Vector Fitting and Passivity
Node role: Method.
Reader task: Describe sampled complex response data with poles and residues, then distinguish what the fitted response says from what a complete physical network would need to say.
Why fit a rational response?
A sampled trace tells us what a network does at measured or simulated frequencies. A rational model gives a compact interpolation with inspectable singularities. This is useful when several resonances overlap and one isolated resonator formula cannot describe the trace. Vector fitting relocates trial poles and solves linear coefficient problems; it does not infer a circuit topology from the shape of a curve. This is a reorganized explanation of the rational approximation method (Gustavsen and Semlyen 1999).
The physical question comes first: is the input one scalar transfer response, or a complete labeled network matrix? The distinction determines what can be inferred, not whether the fit looks smooth.
| Available data | Rational result | Important missing information |
|---|---|---|
| One complex \(S_{21}\) trace | Poles and residues visible in that projection, a reconstructed transmission, and complex residuals. | Reflections, reverse transmission, dark modes, and internal coordinates. |
| Complete labeled multiport \(S\), \(Y\), or \(Z\) | A matrix response with shared poles and residue matrices. | A unique physical realization; network properties still need their own analysis. |
A scalar trace is a legitimate smaller problem. Inventing missing matrix entries does not turn it into a multiport network.
Model and assumptions
For a Laplace variable \(s\), the starting rational form is (Gustavsen and Semlyen 1999)
\[ \mathbf H(s) =\mathbf D_{\rm rat}+s\mathbf E_{\rm rat} +\sum_{k=1}^{K}\frac{\mathbf R_k}{s-p_k}. \]
| Symbol | Meaning | Units |
|---|---|---|
| \(\mathbf H\) | Scalar or matrix response in the declared \(S\), \(Y\), or \(Z\) domain. | Dimensionless, S, or \(\Omega\), respectively. |
| \(p_k\) | Pole of the fitted response. | \(\mathrm{s^{-1}}\) |
| \(\mathbf R_k\) | Scalar or matrix residue. | Response unit \(\times\mathrm{s^{-1}}\) |
| \(\mathbf D_{\rm rat}\) | Constant term, not an input–output emission map. | Response unit |
| \(\mathbf E_{\rm rat}\) | Proportional term. | Response unit \(\times\mathrm s\) |
| \(K\) | Number of retained rational poles. | Dimensionless |
The model assumes a linear time-invariant response on the declared data domain. A real time-domain realization requires real direct terms \(\mathbf D_{\rm rat},\mathbf E_{\rm rat}\), real residues at real poles, and complex-conjugate pole/residue pairs. A complex-envelope model need not obey those real-rational constraints; its signal and realization assumptions must be declared separately. The choice of response domain, fitted band, weighting, model order, and direct terms is an engineering choice: different choices describe different approximation problems. A term permitted algebraically is not automatically appropriate to the physical high-frequency asymptote.
Port order, reference impedances, reference planes, phasor convention, and calibration remain attached to the input. In particular, a change of wave reference changes the numerical scattering response without changing the underlying terminal network (Kurokawa 1965). See Port Reference Impedance Semantics before converting between domains.
From trial poles to fitted coefficients
The central vector-fitting step introduces a common auxiliary denominator: rather than solve directly for every pole in a nonlinear partial-fraction problem, solve a linearized rational problem with the current poles and use its denominator zeros to relocate them. After relocation, solve for the residues and direct terms. This paragraph reorganizes the algorithmic idea in (Gustavsen and Semlyen 1999); it does not prescribe an implementation API or convergence setting.
For several responses, shared poles let each response have its own residues while describing a common dynamical spectrum. A pole can have a zero or small residue in one entry and remain visible in another. The pole count is therefore not a count of visible dips, nor proof that each fitted state is one physical resonator.
