## 2 Design and Analysis
### DGS Structure Analysis  
The concept of the defected ground structure (DGS) originated from the research on the photonic bandgap structure (PBG). The name of the concept \"DGS\" simply means that a defect has been in the ground plane which could be considered as an approximation of an infinite and perfectly conducting current sink. It is of a compact geometry which is commonly known as a unit cell that is etched out in the backside metallic ground plane as a single defect or a periodic configuration. This artificial structure prevents the electromagnetic (EM) waves from propagating through the defects over a range of frequencies (stopband), but allow the EM waves to pass through them (defects) over a range of frequencies (passband). That's why PBG's implemented in EM applications are also referred to as electromagnetic band gap (EBG) structures. However, this nature of influencing the guided wave characteristics produces bandgap properties and slow wave effects which help provide more compact printed circuits.  
In terms of current distribution, the presence of these slots under the printed transmission line disturbs the current distribution in the ground plane. This leads to a reduction in the image current, and therefore, the surface wave reduces. Moreover, due to the reduction of the surface wave, the stored energy as well as the Q factor decreases and results in bandwidth enhancement.  
Previously, a number of DGS structures including unit cells and periodic configurations with 1-D, 2-D, or with uniform or non-uniform structures have been studied for various purposes such as the implementation of filters, suppression of unwanted surface waves, and control of harmonics in microstrip antennas, to achieve compactness and improve performance in terms of stopbands and passbands. In this paper, a 2-D multiperiod EBG is analyzed with the stub-notch configuration to study the effect of this structure on the antenna performance.  
In the proposed design, the effect of loading a 2-D EBG with two periods on the stub-notch configuration has been studied and compared with the dumbbell shaped DGS. The dimensions of the EBG structure are generated by using the conventional equations in [24].  
The equivalent circuit of a pair of rectangular EBG lattice shapes etched in the metallic ground plane parallel with respect to the transmission line can be obtained from the equivalent circuit of a dumbbell-shaped lattice as shown in Figure 4a [25]. The equivalent circuit of a dumbbell-shaped lattice etched perpendicularly in the ground plane with respect to the transmission line is shown in Figure 5 which is a parallel LC resonator. For this type of unit cell, the increased lattice area increases the series inductance of the transmission line and so the resonance shifts to a lower frequency. The variation of the gap distance has little effect on the passband frequency but affects the attenuation pole location. The change in attenuation pole location means that there is a parallel capacitance (C) with a series inductance (L) that is provided by the gap distance [26].  
Figure 3: The current distribution of the antenna without DGS. (**a**) 3.736 GHz; (**b**) 6.0734 GHz.  
The same analysis has been performed on the dumbbell-shaped lattice which is placed parallel with respect to the transmission line as shown in Figure 4b and got the same results, shown in Figures 6 and 7 as for the dumbbell-shaped DGS unit perpendicularly placed with respect to the transmission line shown in Figure 4a.  
Figure 4: Different DGS configurations. (**a**) Parallel dumbbell-shaped unit cell; (**b**) Perpendicular dumbbell-shaped unit cell; (**c**) Proposed electromagnetic band gap (EBG) structure.  
Figure 5: An equivalent circuit to the Dumbbell-shaped DGS. \(Z_{0}\) denotes the scaled impedance level of the in/out terminated ports.  
Figure 6: Effect of the lattice size on the S-parameters of the parallel dumbbell-shaped DGS unit cell with a constant gap width of 1 mm. (**a**) Reflection coefficient (S11) (**b**) Transmission coefficient (S21).