Given S-Parameters (or Y,Z,T,ABCD), how can you calculate \$Q_C\$ and \$Q_L\$ (or, how do you calculate ESR)?

Question

Background

I'm writing a .s2p/Touchstone library for Perl and I'm trying to get the Q value of a component given an S-parameter matrix.

The Goal

In this case I'm trying to get Q for a 10nF muRata capacitor. I'm checking correctness against the Q graph in their SimSurfing app and so far the result isn't lining up to what they provide.

What I've tried

Given that \$ Q_C = \frac{X_C}{ESR} \$, I have already validated that \$X_C\$ is correct (by comparing to muRata's graphs) but I've not found a way to calculate ESR to find \$Q_C\$. Somewhere I read that \$ESR=\frac{-1}{Re(Y_{12})}\$ but the values aren't coming out right, so maybe that is true only under certain conditions? (I know that \$Y_{12}\$ is correct because the capacitance value using \$Y_{12}\$ is close to 10nF.)

This reference provides quite a few from S-parameter conversions, but not ESR. I've read articles about how it relates to tan(dissipation) and how to measure ESR...but nothing on how to get ESR from S-parameters.

The questions

I can convert S-parameters to Y, Z, ABCD, T, or whatever...so if those are better for this purpose then I'm fine with that too.

1. So this question is fundamentally:

   • How do you calculate Q from S-parameters?

2. However as an xyproblem, an answer to this question would also suffice because it provides #1:

   • How do you calculate ESR from S-parameters?

Answer 1

It turns out my math for ESR was wrong.

1. Above I have this which is incorrect, at least for Y-parameters that have complex values:

   • \$ESR=\frac{-1}{Re(Y_{12})}\$

2. The correct form to calculate ESR is:

   • \$ESR=Re(\frac{-1}{Y_{12}})\$

Subtle difference! Now my numbers match muRata's.