# VNA Measurements, Passivity, and K-K Enforcement *Measured S-parameters are complex functions of frequency over a finite band — and real hardware routinely produces data that violates causality or passivity. Kramers–Kronig enforcement is how behavioural models are rescued into something an SI/PI simulator can actually integrate.* This chapter uses the **engineering convention** `e^{+jωt}` throughout, since it is the universal choice in microwave/SI literature. The causal transfer function is analytic for Re s ≥ 0, i.e. in the *lower* half of the ω-plane (see Chapter 16). Every K-K sign below is the engineering one; the physics form is obtained by `j → −i`. ## What a vector network analyser actually measures A vector network analyser (VNA) stimulates a device under test (DUT) with a swept sinusoid at frequencies ω_k and measures the complex amplitude and phase of the incident, reflected, and transmitted waves at its ports. From these it computes the scattering parameters ``` S_ij(ω_k) = b_i(ω_k) / a_j(ω_k) (other ports match-terminated) ``` delivered as a table — typically a Touchstone `.s2p` or `.s4p` file — of complex S_ij over a finite band [ω_min, ω_max] on a discrete grid. For a two-port, S_11 and S_22 are reflection coefficients, S_21 and S_12 are transmission coefficients; for an N-port, S is an N×N matrix. The hardware reality diverges from the ideal in several ways: - **Finite bandwidth.** The VNA covers, say, 10 MHz–50 GHz. Data below and above that is simply missing. - **Calibration residuals.** SOLT / TRL calibration corrects systematic errors but leaves random noise and small residuals, especially at band edges and at ports with large mismatch. - **Trace noise and IF-bandwidth trade-offs.** Lower IF bandwidth averages noise down but lengthens the sweep. - **Cable flex, connector repeatability, temperature drift.** These introduce small uncorrelated phase errors. - **Fixturing and de-embedding errors.** Launching onto a PCB through a connector is never perfect; the de-embedding model is approximate. ## Why raw data can violate causality A physical passive linear network must have a causal impulse response: taking the inverse Fourier transform of S_ij(ω) should give s_ij(t) that is zero for t < 0 and bounded in energy. Raw measured data rarely satisfies this, for three reasons: 1. **Truncation.** The K-K integrals require data from 0 to ∞; missing bands bias the reconstructed time-domain response. 2. **Noise.** Random real and imaginary parts have no causality relationship — their inverse transform is two-sided. 3. **Calibration and de-embedding residuals.** Small phase errors at high frequency produce large t < 0 artefacts because the inverse transform is exponentially sensitive to phase near the band edge. The visible symptoms in time-domain reflectometry (TDR) derived from the measured S-parameters are: - **Pre-ringing** — non-zero response for t < 0, which is physically impossible. - **Non-monotonic energy accumulation** — the step response overshoots in ways inconsistent with a passive network. - **|S_ij(ω)| > 1** for a passive DUT — passivity violation, typically after interpolation or de-embedding. A behavioural model fitted to such data is unstable: an SI/PI simulator (HSPICE, ADS, Sigrity) driving the macromodel in transient will see its time-stepper blow up. ## K-K enforcement: causality correction The K-K relations tell us that for a causal response, real and imaginary parts are not independent — each determines the other up to a real constant and asymptotic subtractions. The engineering form for a (suitably subtracted) scalar S(ω) = S′(ω) + jS″(ω) is ``` S′(ω) = S′(∞) − (1/π) P ∫_{−∞}^{∞} S″(ω′) / (ω′ − ω) dω′ S″(ω) = (1/π) P ∫_{−∞}^{∞} S′(ω′) / (ω′ − ω) dω′ ``` Causality correction re-derives one part from the other over the measured band, explicitly accounting for: - **Extrapolation to DC.** S_21(0) for a passive interconnect is real and finite; S_11(0) is a real DC reflection. These anchor the low-ω tail. - **Extrapolation to ω → ∞.** Physical interconnects have high-frequency roll-off (skin effect, dielectric loss, radiation) that forces S → 0 with a known power law; this supplies the integrand tail. - **Truncation subtractions.** The missing-band contribution is modelled as a correction term rather than ignored. Passivity enforcement is a related but distinct step: the S-matrix is constrained so that I − S^H S ≥ 0 on the band, which ensures no energy is generated. The typical workflow is: 1. Fit measured S to a rational (vector-fitting) macromodel. 2. Enforce causality via K-K on each element. 3. Perturb pole–residue pairs minimally to restore passivity. 4. Hand the passivity-corrected rational model to the SI/PI simulator. ## Applications Causality- and passivity-enforced S-parameter models underlie almost every modern high-speed link analysis: - **IBIS-AMI** — behavioural models of SERDES transmitter/receiver linear parts rely on K-K-compliant S-parameter files for the channel. - **S2P / S4P → Spice-like macromodels** — vector fitting with passivity enforcement generates SPICE-compatible pole–residue subcircuits used in PCIe, DDR, and 400/800 GbE channel simulations. - **Power integrity** — PDN impedance profiles measured on a VNA are K-K-enforced before being used in decoupling optimisation. The definitive review is Triverio, Grivet-Talocia, Nakhla, Canavero, Achar, *"Stability, Causality, and Passivity in Electrical Interconnect Models"*, IEEE Transactions on Advanced Packaging, 2007, which catalogues the checks (time-domain zero-for-t<0 tests, Hilbert-pair consistency, positive-real tests) and the enforcement algorithms. ## Practical checks Before trusting a measured S-parameter file: - **Time-domain causality check.** IFFT to s_ij(t); the energy in t < 0 should be a small fraction of the total. Anything above a few percent indicates bad data or bad extrapolation. - **Hilbert-pair residual.** Compute Im from Re via K-K on the measured band; compare with measured Im. The residual is a one-number causality figure of merit. - **Singular-value passivity.** Check σ_max(S(ω)) ≤ 1 at every sample. These three diagnostics catch the vast majority of problems before the model goes near a simulator. ## See also - [10_causality_LTI.md](10_causality_LTI.md) - [16_sign_conventions.md](16_sign_conventions.md) - [19_minimum_phase_DSP.md](19_minimum_phase_DSP.md) - [30_pitfalls_limitations.md](30_pitfalls_limitations.md) ## References - P. Triverio, S. Grivet-Talocia, M. S. Nakhla, F. G. Canavero, R. Achar, *"Stability, Causality, and Passivity in Electrical Interconnect Models"*, IEEE Trans. Advanced Packaging, vol. 30, no. 4, pp. 795–808, 2007. - D. M. Pozar, *Microwave Engineering*, 4th ed., Wiley, 2011.