Equivalent resistance calculator (series and parallel)

Two resistors can be wired in two ways, and each has its own rule.

Series: resistances add

R = R₁ + R₂ + R₃ + …

In series there is a single path for the current: it crosses one resistor after another, accumulating opposition. So the equivalent is always larger than the largest resistor in the set.

Example: 100 Ω + 220 Ω + 330 Ω = 650 Ω.

Parallel: the reciprocals add

1/R = 1/R₁ + 1/R₂ + 1/R₃ + …

In parallel the current gains several paths at once. More paths, less total opposition — the equivalent is always smaller than the smallest resistor in the set. The result surprises most people the first time; the analogy that helps is exit doors in a theatre: opening a second door cannot make leaving harder.

Step by step with 100 Ω, 220 Ω and 330 Ω:

1. 1/100 = 0.01000
2. 1/220 = 0.004545
3. 1/330 = 0.003030
4. Sum: 0.017576
5. Invert: 1 ÷ 0.017576 ≈ 56.9 Ω

Less than the smallest resistor's 100 Ω, as expected.

Two shortcuts worth memorizing

• Two in parallel: R = (R₁ × R₂) ÷ (R₁ + R₂). Two 10 Ω → (100)/(20) = 5 Ω.
• N equal resistors in parallel: divide the value by the count. Three 30 Ω → 30 ÷ 3 = 10 Ω. Three 1 kΩ → 333.3 Ω.

Mixed circuits: solve from the inside out

Most real circuits mix both connections. The method is always the same: find the innermost group, reduce it to a single equivalent resistor, rewrite the circuit with that value, and repeat until one resistor is left.

Example: a 100 Ω resistor in series with a pair of 220 Ω in parallel. The pair first: 220 ÷ 2 = 110 Ω. Then the series: 100 + 110 = 210 Ω.

The mistake that burns parts: forgetting power

Getting the ohms right and the power rating wrong is the most common beginner trip-up. The resistor has to survive the heat it dissipates:

P = V² ÷ R  or  P = I² × R

A 100 Ω resistor across 12 V dissipates 12² ÷ 100 = 1.44 W. An ordinary 1/4 W (0.25 W) resistor in that spot heats up, darkens and burns out — the circuit needs a 2 W part or better. Common practice is to pick a component rated at least double the calculated power, so it runs cool.

Watch the wiring too: in series every resistor sees the same current, and the largest one heats the most. In parallel every resistor sees the same voltage, and the smallest one dissipates the most.

Why 56.9 Ω is not on the shelf

Commercial resistors follow standardized value series — E12 (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 and their powers-of-ten multiples) and the finer E24. There is no off-the-shelf 333 Ω resistor: you buy 330 Ω, which is in the series.

That is fine, because tolerance follows suit: a 5% 330 Ω resistor may actually measure between 313.5 and 346.5 Ω. Combining several stacks those tolerances — circuits that need precision use 1% resistors or trim with a potentiometer.

The classic example: the LED resistor

A common red LED drops about 2 V and runs at 20 mA. On 5 V, the series resistor must absorb the remaining 3 V:

R = (5 − 2) ÷ 0.02 = 150 Ω, dissipating 3 × 0.02 = 0.06 W — comfortably within a 1/4 W part.