## 2 Theory
### Construction of the T-Matrices of Error Boxes  
The left and right error boxes are described in the form of T-matrix by (1)-(2), which should be solved during the calibration procedure. Unlike the traditional calibration methods, the proposed algorithm constructs the T-matrix of each error box by measuring two independent single-port standards as shown in Figure 2.  
When measuring the single-port standard \(\mathrm{R_{kl}}\) (k, l = 1, 2), the error model simplifies to the error box as shown in Figure 2. The subscript of \(\mathrm{R_{kl}}\) represents the kth (k = 1, 2... ) standard connected to port l (l = 1, 2). The reflection coefficient of \(\mathrm{R_{kl}}\) is defined as \(\Gamma_{\mathrm{Rkl}}\).  
By measurements of two single-port standards \(\mathrm{R_{11}}\) and \(\mathrm{R_{21}}\) at port 1 (Figure 1(a)), two Equations similar to (1) can be obtained, and after combination the result is shown as follows:  
Figure 1: Error model of a two-port vector network analyzer (VNA).  
Figure 2: Single-port calibration. (**a**) Error model for standard \(\mathrm{R_{k1}}\) measurement; (**b**) Error model for standard \(\mathrm{R_{k2}}\) measurement.  
\[\left[\begin{array}{cc}a_{11}^{\rm R11}&a_{11}^{\rm R21}\\ \Gamma_{\rm R11}a_{11}^{\rm R11}&\Gamma_{\rm R21}a_{11}^{\rm R21}\end{array} \right]=\left[T_{10}\right]\left[\begin{array}{cc}a_{\rm m11}^{\rm R11}&a_{ \rm m11}^{\rm R21}\\ b_{\rm m11}^{\rm R11}&b_{\rm m11}^{\rm R21}\end{array}\right]. \tag{4}\]  
Similarly, measurements of two single-port standards R\({}_{12}\) and R\({}_{22}\) at port 2 (Figure 2b) generate the following Equation:  
\[\left[\begin{array}{cc}a_{22}^{\rm R12}&a_{22}^{\rm R22}\\ \Gamma_{\rm R12}a_{22}^{\rm R12}&\Gamma_{\rm R22}a_{22}^{\rm R22}\end{array} \right]=\left[T_{23}\right]\left[\begin{array}{cc}a_{\rm m22}^{\rm R12}&a_{ \rm m22}^{\rm R22}\\ b_{\rm m22}^{\rm R12}&b_{\rm m22}^{\rm R22}\end{array}\right]. \tag{5}\]  
Based on (4) and (5), the T-matrices of error boxes can be rewritten as follows:  
\[\left[T_{10}\right]=a_{11}^{\rm R11}\left[\begin{array}{cc}1&x\\ \Gamma_{\rm R11}&\Gamma_{\rm R21}x\end{array}\right]\left[\begin{array}{cc} a_{\rm m11}^{\rm R11}&a_{\rm m11}^{\rm R21}\\ b_{\rm m11}^{\rm R11}&b_{\rm m11}^{\rm R21}\end{array}\right]^{-1} \tag{6}\]  
\[\left[T_{23}\right]=\frac{a_{11}^{\rm R11}}{z}\left[\begin{array}{cc}1&y\\ \Gamma_{\rm R12}&\Gamma_{\rm R22}y\end{array}\right]\left[\begin{array}{cc} a_{\rm m22}^{\rm R12}&a_{\rm m22}^{\rm R22}\\ b_{\rm m22}^{\rm R12}&b_{\rm m22}^{\rm R22}\end{array}\right]^{-1} \tag{7}\]  
where \(x\), \(y\), and \(z\) are defined as follows:  
\[x=\frac{a_{11}^{\rm R21}}{a_{11}^{\rm R11}} \tag{8}\]  
\[y=\frac{a_{22}^{\rm R22}}{a_{22}^{\rm R12}} \tag{9}\]  
\[z=\frac{a_{11}^{\rm R11}}{a_{22}^{\rm R12}}. \tag{10}\]  
As shown in (8)-(10), \(x\), \(y\), and \(z\) are the ratios of wave parameters; therefore, we define them as wave ratios in this paper. Meanwhile, because \(a_{11}^{\rm R11}\), \(a_{11}^{\rm R21}\), \(a_{22}^{\rm R12}\), and \(a_{22}^{\rm R22}\) are directly inspired by the stimulated signals, the wave ratios \(x\), \(y\), and \(z\) should have nonzero and finite values. Obviously, (6) and (7) contain the information about the standards R\({}_{\rm kd}\) (k, \(l\) = 1, 2), so no more Equations need to be set up for these standards.