Rational-model passivity¤

A passive S-parameter model must satisfy

at every frequency. Checking each S-parameter magnitude separately is not sufficient because several ports can be excited together.

Circulax can apply a fixed-pole coefficient correction:

from circulax.fitting import ModelFitOptions, fit_model

coefficients = fit_model(
    s_parameters,
    frequencies_hz,
    options=ModelFitOptions(enforce_passivity=True),
)

The correction changes residues and the constant term while leaving poles fixed. It minimizes the change to the fitted response subject to singular-value limits on a frequency grid and at the infinite-frequency limit. The requested fitting error limits are checked again afterwards.

What the check establishes¤

A successful correction establishes passivity only on its sampled grid. It does not prove passivity between samples or outside the modeled band. Use independent frequency samples and application-specific source and load conditions when qualifying a model.

Circulax also checks the converted admittance realization. Stable S poles alone do not guarantee stable Y poles because

Zeros of \(\det[I+S(s)]\) become poles of the voltage-driven admittance realization. This is why validate_model checks both S- and Y-domain stability.

Active models¤

Do not enforce passivity on a model whose gain is intentional. Fit it with reciprocal=False when appropriate and validate with expected_passive=False. Stability with a particular source, load, or feedback network still requires analysis of the assembled circuit.

Algorithm comparisons, optimizer diagnostics, and reproducible timing studies belong in the repository's benchmarks/fitting directory rather than this user guide.