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Derived Metrics

Stability, passivity, group delay and the two-terminal passive readouts — every equation, written out.

A derived metric is a quantity the Data Display computes from an S-matrix rather than reading out of it. Stability factors, gain limits, passivity, group delay, and the impedance readouts a capacitor or an inductor is actually specified in — none of these is a matrix element, and all of them are offered on the trace card beside the elements themselves.

This page exists so that you never have to guess which convention was used. Every derived metric below is given as the equation the code evaluates, in the same symbols, so that a number you disagree with can be traced to an assumption rather than to a mystery.

Where they live, and what they will plot on

Add one exactly as you would add a matrix element: select a network source in the trace card's item selector and pick the derived metric from the list beneath the S-parameter elements.

Picking one reveals its own controls — the ordered port pair, and for a passive readout the Fixture row, which is the subject of its own section below.

Two kinds, and they do not mix in one plot:

Picking one on the wrong plot type is not an error and is not silently ignored — the item is offered but disabled, with the reason on it. See Plot Types.

What every derived metric assumes

Three assumptions are shared by everything on this page. They are stated once here rather than repeated under each formula.

1 — The reference impedance is made uniform and real first. Every formula below is derived for a single real Z₀. A simulated S-matrix may be referenced to a different impedance at each port, and those may be complex; a Touchstone file is uniform by construction but need not be 50 Ω. So before any of these is evaluated, the matrix is renormalised — using the full power-wave form, on the whole matrix, never element by element. The target is Re(Z0) of the input port, which is why a 75 Ω part and its Touchstone file give identical answers instead of both being forced to 50 Ω.

2 — The port pair is ORDERED. For a 2-port formula the trace card offers an input and an output selector, and swapping them is a different selection, not a re-spelling of the same one. μ is the load stability factor and μ′ the source one, so (1,2) and (2,1) exchange their meanings. On a 2-port the selectors are hidden — input 1, output 2 is the only sensible reading.

3 — On an N-port, the other ports are assumed terminated in the reference impedance. Extracting a 2×2 sub-matrix from a 4-port is valid only under that assumption. It is the standard one and it is almost always what you want, but if you are comparing against a bench measurement that terminated port 3 in something else, that is where the disagreement comes from. The trace card says so in a note under the port selectors whenever N > 2.

Stability: μ, μ′, K, |Δ|

μ — load stability factor (Edwards–Sinsky)

μ = (1 − |S11|²) / ( |S22 − Δ·S11*| + |S12·S21| )

μ > 1 ⟺ unconditionally stable. Its great advantage over K is that it is a single number with a direct geometric meaning: μ is the distance from the centre of the Smith chart to the nearest unstable load. So a larger μ is unambiguously more stable, and two designs can be compared by it. K cannot do that.

μ′ — source stability factor

μ′ = (1 − |S22|²) / ( |S11 − Δ·S22*| + |S12·S21| )

The same measure taken from the source side: the distance to the nearest unstable source termination. Note that μ and μ′ come from the same trace card control — selecting μ with the ports swapped gives μ′, which is why the port pair being ordered matters.

K — Rollett stability factor

K = (1 − |S11|² − |S22|² + |Δ|²) / (2·|S12·S21|)

Unconditional stability requires K > 1 and |Δ| < 1 — both, always. K alone is not a stability criterion, which is the reason |Δ| is offered as its own trace rather than folded into K's. Plot them together.

And both together are still a two-port test. A two-port that passes K and |Δ| can still oscillate, because these are computed from the reduced two-port and a pole the reduction hides is a pole they cannot see — see the WSProbe page's NDF example.

|Δ|

The magnitude of the determinant above, plotted on its own so that the K > 1 test can be completed.

Maximum gain: MAG and MSG

One trace, one formula, and which of the two classical quantities you get depends on K at that frequency:

                |S21|
ratio     =    -------
                |S12|

K ≥ 1  (MAG)   G = ratio · ( K − √(K² − 1) )      maximum available gain
K < 1  (MSG)   G = ratio                          maximum stable gain

The transition is exactly at K = 1 and it is continuous there, so a single curve runs across the potentially-unstable region into the unconditionally-stable one without a discontinuity — which is what makes the plot readable in the first place.

