## 2 Design of Compact and Multi-Stack Electromagnetic Bandgap (CMS-EBG) Structure  
### Geometrical Design  
The CMS-EBG structure is designed to exhibit a wideband suppression of GHz noise with a compact area and a short source-to-victim distance. To achieve this objective, a multi-stack technique is applied to the CMS-EBG structure. The multi-stack technique is the original configuration of a stepped impedance resonator, which consists of a low-characteristic-impedance (low-\(\mathrm{Z_{o}}\)) part and a high-\(\mathrm{Z_{o}}\) part. As illustrated in Figure 2a, the low-\(\mathrm{Z_{o}}\) part consists of a rectangular-shape low-\(\mathrm{Z_{o}}\) patch in a power layer (P) and two rectangular planes (G) in the ground layers (G), which are located above and below the low-\(\mathrm{Z_{o}}\) patch. The dielectric material is located between the power and ground layers. The high-\(\mathrm{Z_{o}}\) part is formed by two narrow high-\(\mathrm{Z_{o}}\) patches and a via structure. The high-\(\mathrm{Z_{o}}\) patches are placed in different layers and connected through the via structure. The high-\(\mathrm{Z_{o}}\) part is denoted as a vertical-cell-array (VCA) branch. The ground via structures should be placed adjacent to the VCA branch, but the ground via are not shown in Figure 2a for simplicity. The ground via is described in detail in Figure 4. In addition to the VCA branch, the term 'VCA' implies a vertical stack of low-\(\mathrm{Z_{o}}\) patches. The CMS-EBG structure contains four VCAs as shown in Figure 2a. Each VCA includes a stack of four low-\(\mathrm{Z_{o}}\) patches. To explain the CMS-EBG structure, a unit cell is defined as the low-\(\mathrm{Z_{o}}\) parts with half of the VCA branches as shown in Figure 2a. The notation \(\mathrm{UC}_{(i,j)}\) is adopted to express the unit cell in the \(i\)-th VCA and \(j\)-th tier. The VCA branch was originally developed to connect low-\(\mathrm{Z_{o}}\) patches in different tiers or different UCs. Two types of connection between low-\(\mathrm{Z_{o}}\) patches in different tiers are considered in the CMS-EBG structure. First, low-\(\mathrm{Z_{o}}\) patches in different tiers but the same VCA are considered, which is called an intra-VCA branch. In Figure 2a, the intra-VCA branch is highlighted with a blue dashed line, which connects the low-\(\mathrm{Z_{o}}\) patches in \(\mathrm{UC}_{(4,2)}\) to \(\mathrm{UC}_{(4,3)}\). The other type is an inter-VCA branch, which connects low-\(\mathrm{Z_{o}}\) patches in different tiers and different VCAs. In Figure 2a, for instance, the inter-VCA branch is highlighted to connect a low-\(\mathrm{Z_{o}}\) patch in \(\mathrm{UC}_{(3,5)}\) to a low-\(\mathrm{Z_{o}}\) patch in \(\mathrm{UC}_{(4,4)}\). To more clearly explain the intra-/inter-VCA branches, only the low-\(\mathrm{Z_{o}}\) patches and VCA branches are shown in the other view point in Figure 2b, where the red and blue arrows point at the inter-VCA branches and intra-VCA branches, respectively. All low-\(\mathrm{Z_{o}}\) patches have one or two intra-VCA branches. For the inter-VCA branch, the low-\(\mathrm{Z_{o}}\) patches in VCA 1 and VCA 4 contain one inter-VCA branch, whereas each low-\(\mathrm{Z_{o}}\) patch in VCA 2 and VCA 3 contains two inter-VCA branches. For example, \(\mathrm{UC}_{(1,5)}\) has an intra-VCA branch to connect with \(\mathrm{UC}_{(1,5)}\) and an inter-VCA branch to connect with \(\mathrm{UC}_{(2,4)}\). For \(\mathrm{UC}_{(3,5)}\), two inter-VCA branches are used to connect with \(\mathrm{UC}_{(2,4)}\) and \(\mathrm{UC}_{(4,4)}\), whereas one intra-VCA branch connects it to \(\mathrm{UC}_{(3,4)}\). In summary, the intra-VCA branch connects the UCs in different tiers but the same VCA, whereas the inter-VCA branch connects the UCs in different tiers and different VCAs. Due to the original configuration of the UCs and VCA branches, the CMS-EBG structure is able to place numerous EBG cells in a compact horizontal area of MPCBs.