RF Amplifier Analysis — rfplot, Stability, Gain, Polish

This is the second half of the rustlab RF S-parameter walkthrough. The companion notebook sparameters_intro.md covered loading, conversions, cascading, and Smith charts. Here we focus on what a measurement actually tells you about a 2-port device: the standard frequency-domain review panel, port-level metrics (VSWR, return / insertion loss), stability assessment via Rollett K and µ-parameters, simultaneous- conjugate-match design with Γms / Γml, gain limits (MAG / MSG), and graphical stability / gain analysis via circles on the Smith chart.

The final section covers the Phase 6 polish features: frequency-grid interpolation for cross-sweep cascading, time-domain reflectometry via IFFT, Touchstone noise-parameter access, and mixed-mode 4-port conversion for differential designs.

Setup — the same synthetic LNA

We start from the same hand-built 2-port LNA as the intro notebook so the numerical results are comparable:

The 2×2 review panel — rfplot(s)

The default rfplot call gives you the canonical 2×2 "what did the VNA measure" panel: |S11| and |S21| on the top row, |S12| and |S22| on the bottom, all in dB on a log-frequency axis. This is the standard RF first-look.

Layout convention matches every commercial RF tool:

For non-2-port networks the same call falls back to a single |S11| dB trace; the multi-trace 2×2 layout is specifically the 2-port review.

Single-trace forms — phase, unwrap, group delay

Five single-trace variants pull out specific quantities:

For a delay-line or amplifier with non-trivial phase response, wrapped phase is hard to read. The "unwrap" form removes the ±2π jumps:

And group delay \(\tau_g = -d\varphi/d\omega\) is the natural metric for linearity / dispersion:

Group delay uses central differences on the unwrapped phase with forward/backward differences at the endpoints. Constant group delay across the band of interest indicates a linear-phase response.

Port-level metrics

Three quick scalar metrics per frequency: voltage standing-wave ratio at each port, return loss, and insertion loss between any two ports.

vswr caps at \(10^6\) for fully-reflecting ports (|S| → 1); return loss floors at 200 dB for matched ports (|S| → 0). Both keep the results plottable and finite.

Stability — Rollett K and µ-parameters

A 2-port is unconditionally stable at a given frequency if and only if both of these hold:

• \(K > 1\) where \(K = \dfrac{1 - |S_{11}|^2 - |S_{22}|^2 + |\Delta|^2}{2\,|S_{12}\,S_{21}|}\) is Rollett's stability factor and \(\Delta = S_{11}\,S_{22} - S_{12}\,S_{21}\).
• \(|\Delta| < 1\).

Equivalently, a single-number test exists: \(\mu_1 > 1\), where \(\mu_1 = \dfrac{1 - |S_{11}|^2}{|S_{22} - \Delta\,S_{11}^*| + |S_{12}\,S_{21}|}\).

stabilitymu returns both µ1 and µ2; either > 1 is necessary and sufficient for unconditional stability.

K > 1 and µ1 > 1 across the entire 1–6 GHz band — this amplifier is unconditionally stable, so the conjugate-match design that follows is valid.

Maximum available gain (MAG) and the conjugate match

For an unconditionally-stable amplifier, the optimum source and load terminations satisfying \(\Gamma_{\text{in}}(\Gamma_L) = \Gamma_S^*\) and \(\Gamma_{\text{out}}(\Gamma_S) = \Gamma_L^*\) exist in closed form:

\[\Gamma_{ms} = \frac{B_1 - \text{sign}(B_1)\sqrt{B_1^2 - 4|C_1|^2}}{2\,C_1}\]

where \(B_1 = 1 + |S_{11}|^2 - |S_{22}|^2 - |\Delta|^2\) and \(C_1 = S_{11} - \Delta\,S_{22}^*\). Γml has the symmetric form with ports swapped. With those terminations, the transducer power gain equals the maximum available gain:

\[\text{MAG} = \frac{|S_{21}|}{|S_{12}|}\left(K - \sqrt{K^2 - 1}\right)\]

For K ≤ 1 (potentially unstable), MAG is undefined and gainmax returns instead the maximum stable gain \(\text{MSG} = |S_{21}/S_{12}|\).

Both Γms and Γml lie inside the unit disk (their magnitudes < 1) — expected for an unconditionally-stable network. MAG ranges from about 8 dB at the band edges to 12 dB at the peak around 3 GHz.

Reflection coefficient with termination

gammain(s, Γ_load) and gammaout(s, Γ_source) compute the input or output reflection for a given termination. The defining identity for the simultaneous conjugate match is that

\[\Gamma_{\text{in}}(\Gamma_{ml}) = \Gamma_{ms}^* \quad \text{and} \quad \Gamma_{\text{out}}(\Gamma_{ms}) = \Gamma_{ml}^*\]

gammain accepts a scalar (broadcast across frequencies) or a per- frequency vector matching n_freqs.