Translate poles into decay, without changing conventions
The following is this page’s algebraic translation of a continuous-time pole, not a new physical model. A decaying time dependence \(e^{pt}\) has \(\operatorname{Re}p<0\). For a real-rational resonant pair,
\[ p_\pm=-\sigma\pm i\omega_0,\qquad \sigma>0. \]
Under the physical \(e^{-i\omega t}\) convention,
\[ s=-i\widetilde\omega,\qquad p_-=-\sigma-i\omega_0 \longleftrightarrow \widetilde\omega=\omega_0-i\sigma. \]
For the same Laplace transfer function, canonical \(e^{-i\omega t}\) data are \(\mathbf H_L(-i\omega)\), not unchanged samples of \(\mathbf H_L(+i\omega)\). A representation sampled at \(s=+i2\pi f\) commonly displays \(p_+\) instead. For a real-rational model, the two axis values are complex conjugates; a complex-envelope model has no automatic conjugation identity. Record the data/evaluator conversion rather than treating its sign as a second physical convention.
The corresponding residue substitution is also this page’s algebra:
\[ \frac{\mathbf R_s}{s-p} =\frac{i\mathbf R_s}{\omega-\widetilde\omega}, \qquad s=-i\omega,\quad p=-i\widetilde\omega, \qquad \mathbf R_\omega=i\mathbf R_s. \]
For this isolated simple decaying pair,
\[ f_0=\frac{|\operatorname{Im}p_\pm|}{2\pi},\qquad \kappa_{\rm tot}=2\sigma,\qquad Q=\frac{\omega_0}{2\sigma},\qquad \frac{\kappa_{\rm tot}}{2\pi}=\frac{2\sigma}{2\pi}. \]
The amplitude/energy distinction follows the damped-mode input–output model (Gardiner and Collett 1985); its full derivation and units are in Resonator Decay, Linewidth, and Quality Factor. An isolated-pole linewidth is not generally the width of an interfering transmission dip.
Residuals: what did the model leave unexplained?
For one dimensionless scalar transmission, an engineering diagnostic is
\[ \epsilon_{21} =\sqrt{\frac1N\sum_{n=1}^{N} \left|S_{21}^{\rm fit}(f_n)-S_{21}^{\rm data}(f_n)\right|^2}. \]
This is a definition chosen here, not a physical acceptance threshold. Inspect the real and imaginary residual versus frequency as well as its norm: a small average can hide structured error around a narrow feature. For a matrix fit, report each response and any aggregate weighting. Duplicating one trace into artificial matrix entries changes the norm without adding evidence.
Model-order or fit-band comparisons, sampling refinement, pole movement, residue visibility, and parameter continuation are useful diagnostics of sensitivity. They do not have universal sample counts, \(Q\) cutoffs, windows, or tolerances. An under-resolved feature can admit a plausible interpolation without determining its pole reliably.
Stability, reciprocity, and passivity answer different questions
| Property | Mathematical question | Limit of scalar \(S_{21}\) |
|---|---|---|
| Stability | Does the realization’s free response decay under its stated time convention? | Only the poles represented by that scalar fit are inspected; hidden dynamics are not determined. |
| Reciprocity | Does the complete response obey the applicable transpose symmetry in compatible port coordinates? | Reverse transmission is absent. |
| Passivity | Can any incident-wave combination extract net power from the network? | Reflection and other incident-wave directions are absent. |
For a complete power-wave scattering matrix with consistent references, nonnegative absorbed power means \(a^\dagger a-b^\dagger b\geq0\) for every incident vector, with \(b=\mathbf S a\) (Kurokawa 1965). This page derives the equivalent condition
\[ \mathbf I-\mathbf S^\dagger\mathbf S\succeq0 \quad\Longleftrightarrow\quad \sigma_{\max}(\mathbf S)\leq1. \]
This condition applies to the declared frequency response and wave convention; a finite set of samples does not establish passivity between samples or outside the observed band. Rational-model passivity assessment can address the fitted model itself (Gustavsen and Semlyen 2008). Stable poles alone do not imply passivity, and \(|S_{21}|\leq1\) does not prove two-port passivity. A globally bounded passive scattering realization cannot retain a nonzero \(s\mathbf E_{\rm rat}\) term: that term grows without bound at high frequency. This is an asymptotic consequence of the stated rational form and contractivity, not an extra fit-window threshold.
