## 2 EBG for Microstrip Patch Antenna  
The previous section provides a general definition of the electromagnetic band gap (EBG), which originates from the photonic band gap (PBG) phenomenon observed in optics [14], and the term 'EBG' was introduced in [15]. There are various types of EBGs but they are generally classified into three main groups: (1) 1D transmission lines, (2) 2D planar surfaces and (3) 3D volumetric structures [16]. Our study focuses on the 2D planar surfaces EBG group, specifically the mushroom-like EBG structure, which was first introduced in 1999 by [PERSON] et al. [17]. This structure consists of a ground plane, a dielectric substrate on which metallic plates are mounted, and metallic vias to connect these plates through the substrate to the ground. The operation mechanism of the mushroom-like EBG structure can be explained by referring to an equivalent parallel resonant LC circuit, as described in Figure 1.  
As can be seen from Figure 1a, two adjacent metallic plates have connecting vias (denoted as black dots in the top view) to the ground plane. The gap between the plates can be considered as a capacitive component, while the vias provide inductive paths to the ground plane. In Figure 1b, when the EBG structure interacts with the electromagnetic wave, a voltage applied along the top surface leads to an accumulation of charges at the edges of the plates, thus forming a capacitance. As the charges travel back and forth between the metallic plates and the ground plane through the inductive paths created by the vias, an inductance is created [18]. The bandgap formation can be explained by considering the resonance of the parallel LC circuit at a specific frequency, where a stopband is created and prevents the surface waves from propagating. The values of \(\mathbf{L}\) and \(\mathbf{C}\) can be varied by modifying the physical dimensions such as the size of the EBG elements, the gap between them, the length of the ground connecting vias, as well as the dielectric constant of the substrate. Therefore, the frequency band gap of the structure can be tuned. In further detail, larger EBG element dimensions increase the capacitance, and smaller gaps enhance the interactions of the capacitive components, leading to a decrease in the bandgap frequency. Meanwhile, increasing the via lengths will cause a higher inductance which reduces the bandgap frequency. Furthermore, when increasing the dielectric substrate thickness, enhanced fringing fields improve the antenna radiation efficiency that leads to a  
Figure 1: (**a**) A demonstration of two adjacent elements of the mushroom-like EBG, a top view and side view; (**b**) an illustration of the capacitive and inductive components, and the equivalent parallel LC circuit of this EBG structure. The red arrow indicates the inductive path between the ground and the EBG patches. higher gain. However, excessive thickness can cause stronger surface wave propagation, which deteriorates the radiation performance. The advantages of the mushroom-like type over other EBG types are the good band gap performance, compact design, and the ease of fabrication and integration on printed circuit boards (PCBs), making it a good candidate for planar and microstrip patch antenna applications. The mushroom-like EBG can be applied to single-element MPAs [9; 10; 19; 20; 21; 22; 23], as well as MPAs in an array configuration [24; 25; 26; 27]. Figure 2 shows two examples of the deployment of mushroom-like EBG structures for these two cases. The enhancement of antenna gain and radiation efficiency due to surface wave suppression can be explained for these two cases as follows: For the single-element MPA, the surface waves travel along the substrate and radiate from the edges of the antenna, contributing to the unwanted edge radiation that degrades the antenna efficiency. By filling up the non-metal space from the center radiator to the edges of the MPA by EBG elements, stopband 'rings' are created and prevent the surface wave from going beyond the patch area.