*2.2.3 Equivalent circuit parameters*

Extracted equivalent-circuit parameters (equivalent inductance, L and capacitance, C) of the proposed DGS unit section for different dimensions are calculated by (Eq. (1)) [4] and Eq. (2) [4] and listed in **Tables 4** and **5** given below.

**Figure 22.** *Tuning of S-parameter of fc and f0 with a split gap (g2).*

*Filter Designs Based on Defected Ground Structures DOI: http://dx.doi.org/10.5772/intechopen.103065*

#### **Figure 23.**

*Variation of the ratios responses with the split gap (g2).*


#### **Table 4.**

*Extracted equivalent-circuit parameters of the proposed split ring DGS unit section having dimensions g1 and g2 kept fixed (*g*1 =0.3 mm;* g*2 =0.2 mm).*


#### **Table 5.**

*Extracted equivalent-circuit parameters of the proposed split ring DGS unit section having dimensions r2 and g2 kept fixed (*r*2 = 3.3 mm;* g*2 = 0.2 mm).*

$$\mathbf{C} = \frac{a\_{\mathbf{c}}}{\mathbf{Z}\_{\mathbf{0}} \mathbf{g}\_1 \left(a\_0^2 - a\_{\mathbf{c}}^2\right)}\tag{1}$$

$$L = \frac{1}{4\pi^2 f\_0^2 C} \tag{2}$$

#### *2.2.4 Improvement of the response*

To increase the order of the bandstop filter, two split ring type DGS units are introduced in the ground plane under the 50-Ohm microstrip line, as shown in **Figure 24**. Here in this design, both the DGS units have equal dimensions. The dimensions of the DGS units are: - outer radii of the rings, *r*<sup>2</sup> = 3.3 mm, inner radii of the rings, *r*<sup>1</sup> = 3.0 mm, the widths of the rings, *g*<sup>1</sup> = 0.3 mm, the widths of the gaps, *g*<sup>2</sup> = 0.2 mm, the length between the two split ring units, L = 17.4 mm, widths of the 50 Ohm microstrip line, W = 1.92 mm.

**Figure 24.** *Schematic diagram of the lowpass filter using double unit split ring DGS.*

#### *2.2.5 Simulated result of dual units*

The S-parameter response of this structure is given in **Figure 25**. It is clear from the figure that the filter has a cutoff frequency of 3.9 GHz and insertion loss of 0.1 dB. The sharpness of this filter is quite good, about 54 dB/GHz. The main disadvantage of the filter is the poor out-of-band performances, i.e., the stopband outside the passband only ranges from 4.38 GHz to 4.76 GHz at the attenuation level of –20 dB.

#### **2.3 Improvement of out of band performance using open stub**

This disadvantage can be overcome by introducing another transmission zero at the region of the stopband [15, 16]. An open circuited stub employed at the top plane creates this additional transmission zero, as shown in **Figure 26**. A compensated high-low impedance line is employed in the microstrip line above the DGS structure for matching purposes. The physical dimensions of the modified structure with open stub are: *W*<sup>1</sup> = 8.5 mm, *W* = 1.92 mm, *g*<sup>1</sup> = 0.3 mm, *g*<sup>2</sup> = 0.2 mm, L = 9.1 mm, *r*<sup>1</sup> = 3 mm, *r*<sup>2</sup> = 3.3 mm. The shunt capacitor due to the stub is given by the following (Eq. (3)) [9].

**Figure 25.** *S-parameter result of the lowpass filter using double unit split ring DGS.*

*Filter Designs Based on Defected Ground Structures DOI: http://dx.doi.org/10.5772/intechopen.103065*

**Figure 26.** *Schematic diagram of the bandstop filter using a split ring DGS section and open-ended stub.*

Here the left side of the equation is the susceptance of the shunt capacitor, and the right side of the equation represents the input susceptance of the open-circuited stub, which has the characteristic impedance "*ZOC*," and the physical length "*W*1" (*W*<sup>1</sup> < λg/4) of the stub. The physical length of the open circuited stub is determined by the (Eq. (4)) below [9].

$$\mathcal{W}\_1 = \frac{\lambda\_{\rm gc}}{2\pi} \tan^{-1} \mathcal{w}\_\ell \mathcal{C}\_2 \mathcal{Z}\_{\rm OC} \tag{4}$$

#### *2.3.1 Simulated response of the structure*

The simulated S-parameter response is shown in **Figure 27**. Two prominent resonances are found in the response. The first one is at 4.7 GHz and is responsible for the DGS unit with the same dimensions as given above, and the second pole is at 5.1 GHz and is responsible for the stub. The upper and lower cutoff frequencies are 4.1 GHz and 5.9 GHz, respectively. The filter provides a –20 dB bandwidth of 0.9 GHz and a rejection level of 30 dB. The sharpness of the lower and upper edge of the band are

**Figure 27.** *S-parameter response of the lowpass filter using double unit split ring DGS sections and open-ended stub.*

60.1 dB/GHz and 54.8 dB/GHz, respectively. Maximum insertion loss at the passband is 0.2 dB.

#### *2.3.2 Equivalent circuit model of the filter*

The equivalent circuit of the proposed filter is given in **Figure 28**, where circular split ring DGS is represented by a parallel resonant circuit connected in series with the source, and an open stub is referred to as a series resonant circuit connected in shunt with the source. The inductance (L2) is obtained due to the high value of *Z*0*c*. The comparison of the simulated and circuit response is given in **Figure 29**.

#### *2.3.3 Parametric study of the filter*

By changing either the dimension (both inner radius "*r*1" and outer radius "*r*2") of the split ring DGS or the dimension of the stub (stub length "*W*1"), the stopband of the filter can be controlled as shown in **Figures 30** and **31**. The changes of the ring radius and width of the stub with respect to the cutoff and pole frequency are clearly explained by **Figures 32** and **33**, respectively.

The corresponding mathematical expressions for the cutoff and pole frequency of the filter varying with inner radius "*r*1" and stub length "*W*1" can be obtained from the Eqs. (5)–(8).

$$f\_{c1} = -0.72r\_1^2 + 3.88r\_1 - 0.86\tag{5}$$

$$f\_{p1} = -0.83r\_1^2 + 4.35r\_1 - 0.8\tag{6}$$

**Figure 28.** *Equivalent circuit of proposed bandstop filter.*

**Figure 29.** *Comparison of the simulated and circuit response.*

*Filter Designs Based on Defected Ground Structures DOI: http://dx.doi.org/10.5772/intechopen.103065*

**Figure 30.** *Variation of S-parameters magnitudes w.r.t outer and inner radii (r2, r1).*

**Figure 32.** *Change of cutoff and pole frequencies with outer and inner radii (r2, r1).*

$$f\_{c2} = -0.006W\_1^2 + 0.07W\_1 + 3.97\tag{7}$$

$$f\_{p2} = 0.08W\_1^2 - 2.07W\_1 + 16.86\tag{8}$$

#### *2.3.4 Surface current distribution of the filter*

**Figures 34** and **35** show the surface current distribution of the filter. At the first resonance, the magnetic current distribution in the split ring DGS is maximum, whereas the electric current density at the stub is negligible.

**Figure 33.** *Change of cutoff and pole frequencies with length (W1) of the open-ended stub.*

**Figure 34.** *Surface electric and Magnetic current distribution at 4.71 GHz.*

During the second resonance, the split ring DGS's magnetic current is minimal, and the surface electric current distribution is maximum through the stub. The conclusion for the phenomena can be explained, such as that the split ring DGS is responsible for the first resonance whereas the stub provides the second resonance.
