## 3 LRM/LRMM/LRRM Deviation
### LRRM Derivation  
Based on the above LRM and LRMM calibration algorithms, a similar algorithm is developed for LRRM calibration. As shown in Table 1, a two-port standard L (L\({}_{1}\)) and single-port standards R\({}_{1}\) (R\({}_{11}\)/R\({}_{12}\)), R\({}_{2}\) (R\({}_{21}\)/R\({}_{22}\)), and M (R\({}_{31}\)) are used. Here, R\({}_{1}\) and R\({}_{2}\) are assumed to be the short with the unknown reflection coefficient \(\Gamma_{\rm R1}\) and the open standard with \(\Gamma_{\rm R2}\), respectively. With \(\Gamma_{\rm R11}=\Gamma_{\rm R12}=\Gamma_{\rm R1}\) and \(\Gamma_{\rm R21}=\Gamma_{\rm R22}\)= \(\Gamma_{\rm R2}\), R\({}_{1}\) and R\({}_{2}\) are used to construct the T-matrices of error boxes by (6) and (7). Different from LRM, the reflection coefficient \(\Gamma_{\rm M}\) of the match standard M is not fully known. With \(\Gamma_{\rm Rk1}=\Gamma_{\rm M}\) (k = 3), applying (13) results in the following Equation:  
\[\Gamma_{\rm M}=\frac{w_{1}^{\rm M}\Gamma_{\rm R1}+w_{2}^{\rm M}\Gamma_{\rm R2 }x}{w_{1}^{\rm M}+w_{2}^{\rm M}x} \tag{47}\]  
where the measurement terms \(w_{1}^{\rm M}\) and \(w_{2}^{\rm M}\) are defined similar to (14). Meanwhile, based on (20), the measurement of the two-port standard L can generate the following Equation:  
\[\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right][T_{12}^{\rm L}]=z\left[ \begin{array}{cc}1&1\\ \Gamma_{\rm R1}&\Gamma_{\rm R2}\end{array} \right]\left[ \begin{array}{cc}1&0 \\ 0&x\end{array} \right][W_{\rm L}]\left[ \begin{array}{cc}1&0\\ 0&y\end{array} \right]^{-1}\left[ \begin{array}{cc}1&1\\ \Gamma_{\rm R1}&\Gamma_{\rm R2}\end{array} \right]^{-1}. \tag{48}\]  
From (21), we know that the matrix \([W_{\rm L}]\) can be gained by measurement. Also, the T-matrix \([T_{12}^{\rm L}]\) is known from the definition of the standard L. However, for the unknowns, not only the wave ratios \(x\), \(y\), and \(z\), but also the reflection coefficients \(\Gamma_{\rm R1}\), \(\Gamma_{\rm R2}\), and \(\Gamma_{\rm M}\) need to be solved before the error correction. Similar to the analysis in LRM calibration, we can obtain the same results as shown in (32)-(37) by using the determinant and trace conservation for (48). To ensure that the analysis is complete and concise, only two formulas are repeated and given below:  
\[z=\frac{-c_{1}\pm\sqrt{c_{1}^{2}-4c_{2}c_{0}}}{2c_{2}} \tag{49}\]  
\[\frac{x}{y}z=-\frac{\left|[T_{12}^{\rm L}]\right|}{\left|[W_{\rm L}\right| \mid z}. \tag{50}\]  
The variables on the right side of (49) have been defined in (35). Equations (49) and (50) are plugged into (48), and after a similar expansion as in (22), the resulting matrix Equation is shown as follows:  
\[\left[ \begin{array}{cc}t_{22}^{\rm L}&-w_{21}^{\prime\rm L}\\ 0&t_{21}^{\rm L}+t_{22}^{\rm L}\Gamma_{\rm R2}-w_{22}^{\prime\rm L}\\ w_{11}^{\prime\rm L}-t_{12}^{\rm L}&w_{21}^{\prime\rm L}\Gamma_{\rm R2}\\ w_{12}^{\prime\rm L}&w_{22}^{\prime\rm L}\Gamma_{\rm R2}-t_{11}^{\rm L}-t_{12}^ {\rm L}\Gamma_{\rm R2}\end{array} \right]\left[ \begin{array}{c}\Gamma_{\rm R 1}\\ x\end{array} \right]=\left[ \begin{array}{c}w_{11}^{\prime\rm L}-t_{21}^{\rm L }\\ w_{12}^{\prime\rm L}\\ t_{11}^{\rm L}\\ 0\end{array} \right] \tag{51}\]  
where the known variables \(w_{\rm ij}^{\prime\rm L}\) (i, j = 1, 2) are defined in (39).