{
  "id": "rc-filter",
  "version": "668326a77ffa",
  "status": "published",
  "name": "RC Filter Calculator",
  "question": "What is the RC filter cutoff?",
  "summary": "RC filter calculator: the cutoff frequency f_c = 1 ÷ (2πRC) of a low-pass or high-pass RC filter, or the R or C for a cutoff, with the gain in dB and the phase at a signal frequency.",
  "category": "physics",
  "subcategory": "electricity",
  "url": "https://www.acalculator.org/physics/rc-filter-calculator",
  "markdown": "https://www.acalculator.org/physics/rc-filter-calculator.md",
  "kind": "function",
  "method": "f_c = 1 ÷ (2πRC); R = 1 ÷ (2πf_cC); C = 1 ÷ (2πf_cR); with x = f ÷ f_c: low-pass gain 1 ÷ √(1 + x²), phase −arctan x; high-pass gain x ÷ √(1 + x²), phase +arctan(1 ÷ x); dB = 20 log₁₀ gain.",
  "assumptions": [
    "An ideal first-order RC filter: an ideal resistor and capacitor in series, driven by a sine wave from a source with no resistance of its own, and an output that draws no current.",
    "The cutoff (corner, −3 dB, half-power) frequency is where the capacitive reactance equals the resistance."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Which value to work out from f_c = 1 ÷ (2πRC).",
        "type": "string",
        "enum": [
          "cutoff",
          "resistance",
          "capacitance"
        ]
      },
      "type": {
        "title": "Filter",
        "description": "Low-pass takes the output across the capacitor; high-pass takes it across the resistor.",
        "type": "string",
        "enum": [
          "low",
          "high"
        ]
      },
      "R": {
        "title": "Resistance (R)",
        "description": "The resistor in series with the capacitor.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "resistance",
        "minimum": 0.001,
        "maximum": 1000000000000
      },
      "C": {
        "title": "Capacitance (C)",
        "description": "The capacitor, in the unit chosen below.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000
      },
      "Cu": {
        "title": "Capacitance unit",
        "description": "The unit of the capacitance you type.",
        "type": "string",
        "enum": [
          "pF",
          "nF",
          "uF",
          "mF",
          "F"
        ]
      },
      "fc": {
        "title": "Cutoff frequency (f_c)",
        "description": "The frequency where the output falls to 1 ÷ √2 of the input (−3 dB).",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "frequency",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "f": {
        "title": "Signal frequency (f)",
        "description": "Optional: a frequency to see the gain and phase of the filter at.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "frequency",
        "minimum": 0.000001,
        "maximum": 1000000000000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The value worked out, to 6 significant figures, in the largest unit that keeps it at 1 or more.",
      "format": "text"
    },
    "cutoffHz": {
      "label": "Cutoff frequency (Hz)",
      "description": "f_c = 1 ÷ (2πRC), in hertz.",
      "format": "number"
    },
    "omega": {
      "label": "Cutoff angular frequency (rad/s)",
      "description": "ω_c = 2πf_c = 1 ÷ (RC), in radians per second.",
      "format": "number"
    },
    "tau": {
      "label": "Time constant RC (s)",
      "description": "τ = R × C, in seconds.",
      "format": "number"
    },
    "resistance": {
      "label": "Resistance (Ω)",
      "description": "R in ohms.",
      "format": "number"
    },
    "capacitance": {
      "label": "Capacitance (F)",
      "description": "C in farads.",
      "format": "number"
    },
    "gain": {
      "label": "Gain at f (Vout ÷ Vin)",
      "description": "The output amplitude over the input amplitude at the signal frequency.",
      "format": "number"
    },
    "db": {
      "label": "Gain at f (dB)",
      "description": "20 × log₁₀ of the gain: −3.01 dB at the cutoff.",
      "format": "number"
    },
    "phase": {
      "label": "Phase at f (°)",
      "description": "How far the output leads (+) or lags (−) the input, in degrees.",
      "format": "number"
    },
    "reactance": {
      "label": "Capacitive reactance at f (Ω)",
      "description": "X_C = 1 ÷ (2πfC), in ohms.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "cutoff",
      "type": "low",
      "R": "1 kΩ",
      "C": 100,
      "Cu": "nF",
      "fc": "1 kHz",
      "f": "10 kHz"
    },
    "outputs": {
      "result": "f_c = 1.59155 kHz",
      "cutoffHz": 1591.5494309189535,
      "omega": 10000,
      "tau": 0.0001,
      "resistance": 1000,
      "capacitance": 1e-7,
      "gain": 0.15717672547758985,
      "db": -16.072235265805517,
      "phase": -80.95693892096232,
      "reactance": 159.15494309189535
    },
    "text": "f_c = 1.59155 kHz; the time constant RC is 0.0001 s."
