{
  "schemaVersion": 1,
  "id": "cdr-natural-frequency",
  "kind": "cdr",
  "model": "cdr",
  "modelVersion": "1.1",
  "contractVersion": "2.1",
  "params": {
    "fn": 4,
    "zeta": 0.5,
    "margin": 0.3,
    "ceiling": 20,
    "jitterFrequency": 4,
    "jitterAmplitude": 0.4
  },
  "topology": "Declared linear type-2 second-order CDR teaching loop.",
  "planes": "Input phase, recovered phase and residual phase at the same time; phase measured in UI.",
  "evidence": {
    "type": "analytical",
    "provenance": "First-principles boundary conditions, linear circuit laws or the declared teaching-loop equation; no measured or licensed data."
  },
  "checks": [
    {
      "metric": "inputAtZero",
      "expected": 0,
      "absoluteTolerance": 1e-10,
      "unit": "UI",
      "derivation": "Input is a sine with zero phase at t=0."
    },
    {
      "metric": "recoveredAtZero",
      "expected": -0.2,
      "absoluteTolerance": 1e-10,
      "unit": "UI",
      "derivation": "At f=fn and ζ=.5, H=1−j. A .4 UI peak-to-peak sine has .2 UI amplitude; the quadrature term at t=0 is −.2 UI."
    },
    {
      "metric": "residualAtZero",
      "expected": 0.2,
      "absoluteTolerance": 1e-10,
      "unit": "UI",
      "derivation": "Residual=input−recovered=0−(−.2) UI."
    },
    {
      "metric": "recoveredAtQuarter",
      "expected": 0.2,
      "absoluteTolerance": 1e-10,
      "unit": "UI",
      "derivation": "At a quarter period, sin=1 and cos=0; Re(H)=1 gives .2 UI."
    },
    {
      "metric": "residualAtQuarter",
      "expected": 0,
      "absoluteTolerance": 1e-10,
      "unit": "UI",
      "derivation": "Input and recovered phase coincide at a quarter period in this example."
    }
  ],
  "lesson": {
    "prediction": "If the recovered phase amplitude exceeds the input amplitude, can the residual still be found by subtracting magnitudes?",
    "experiment": "Open the scenario, keep its network and stimulus fixed, and inspect the stated observation. Compare with the derived answer before changing one variable.",
    "explanation": "No. At f=fn the real part of the denominator vanishes; the declared loop reduces to H=1−j/(2ζ). At ζ=.5 the residual transfer is j, so the phase relation determines instantaneous error.",
    "transferQuestion": "Does this frequency equal the closed-loop −3 dB bandwidth?",
    "answer": "No. It is the natural frequency of the declared denominator; the closed-loop bandwidth depends on damping and the numerator."
  },
  "limitations": [
    "Steady-state linear teaching loop, not acquisition, cycle slips, noise, or a BER-qualified tolerance mask.",
    "Peak-to-peak input amplitude is twice its sine amplitude.",
    "Quarter-period sample is checked against its exported time coordinate."
  ],
  "scenario": "/topics/signal-integrity/receiver-and-clock.html#lab=cdr;v=2.1;cdr-fn=4;cdr-zeta=50;cdr-margin=30;cdr-jitter-f=4;cdr-jitter-a=40",
  "parameterUnits": {
    "fn": "MHz",
    "zeta": "-",
    "margin": "UI",
    "ceiling": "UI pp",
    "jitterFrequency": "MHz",
    "jitterAmplitude": "UI pp"
  },
  "stimulus": "Steady-state sinusoidal input phase, 0.4 UI peak-to-peak at 4 MHz."
}