## 2 Theory
### Establishment of the Calibration Equation  
Having obtained the description for \(\left[T_{10}\right]\), when a single-port standard R\({}_{\rm kd}\) (k \(>\)2) is connected to port 1 and is measured by the VNA (Figure 2a), the following Equation is obtained:  
\[\left[ \begin{array}{cc}a_{11}^{\rm Rk1}\\ \Gamma_{\rm Rk1}a_{11}^{\rm Rk1}\end{array} \right]=\left[T_{10}\right]\left[ \begin{array}{cc}a_{\rm m11}^{\rm Rk1}\\ b_{\rm m11}^{\rm Rk1}\end{array} \right]. \tag{11}\]  
Then, substituting (6) into (11) results in the following Equation:  
\[\left[ \begin{array}{cc}a_{11}^{\rm Rk1}\\ \Gamma_{\rm Rk1}a_{11}^{\rm Rk1}\end{array} \right]=a_{11}^{\rm R11}\left[ \begin{array}{cc}1&x\\ \Gamma_{\rm R11}&\Gamma_{\rm R21}x\end{array} \right]\left[ \begin{array}{cc} a_{\rm m11}^{\rm R11}&a_{\rm m11}^{\rm R21}\\ b_{\rm m11}^{\rm R11}&b_{\rm m11}^{\rm R21}\end{array} \right]^{-1}\left[ \begin{array}{cc}a_{\rm m11}^{\rm Rk1}\\ b_{\rm m11}^{\rm Rk1}\end{array} \right]. \tag{12}\]  
Based on (12), the calibration Equation for R\({}_{\rm kd}\) (k \(>\)2) is given as follows:  
\[\Gamma_{\rm Rk1}=\frac{w_{1}^{\rm Rk1}\Gamma_{\rm R11}+w_{2}^{\rm Rk1}\Gamma_{ \rm R21}x}{w_{1}^{\rm Rk1}+w_{2}^{\rm Rk1}x} \tag{13}\]  
where the measurement terms \(w_{1}^{\rm Rk1}\) and \(w_{2}^{\rm Rk1}\) are defined as \[\left[ \begin{array}{c}w_{1}^{\text{Rk1}}\\ w_{2}^{\text{Rk1}}\end{array} \right]=\left[ \begin{array}{cc}a_{\text{m11}}^{ \text{R11}}&a_{\text{m11}}^{\text{R21}}\\ b_{\text{m11}}^{\text{R11}}&b_{\text{m11}}^{\text{R21}}\end{array} \right]^{-1 }\left[ \begin{array}{c}a_{\text{m11}}^{\text{Rk1}}\\ b_{\text{m11}}^{\text{Rk1}}\end{array} \right]. \tag{14}\]  
Similarly, measurement of a single-port standard R\({}_{\text{k2}}\) (k >2) (Figure 2b) can generate the following Equation:  
\[\Gamma_{\text{Rk2}}=\frac{w_{1}^{\text{Rk2}}\Gamma_{\text{R12}}+w_{2}^{\text{ Rk2}}\Gamma_{\text{R22}}y}{w_{1}^{\text{Rk2}}+w_{2}^{\text{Rk2}}y} \tag{15}\]  
where the variables \(w_{1}^{\text{Rk2}}\) and \(w_{2}^{\text{Rk2}}\) are defined by using measurements as shown below. \[\left[ \begin{array}{c}w_{1}^{\text{Rk2}}\\ w_{2}^{\text{Rk2}}\end{array} \right]=\left[ \begin{array}{cc}a_{\text{m22}}^{ \text{R12}}&a_{\text{m22}}^{\text{R22}}\\ b_{\text{m22}}^{\text{R12}}&b_{\text{m22}}^{\text{R22}}\end{array} \right]^{-1 }\left[ \begin{array}{c}a_{\text{m22}}^{\text{Rk2}}\\ b_{\text{m22}}^{\text{R22}}\end{array} \right] \tag{16}\]  
Besides the calibration Equations (13) and (15) for single-port standards, at least a two-port standard must be introduced in order to complete the two-port VNA calibration. The kth two-port standard L\({}_{\text{k}}\) (k = 1, 2... ) with T-matrix \([T_{12}^{\text{Lk}}]=\left[ \begin{array}{cc}t_{11}^{\text{Lk}}&t_{12}^{ \text{Lk}}\\ t_{21}^{\text{Lk}}&t_{22}^{\text{Lk}}\end{array} \right]\), is measured to continue the calibration as shown in Figure 3.  
Figure 3 shows the same model as in Figure 1, but the superscript 'Lk' is added to represent the standard L\({}_{\text{k}}\). According to (1) and (2), the waves at the ports of L\({}_{\text{k}}\) in Figure 3 can be combined and expressed by  
\[\left[ \begin{array}{cc}a_{11}^{\text{Lk}}&a_{12}^{\text{Lk}}\\ b_{11}^{\text{Lk}}&b_{12}^{\text{Lk}}\end{array} \right]=\left[T_{10}\right] \left[ \begin{array}{cc}a_{\text{m11}}^{\text{Lk}}&a_{\text{m12}}^{\text{Lk}} \\ b_{\text{m11}}^{\text{Lk}}&b_{\text{m12}}^{\text{Lk}}\end{array} \right] \tag{17}\]  
\[\left[ \begin{array}{cc}a_{21}^{\text{Lk}}&a_{22}^{\text{Lk}}\\ b_{21}^{\text{Lk}}&b_{22}^{\text{Lk}}\end{array} \right]=\left[T_{23}\right] \left[ \begin{array}{cc}a_{\text{m21}}^{\text{Lk}}&a_{\text{m22}}^{\text{Lk}} \\ b_{\text{m21}}^{\text{Lk}}&b_{\text{m22}}^{\text{Lk}}\end{array} \right]. \tag{18}\]  
Based on (3), we can use (17) and (18) to get the following matrix Equation for the two-port standard:  
\[\left[T_{10}\right]\left[ \begin{array}{cc}a_{\text{m11}}^{\text{Lk}}&a_{ \text{m12}}^{\text{Lk}}\\ b_{\text{m11}}^{\text{Lk}}&b_{\text{m12}}^{\text{Lk}}\end{array} \right]=\left[ \begin{array}{cc}0&1\\ 1&0\end{array}