  },
  "examples": [
    {
      "given": {
        "find": "cutoff",
        "type": "low",
        "R": 1000,
        "C": 100,
        "Cu": "nF"
      },
      "expect": {
        "cutoffHz": 1591.5494309189532,
        "omega": 10000,
        "tau": 0.0001,
        "result": "f_c = 1.59155 kHz"
      },
      "source": "OpenStax, University Physics Volume 2, §15.3 RLC Series Circuits with AC (Equations 15.9 to 15.11: φ, I₀ = V₀ ÷ Z, Z = √(R² + (X_L − X_C)²); V_R = I₀R, V_C = I₀X_C), https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac (retrieved 2026-10-19); hand calculation in content.mdx: f_c = 1 ÷ (2π × 1,000 × 10⁻⁷) = 1,591.55 Hz"
    },
    {
      "given": {
        "find": "resistance",
        "type": "high",
        "C": 10,
        "Cu": "uF",
        "fc": 60,
        "f": 60
      },
      "expect": {
        "R": 265.25823848649225,
        "result": "R = 265.258 Ω",
        "gain": 0.7071067811865476,
        "db": -3.010299956639812,
        "phase": 45,
        "reactance": 265.25823848649225
      },
      "source": "OpenStax, University Physics Volume 2, §15.2 Simple AC Circuits (Equation 15.3, X_C = 1 ÷ ωC; ω = 2πf; Example 15.1: a 10 µF capacitor at 60 Hz has X_C = 265 Ω), https://openstax.org/books/university-physics-volume-2/pages/15-2-simple-ac-circuits (retrieved 2026-10-19): X_C = 265 Ω for 10 µF at 60 Hz, so R = X_C there; OpenStax, University Physics Volume 2, §15.3 RLC Series Circuits with AC (Equations 15.9 to 15.11: φ, I₀ = V₀ ÷ Z, Z = √(R² + (X_L − X_C)²); V_R = I₀R, V_C = I₀X_C), https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac (retrieved 2026-10-19); hand calculation in content.mdx"
    },
    {
      "given": {
        "find": "capacitance",
        "type": "low",
        "R": 10000,
        "fc": 1000
      },
      "expect": {
        "C": 1.5915494309189534e-8,
        "result": "C = 15.9155 nF"
      },
      "source": "OpenStax, University Physics Volume 2, §15.3 RLC Series Circuits with AC (Equations 15.9 to 15.11: φ, I₀ = V₀ ÷ Z, Z = √(R² + (X_L − X_C)²); V_R = I₀R, V_C = I₀X_C), https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac (retrieved 2026-10-19); hand calculation in content.mdx: C = 1 ÷ (2π × 1,000 × 10,000) = 15.9155 nF"
    },
    {
      "given": {
        "find": "cutoff",
        "type": "low",
        "R": 1000,
        "C": 100,
        "Cu": "nF",
        "f": 10000
      },
      "expect": {
        "gain": 0.15717672547758985,
        "db": -16.072235265805517,
        "phase": -80.95693892096232
      },
      "source": "OpenStax, University Physics Volume 2, §15.3 RLC Series Circuits with AC (Equations 15.9 to 15.11: φ, I₀ = V₀ ÷ Z, Z = √(R² + (X_L − X_C)²); V_R = I₀R, V_C = I₀X_C), https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac (retrieved 2026-10-19); hand calculation in content.mdx: gain = 1 ÷ √(1 + 4π²), phase = −arctan 2π"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 2, §15.2 Simple AC Circuits: Equation 15.3, capacitive reactance X_C = 1 ÷ (ωC), ω = 2πf; Example 15.1 (a 10 µF capacitor at 60 Hz: X_C = 265 Ω), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/15-2-simple-ac-circuits",
    "OpenStax, University Physics Volume 2, §15.3 RLC Series Circuits with AC: Equations 15.9 (phase angle), 15.10 (I₀ = V₀ ÷ Z) and 15.11 (Z = √(R² + (X_L − X_C)²)); V_R = I₀R and V_C = I₀X_C, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/15-3-rlc-series-circuits-with-ac",
    "OpenStax, University Physics Volume 2, §10.5 RC Circuits: Equation 10.9, the time constant τ = RC; 63.2% of full charge after one τ, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits",
    "OpenStax, University Physics Volume 2, §9.5 Electrical Energy and Power: Equation 9.13, P = V² ÷ R, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-2/pages/9-5-electrical-energy-and-power",
    "OpenStax, University Physics Volume 1, §17.3 Sound Intensity: Equation 17.12, a level in decibels is 10 log₁₀ of a power ratio; twice the power is 3.01 dB more, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/17-3-sound-intensity",
    "HyperPhysics (Georgia State University), RC High Pass Filter: breakpoint angular frequency ω = 1 ÷ (RC); a voltage ratio in decibels is 20 log₁₀(Vout ÷ Vin), retrieved 2026-10-19. https://hyperphysics.gsu.edu/hbase/Electric/filcap.html"
  ],
  "related": [
    "impedance",
    "ohms-law",
    "frequency",
    "voltage-divider"
  ],
  "changelog": []
}