## 4 The Design of the Proposed Filter
### Proposed Low-Pass Filter  
A schematic of the proposed LPF is depicted in Figure 15. The presented LPF is used in the power divider structure. Therefore, the designed LPF should have a small size. An RT/Duroid 5880 substrate is used to design the LPF; using the aforementioned substrate, the overall size of the LPF is only 19.3 mm \(\times\) 9.9 mm (0.15 \(\lambda\)g \(\times\) 0.081 \(\lambda\)g), and with this small size, the designed LPF may be used in the divider structure.  
The proposed LPF consists of designed resonators and suppressed cells. The dimensions of the applied stubs are listed as follows, in which all dimensions are in millimeters: L\({}_{1}\) = 1.2, L\({}_{2}\) = 6.5, L\({}_{3}\) = 9.8, L\({}_{4}\) = 4.2, L\({}_{5}\) = 3, L\({}_{6}\) = 1.8, L\({}_{7}\) = 0.8, L\({}_{8}\) = 0.8, W\({}_{1}\) = 1.56, W\({}_{2}\) = 0.3, W\({}_{3}\) = 0.8, W\({}_{4}\) = 1, W\({}_{5}\) = 0.1, W\({}_{6}\) = 1.4, W\({}_{7}\) = 9.9, W\({}_{8}\) = 2.2, W\({}_{9}\) = 0.1, W\({}_{10}\) = 0.2, S = 0.2, L\({}_{7}\) = 19.3, W\({}_{7}\) = 9.9.  
The frequency response (S\({}_{11}\) and S\({}_{21}\) parameters) of the proposed LPF is depicted in Figure 16. As results shown, the proposed LPF has a 1.8 GHz cut of frequency, and passes signals below this frequency with a very low insertion loss that is close to zero.  
Figure 14: The S-parameter results of the suppressor cells.  
Figure 15: The structure of the proposed low-pass filter.  
The performance of the proposed LPF is listed and compared with similar related filters in Table 3. The results show the good performance of the proposed LPF. In this table, the roll-off rate (\(\zeta\)) parameter is defined as in Equation (3):  
\[\zeta=\frac{\alpha_{max}-\alpha_{min}}{f_{\rm s}-f_{\rm c}}\ (\rm dB/GHz) \tag{3}\]  
where \(\alpha_{max}\) and \(\alpha_{min}\) are the 40 dB and 3 dB attenuation values, and \(f_{\rm s}\) and \(f_{\rm c}\) are the corresponding frequencies at 40 dB and 3 dB, respectively. In the proposed LPF, at a frequency of 1.8 GHz, the value of parameter S\({}_{21}\) is equal to \(-\)3 dB, and at a frequency of 2 GHz, it is equal to \(-\)40 dB. Therefore, according to Equation (3), the obtained roll-off rate (\(\zeta\)) parameter is about 185 dB/GHz. The related stopband bandwidth (RSB) is defined in Equation (4):  
\[\rm RSB=\frac{stopband\ (-20\ dB)}{stopband\ centre\ frequency} \tag{4}\]  
The proposed LPF has an ultra-wide stopband from 2 GHz to 20 GHz, with attenuation of more than 20 dB. Therefore, the RSB parameter for the designed LPF is equal to 1.63. The suppression factor (SF) parameter is defined in Equation (5). With a 20 dB attenuation level in the stop band, the suppression factor (SF) parameter is equal to 2 in the proposed LPF.  
\[\rm SF=\frac{rejection\ level\ in\ stopband}{10} \tag{5}\]  
Figure 16: The frequency response of the proposed low-pass filter.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**FOM** & **AF** & **NCS** & **SF** & **RSB** & \(\zeta\) & **Refs.** \\ \hline
62,520 & 1 & 0.126 \(\rm\,\rm\lambda g\times 0.055\,\rm\lambda g\) & 2.4 & 1.73 & 103.9 & [10] \\
11,221 & 1 & 0.12 \(\rm\,\rm\lambda g\times 0.071\,\rm\lambda g\) & 1.7 & 1.52 & 37 & [50] \\  
The normalized circuit size (NCS) is defined in Equation (6):  
\[\text{NCS}=\frac{\text{physical size (length $\times$ width)}}{\lambda\text{g}^{2}} \tag{6}\]  
The proposed LPF has a small size of 19.3 mm \(\times\) 9.9 mm, which is equal to 0.15 \(\lambda\)g \(\times\) 0.081 \(\lambda\)g. The architecture factor (AF) parameter corresponds to the planar or 3D structure, which is equal to 1 for two-dimensional (2D) circuits and equal to 2 for three-dimensional (3D) circuits. For the proposed LPF, the AF is defined as 1. Finally, the figure of merit (FOM) parameter is calculated as written in Equation (7):  
\[\text{FOM}=\frac{\zeta\times KSB\times SF}{NCS\times AF} \tag{7}\]