{
  "id": "parallel-resistor",
  "version": "c61360fb952b",
  "status": "published",
  "name": "Parallel Resistor Calculator",
  "question": "What is the total of parallel resistors?",
  "summary": "Computes the total resistance of resistors in parallel, 1 ÷ (1/R₁ + 1/R₂ + …), or the resistor to add in parallel to reach a target, with the total current and power at a voltage.",
  "category": "physics",
  "subcategory": "electricity",
  "url": "https://www.acalculator.org/physics/parallel-resistor-calculator",
  "markdown": "https://www.acalculator.org/physics/parallel-resistor-calculator.md",
  "kind": "function",
  "method": "Total = 1 ÷ (1/R₁ + 1/R₂ + … + 1/Rₙ); resistor to add = 1 ÷ (1/target − (1/R₁ + … + 1/Rₙ)); current = V ÷ total; power = V² ÷ total.",
  "assumptions": [
    "The resistors are ideal and all connected between the same two points.",
    "The voltage is DC, or the RMS value of AC across resistors (no reactance).",
    "Each resistance is from 1 µΩ to 1 TΩ."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "The total of the resistors, or the resistor to add in parallel to reach a target total.",
        "type": "string",
        "enum": [
          "total",
          "missing"
        ]
      },
      "resistors": {
        "title": "Resistors in parallel",
        "description": "Each resistor connected in parallel, one per row.",
        "type": "array",
        "items": {
          "type": "object",
          "properties": {
            "r": {
              "title": "Resistance",
              "description": "The resistance of one resistor.",
              "type": [
                "number",
                "string"
              ],
              "x-quantity": "resistance",
              "minimum": 0.000001,
              "maximum": 1000000000000
            }
          }
        }
      },
      "target": {
        "title": "Target total",
        "description": "The total resistance you want, lower than the resistors you have.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "resistance",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "v": {
        "title": "Voltage (optional)",
        "description": "The voltage across the resistors, for the total current and power; clear it to leave them out.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "voltage",
        "minimum": 0,
        "maximum": 1000000
      }
    }
  },
  "outputs": {
    "needed": {
      "label": "Resistor to add (Ω)",
      "description": "The resistor that brings the parallel total down to the target, in ohms.",
      "format": "number"
    },
    "total": {
      "label": "Total resistance (Ω)",
      "description": "The resistance of all the resistors together in parallel, in ohms (with a resistor to add: the target).",
      "format": "number"
    },
    "conductance": {
      "label": "Total conductance (S)",
      "description": "The sum of 1/R for every resistor, in siemens: 1 ÷ the total resistance.",
      "format": "number"
    },
    "current": {
      "label": "Total current (A)",
      "description": "The current drawn at the voltage: V ÷ the total resistance.",
      "format": "number"
    },
    "power": {
      "label": "Total power (W)",
      "description": "The power the resistors turn into heat at the voltage: V² ÷ the total resistance.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "total",
      "resistors": [
        {
          "r": 100
        },
        {
          "r": 220
        },
        {
          "r": 470
        }
      ],
      "target": 50,
      "v": 12
    },
    "outputs": {
      "total": 59.97679814385151,
      "conductance": 0.01667311411992263,
      "current": 0.20007736943907156,
      "power": 2.400928433268859
    },
    "text": "With every resistor in parallel, the total resistance is 59.976798 Ω."
  },
  "examples": [
    {
      "given": {
        "find": "total",
        "resistors": [
          {
            "r": 100
          },
          {
            "r": 220
          },
          {
            "r": 470
          }
        ]
      },
      "expect": {
        "total": 59.97679814385151,
        "conductance": 0.01667311411992263
      },
      "source": "OpenStax, University Physics Volume 2, §10.2 Resistors in Series and Parallel, https://openstax.org/books/university-physics-volume-2/pages/10-2-resistors-in-series-and-parallel; hand calculation in content.mdx: 1 ÷ (1/100 + 1/220 + 1/470) = 25,850/431 = 59.98 Ω"
    },
    {
      "given": {
        "find": "total",
        "resistors": [
          {
            "r": 1000
          },
          {
            "r": 1000
          }
        ],
        "v": 12
      },
      "expect": {
        "total": 500,
        "current": 0.024,
        "power": 0.288
      },
      "source": "OpenStax, University Physics Volume 2, §10.2 (equal resistors in parallel: R ÷ n) and §9.5 (P = V²/R); hand calculation in content.mdx"
    },
    {
      "given": {
        "find": "total",
        "resistors": [
          {
            "r": 10
          },
          {
            "r": 20
          },
          {
            "r": 30
          }
        ]
      },
      "expect": {
        "total": 5.454545454545454
      },
      "source": "OpenStax, University Physics Volume 2, §10.2; hand calculation in content.mdx: 1 ÷ (6/60 + 3/60 + 2/60) = 60/11 Ω"
    },
    {
      "given": {
        "find": "missing",
        "resistors": [
          {
            "r": 150
          }
        ],
        "target": 100
      },
      "expect": {
        "needed": 300
      },
      "source": "OpenStax, University Physics Volume 2, §10.2; hand calculation in content.mdx: 1 ÷ (1/100 − 1/150) = 300 Ω"
    },
    {
      "given": {
        "find": "total",
        "resistors": [
          {
            "r": 4.5
          },
          {
            "r": 4.5
          }
        ]
      },
      "expect": {
        "total": 2.25
      },
      "source": "OpenStax, University Physics Volume 2, §10.2; hand calculation in content.mdx: 4.5 ÷ 2 = 2.25 Ω"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 2, §10.2 Resistors in Series and Parallel (1/R = 1/R₁ + 1/R₂ + …). https://openstax.org/books/university-physics-volume-2/pages/10-2-resistors-in-series-and-parallel",
    "OpenStax, University Physics Volume 2, §9.5 Electrical Energy and Power (P = V²/R). https://openstax.org/books/university-physics-volume-2/pages/9-5-electrical-energy-and-power",
    "NIST Special Publication 330 (2019), The International System of Units (SI): the ohm and the siemens (S = 1/Ω). https://www.nist.gov/pml/special-publication-330"
  ],
  "related": [
    "ohms-law",
    "power"
  ],
  "changelog": []
}