Extended Capabilities of the 3-D Smith Chart With Group Delay and Resonator Quality Factor

This paper extends the capabilities of the 3-D Smith chart for representing positive and negative differential-phase group delay and the associated loaded resonator quality factor, displayed simultaneously with scattering (S)-parameters. Here, mathematical concepts, inspired from elementary differential geometry and topology, are used to implement 3-D projections. It is shown that a condition for a circuit to exploit negative differential-phase group delay is that its S-parameter winding number should be ≥ 0 (relative to its origin). Finally, exemplar network responses that exhibit both positive and negative differential-phase group delay and loaded resonator quality factor are shown with the 3-D Smith chart. The convenience of being able to simultaneously display a wider range of parameters on one visualization platform, with the 3-D Smith chart, may help to speed-up the design and analysis of microwave circuits by the user.