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  "assumptions": [
    "Lumped series first-order RC.",
    "When R and C are both given, tau is taken as R·C."
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  "limitations": [
    "Series first-order RC only — not parallel networks or distributed RC",
    "f_c is the ideal LPF −3 dB corner; not a filter design tool",
    "Fewer than two inputs → NEEDS_TWO_INPUTS; non-positive → VALUE_MUST_BE_POSITIVE",
    "When R and C are both provided, tau is recomputed from R·C"
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    "Non-finite → INVALID_NUMBER",
    "≤ 0 → VALUE_MUST_BE_POSITIVE"
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      "title": "IEC 60050 — International Electrotechnical Vocabulary",
      "section": "Time constant of an exponential quantity",
      "supports": "τ definition for exponential RC response",
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      "title": "Horowitz & Hill — The Art of Electronics",
      "section": "RC circuits, time constants, and single-pole filters",
      "supports": "τ = RC; ≈2.2τ for 10%–90%; f_c = 1/(2πRC)",
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    {
      "title": "NIST Guide to the SI (SP 811)",
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      "url": "https://www.nist.gov/pml/special-publication-811",
      "supports": "Ω, F, s as coherent units for τ = RC",
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        "t_5tau": 0.05
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        "C": 0.000001,
        "tau": 0.01,
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        "t_10_90": 0.022000000000000002,
        "t_5tau": 0.05
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        "tau": 0.001
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        "R": 4700,
        "C": 2.1276595744680852e-7,
        "tau": 0.001,
        "f_c": 159.15494309189532,
        "t_10_90": 0.0022,
        "t_5tau": 0.005
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      "tol": 1e-12
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        "C": 1e-7,
        "tau": 0.000047
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        "C": 1e-7,
        "tau": 0.000047,
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        "t_10_90": 0.0001034,
        "t_5tau": 0.000235
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      "tol": 1e-9
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        "R": 1000
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      "expectError": "NEEDS_TWO_INPUTS",
      "code": "NEEDS_TWO_INPUTS",
      "error": "RC time constant needs any two of R, C, tau"
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      "id": "empty",
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      "code": "NEEDS_TWO_INPUTS",
      "error": "RC time constant needs any two of R, C, tau"
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    {
      "id": "zero-R",
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      "inputs": {
        "R": 0,
        "C": 0.000001
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      "expectError": "VALUE_MUST_BE_POSITIVE",
      "code": "VALUE_MUST_BE_POSITIVE",
      "error": "R must be greater than zero"
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    {
      "id": "nan-C",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "R": 1000,
        "C": "n/a"
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      "expectError": "INVALID_NUMBER",
      "code": "INVALID_NUMBER",
      "error": "C must be a finite number"
    },
    {
      "id": "boundary-rc-10ms",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "R": 10000,
        "C": 0.000001
      },
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        "tau": 0.01,
        "R": 10000,
        "C": 0.000001,
        "f_c": 15.915494309189533
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      "actual": {
        "R": 10000,
        "C": 0.000001,
        "tau": 0.01,
        "f_c": 15.915494309189533,
        "t_10_90": 0.022000000000000002,
        "t_5tau": 0.05
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      "tol": 1e-9
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        "tau": 0.001
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        "C": 2.1276595744680852e-7,
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        "f_c": 159.15494309189532,
        "t_10_90": 0.0022,
        "t_5tau": 0.005
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      "tol": 1e-12
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