Elec Class
learn electricity hands-on
Free calculator · series, parallel, combination
In series, resistances add up: Réq = R₁ + R₂. In parallel, their reciprocals add up: 1/Réq = 1/R₁ + 1/R₂, and the equivalent resistance is always smaller than the smallest one.
Choose the circuit, enter your resistances: the calculation is written out line by line.
1I choose the circuit
2I enter the resistances
Values from the E12 series, the ones you find in shops.
3The result
Equivalent resistance Réq56.9 Ω
Réq is smaller than the smallest resistance (100 Ω).
These results come up in almost every problem.
| Circuit | Equivalent resistance | Example |
|---|---|---|
| n identical resistors in series | Réq = n · R | 3 × 100 Ω = 300 Ω |
| n identical resistors in parallel | Réq = Rn | 2 × 220 Ω in parallel = 110 Ω |
| Two resistors in parallel | Réq = R₁ · R₂R₁ + R₂ | 100 Ω and 400 Ω: 40,000 / 500 = 80 Ω |
| A very large one in parallel with a small one | Réq ≈ the small one | 10 Ω and 1 MΩ: Réq ≈ 10 Ω |
| A wire (0 Ω) in parallel | Réq = 0: short circuit | all the current flows through the wire |
In parallel, adding a resistor always lowers Réq: the current has one more path.
Replace simple groups with a single resistor, until only one is left.
Circle the resistors that are strictly in series (same wire, no node) or in parallel (same two nodes).
Replace each parallel group with its Réq (product over sum for two).
Resistors that follow each other on the same wire add up.
Redraw the simplified diagram and carry on until only one resistor is left.
ExampleR₁ = 100 Ω in series with R₂ = R₃ = 220 Ω in parallel: R₂₃ = 220 / 2 = 110 Ω, then Réq = 100 + 110 = 210 Ω.
Add the reciprocals: 1/Réq = 1/R₁ + 1/R₂ + … then take the reciprocal of the result. For 100 Ω and 400 Ω: 1/Réq = 0.01 + 0.0025 = 0.0125, so Réq = 80 Ω.
Product over sum: Réq = R₁ · R₂ / (R₁ + R₂). It only works for two resistors; beyond that, take them two at a time or add the reciprocals.
Each branch adds a path for the current: for the same voltage, more current flows, as if the resistance had dropped. So Réq is smaller than the smallest branch.
In series, they follow each other on the same wire with no node between them. In parallel, they are connected between the same two nodes. Trace the current with your finger: if it has to choose a path, it's a parallel circuit.
In series, the current is the same in every resistor and the total voltage is shared: U = U₁ + U₂. The largest resistance gets the largest voltage.
In parallel, each branch has the same voltage and the total current is shared: I = I₁ + I₂. The smallest resistance lets the largest current through.
The problems from the Resistor combinations chapter, from the simplest to combination circuits.