{
  "schemaVersion": 1,
  "id": "asymmetric-two-port-passivity",
  "title": "Passivity is the largest singular value, not the largest element",
  "kind": "two-port-sigma",
  "availability": "helper-only",
  "inputs": {
    "referenceOhms": 50,
    "entryMagnitude": 0.7071067811865476,
    "ports": 2
  },
  "inputUnits": {
    "referenceOhms": "ohm",
    "entryMagnitude": "-",
    "ports": "-"
  },
  "prediction": "This two-port is passive and lossless. Before looking: which of the three common tests — the largest matrix element, the column norm, or |S11 ± S21| — will agree with that, and which will not?",
  "explanation": "A two-port is passive when the largest singular value of S is at most one, because that is the largest possible ratio of reflected-plus-transmitted power to incident power over every combination of incident waves. For a RECIPROCAL and PORT-SYMMETRIC two-port the singular values happen to be |S11 + S21| and |S11 − S21|, which is where that shortcut comes from. This network is reciprocal but not port-symmetric — S22 = −S11 — so the shortcut does not apply, and applied anyway it reports 1.414 and calls a lossless network active. The largest element is 0.707, which passes everything and therefore discriminates nothing. The column norm reaches 1 and cannot separate this case from a marginal one.",
  "limitations": [
    "One frequency-independent matrix, two ports, 50 ohm reference.",
    "The singular value bound is a necessary condition for passivity at each frequency; it is not a statement about causality or about stability under feedback."
  ],
  "evidence": {
    "type": "analytical",
    "provenance": "S^H S = I by construction for this matrix, so every singular value is exactly 1. The fixture and its counterexample come from Astra’s review of 12 September 2026, appendix A; the comparison values here are computed from the same matrix."
  },
  "relatedPage": "/topics/fundamentals/s-parameters.html",
  "checks": [
    {
      "metric": "sigmaMax",
      "expected": 1.0,
      "absoluteTolerance": 1e-12,
      "unit": "-",
      "derivation": "S^H S = I, so all singular values are 1."
    },
    {
      "metric": "symmetricShortcut",
      "expected": 1.4142135623730951,
      "absoluteTolerance": 1e-12,
      "unit": "-",
      "derivation": "max(|S11+S21|, |S11-S21|) with S11 = S21 = 1/sqrt(2) gives sqrt(2); the shortcut is only valid when S22 = S11, which it is not here."
    },
    {
      "metric": "largestElement",
      "expected": 0.7071067811865476,
      "absoluteTolerance": 1e-12,
      "unit": "-",
      "derivation": "Every entry has magnitude 1/sqrt(2)."
    },
    {
      "metric": "columnNorm",
      "expected": 1.0,
      "absoluteTolerance": 1e-12,
      "unit": "-",
      "derivation": "Each column has two entries of 1/sqrt(2), so its 2-norm is 1 — equal to the true bound here, which is why it cannot discriminate."
    }
  ],
  "sourceFile": "tests/fixtures/passive-unitary-asymmetric.s2p"
}