The card offers this in dB (10·log10 G) or as a linear power ratio. The two come from the same implementation rather than the display taking a logarithm of the other, so they cannot disagree.

Stability circles

Loci in the Γ plane, one circle per frequency. The load circle is the set of Γ_L that put |Γ_in| = 1; the source circle is the set of Γ_S that put |Γ_out| = 1.

Load    CL = (S22 − Δ·S11*)* / (|S22|² − |Δ|²)          rL = |S12·S21| / | |S22|² − |Δ|² |

Source  CS = (S11 − Δ·S22*)* / (|S11|² − |Δ|²)          rS = |S12·S21| / | |S11|² − |Δ|² |

circuitRF also decides, per frequency, whether the stable region is inside or outside the circle and renders it accordingly — that is not a cosmetic choice, and reading a circle without it is how a "stable" design gets built around the unstable half of the chart.

Passivity, σ_max

σmax = the largest singular value of S

The network is passive at that frequency iff σ_max ≤ 1. The value says how far from passive, not merely whether: 1 is the boundary, 1.05 is 5 % of excess gain, and a well-behaved passive file sits just below 1 everywhere.

Two things make this different from everything else on this page:

• It is not a 2-port formula. It is defined for any N ≥ 1 (at N = 1 it is just |S₁₁|), so it needs no port pair.
• Scope is a choice above 2 ports. The card offers whole network (the default) or the extracted input/output pair. They answer different questions, and a sub-matrix can test passive while the full network is not — so if you are checking a de-embedding or an EM result for physicality, you want the whole network.

Group delay

τ(ω) = −dφ/dω        φ = the UNWRAPPED phase of S21 (input → output)

Plotted in nanoseconds. Three details that decide whether the number is meaningful:

• The phase is unwrapped first. A raw atan2 phase is a principal value in (−π, π], so any network whose transmission runs through more than half a cycle wraps somewhere inside its own passband. A difference taken across that wrap is a spike one grid-step wide that moves when you change the grid.
• The derivative is a central difference on the sweep's own grid (one-sided at the two ends), so the result has exactly as many points as the sweep. The grid need not be uniform.
• The sweep must resolve the phase to better than half a cycle per step. That is the one real limit: a sweep too coarse for that cannot have its group delay measured by any method, and unwrapping will quietly guess wrong rather than fail.

The two-terminal passive readouts

Six quantities, and they exist for a specific reason: a capacitor, an inductor or a ferrite bead is specified, bought and compared in ESR, C, L, Q and self-resonance — never in S-parameters. When a vendor ships you both an S-parameter file and a model, the question you actually want to ask is whether the two agree, and answering it from S₁₁ in dB is possible but nobody does it.

All six are point functions of one impedance Z(f) = R + jX, extracted from the S-matrix by the fixture you select. With ω = 2πf:

The traces are left in base SI — ohms, farads, henries. Unlike group delay, where nanoseconds are universal, the capacitors in one plot can run from picofarads to millifarads and there is no single scale that would not be wrong for half of them. The unit is on the trace's own label.

The fixture: the one thing you must tell it

Nothing in a Touchstone file records how the part was measured, and the readings differ by orders of magnitude. So the trace card carries a Fixture selector, and it is an explicit choice rather than something circuitRF tries to detect.

Shunt-through is the default because it is how vendor decoupling-capacitor and inductor files are measured. It is the fixture with usable dynamic range on a part that is milliohms at resonance: there, S₁₁ sits within a part in 10⁴ of −1, where a network analyser's directivity and calibration residuals dominate, while the same impedance appears in S₂₁ as a well-conditioned transmission measurement. A file with only one port has no S₂₁ to read and is always taken as a 1-port, whatever the selector remembers.

