EBG unit-cell design and simulation

In this section, the EBG substrate is designed. The equivalent circuit of conventional mushroom EBG structures is shown in Fig. 1. In this EBG structure, vias and the patches play the roles of parallel and series inductors, respectively. The gap between the two elements of the single cell constitutes the series capacitor, while the ground plane and the patch are modeled as a parallel capacitor.

Fig. 1. The equivalent circuit of the EBG unit cell: (a) conventional unit cell, (b) ring resonator unit cell.

The equivalent circuit [Reference Gangwar and Gupta27] of the structure used in this article is the same except the EBG structure can better control the inductances [Reference Chen, Ran, Huangfu, Grzegorczyk and Kong28, Reference Islam, Faruque and Islam29].

The bandwidth of the planar antenna associated with the inductance and capacitance of the equivalent circuit is [Reference Yousefi, Mohajer-Iravani and Ramahi30]

(1)$$BW\propto \sqrt {\displaystyle{L \over C}} \comma \;$$

where inductances are given by [Reference Patel, Argyropoulos and Kosta8, Reference Jafargholi and Mazaheri31, Reference Paul32]

(2)$$\eqalign{\mathop L\nolimits_R \,& = \displaystyle{{\mu _0} \over \pi }\sqrt {\mathop e\nolimits^2 + \mathop f\nolimits^2 } \,\left[{\sqrt 2 \,Ln\displaystyle{{2\,\sqrt {\mathop e\nolimits^2 + \mathop f\nolimits^2 } } \over w}\,-1} \right]\cr L_{via}\,& = 5\cdot 08\,t\left({Ln\displaystyle{{2t} \over r}-1} \right).} $$

In which “e” and “f” are ring dimensions and “w” is the width of its strip as shown in Fig. 2. Therefore,

(3)$$BW\propto \sqrt {\displaystyle{{\mu _r} \over {\varepsilon_r}}}.$$

Fig. 2. Ring resonator unit cell.

According to equation (2), in addition to the unit-cell dimensions, the ring width, w, is also one of the parameters affecting the inductive property and consequently the resonant frequency of the structure.

To calculate the relative permeability, relative permittivity, and the refractive index of the EBG unit cell, the ring resonator unit cell with boundary conditions given in Fig. 3 is simulated in CST. PEC boundary conditions are used on two sides and PMC was used on the other two, indicating the periodic structure. To excite the structure, two wave ports were used and the relative permeability, the relative permittivity, and the refractive index were extracted from the S parameters [Reference Chen, Grzegorczyk, Wu, Pacheco and Kong33].

(4)$$S_{11} = S_{22} ={-}\displaystyle{1 \over 2}\left({z-\displaystyle{1 \over 2}} \right)\sin \lpar nkh\rpar \comma \;$$

(5)$$S_{12} = S_{21} = \displaystyle{1 \over {\cos \lpar nkh\rpar -\displaystyle{i \over 2}\lpar z + \displaystyle{1 \over z}\rpar \sin \lpar nkh\rpar }}.$$

Fig. 3. Boundary conditions of the simulated ring resonator of the EBG unit cell.

Equations (4) and (5) result in:

(6)$$z = \sqrt {\displaystyle{{{\left( {1 + S_{11}} \right)}^2-S_{21}^2 } \over {{\left( {1-S_{11}} \right)}^2-S_{21}^2}}},$$

(7)$$n = \displaystyle{1 \over {kh}}\cos ^{{-}1}\left[{\displaystyle{1 \over {2S_{21}}}\lpar {1-S_{11}^2 + S_{21}^2 } \rpar } \right]\comma \;$$

(8)$$\varepsilon = \displaystyle{n \over z}\,\comma \;\,\mu = nz.$$

With an increased relative permeability and a decreased relative permittivity of a material, wider operation bandwidth can be achieved. The EBG structure can increase the inductance of the circuit and therefore results in bandwidth improvement. Relative permeability of the ring resonator unit cell and a conventional unit cell of the same area are shown in Fig. 4. As can be observed, the relative permeability of the ring resonator unit cell can be changed simply by varying its width, w. This can be ascribed to yet another degree of freedom which leads to increased inductive properties and, as a result, an improved bandwidth.

Fig. 4. Relative permeability of the conventional mushroom unit cell and the ring resonator unit cell for different widths of the ring, w (mm).

A parametric study is performed to select the dimensions of the unit cell for a resonant frequency of 9.7 GHz. The parameter “d” is the periodicity of the neighboring cells. The results are illustrated in Tables 1–5. As can be seen, the parameters “e”, “f”, and “d” do not significantly change the resonant frequency while “t” and “w” affect the resonant frequency of the structure. The final values of these parameters for the resonant frequency of 9.7 GHz are shown in Table 6. The constitutive parameters of the optimized EBG unit cell are illustrated in Table 7.

Table 1. Effect of parameter “e” on the resonant frequency

Table 2. Effect of parameter “f” on the resonant frequency

Table 3. Effect of parameter “t” on the resonant frequency

Table 4. Effect of parameter “d” on the resonant frequency

Table 5. Effect of parameter “w” on the resonant frequency

Table 6. Dimensions of the unit cell

Table 7. Constitutive parameters of the EBG unit cell

EBG substrate design and simulation

To obtain an improved bandwidth for a transmission coefficient, S ₂₁, lower than −10 dB, the rings and vias dimensions and vias locations are the parameters to be optimized. The EBG structure used in this paper (Fig. 5) includes a 4 × 9 array of square rings on a layer of Rogers 4003C (ɛ_r = 3.58) with 0.8 (mm) thickness and a ground plane of 40 mm × 30 mm. The square patch antenna is designed to operate at 9.7 GHz.

Fig. 5. The schematic configuration of the electromagnetic band gap (EBG) layer.

To find the stopband of the EBG substrate, the two ports are assumed at the opposite edges of the bottom and a transmission line is located on the top of the EBG structure. The scattering parameters can then be extracted as shown in Fig. 6. The transmission coefficient, S ₂₁, is calculated by exiting the ports. The stopband, defined for S ₂₁ lower than −10 dB, is obtained as 8.4–10.7 GHz and the resonant frequency is 9.7 GHz as expected.

Fig. 6. S ₂₁ versus frequency for the EBG and the schematic representation of the transmission line on the EBG structure.

Patch antenna simulation

A conventional patch antenna is designed at 9.7 GHz. To excite the antenna, a microstrip line with a width of 1.9 mm is used corresponding to an input impedance of 50 Ω. Two gaps with the dimensions of 3.5 (mm) × 0.26 (mm) are used for proper impedance matching as shown in Fig. 7. The 10 dB return loss bandwidth is 0.2 GHz and the gain at resonance is 7.71 dBi. The dimensions of the patch antenna are obtained by antenna design equations [Reference Bhalla and Bansal34]. For impedance matching, the inset feed is used as shown in Fig. 7.

Fig. 7. The schematic representation of the conventional antenna.

The dimensions of the designed antenna at 9.7 GHz are given in Table 8.

Table 8. Dimensions of the antenna