## 2 Analysis of the PCB's Vertical Interconnection Structure
The quasi-coaxial inductance \(L_{\mathrm{QC}}\) includes the self-inductance \(L_{\mathrm{self}}\) of each metal via, as well as the mutual inductance between them, which can be calculated according to their size and spacing in the quasi-coaxial structure, according to the equations [23]:  
\[L_{\mathrm{self}}=\frac{\mu_{0}h_{\mathrm{via}}}{2\pi}\left[\ln\left(\frac{2h _{\mathrm{via}}}{\phi_{\mathrm{via}}}+\sqrt{\left(\frac{2h_{\mathrm{via}}}{ \phi_{\mathrm{via}}}\right)^{2}+1}\right)-\sqrt{1+\left(\frac{\phi_{\mathrm{ via}}}{2h_{\mathrm{via}}}\right)^{2}}+\frac{\phi_{\mathrm{via}}}{2h_{\mathrm{via}}}\right] \tag{5}\]  
\[M_{1}=\frac{\mu_{0}h_{\mathrm{via}}}{2\pi}\left[\ln\left(\frac{h_{\mathrm{via }}}{d_{\mathrm{via}}}+\sqrt{\left(\frac{h_{\mathrm{via}}}{d_{\mathrm{via}}} \right)^{2}+1}\right)-\sqrt{1+\left(\frac{d_{\mathrm{via}}}{h_{\mathrm{via}}} \right)^{2}}+\frac{d_{\mathrm{via}}}{h_{\mathrm{via}}}\right] \tag{6}\]  
Figure 2.0  
where \(M_{1}\) is the mutual inductance between the signal via and one of the ground vias, \(M_{\mathrm{m}}\) is the mutual inductance between any two ground vias, and \(d_{\mathrm{m}}\) is the distance between the \(m\)th ground via and the first ground via. The entire quasi-coaxial structure is a complete signal circuit, and the current \(I\) flowing in from the signal via is evenly divided into \(I/n\) flowing back from the \(n\) ground vias. The relationship between inductance, current, and voltage in the quasi-coaxial structure can be used to deduce the inductance on the signal and ground vias, which can be expressed as:  
Figure 2.1Figure 2.2  
Owing to the symmetry of the ground via, the inductance on \(n\) ground via can be considered equal. The quasi-coaxial inductance \(L_{\text{QC}}\) is composed of \(n\) ground via inductance values \(L_{\text{G}}\) connected first in parallel, then in series with the signal via inductance \(L_{\text{S}}\), yielding:  
\[ \begin{split} L_{\text{QC}}&=L_{\text{S}}+\frac{L_ {\text{G}}}{n}\\ &=\left(1+\frac{1}{n}\right)L_{\text{self}}-2M_{1}+\frac{1}{n} \sum_{m=2}^{n}M_{\text{m}}\end{split}  \tag{10}\]  
From this equation, it can be seen that quasi-coaxial inductance is primarily determined by the height, diameter, and spacing of the vias. According to the lumped parameters of the quasi-coaxial structure, the characteristic impedance of which can be calculated using:  
\[Z_{\text{QC}}=\sqrt{\frac{R_{\text{QC}}+i\omega L_{\text{QC}}}{j\omega(C_{ \text{via\_1}}+\cdots+C_{\text{via\_n}})}} \tag{11}\]  
However, in reality, owing to inherent limitations to the PCB manufacturing process, the signal via connected to the inner line often has a copper stub, which is difficult to completely eliminate even with the back drill process, as shown in Figure 3a. This copper stub can generate parallel parasitic capacitance, thereby affecting the impedance matching of the quasi-coaxial. The equivalent \(\pi\) type circuit model of the quasi-coaxial with a stub is shown in Figure 3b. Its characteristic impedance can be calculated as:  
\[Z_{\text{QC\_stub}}=\sqrt{\frac{R_{\text{QC}}+j\omega L_{\text{QC}}}{j\omega( C_{\text{via\_1}}+\cdots+C_{\text{via\_n}}+C_{\text{stub}})}} \tag{12}\]  
where \(C_{\text{stub}}\) is the parasitic capacitance generated by the stub, which is primarily determined by its diameter, length, and material. To make quasi-coaxials work well even in the presence of such stubs, LC circuits are usually used for impedance matching nearby [24], as is shown in Figure 4a. Two segments of line are used for impedance matching between the stripline and signal via, where the narrow line can be equivalent to a series inductance and the wide line can be equivalent to a parallel capacitance. The equivalent circuit model of the entire transmission line is shown in Figure 4b. The characteristic impedance of a stub with a matching LC circuit can then be calculated as:  
Figure 3: Quasi-coaxial structure with a copper stub in the PCB. (**a**) PCB stackup. (**b**) Equivalent \(\pi\) type circuit model.