# Half-Elliptical Resonator Lowpass Filter with a Wide Stopband for Low Band 5G Communication Systems
## 2 LPF Design Process
### The Half-Elliptical Shaped Resonator Design  
The layout of the half-elliptical shaped resonator is demonstrated in Figure 3a. The response of this resonator is shown in Figure 3b. This resonator creates a TZ at 2.5 GHz. To increase the width of the stopband and eliminate more harmonics, this zero must be located at 3.5 GHz. An LC equivalent circuit can be used to set this transmission zero. Figure 3: The half-elliptical shaped resonator (**a**) structure and (**b**) its frequency response. Figure 2: The layout of the initial low-pass filter. The half-elliptical shaped resonator and the rectangular resonator are shown in Figure 4. A new length \"L\" is calculated in Equation (3) and used to calculate the LCE circuit of the half-elliptical resonator. The \"a\" and \"b\" are the diameters of the half-elliptical shape. The semi-elliptical area is calculated in Equation (1), and the area of the rectangular shape is calculated in Equation (2). The new \"L\" is calculated in Equation (3) and used in Equations (4) and (5). The LC equivalent circuit of the semi-circular resonator is calculated in Equations (6) and (7). \[\text{area}_{\text{half-elliptical}}=\frac{\pi\text{ab}}{2} \tag{1}\] \[\text{area}_{\text{rectangular}}=\text{L}\times 2\text{a} \tag{2}\]  
\[\text{L}=\frac{\pi\text{b}}{4} \tag{3}\]  
\[\text{L}_{\text{s}}=\frac{1}{2\pi\text{f}}\times\text{Z}_{\text{s}}\times \text{Sin}\bigg{(}\frac{2\pi\text{L}}{\lambda_{\text{g}}}\bigg{)} \tag{4}\]  
\[\text{C}_{\text{s}}=\frac{1}{2\pi\text{f}}\times\frac{1}{\text{Z}_{\text{s}}} \times\text{Tang}\bigg{(}\frac{\pi\text{L}}{\lambda_{\text{g}}}\bigg{)} \tag{5}\]  
\[\text{L}_{\text{s}}=\frac{1}{2\pi\text{f}}\times\text{Z}_{\text{s}}\times \text{Sin}\bigg{(}\frac{\pi^{2}\text{b}}{2\lambda_{\text{g}}}\bigg{)} \tag{6}\]  
\[\text{C}_{\text{s}}=\frac{1}{2\pi\text{f}}\times\frac{1}{\text{Z}_{\text{s}}} \times\text{Tang}\bigg{(}\frac{\pi^{2}\text{b}}{4\lambda_{\text{g}}}\bigg{)} \tag{7}\]  
Figure 2:0  
Equations (4) and (5) can be used to obtain the LC equivalent circuit. However, to obtain the equivalent half-elliptical circuit, a new formula must be obtained with the help of the half-elliptical resonator area and the simple rectangular resonator area, as shown in Equations (6) and (7). According to Equations (4)-(7), the LCE circuit of the half-circular resonator is extracted and illustrated in Figure 5a. Different approaches to the extraction of equivalent circuits from microstrip filters are explained in [43; 44; 45; 46]. The comparison of its response with the EM simulation results is shown in Figure 5b. As seen, this resonator produces a transmission zero (TZ) at 2.5 GHz. To tune this TZ at 3.5 GHz, the equation of TZ should be extracted. The transfer function (TF) can be used to obtain the equation of TZ. Figure 2:1 The comparison between the structures of the half-elliptical shaped resonator area and the simple rectangular resonator. To calculate the transfer function, the LC-equivalent circuit is divided to three parts, as shown in Figure 5a. In part 1, \(\mathrm{C_{e3}}\) and \(\mathrm{L_{e3}}\) are the series and corresponding impedance of part 1, calculated in Equation (9). Figure 2:2  
Figure 2:3 The (**a**) LC-equivalent circuit and (**b**)the comparison of the LC-equivalent circuit response and EM structure of the half-circular resonator. Part 1 is parallel with \(\rm C_{e2}\); therefore, the impedance of part 2 is calculated in Equation (10). Figure 2:4  
Part 2 is the series with \(\rm L_{e2}\); therefore, the impedance of part 3 is calculated in Equation (11). Figure 2:5  
Finally, part 3 is parallel with \(\rm C_{e1}\); therefore, the impedance of the node \"m\" is calculated in Equation (12). Figure 2:6  
The TF is shown in Equation (13), where parameter \"\(Z_{\rm m}\)\" is defined in Equation (12). In order to obtain the equation of transmission zero, the numerator of TF should be set to zero, where the equation of \(\rm TZ\) is penned in Equation (14). As can be seen in Equation (14), many elements are important in the equation of \(\rm TZ\).