## 2 Design and Simulation
\[Q_{e1}=\frac{g_{0}g_{1}}{FBW} \tag{2}\]  
\[Q_{en}=\frac{g_{0}g_{n+1}}{FBW} \tag{3}\]  
\[M_{i,i+1}=\frac{FBW}{\sqrt{g_{i}g_{i+1}}}\quad i=1,2,\ldots,n-1 \tag{4}\]  
where \(Q_{e1}\) and \(Q_{en}\) represent the external quality factors of resonators at the input and output, respectively, and \(M_{(i,i+1)}\) represents the coupling coefficients between the adjacent resonators. The standard bandpass filters produce harmonics, which is why the design of microwave bandpass filters starts with designing a low-pass filter, then transforming it into a bandpass filter using numerous calculations. The lumped element bandpass filter is then transformed into half-wave coupled line resonators. Their respective length and widths for inductance and capacitance are calculated using the LineCalc tool in advanced design system (ADS) software. For 0.5 dB equal ripple response, the order of all bandpass filters is the sixth order for significant rejection, i.e., >15 dB. Resonance will occur when both inductive and capacitive reactance become equal. \[X_{L}=2\pi fL \tag{5}\]  
\[X_{C}=\frac{1}{2\pi fC} \tag{6}\]where \(L\) and \(C\) represent inductance and capacitance, respectively; \(X_{L}\) represents inductive reactance; \(X_{C}\) represents capacitive reactance; and \(f\) represents frequency. The inductor and capacitor values will remain constant, whereas frequency varies the reactance. The schematic of a hairpin filter designed in ADS is shown in Figure 4. Figure 5 illustrates the approximate equivalent L-C model for the fundamental hairpin resonator, while Figure 6 displays the approximate equivalent circuit for the entire hairpin filter [43; 44]. It is important to emphasise that this LC model is an approximation serving solely to describe the microstrip resonator's behavior and is not capable of capturing all the intricacies of its response. The interaction between two resonators is represented by a series capacitor and inductor. Additionally, a T-shaped model is employed to characterise each side of the hairpin resonator, incorporating two inductors \((L_{R})\) and one grounded capacitor \((C_{R})\). Table 1 provides the values of inductors and capacitors corresponding to each hairpin filter. Figure 4: ADS schematic view of hairpin filter. Figure 3:0 Approximate L-C model of basic hairpin resonator. The design presented in Figure 7 features a compact filter utilising a three-dimensional stacking arrangement. It consists of five layers, ranging from a top masking layer to a bottom masking layer, with two metal layers in between. The top metal layer features a patterned hairpin design, while the bottom layer serves as a ground plane. Upon completion of the PCB manufacturing process, the copper traces on the board are prone to oxidation and corrosion due to exposure to the environment. To prevent these issues and prolong the life of the PCB, a protective covering known as a solder mask layer is applied. This not only prevents oxidation but also shields the PCB from environmental factors like dirt and pollutants that could cause shorts over time. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **BPF 1** & **BPF 2** & **BPF 3** & **BPF 4** \\ \hline Centre Frequency & 2.1 GHz & 2.4 GHz & 3.2 GHz & 4.0 GHz \\ \(L_{f}\) & 1.33 nH & 1.33 nH & 1.33 nH & 1.33 nH \\ \(L_{1}\) & 0.70 nH & 0.70 nH & 0.70 nH & 0.70 nH \\ \(L_{2}\) & 0.48 nH & 0.48 nH & 0.48 nH & 0.48 nH \\ \(L_{3}\) & 0.12 nH & 0.12 nH & 0.12 nH & 0.12 nH \\ \(L_{4}\) & 1.20 nH & 1.02 nH & 0.69 nH & 0.50 nH \\ \(C_{1}\) & 2.72 pF & 2.72 pF & 2.72 pF & 2.72 pF \\ \(C_{2}\) & 0.94 pF & 0.94 pF & 0.94 pF & 0.94 pF \\ \(C_{3}\) & 0.47 pF & 0.47 pF & 0.47 pF & 0.47 pF \\ \(C_{4}\) & 4.61 pF & 3.93 pF & 2.68 pF & 1.94 pF \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LC values of given equivalent circuit. Figure 6: Approximate L-C model of hairpin filter. To build the switchable filter, four hairpin BPFs are placed on a single substrate, along with SP4T switches used throughout the switching process.