{\rm L}\)) in the same tier are not connected with one another. Each low-Z\({}_{\rm o}\) part connects to the low-Z\({}_{\rm o}\) parts in different tiers through the high-Z\({}_{\rm o}\) parts. All connections between the low-Z\({}_{\rm o}\) parts are implemented by the same VCA branches represented as Z\({}_{\rm H}\). Thereby, the stepped impedance characteristics of the CMS-EBG structure are symmetric and periodical, whereas the physical structure is three-dimensionally complex for the compact form factor. The 2-D periodic arrangement without a defect ensures a broadband stopband and a simple estimation of the stopband using a dispersion analysis for the periodic structure. This distinguished characteristic of the CMS-EBG structure is accomplished by a multi-stack technique based on the original configuration with VCA branches.  
Figure 3: A lattice diagram of the characteristic impedance of the CMS-EBG structure.  
To obtain the wideband GHz noise suppression with an extremely short source-to-victim distance using the CMS-EBG structure, the port configuration needs to be further explained. In the CMS-EBG structure, a noise source and a victim can be connected to the unit cells in different tiers. Suppose that a noise source device and a victim device are mounted on the top or bottom layer of the CMS-EBG structure. Then, the noise source and victim can be vertically connected to UC\({}_{(5,1)}\) and UC\({}_{(1,4)}\), respectively, through the via structures. The lateral distance between the noise source and a victim is extremely short because UC\({}_{(5,1)}\) and UC\({}_{(1,4)}\) are closely placed in the horizontal view. However, the noise source and a victim are sufficiently separated in the view of the \(Z_{\text{o}}\)-lattice diagram, as shown in Figure 3. Figures 2 and 3 depict the locations of the noise source and victim, which are represented as Port 1 and Port 2. This configuration is fairly reasonable, and the usage of the CMS-EBG structure is not severely limited.