## 2 Methodology of the Proposed SWB-BPF  
Figure 1 depicts the SWB-BPF architectural layout. It is made up of a UTL that has three-quarter wavelength stubs loaded onto it; one of these stubs is attached to the upper portion of the UTL, and the other two are positioned symmetrically on the lower portion. The SWB-BPF has been investigated by considering a lossless transmission line, and ignores the effect of inductance and capacitance at short-circuited stubs and at the edges of the junctions, which appears due to the step discontinuities. The equivalent configuration of the initial prototype is shown in Figure 2. \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) represent the characteristic impedances of the shorted stubs with electrical length of \(\theta_{s}\), separated with connecting lines of electrical lengths of \(\theta_{c}=2\theta_{s}\). All impedances (\(Z_{1}=Z_{2}=Z_{3}\)) should be equal for design simplicity. The following steps are used to achieve the extreme broadband response. At first, the resonator with open- or short-circuited stubs, as shown in Figure 3a,b, is connected in the middle of the UTL. Figure 1: Proposed SWB-BPF architecture. Figure 3: (**a**) Open-stub configuration connected to UTL; (**b**) short-circuited stub connected to UTL. Figure 2: Equivalent configuration of the proposed prototype. The input admittance (\(Y_{in}\)) can be found by using the ABCD matrix method to obtain the network resonance condition, assuming no losses in the transmission line [41; 42]. \[Y_{in}=\frac{1}{Z_{11}}=\frac{C}{A}=0 \tag{1}\]  
where \(Z_{11}\) shows the input impedance of the resonator. In Equation (1), when elements \(C\) and \(A\) in the transfer matrix have different zeros and \(A\) does not have poles that are different from those in \(C\), this condition ensures the resonance requirement for the resonators in this context. \[C=0 \tag{2}\]  
when \(C=0\), then \(Y_{out}\) also becomes zero. Therefore,  
\[Y_{out}=\frac{1}{Z_{22}}=\frac{C}{D}=0 \tag{3}\]  
The generalized equation for the left portion of Figure 3a in terms of the matrix elements \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), \(D^{\prime}\) is as below [43]. \[ \begin{pmatrix}A&B\\ C&D\end{pmatrix} = \begin{pmatrix}1+2B^{\prime}C^{\prime}&2A^{\prime}B^{\prime} \\ 2C^{\prime}D^{\prime}&1+2B^{\prime}C^{\prime}\end{pmatrix}  \tag{4}\]  
The resonance conditions in Equation (4) are same as in Equation (2), and fulfil the following two criteria:  
\[C^{\prime}=0 \tag{5}\]  
\[D^{\prime}=0 \tag{6}\]  
The above criteria are easier to analyse as compared to the conventional method, such as \(Y_{in}=0\). Now, formulate the resonance equations (\(A^{\prime}B^{\prime}C^{\prime}D^{\prime}\)) for Figure 3a,b, which is equal to the input admittance (\(Y_{s}\)) of the stub and the transfer matrix (_abcd_) of the left section of the TL segment [44; 45]:  
\[ \begin{pmatrix}A^{\prime}&B^{\prime}\\ C^{\prime}&D^{\prime}\end{pmatrix} = \begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}1&0\\ \frac{Y_{s}}{2}&1\end{pmatrix} = \begin{pmatrix}a+\frac{BY_{s}}{2}&b\\ c+d\frac{Y_{s}}{2}&d\end{pmatrix}  \tag{7}\]  
Here, the matrix (_abcd_) denotes a portion of a non-uniform segment. Using Equation (3), the above equation becomes  
\[d=0\]  
\[Y_{s}=\frac{-2c}{d}\]  
When a segment with a uniform TL is used, its \(\theta\) and \(Z_{o}\) become  
\[ \begin{pmatrix}a&b\\ c&d\end{pmatrix} = \begin{pmatrix}\cos\frac{\theta}{2}&jZ_{o}\sin\frac{\theta}{2} \\ Z_{o}{}^{-1}j\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{pmatrix}  \tag{8}\]  
By applying resonance conditions, it becomes  
\[\cos\frac{\theta}{2}\text{ where }\theta_{n}=\pi,3\pi,5\pi \tag{9}\]  
\[Y_{s}=-j2Z_{o}{}^{-1}\tan\frac{\theta}{2} \tag{10}\]Equation (10) determines the resonant electrical lengths (\(\theta\)) for the other component, which relies on the stub parameters. The same analysis can also be used for Figure 3b.