## 2 Design and Fabrication
### Design and Circuit Analysis
#### 2.1.1 Model inside the Segment Box  
The intertwined spiral metal track is divided into 10 segments by four air-bridges, as shown in Figure 2. Then, the equivalent circuit model was established as exhibited in Figure 2. As shown in the Figure 2, each segment (Seg i) can be analyzed using the \(\pi\)-type lumped model, where \(R_{T}\) and \(Lt\) represent the series resistance and series inductance of this segment, respectively, \(C_{SiNx}\) represents the capacitance associated with SiN\({}_{x}\), and \(C_{SUB}\) and \(G_{SUB}\) represent the capacitance and conductance associated with GaAs, respectively.  
Figure 1: Design of the proposed bandpass filter (BPF): (a) 3D view of the packaged BPF on a printed circuit board (PCB) board; (b) enlarged view; (c) side view; (d) three metal layers (leads, text, and bond).  
Figure 2: _Cont._  
The series inductance and series resistance can be determined by the geometry of the metal track and low-frequency resistivity. The current distribution on the track is uneven as the current of the spiral inductor has a superposition effect under high-frequency excitation. Therefore, the inductance and resistance of each segment are related to the instantaneous frequency [20]. Hence, the concept of frequency-dependent effective linewidth \(W_{\textit{eff}}\) was employed to model the equivalent circuit, which can be calculated as follows:  
\[W_{\textit{eff}}=W_{0,i}\left(1-exp\left(-\frac{w}{W_{0,i}}\right)\right), \tag{2}\]  
\[W_{0,i}=c_{1}\times c_{2}^{i-1}\sqrt{\frac{1}{f}}\,, \tag{3}\]  
where \(w\) is the physical metal line width, \(i\) is the number of turns of the coil, and \(c_{1}\) and \(c_{2}\) are the fitting parameters to match the resistance and inductance with the measurement results, respectively. The resistance of each metal line (R-line) can be obtained using \(W_{\textit{eff}}\) as given in Equation (4) to evaluate the signal losses [20],  
\[R_{\textit{line}}=\frac{\rho\times I}{W_{\textit{eff}}\times t_{m}}\,, \tag{4}\]  
where \(\rho\) is the resistivity (\(\Omega\)-cm) of the metal, and \(tm\) and \(l\) are the thickness and length of the metal line, respectively. The inductance of each segment consists of the self-inductance of the metal line and the mutual inductance with other metal segments. The self-inductance depends on the changing frequency and can be expressed as follows based on \(W_{\textit{eff}}\)[20]:  
\[L_{\textit{self}}=2l\left(ln\frac{2l}{W_{\textit{eff}}+t_{m}}-0.5\right). \tag{5}\]  
The mutual inductance can be approximated as expressed below, as the metal tracks of each segment are almost parallel to each other [20]:  
\[L_{M}=2\times 10^{-4}l\left[ln\left(\frac{l}{d}+\sqrt{1+\frac{l^{2}}{d^{2}}} \right)-\sqrt{1+\frac{l^{2}}{d^{2}}}+\frac{d}{l}\right], \tag{6}\]  
where \(d\) represents the distance between the center positions of every two wires. It should be emphasized that the mutual inductance is positive when the current directions of the two metal tracks are the same, whereas metal tracks with opposite current directions have a negative mutual inductance.  
Figure 2: Equipment circuit model of the proposed BPF: (**a**) number of segment boxes of the proposed BPF; (**b**) \(\pi\)-type lumped-element model inside the segment box; (**c**) equivalent circuit model of the proposed BPF.  
Hence, the total inductance associated with length \(l\) and the desired metal track length can be calculated as given in Equations (4) and (5).