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What is the relationship, if any, between a Smith Chart and a frequency response (magnitude and phase) graph?

If there is any relationship, what additional constraints or information (topology, port characteristic(s), electrical length(s)/size(s), etc.) would be needed to transform or derive one from the other (over some given frequency range).

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  • \$\begingroup\$ You might take a look at the NanoVNA PC software. It plots Smith Charts and magnitude/phase charts from the same data. You might be able to piece it back together from the source code. \$\endgroup\$
    – JRE
    Commented Jul 2, 2021 at 14:39
  • \$\begingroup\$ Smith Chart is a normalized circular plot of s11 which is insufficient alone to plot s21 \$\endgroup\$
    – D.A.S.
    Commented Jul 3, 2021 at 12:39

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There is none, really. A Smith chart is a mapping of complex impedance plane (Real Resistance, Positive and negative reactance) into a circle, while a frequency response is gain and phase measured between two points in a circuit.

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  • \$\begingroup\$ Are you saying that there is zero relationship between complex impedance and frequency response in any circuit or system topology? \$\endgroup\$
    – hotpaw2
    Commented Jul 3, 2021 at 3:28
  • \$\begingroup\$ Think about what a Smith chart represents. It's an impedance. Nothing more, nothing less. You could plot the impedance of a series RLC over frequency on a Smith chart, from one point to ground. Now, if you assume two ports (somewhere on that network) you can calculate a frequency response plot (Vout/Vin or Iout/In) showing gain and phase between those two points. However, those two points aren't represented on single smith chart, you need two (One for the input impedance, one for the output impedance) Now, consider active circuitry with gain and loss. How does that fit in a Smith chart? \$\endgroup\$
    – rfdave
    Commented Jul 3, 2021 at 3:37

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