Both diffs come out at machine precision — the identity holds exactly.

Stability circles on the Smith chart

Even for unconditionally-stable networks it's useful to visualize the stability boundary: the locus of source (or load) reflections at which the other port's reflection touches the unit disk. stability_circles returns a tagged struct containing per-frequency centre and radius vectors; render them with smith_circle() in a loop:

For this LNA the input stability circles lie entirely outside the unit disk — the geometric witness of unconditional stability. (If a circle crossed into the unit disk, sources with reflections inside that intersection would drive |Γout| > 1 at that frequency, i.e. oscillation.)

Constant-gain circles

gain_circles(s, gain_db) returns the locus of load reflections that achieve a specified operating-power gain. Useful for amplifier matching design: pick an operating gain a couple of dB below MAG, draw the circle, and any load termination on that locus delivers that gain (trading off bandwidth, noise figure, etc.).

As the requested gain approaches MAG the circle shrinks to a single point at Γml; gains beyond MAG return NaN radii (no real solution).

Phase 6 — polish features

The remaining features cover four common practical needs.

Cross-grid cascading via interp_freq

When two measurements live on different VNA sweeps, you can't cascade them directly — the frequency grids must match. interp_freq resamples linearly onto a new monotonic grid:

Extrapolation past the original sweep range is rejected with a clear error — RF data is bandlimited; extrapolated S-parameters are garbage.

Time-domain reflectometry — s2td

s2td(s, i, j) IFFTs a single Sij(f) trace into the time domain. Useful for TDR (locating impedance discontinuities along a cable) and impulse- response inspection. The frequency grid must be uniform — that's exactly what interp_freq produces.

Two modes available: "step" (default, integrates the impulse response — the standard TDR view) and "impulse" (the raw impulse response). The internal IFFT uses a conjugate-symmetric 2N-point spectrum so the time- domain signal is real, zero-padded to the next power of two for finer time resolution.

Touchstone noise parameters

Many real .s2p files include a noise-parameter block after the S-parameter rows: five columns per row with frequency, \(F_{\min}\) in dB, \(|\Gamma_{\text{opt}}|\), ∠Γopt in degrees, and the normalised equivalent noise resistance \(R_n / Z_0\).

The reader picks them up automatically when present and attaches them as extra fields on the sparameters struct. The LNA in this notebook is built from S only, so it has no noise block. A file that does, such as examples/rf/data/lna_with_noise.s2p, loads with sparameters(path) from rustlab run (a notebook jails paths to the notebook directory). The Touchstone writer emits S only; a noise block is not round-tripped.

When loading a real noise-bearing .s2p, the accessor builtins return the per-noise-frequency vectors:

nfmin(s_noise)      → real Vector, dB
gamma_opt(s_noise)  → complex Vector
rn(s_noise)         → real Vector (Rn/Z0)
noise_freqs(s_noise) → real Vector, Hz
has_noise(s_noise)  → Bool

The noise frequencies need not match the S grid — VNAs commonly sweep many fewer noise points than S points.

Mixed-mode for 4-port differential designs

For high-speed differential designs (SerDes, USB, PCIe, RF balanced amps) the single-ended 4-port S-parameter matrix isn't the natural representation — you want differential / common-mode (Sdd, Sdc, Scd, Scc) instead. s2smm transforms a 4-port single-ended network to its mixed-mode form via the standard Bockelman/Eisenstadt orthogonal transform; smm2s is its inverse.

Port pairing convention: ports 1 (positive) and 3 (negative) form differential pair 1; ports 2 and 4 form differential pair 2. The result is organised as the block matrix [Sdd | Sdc; Scd | Scc] with port order [d1, d2, c1, c2].

A round-trip through smm2s recovers the single-ended network exactly because the transform is orthogonal:

Summary

Across the two notebooks, the toolbox surface is:

• Data + plotting (intro): sparameters, nports/freqs/sij/s11..s22, parameter conversions s2z/y/t/abcd and reverses, cascade, deembed, newref, save, smith, marker.
• Analysis (this notebook): rfplot 2×2 + single-trace variants, vswr, return_loss, insertion_loss, gammain/gammaout, stabilityk, stabilitymu, gammams/gammaml, gainmax, stability_circles, gain_circles, smith_circle.
• Polish (this notebook): interp_freq, s2td, noise accessors (nfmin, gamma_opt, rn, noise_freqs, has_noise), mixed-mode s2smm/smm2s, Touchstone v2 keyword tolerance.

The standalone runnable scripts in examples/rf/ are organised by phase if you want a smaller starting point for any single topic: load_s2p.rlab, cascade_attenuator.rlab, smith_chart.rlab, measurement_review.rlab, amplifier_stability.rlab, polish_features.rlab.