{
  "circuit": "Rs + (CPE || (Rct + semi-infinite Warburg))",
  "formula": "Z=Rs+1/(Q*(j*omega)^alpha+1/(Rct+sigma*(1-j)/sqrt(omega))); omega=2*pi*f",
  "parameter_units": {
    "Rs_ohm": "ohm",
    "Rct_ohm": "ohm",
    "CPE_Q": "S s^alpha",
    "CPE_alpha": "1",
    "Warburg_sigma": "ohm s^(-1/2)"
  },
  "parameters": {
    "A": {
      "Rs_ohm": 4.2,
      "Rct_ohm": 23,
      "CPE_Q": 0.00085,
      "CPE_alpha": 0.86,
      "Warburg_sigma": 3.0
    },
    "B": {
      "Rs_ohm": 5.1,
      "Rct_ohm": 43,
      "CPE_Q": 0.00067,
      "CPE_alpha": 0.83,
      "Warburg_sigma": 4.0
    }
  },
  "scientific_basis": {
    "schema_version": 1,
    "model_class": "analytic_model",
    "data_origin": "original_synthetic",
    "parameter_origin": "self-chosen teaching values; never paper measurements",
    "validation_scope": "Declared teaching model; no experimental/material certification",
    "equations": [
      "omega=2*pi*f; ZW=sigma*(1-j)/sqrt(omega)",
      "Y_CPE=Q*(j*omega)^alpha",
      "Z=Rs+1/(Y_CPE+1/(Rct+ZW))",
      "Nyquist=(Re Z,-Im Z); Bode=(abs(Z),arg(Z)) from the same stored complex values"
    ],
    "assumptions": [
      "Linear small-signal stationary passive circuit, f>0, Rs>=0, Rct/Q>0, 0<alpha<=1, sigma>=0.",
      "Semi-infinite planar diffusion is an assumed boundary; finite diffusion or nonstationary spectra require another model.",
      "CPE is empirical; its Q with units S*s^alpha is not a measured capacitance unless alpha=1."
    ],
    "parameters": {
      "frequency_Hz": [
        0.01,
        100000.0
      ],
      "A": {
        "Rs_ohm": 4.2,
        "Rct_ohm": 23,
        "CPE_Q": 0.00085,
        "CPE_alpha": 0.86,
        "Warburg_sigma": 3.0
      },
      "B": {
        "Rs_ohm": 5.1,
        "Rct_ohm": 43,
        "CPE_Q": 0.00067,
        "CPE_alpha": 0.83,
        "Warburg_sigma": 4.0
      }
    },
    "references": [
      {
        "url": "https://www.gamry.com/application-notes/EIS/basics-of-electrochemical-impedance-spectroscopy/",
        "title": "Basics of Electrochemical Impedance Spectroscopy",
        "supports": "Complex impedance, series/parallel Randles topology, CPE and semi-infinite Warburg conventions; numbers self-chosen.",
        "scope": "model_definition_only",
        "checked_date": "2026-10-01"
      }
    ],
    "limitations": [
      "No circuit fit, experimental validation, transport coefficient or microscopic mechanism."
    ]
  },
  "status": "original synthetic teaching model; no fitted experimental or material certification"
}