Generalized Mixed-Mode S-Parameter Framework for Accurate Multipair Crosstalk Analysis in High-Speed Digital Channels

Abstract

The accuracy of mixed-mode S-parameter conversion is important for crosstalk mitigation in high-speed digital systems. However, the conventional mixed-mode S-parameter formulation assumes equal even-mode and odd-mode impedances$\ ( {\ {{{{Z}}}_{{{oo}}}} = {{{{Z}}}_{{{oe}}}}\ } )$, which limits its applicability, particularly in tightly coupled differential structures. In this article, we propose a novel mixed-mode S-parameter generalization (generalized M1/M2 approach) using an N-differential port network, which allows for multipair (i.e., pair-to-pair) crosstalk analysis on coupled differential systems, given by:$\ {{[ {{{{{S}}}_{{{mm}}}}} ]}_{{{i}} \times {{i}}}} = \ ({{[ {{{{{M}}}_1}} ]}_{{{i}} \times {{i}}}} \times {{[ {{{{{S}}}_{{s}}}} ]}_{{{i}} \times {{i}}}} + {{[ {{{{{M}}}_2}} ]}_{{{i}} \times {{i}}}}) \times {{( {{{{[ {{{{{M}}}_1}} ]}}_{{{i}} \times {{i}}}} + \ {{{[ {{{{{M}}}_2}} ]}}_{{{i}} \times {{i}}}} \times [ {{{{{S}}}_{{s}}}} ]} )}^{ - 1}}$. The proposed M1/M2 formulation eliminates the need for renormalization by integrating mode-dependent coupling factors ${{{{k}}}_{{{oo}}}}\ {and}\ {{{{k}}}_{{{oe}}}},$ ensuring a more physically meaningful representation of mixed-mode S-parameters, thereby improving the accuracy of both intrapair and interpair crosstalk analysis in high-speed digital systems. The effectiveness of the proposed M1/M2 approach is demonstrated through intrapair and interpair analysis on tightly coupled striplines, revealing peak-to-peak variations in differential return loss, interpair near-end crosstalk, and far-end crosstalk. Validation using a differential setup with commercial tools (Balun approach) confirmed the formulation's accuracy, with errors below 1%. In addition, measurement validation on a microstrip differential pair highlighted the model's scalability and precision, emphasizing the importance of incorporating mode-dependent impedance variations.