# What is the total of parallel resistors?

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.

- Page: https://www.acalculator.org/physics/parallel-resistor-calculator
- JSON spec: https://www.acalculator.org/physics/parallel-resistor-calculator.json
- Version: c61360fb952b

## Default answer

Example with the default inputs (Find Total resistance, Resistors in parallel [Resistance 100 Ω; Resistance 220 Ω; Resistance 470 Ω], Voltage (optional) 12 V): With every resistor in parallel, the total resistance is 59.976798 Ω.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| find | Find | The total of the resistors, or the resistor to add in parallel to reach a target total. |
| resistors | Resistors in parallel | Each resistor connected in parallel, one per row. |
| target | Target total | The total resistance you want, lower than the resistors you have. |
| v | Voltage (optional) | The voltage across the resistors, for the total current and power; clear it to leave them out. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| needed | Resistor to add (Ω) | The resistor that brings the parallel total down to the target, in ohms. |
| total | Total resistance (Ω) | The resistance of all the resistors together in parallel, in ohms (with a resistor to add: the target). |
| conductance | Total conductance (S) | The sum of 1/R for every resistor, in siemens: 1 ÷ the total resistance. |
| current | Total current (A) | The current drawn at the voltage: V ÷ the total resistance. |
| power | Total power (W) | The power the resistors turn into heat at the voltage: V² ÷ the total resistance. |

## 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.

## Worked examples

1. find = total, resistors = {"r":100} or {"r":220} gives total = 59.976798, conductance = 0.016673. 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.
2. find = total, resistors = {"r":1000} or {"r":1000}, v = 12 gives total = 500, current = 0.024, power = 0.288.
3. find = total, resistors = {"r":10} or {"r":20} gives total = 5.454545.
4. find = missing, resistors = {"r":150} or undefined, target = 100 gives needed = 300.
5. find = total, resistors = {"r":4.5} or {"r":4.5} gives total = 2.25.

## FAQ

### How do I calculate resistors in parallel?

Add up the reciprocals and take the reciprocal of the sum: 1/R = 1/R₁ + 1/R₂ + … For 100 Ω, 220 Ω and 470 Ω: 1/100 + 1/220 + 1/470 = 0.016673, so R = 1 ÷ 0.016673 = 59.98 Ω.

### What is the formula for two resistors in parallel?

For two resistors, R = (R₁ × R₂) ÷ (R₁ + R₂), the product over the sum. For 100 Ω and 220 Ω: 22,000 ÷ 320 = 68.75 Ω. For more than two, use the sum of reciprocals.

### Why is the total less than the smallest resistor?

Each resistor in parallel is one more path for the current, so the circuit passes more current at the same voltage and the total resistance falls. The total is always below the smallest resistor. Two equal resistors give half of one: 1 kΩ ∥ 1 kΩ = 500 Ω.

### How do I find the resistor to add in parallel?

Subtract the conductance you have from the conductance you want: R = 1 ÷ (1/target − 1/R₁ − …). To bring 150 Ω down to 100 Ω, add 1 ÷ (1/100 − 1/150) = 300 Ω. The target must be below the total you already have.

### What current flows through resistors in parallel?

Every resistor has the same voltage across it, so the total current is V ÷ total resistance, and each resistor takes V ÷ its own resistance. Two 1 kΩ resistors at 12 V draw 12 ÷ 500 = 0.024 A (24 mA) together, 12 mA each, and turn 12² ÷ 500 = 0.288 W into heat.

### What is conductance?

Conductance is 1 ÷ resistance, measured in siemens (S). In parallel, conductances simply add, which is why the formula adds reciprocals.

## 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