For terminal admittance or impedance, do not apply the scattering singular- value test directly. Under the voltage/current power pairing, this page’s dissipated-power algebra instead gives the frequency-local conditions
\[ \frac{\mathbf Y+\mathbf Y^\dagger}{2}\succeq0 \quad\text{or}\quad \frac{\mathbf Z+\mathbf Z^\dagger}{2}\succeq0, \]
in the applicable terminal representation. Pointwise dissipativity does not by itself certify a causal globally passive realization. The cited rational S-model assessment (Gustavsen and Semlyen 2008) has its own scattering-model assumptions; it is not a domain-independent algorithm for every \(Y/Z\) model.
Passivity enforcement changes model coefficients. It can alter residuals and observables and should not be described as a cosmetic operation. Reciprocity must be assessed only for a model whose physical boundary is expected to be reciprocal; it is not supplied merely by choosing stable poles.
Compare a rational fit with a physical model
A topology-constrained model and a rational fit can describe the same complex data while assigning very different meanings to their parameters:
\[ S_{ij}^{\rm data} \xrightarrow{\text{physical fit}}\boldsymbol\theta_{\rm phys} \longrightarrow\{\widetilde\omega_\mu^{\rm phys},R_{ij,\mu}^{\rm phys}\}, \qquad S_{ij}^{\rm data} \xrightarrow{\text{rational fit}} \{\widetilde\omega_\nu^{\rm rat},R_{ij,\nu}^{\rm rat}\}. \]
This is an engineering comparison proposed here, not two independent experiments. Compare corresponding open-system poles and projected residues under the same calibration, units, boundary, and background convention. The rational fit does not supply the physical model’s retained-coordinate diagonals, coherent coupling coefficients, or separate loss channels. A physical model can contain a pole dark in the selected projection; its absence from the scalar fit is not by itself a contradiction. See Poles, Zeros, and Residues.
Use and limits
Use a rational approximation to inspect response structure, interpolate a declared band, or build a candidate network representation from complete network data. Keep three conclusions separate: agreement with sampled data, properties of the fitted rational model, and validity of its physical interpretation.
Too few poles can leave structured residuals; too many can fit noise or create weak, unstable, or poorly determined features. Out-of-band poles may influence the in-band response. A good scalar reconstruction cannot identify internal versus external loss, left versus right emission, a unique Hamiltonian, or a fabrication-ready circuit. Fitting \(S\), \(Y\), and \(Z\) involves different numerical conditioning and reference assumptions. No numerical fit criterion here is a universal scientific or design acceptance rule.
Connections
- Network Trace Views: identify the data and boundary before fitting.
- Poles, Zeros, and Residues: separate existence, visibility, and interference.
- Multimode Input–Output Scattering: derive a topology-aware response rather than infer internal matrices from a scalar trace.
- Port Reference Impedance Semantics: preserve wave normalization and reference metadata.
References
The rational starting model and fitting algorithm are supported by (Gustavsen and Semlyen 1999); rational scattering passivity assessment by (Gustavsen and Semlyen 2008); power-wave meaning by (Kurokawa 1965). Pole translation, the RMSE definition, and the physical-model comparison are identified above as this-page algebra or engineering diagnostics, not source-wide guarantees.
| Field | Value |
|---|---|
| Status | Source-backed |
| Evidence boundary | Primary sources support rational fitting and power-wave/passivity definitions; this page does not certify a current software implementation or physical device. |
| Open review question | Can a reader distinguish a scalar rational response from a complete passive network model, and keep fit residual, stable poles, reciprocity, and passivity separate? |