## 3 Mathematical Modelling of the Dual-Notch Filter  
In this section, the dual-notch characteristics using a short-circuited, T-shaped, dual-mode resonator are achieved. A symmetrical quarter-wavelength resonator is used instead of a half-wavelength resonator to make the structure compact, as shown in Figure 6. To analyse the behaviour of the structure, the design utilizes a classical method called even-odd mode analysis. This involves separating the circuit into its even and odd modes and analysing each mode separately. This method helps to predict the controllable operating bands of the filter. The even mode will have a symmetric current distribution and can be represented by even terms in the analysis, as shown in Figure 6b, while the odd mode will have an anti-symmetric current distribution and can be represented by odd terms, as shown in Figure 6c, respectively. By applying this analysis and using the basic concept of microwave transmission lines, the equations of the characteristic input admittance of each resonator can be derived, taking into account the lengths of the half-wavelength line (\(Y_{1},L_{1}\)) and quarter-wavelength line (\(Y_{2},L_{2}\)), with a load admittance (\(Y_{L}\)) [46; 47]. \[Y_{int}=Y_{0}\frac{Y_{L}+jY_{0}\tan\theta}{Y_{0}+jY_{L}\tan\theta} \tag{15}\]  
Here, \(\theta=\beta L\), so the above equation becomes  
\[Y_{in}=Y_{0}\frac{Y_{L}+jY_{0}\tan\beta L}{Y_{0}+jY_{L}\tan\beta L} \tag{16}\]  
The next discussion is about how the stopbands are formed. Using the mathematical analysis discussed above, the theoretical resonant frequencies for each stopband are calculated. The first stopband, formed through the fundamental even mode and its resonance frequency, are calculated using the parameters provided in the Equation (19) denominator. Substituting the relevant length values from Figure 7 into Equation (19) and referencing Figure 6b yields a theoretical frequency of 4.18 GHz for the even mode, while the simulation shows 4.4 GHz. The second stopband is generated through the fundamental odd mode, and its resonance frequency is calculated by inserting the parameter values from Equation (21). Substituting the corresponding length values from Figure 7 into Equation (21) and referencing Figure 6c yields a theoretical frequency of 6.9 GHz for the second stopband, while the simulation shows 7.5 GHz. The slight deviations betw  
Figure 6: (a) Quarter-wavelength T-shaped resonator; (b) equivalent even mode; (c) equivalent odd-mode model. simulated frequencies stem from the magnetic coupling between the resonators, which are adjusted for specific band applications through parametric analysis in the HFSS software. Now, for the even mode, the input admittance can be calculated by making the stub short-circuit due to already being present; thus, \(Y_{L}=\alpha\) and \(Y_{0}=(\frac{Y_{6}}{2})\). Now, the equation above can be rewritten as  
\[Y_{in\text{-}aborted}=-j(\frac{Y_{6}}{2})\cot\theta_{6} \tag{17}\]  
Putting Equation (17) in (15), \(Y_{\text{in-even}}\) can be obtained in the following equation:  
\[Y_{in\text{,}even}=-jY_{1}\frac{Y_{6}-2Y_{1}\tan(\theta_{5}+\theta_{7})\tan( \theta_{6})}{2Y_{1}\tan(\theta_{6})+Y_{6}\tan(\theta_{5}+\theta_{7})} \tag{18}\]  
At resonance, \(Y_{in\text{-}even}=0\), so the first stopband frequency (\(f_{even}\)) equation is obtained using Figure 6a. \[f_{even}=\frac{(2n-1)c}{4(L_{5}+L_{6}+L_{7})\sqrt{\varepsilon_{eff}}} \tag{19}\]  
In the above equation, \(c=3\times 10^{8}\) m/s, and n is the integer, which is 1, 2, 3,....... in this case, and \(\varepsilon_{eff}\) is the PCB permittivity with the corresponding width (\(w\)) and height (\(h\)) which is  
\[\varepsilon_{eff}=\frac{1+\varepsilon_{r}}{2}+\frac{\varepsilon_{r}-1}{2} \times\frac{1}{\sqrt{1+12\frac{h}{w}}}\]  
Similarly, in the case of the odd mode, the stub \(L_{1}\) is short-circuited at AA\({}^{\prime}\).