Can admittance (Y) be used to calculate reactance when L and C values are calculated from the Y12 and Y11 parameters like normal as X=XL-XC?

Question

My original question sought to understand the relationship between S-parameters and reactance. In the process of trying to figure that out, this second question arose. The actual question is at the bottom, and first here is some background:

According to this video, Y12 can be used to find L, R, and Q as shown in this frame:

Additionally (and correct me if I am wrong), C can be calculated from Y11 as:

\$C = \frac{Im(Y_{11})}{2 \pi f}\$

Given L and C values calculated from the admittance matrix as shown above, reactance is calculated as:

• \$X_L=2\pi f L\$
• \$X_C=\frac{1}{2\pi f C}\$
• \$X = X_L - X_C\$

Using S2P measurements of this 120 nH inductor to create a Y matrix from the S parameters using the equations above will yield the following values:

• L=120.243566 nH

• C=-20.970845 pF (ok, so negative capacitance is inductance, but is this right?)

• R=5.084455

• Q=14.859263

• \$X_L=75.551260\$

• \$X_C=-75.893435\$

• \$X=X_L-X_C = 151.444695\$

When I run this calculation at different frequencies for different parts I always get a \$X_C\$ and \$X_L\$ that is nearly the same, but negative so \$X \approx 2X_L\$. When I run this against an S2P representing a 10pF cap I get the same behavior but reactance is negative instead of positive (as expected).

Questions:

• Why does \$X_L \approx -X_C\$?
  • Is it expected that \$X_L\$ and \$X_C\$ calculated from S2P files should be similar?
• The \$X_L\$ and \$X_C\$ numbers are so close that it makes me doubt the math somewhere along the line and triggers these follow-up questions:
  • Is the reactance for this component only \$X_L\$ using the admittance calculations above, or is it really twice that as \$X=X_L-X_C\approx 2X_L\$ as you would expect?
  • C=-20.970845 pF: Ok, so negative capacitance is inductance, but is this the correct capacitance value for calculating \$X_C\$?

Answer 1

S parameters, Y parameters, and all the other parameters, are model-free. They just represent the voltages and currents at the device ports.

At any one frequency, a one port measurement has two degrees of freedom, conveniently represented by the real and imaginary parts of the complex number measurement.

If our model is that there is an ideal lossless L or C on the measurement port, then by definition, if L is positive then C will be negative, and vice versa, and the reactances will be of equal magnitude. This will be easy to see if you synthesise S or Y parameters from ideal components in a simulator or spreadsheet.

When you model a non-ideal component, the loss has to be represented somewhere. We typically assume a series R with an L, it tends to be more physically meaningful than a parallel R. Loss with a C tends not to have such a preferred position.

Still with the synthetic S/Y parameters, add a loss term, with a lossy R in either series or parallel position. As the modeled loss changes position, a low loss component will change value slightly, a lossy one more so. Synthesise the parameters measurements for both an L and a C, with small losses either in series or parallel. Now compare the values for L and C inferred from the 'wrong' loss model.

When we take a measurement of a real component, that's going to be contaminated by measurement noise, calibration error, the component's non-idealities (self C for an inductor, lead inductance for a C, electrical length, frequency dependence), and how we model the loss terms.

At any single frequency, we cannot disambiguate the electrical length and reactive strays from the basic value of a reactive component model. If we have a good model of the component under measurement, then we can attempt to fit our model to the values from a frequency sweep. The art in this sort of process is finding a circuit topology that keeps the number of model components low, while still matching the measured behaviour adequately. For instance at low frequency, the impedance of an inductor tends to increase as frequency increases. That due to self capacitance is quite easy to account for with an extra component in the model. The change of skin depth altering the geometry of the current flow is less easy to model, but may be less significant to the final use of the model.