## 2 Analysis of the PCB's Vertical Interconnection Structure  
Owing to the large number of layers in the PCB, the distance for RF signals to be transmitted from the bottom chip pins to the top antenna is relatively long. If only a single metal via is used for this interconnection, significant impedance mismatch may occur as a result of the large inductance characteristics introduced in series. In the PCB, quasi-coaxials are commonly used as vertical interconnect structures to connect surface microstrip lines to inner striplines, as is shown in Figure 1. The quasi-coaxial structure includes a central metal signal via, as well as a peripheral grounding metal via used as a shielding layer. The metal vias are interconnected with a layer of metal ground, which allows for better shielding. Adopting a quasi-coaxial can make the signal transmission more matching and less loss. In the quasi-coaxial structure, the diameter of the cylindrical metal vias is \(\phi_{\mathrm{via}}\) and their height is \(h_{\mathrm{via}}\). The distances between the signal vias and \(n\) ground vias are all \(d_{\mathrm{via}}\). The two RF ports, P1 and P2, are located at the two ends of the structure. By representing the electrical characteristics of the quasi-coaxial with lumped parameters, the equivalent \(\pi\)-type circuit model can be obtained as shown in Figure 2.  
The quasi-coaxial resistance \(R_{\mathrm{QC}}\) is composed of \(n\) ground via resistances \(R_{\mathrm{G}}\) connected in parallel, then connected in series with the signal via resistance \(R_{\mathrm{S}}\), to yield:  
\[R_{\mathrm{QC}}=R_{\mathrm{S}}+\frac{1}{n}R_{\mathrm{G}} \tag{1}\]  
Figure 1: Typical quasi-coaxial interconnect structures to be modeled with analytical equation-based methodologies. (**a**) Top view. (**b**) Side view. Figure 2: Equivalent \(\pi\)-type circuit model of the quasi-coaxial with lumped parameters. As the resistance on the via is primarily caused by the skin effect of the RF signal, this then becomes:  
\[R_{\mathrm{S}}=R_{\mathrm{G}}=\frac{h_{\mathrm{via}}}{\pi(\phi_{\mathrm{via}} \delta-\delta^{2})} \tag{2}\]  
where \(\delta=\frac{1}{\sqrt{\pi_{H}/\sigma_{\mathrm{via}}}}\) is the skin depth of the metal via, \(\mu\) is the magnetic permeability, \(f\) is the signal frequency, and \(\sigma_{\mathrm{via}}\) is the electrical conductivity. The resistance of the quasi-coaxial structure can therefore be calculated using the equation:  
\[R_{\mathrm{QC}}=\left(1+\frac{1}{n}\right)\frac{h_{\mathrm{via}}}{\pi(\phi_{ \mathrm{via}}\delta-\delta^{2})} \tag{3}\]  
This shows that the resistance of the quasi-coaxial is primarily determined by the height, diameter, material and operating frequency of the vias. The quasi-coaxial capacitance \(C_{\mathrm{via}}\) is the capacitance formed between the signal via and one of the ground vias. It can be calculated as [22]  
\[C_{\mathrm{via}}=\frac{\pi\epsilon_{0}\epsilon_{r}h_{\mathrm{via}}}{\mathrm{ arcosh}\left(\frac{\delta_{\mathrm{via}}}{\phi_{\mathrm{via}}}\right)} \tag{4}\]  
where \(\epsilon_{0}\) is the vacuum dielectric constant, and \(\epsilon_{r}\) is the relative dielectric constant of the substrate in the middle of the vias. Because the ground vias are symmetrically distributed around the signal via, the capacitance values between each ground via and the signal via are equivalent (i.e., \(C_{\mathrm{via},1}=\cdots=C_{\mathrm{via},n}\)). It can be seen that the capacitance of the quasi-coaxial capacitance is mainly determined by the height and diameter of the vias, as well as the spacing of the vias and the substrate in the middle.