The formulas
- Series: R = R1 + R2 + R3 + … One path, so the same current flows through every resistor and the resistances add.
- Parallel: 1 ÷ R = 1 ÷ R1 + 1 ÷ R2 + 1 ÷ R3 + … Every branch connects across the same two points, so each sees the full voltage. For two resistors this reduces to R = (R1 × R2) ÷ (R1 + R2).
- Combination: add the groups. Reduce each parallel bank to one equivalent resistance, then add it to the series resistors in line with it.
- Ohm's law: I = V ÷ R, V = I × R. Use it on the total to get the total current, then on each part to get its voltage or current.
Two checks catch most mistakes. A parallel total is always smaller than the smallest branch, and a series total is always larger than the largest resistor. The voltage drops in a series string add up to the source voltage, and the branch currents in a parallel bank add up to the total current.
Worked example 1: three resistors in series
100, 220 and 330 Ω on 12 V. R = 100 + 220 + 330 = 650 Ω. I = 12 ÷ 650 = 18.46 mA. The drops are 1.846 V, 4.062 V and 6.092 V, which add to 12 V.
Worked example 2: two resistors in parallel
100 and 220 Ω on 12 V. R = (100 × 220) ÷ (100 + 220) = 22,000 ÷ 320 = 68.75 Ω. Branch currents are 12 ÷ 100 = 120 mA and 12 ÷ 220 = 54.55 mA, so the total is 174.55 mA, and 12 ÷ 68.75 gives the same figure. Three equal 300 Ω resistors in parallel give 300 ÷ 3 = 100 Ω.
Worked example 3: a series group and a parallel group
47 Ω and 100 Ω in series with a bank of 220 Ω and 330 Ω in parallel, on 12 V. The series group is 147 Ω. The bank is (220 × 330) ÷ 550 = 132 Ω. The total is 147 + 132 = 279 Ω and I = 12 ÷ 279 = 43.01 mA. The series drops are 2.022 V and 4.301 V. The bank drop is 43.01 mA × 132 Ω = 5.677 V, so the branches carry 25.81 mA and 17.20 mA, which add to the total.
These are original arithmetic examples, not exam questions.
Where this fits
Series and parallel arithmetic is the base for the Ohm's law work in the Calculations portion. For power, the power formulas and three-phase work, use the Ohm's law calculator. The Calculations portion guide shows where theory sits in the whole portion, and the transformer calculator covers current on each side of a transformer.
Series and parallel resistance questions
How do I find total resistance in series?
Add the resistances. Three resistors of 100, 220 and 330 ohms in series total 650 ohms.
How do I find total resistance in parallel?
Add the reciprocals of the resistances, then take the reciprocal of that sum. For two resistors, multiply them and divide by their sum. 100 and 220 ohms in parallel give 68.75 ohms.
Why is parallel resistance smaller than the smallest resistor?
Each extra branch gives the current another path, so the source can push more current at the same voltage. More current at the same voltage means less total resistance.
How do I get the voltage across each resistor?
Find the total resistance and the total current with Ohm's law, I = V ÷ R. In a series string the same current flows through every resistor, so each drop is I × R. In a parallel bank every branch has the full source voltage instead.