Library / Electrical / ELE-07
Electrical
Resistors in parallel
Single equivalent resistance of two resistors connected side by side across the same voltage.
- Ref
- ELE-07
- Rev
- v1
- Field
- Electrical
- Result
- RT [Ω]
Product over sum is for exactly two resistors. For more, add the reciprocals: 1/RT = 1/R1 + 1/R2 + 1/R3 + …
Each extra branch gives the current another path, so the combined resistance is always smaller than the smallest single resistor. Conductances (1/R) add, which is why the reciprocal form works.
Calculator
- Conductance 1/RT0.01455 S
- Series total R1 + R2320 Ω
- RT ÷ smaller resistor0.6875 ×
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- RT is always smaller than the smaller of the two resistors.
- If one resistor is much larger than the other, RT is close to the smaller one. Adding a 10 kΩ resistor in parallel with 100 Ω changes the result very little.
- Two equal resistors in parallel give half the value, R/2.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Ω) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R1 First resistor | 10 to 1,000 Ω | 9.565 | 180.3 | 170.8 | +10% | +6.67 % |
| R2 Second resistor | 10 to 1,000 Ω | 9.091 | 90.91 | 81.82 | +10% | +2.92 % |
Sweep of Second resistor
| Second resistor (Ω) | Total resistance (Ω) |
|---|---|
| 10 | 9.091 |
| 17.78 | 15.1 |
| 31.62 | 24.03 |
| 56.23 | 35.99 |
| 100 | 50 |
| 177.8 | 64.01 |
| 316.2 | 75.97 |
| 562.3 | 84.9 |
| 1,000 | 90.91 |
Sweep one variable
Variables and how each one contributes
Total resistance
What the equation gives you. Everything else feeds it.
First resistor
One of the two parallel resistors. Raising R1 well above R2 changes RT very little. The smaller resistor dominates.
Second resistor
The other parallel resistor. Raising R2 well above R1 changes RT very little. The smaller resistor dominates.
Worked example with the values above
- Product: R1R2 = 100 × 220 = 22,000 Ω².
- Sum: R1 + R2 = 320 Ω.
- RT = 22,000 / 320 = 68.75 Ω.
- Check with reciprocals: 1/100 + 1/220 = 0.01455 S, and 1/0.01455 = 68.75 Ω.
Assumptions and limits
- Ideal resistors with negligible lead and contact resistance.
- Both resistors are connected across the same voltage.
- The product-over-sum form applies to exactly two resistors.
Common mistakes
- Adding the resistances (the series rule) instead of the reciprocals.
- Forgetting to invert at the end of the reciprocal method. 1/RT is not RT.
- Using product-over-sum for three or more resistors.
Statistics and uncertainty
Resistor tolerances carry through. If both resistors are within ±t of nominal, RT is also within ±t in the worst case. For independent random errors the typical error is smaller: about t/√2 for two equal resistors.
Where it is used
- Making a non-standard resistance from two preferred values.
- Current sharing between parallel loads.
- Finding the effect of a load connected in parallel with a divider resistor.
Maths you need
- Fractions and reciprocals: adding 1/R terms with a common denominator
- Series and parallel: voltage is shared in series, current is shared in parallel