## 3 LRM/LRMM/LRRM Deviation
### LRM Derivation  
As shown in Table 1, the LRM calibration uses a known two-port standard L (L1), unknown symmetrical reflection standards R (R\({}_{11}\)/R\({}_{12}\)), and known symmetrical match standards M (R\({}_{21}\)/R\({}_{22}\)). The reflection coefficients of R and M are defined as \(\Gamma_{\text{R}}\) and \(\Gamma_{\text{M}}\), respectively. With \(\Gamma_{\text{R11}}=\Gamma_{\text{R12}}=\Gamma_{\text{R}}\) and \(\Gamma_{\text{R21}}=\Gamma_{\text{R22}}=\Gamma_{\text{M}}\), the T-matrices \([T_{10}]\) and \([T_{23}]\) can be constructed by using (6) and (7) through the measurements of R and M. After that, only the following Equation needs to be built by using (20) to implement the LRM algorithm. \[\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right][T_{12}^{\text{L}}]=z\left[ \begin{array}{cc}1&1\\ \Gamma_{\text{R}}&\Gamma_{\text{M}}\end{array} \right]\left[ \begin{array}{ cc}1&0\\ 0&x\end{array} \right][W_{\text{L}}]\left[ \begin{array}{cc}1&0\\ 0&y\end{array} \right]^{-1}\left[ \begin{array}{cc}1&1\\ \Gamma_{\text{R}}&\Gamma_{\text{M}}\end{array} \right]^{-1} \tag{31}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Algorithm**Standard** & **Single-Port Standard** & **Two-Port** \\ \cline{2-4}  & **R\({}_{\text{k1}}\) at Port 1** & **R\({}_{\text{k2}}\) at Port 2** & **Standard L\({}_{\text{k}}\)** \\ \hline LRM & R (\(\times\)), M (\(\surd\)) & R (\(\times\)), M (\(\surd\)) & L (\(\surd\)) \\ LRMM & R (\(\times\)), M1 (\(\surd\)) & R (\(\times\)), M2 (\(\surd\)) & L (\(\surd\)) \\ LRRM & R1 (\(\times\)), R2 (\(\times\)), M (?) & R1 (\(\times\)), R2 (\(\times\)) & L (\(\surd\)) \\ \hline LRM: line-reflect-match; LRMM: line-reflect-match-; LRRM: line-reflect-match-; LRRM: line-reflect-match-; \(\surd\): known; \(\times\): unknown;?: partially known. & \\ \end{tabular}
\end{table} 
Table 1: Standards used for LRM/LRMM/LRRM. The T-matrix \([T_{12}^{\rm L}]\) of standard L and the reflection coefficient \(\Gamma_{\rm M}\) are known from their definition, meanwhile, the matrix \([W_{\rm L}]\) with the same form of (21) can be obtained by measurement. The unknown reflection coefficient \(\Gamma_{\rm R}\) and the wave ratios \(x\), \(y\), and \(z\) need to be solved by (31) before the error correction. To simplify the calculation, the determinant and trace conservation of (31) result in the following Equations:  
\[\frac{x}{y}z^{2}=-\frac{\left|[T_{12}^{\rm L}]\right|}{\left|[W_{\rm L}]\right|} \tag{32}\]  
\[w_{11}^{\rm L}z+w_{22}^{\rm L}\frac{x}{y}z=t_{21}^{\rm L}+t_{12}^{\rm L} \tag{33}\]  
where \(w_{\rm ij}^{\rm L}\) and \(t_{\rm ij}^{\rm L}\) are defined as the (i, j)th element of \([W_{\rm L}]\) and \([T_{12}^{\rm L}]\), respectively. If any reciprocal device is used as the standard L, the determinant \(\left|[T_{12}^{\rm L}]\right|\) equals one and (32) could be further simplified. Multiplying both sides of (33) by \(z\) and substituting (32) into the result leads to the following quadratic Equation about \(z\):  
\[c_{2}z^{2}+c_{1}z+c_{0}=0. \tag{34}\]  
The coefficients are known and expressed by  
\[\left\{ \begin{array}{l}c_{2}=w_{11}^{\rm L}\\ c_{1}=-t_{21}^{\rm L}-t_{12}^{\rm L}\\ c_{0}=-\frac{w_{21}^{\rm L}[[T_{12}^{\rm L}]]}{||W_{\rm L}||}\end{array} \right.. \tag{35}\]  
As a result, the wave ratio \(z\) can be solved directly from (34) as follows:  
\[z=\frac{-c_{1}\pm\sqrt{c_{1}^{2}-4c_{2}c_{0}}}{2c_{2}}. \tag{36}\]  
The root of \(z\) is selected later by the corrected \(\Gamma_{\rm R}\). Besides, if any uniform line is selected as the standard L, the right term in (33) equals zero and (34) could be reduced. However, in this work, the algorithm can use an arbitrary known two-port device as the line standard L, rather than the perfect thru or line, which adds to the versatility of the proposed theory. Substituting (36) into (32) can calculate  
\[\frac{x}{y}z=-\frac{\left|[T_{12}^{\rm L}]\right|}{|[W_{\rm L}]|z}. \tag{37}\]  
Based on (36) and (37), and similar to (22), (31) is expanded and the following Equations are obtained:  
\[\left[