Self-resonance

The self-resonant frequency is reported on the trace card whenever a passive readout is selected, and it is defined as:

the lowest frequency at which the reactance crosses zero going capacitive → inductive, interpolated in log f.

The direction is the whole definition and an unsigned zero-crossing search is wrong. A real part has several crossings: the series resonance where |Z| is a minimum — the number everyone means by "SRF" — and above it the mounting or package parallel resonance where |Z| is a maximum and the reactance crosses back the other way. The sign of the crossing is what separates them.

If the sweep contains no such crossing, the card reports nothing rather than a band edge. "This sweep does not contain the resonance" is the honest answer; a clamped band edge would be read as a measurement.

Interpolation is in log f because these sweeps are logarithmic, and interpolating linearly across a decade-wide gap places the crossing visibly wrong.

Checking data against a model

The workflow these readouts were built for:

1. Add the vendor's .sNp as one trace source, and a schematic built from the vendor's model as another. A Data Display reads files, not runs, so both live in one plot without either knowing about the other.
2. Add the same readout — ESR, or C effective — from each, on one rectangular plot with a log frequency axis.
3. Set the same Fixture on both.

Where the two curves separate is where the model stops being the part. In practice that is almost always above the band the model was fitted over, and almost always in ESR before it shows in |Z| — the reactance dominates |Z| everywhere except within a whisker of resonance, so a model with the loss wrong by a factor of three still overlays convincingly on a |Z| plot. Compare ESR, not |Z|.

Stability from inside the circuit

Everything on this page is computed from an S-matrix — a terminal view of a network. That is the right view for a part you are buying and the wrong one for a node inside an amplifier you are drawing, where the question is what the two halves of that node present to each other.

The WSProbe answers that one. It is a component you drop into a node; it perturbs nothing and costs no extra sweep, and what it produces is the driving-point impedance and admittance at that node, the impedance looking each way out of it, every classical loop gain, and a bounded stability margin that says how close the node is rather than only whether it is stable. It also brings the normalized determinant function, which is the rigorous N-port answer that K and |Δ| are an approximation to.

The two pages meet in both directions: a probe's reduced two-port is an ordinary network source, so every metric above — μ, K, |Δ|, MAG/MSG and the stability circles — applies to a stage measured inside the circuit, loaded by what actually loads it.

Asking the same questions headless

Two CLI verbs answer without opening a plot. Both are read-only.

Is this file sound?

circuitrf check part.s2p

Reports the port count, the sweep, the reference impedance, and then: a frequency axis that is not strictly increasing (an error — every interpolator here assumes a sorted axis and returns a plausible wrong number on one that is not), a reference impedance no renormalisation can use, a reference that is not 50 Ω (worth knowing, because it is carried silently into every comparison), σ_max > 1, a reciprocity error, and a causality measurement.

Passivity and reciprocity are reported as warnings, never errors, and they carry the number and the frequency. Nothing in a Touchstone file says what the part is: an amplifier is supposed to have gain and a circulator is supposed to be non-reciprocal. For a capacitor, either one is a defect. Only you know which you have.

What is this part?

circuitrf explain part.s2p

Reports the ports, the sweep, the reference impedance, and then — under every applicable fixture, side by side — the self-resonance and the impedance floor with the ESR there. Reporting all of them is not indecision: it is the fastest way to identify an unlabelled file, because only the physical reading gives a sensible resonance and a milliohm-scale floor, and the others are obviously nonsense next to it.

part.s2p  (touchstone)
  ports        2
  sweep        401 points, 1 MHz to 1 GHz, non-uniform
  reference impedance 50 Ω
  SRF (shunt-through)     22.507906 MHz
  |Z| min (shunt-through) |Z| = 5.057 mΩ at 22.387211 MHz, ESR 5 mΩ
  SRF (series-through)    (nothing)
  |Z| min (series-through) |Z| = 796.177 Ω at 1 GHz, ESR 1.268 Ω

Both verbs take --json, and every finding carries a stable diagnostic id. See the CLI reference.