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

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<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_oo</i> = Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_oe</i>), which limits its applicability, particularly in tightly coupled differential structures. In this paper, we propose a novel mixed-mode S-parameter generalization (generalized M1/M2- approach) using an N-differential port network, which allows for multi-pair (i.e., pair-to-pair) crosstalk analysis on coupled differential systems, given by: [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mm</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub>= ([<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub> × [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub>+[<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub>) × ([<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub> + [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub>]<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i×i</sub> × [<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sub>])<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> The proposed M1/M2 formulation eliminates the need for renormalization by integrating mode-dependent coupling factors <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_oo</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_oe</i> ensuring a more physically meaningful representation of mixed-mode S-parameters, thereby improving the accuracy of both intra-pair and inter-pair crosstalk analysis in high-speed digital systems. The effectiveness of the proposed M1/M2 approach is demonstrated through intra-pair and inter-pair analysis on tightly coupled striplines, revealing peak-to-peak variations in differential return loss, inter-pair near-end crosstalk (NEXT), and far-end crosstalk (FEXT). Validation using a differential setup with commercial tools (Balun approach) confirmed the formulation's accuracy, with errors below 1%. Additionally, measurement validation on a microstrip differential pair highlighted the model's scalability and precision, emphasizing the importance of incorporating mode-dependent impedance variations.