A Study of Differential Signaling: Stable and Accurate Mixed-Mode Conversion and Extraction of Differential S-Parameters

Low voltage differential signaling (LVDS) in high-speed digital systems is utilized to effectively reduce EMI and improve signal quality. Mixed-mode S-parameters are a more general way to characterize a differential network. Therefore, an accurate extraction of mixed-mode S-parameters from single-ended S- parameters is critical for Signal and Power Integrity co-simulation where SSN is generated mainly by high-frequency interconnects. The standard conversion between mixed-mode and single-ended S-parameters involves inversion of a transformation matrix. If there is no coupling, this transformation matrix is orthogonal and numerical inversion can be done accurately.

In the presence of coupling, the transformation matrix depends on S-parameters and may become ill-conditioned, i.e. has high condition number, for some values of physical parameters resulting in unstable inversion of the transformation matrix and leading to highly inaccurate converted mixed-mode S-parameters. To analyze the possibility of ill-conditioning, we consider two cases: broadside coupled striplines and coupled microstrip pairs. We find that in both cases when two transmission lines are strongly coupled, the condition number becomes very large. In this case, regularized methods from the theory of ill-posed problems should be used, for example, the truncated SVD method, to obtain accurate mixed-mode S- parameters.

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