Modal TRL De-Embedding of Symmetric Differential Transmission Lines with Proper Reference Impedance Matrix Transformations

This paper addresses the question of how to place 50-ohm-referenced measurement reference planes on coupled transmission lines (TLs) using multimode thru-reflect-line (TRL). To that end, the theory of mode transformation is presented for S-matrices and their associated reference impedances. It is shown that when an S-matrix is mode-transformed, the associated reference impedance matrix must also be transformed commensurately. Modal TRL de-embedding of 4-port symmetric differential TLs is presented as an example. It is shown that reference impedance matrices may become nondiagonal, which goes beyond the scope of the assumption made by the conventional S-parameter theory. The mode transformation theory is validated by using synthesized data via applying 2-port TRL and calibration comparison (for characteristic impedance estimation) to the two eigenmodes, followed by inverse mode transformation and renormalization transformation back to the single-ended nodal representation. Measurement data for differential TLs with GSGSG pads on a 65-nm CMOS chip are also presented.