# Space-Borne System-in-Package Based on High Reliability Microwave Interconnections
## 3 A Novel High Reliability Interconnection Structure
The characteristic impedance of coaxial seal \(Z_{1}\) is:  
\[Z_{1}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}\varepsilon_{r1}}}\frac{\ln(D/d)}{ 2\pi}=Z_{0} \tag{1}\]  
where \(Z_{0}\) is the standard 50 ohms. The characteristic impedance \(Z_{2}\) of the air coaxial end towards the seal is:  
\[Z_{2}=\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\frac{\ln(dd/d)}{2\pi} \tag{2}\]  
From \(Z_{0}=Z_{1}=Z_{2}\), it can be seen that:  
\[dd=\frac{D}{e^{\sqrt{\varepsilon_{r1}}}} \tag{3}\]  
Figure 7:0 The novel horizontal interconnection structure. The other end of the gradually opening air coaxial line is a quasi TEM mode, so it can be equivalent to a curved microstrip line, whose dielectric is air and impedance \(Z_{3}\), which can be approximated by the microstrip line:  
Figure 7:1  
where \(W_{e}\) and \(h_{e}\) are the width and dielectric thickness of the equivalent microstrip line, and \(\varepsilon_{e}\) is the equivalent relative dielectric constant:  
Figure 7:2  
Figure 7:3  
Figure 7:4  
The ABCD matrix [13] can be used to analyze the transmission characteristics of the gradually opening part in the air coaxial structure. It is set to be divided into \(n\) parts. The length of each section is \(L_{2}/n\), and the characteristic impedance is \(Z_{2i}\) (\(i\) = 1,2,3... \(n\)), where the transmission matrix \(M_{i}\) can be calculated as:  
\[M_{i}= \begin{pmatrix}A_{i}&B_{i}\\ C_{i}&D_{i}\end{pmatrix} = \begin{pmatrix}\cos\beta\frac{L_{2}}{n}&jZ_{2i}\sin \beta\frac{L_{2}}{n}\\ j\frac{1}{Z_{2i}}\sin\beta\frac{L_{2}}{n}&\cos\beta\frac{L_{2}}{n}\end{pmatrix}  \tag{8}\]  
The total transmission matrix \(M\) is:  
\[M=\underset{n\rightarrow\infty}{\underset{i=1}{\overset{n}{\prod}}} \begin{pmatrix} \cos\beta\frac{L_{2}}{n}&jZ_{2i}\sin\beta\frac{L_{2}}{n}\\ j\frac{1}{Z_{2i}}\sin\beta\frac{L_{2}}{n}&\cos\beta\frac{L_{2}}{n}\end{pmatrix}  \tag{9}\]  
As the length of the air coaxial part is usually very short, sin (\(\beta L_{2}/n\)) is approximately equal to \(\beta L_{2}/n\), where \(\beta\) is the transmission coefficient. In addition, it can be approximately considered that the equivalent characteristic impedance of the gradually opening part is uniformly transformed from \(Z_{2}\) to \(Z_{3}\), i.e.,:  
\[Z_{2i}=Z_{2(i-1)}+\frac{Z_{3}-Z_{2}}{n},i=1,2,\ldots,n \tag{10}\]  
Therefore, the total transmission matrix \(M\) can be approximately expressed as:  
\[M\approx\left( \begin{array}{ccc}\cos\beta L_{2}&j\frac{\sum\limits_{i=1}^{n }Z_{2i}}{n}\sin\beta L_{2}\\ j\frac{\sum\limits_{i=1}^{n}Z_{2i}}{n}\sin\beta L_{2}&\cos\beta L_{2}\\ \end{array} \right) \tag{11}\]  
The equivalent characteristic impedance \(Z_{e}\) of the gradually opening coaxial structure can be derived from Equation (11). \[Z_{e}\approx\frac{Z_{2}+Z_{3}}{2} \tag{12}\]Moreover, the characteristic impedance \(Z_{4}\) of the gold ribbon crossing the gap can be approximated as:  
\[Z_{4}\approx\frac{120\pi}{\frac{W_{h}}{h+\frac{h_{b}}{2}}+2.42-0.44\frac{h+\frac{ h_{b}}{2}}{W_{b}}+(1-\frac{h+\frac{h_{b}}{2}}{W_{b}})^{6}} \tag{13}\]  
where, \(h_{b}\) is the ribbon arch height and \(h\) is the thickness of the dielectric substrate. In order for this structure to achieve excellent matching performance at microwave frequency, it is necessary to:  
\[Z_{e}\frac{Z_{1}+jZ_{e}\tan\beta L_{2}}{Z_{e}+jZ_{1}\tan\beta L_{2}}=Z_{4}\frac {Z_{0}+jZ_{4}\tan\beta del}{Z_{4}+jZ_{0}\tan\beta del} \tag{14}\]  
Therefore, \(L_{2}\) can be obtained as:  
\[L_{2}\approx\frac{1}{\beta}\text{arctan}\left(\frac{\frac{Z_{0}}{Z_{4}}-\frac {Z_{4}}{Z_{0}}}{\frac{Z_{0}}{Z_{e}}-\frac{Z_{e}}{Z_{0}}}\tan\beta del\right),\ \beta=\frac{2\pi f\sqrt{\epsilon_{r}}}{c} \tag{15}\]  
Therefore, the initial values of the diameter \(dd\) and length \(L_{2}\) of the gradually opening air coaxial line can be calculated based on Equations (2)-(4), (12), (13), and (15), and then the design can be completed by further optimization in the 3D electromagnetic simulation software, such as HFSS.