A Unified Approach for Reformulations of LRM/LRMM/LRRM Calibration Algorithms Based on the T-Matrix Representation  
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1 Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi'an 710071, China; [EMAIL_ADDRESS] (W.Z. ); qhb0920 [EMAIL_ADDRESS] (H.Q.) Science and Technology on Metrology and Calibration Laboratory, Beijing Institute of Metrology and Measurement, Beijing 100854, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (S.Z. ); [EMAIL_ADDRESS] (C.C. ); [EMAIL_ADDRESS] (K.F.) School of Aerospace Science and Technology, Xidian University, Xi'an 710071, China; [EMAIL_ADDRESS] 1  
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3 School of Aerospace Science and Technology, Xidian University, Xi'an 710071, China; [EMAIL_ADDRESS] 3  
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1 Key Laboratory of Electronic Equipment Structure Design, Ministry of Education, Xidian University, Xi'an 710071, China; [EMAIL_ADDRESS] (W.Z. ); qhb0920 [EMAIL_ADDRESS] (H.Q.) Science and Technology on Metrology and Calibration Laboratory, Beijing Institute of Metrology and Measurement, Beijing 100854, China; [EMAIL_ADDRESS] (S.L. ); [EMAIL_ADDRESS] (S.L. ); [PERSON]L_ADDRESS]m (C.C. ); [EMAIL_ADDRESS] (K.F.) School of Aerospace Science and Technology, Xidian University, Xi'an 710071, China; [EMAIL_ADDRESS] 1  
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short-open-load-reciprocal (SOLR) algorithms, etc. [7; 8; 9; 10; 11]. The T-matrix form of error boxes can be utilized to establish the direct mapping of the waves at the ports of the DUT and the detected waves, but a perfect thru is often required to simplify the derivation of the calibration algorithms. In recent years, ABCD parameters have been used to reformulate calibration algorithms, because the impedance information about single-port standards is formulated easily [12; 13; 14]. Besides, ABCD parameters have the same advantage as T-parameters in that they allow matrix cascading. Since the introduction of microwave probing, self-calibration techniques with fewer standards have progressed to achieve higher accuracy and more versatility. Around the late 1980s, the LRM calibration was proposed and has developed until present. The original LRM used a pair of fully known and symmetrical loads as the match standards M [8]. As a particular case of the Lxx calibration family studied in [15; 16], the LRM technique exploited redundancy to determine the reflection coefficient of symmetrical reflection standards R, and then calculated all the seven error terms. However, the match standards M may be frequency dependent and non-identical in practice [17], so the LRM calibration has been constantly improved. Some examples of improvement in the literature are Ref. [14], which uses non-ideal match standards, and the line-reflect-match-match (LRMM) algorithm, which uses non-symmetrical load standards [14; 18]. Meanwhile, the line-reflect-reflect-match (LRRM) calibration technique was proposed as an improvement to LRM. The advantage of LRRM is that only one partially known match standard is required [9; 12; 19]. Although the present LRM/LRMM/LRRM techniques are relatively mature, there is still a lack of a general theory which generalizes these three calibration algorithms. A theory to formulate the SOLT and SOT-Line algorithms has been proposed in our early work [20]. The idea is originally from the principle of coordinate transformation, which regards the T-matrix of an error box as the transformation matrix in mathematics. Based on this idea, the T-matrix of an error box is defined to transform the waves at the test port to the detected waves. Meanwhile, two characteristic variables, which are essentially the ratios of the wave parameters, are newly introduced into the derivation of the algorithm. The de-embedding formulas are finally derived to obtain the corrected DUT parameters. However, the previous work is based on the 10-term error model, and is thus inapplicable for most self-calibration techniques [1]. In addition, the T-matrices of error boxes are not calculated.