Series and parallel — the two ways to combine resistors
Every resistor network reduces to some combination of series and parallel connections. Getting the totals right is fundamental to circuit analysis, and it’s a calculation you’ll do constantly in electronics — from LED current-limiting to voltage dividers to sensor conditioning.
How to use this calculator
Two modes cover the two questions people usually ask:
- Calculate total — enter each resistor value in your network. The calculator shows both the series total (all in a line) and the parallel total (all across the same two nodes). Add as many resistors as you need. If you supply a voltage or current, it also computes the current, voltage, and power dissipation for each resistor in both arrangements.
- Find combination — enter a target resistance and pick a standard E-series. The calculator finds the closest achievable series and parallel pairs from that E-series. Useful when your bench only has certain values and you need to hit a specific target.
The formulas
Series: resistors in a line add straight up.
R_total = R1 + R2 + R3 + …
The total is always larger than the largest resistor in the chain. Current is the same through every resistor; voltage divides in proportion to each resistor’s value.
Parallel: resistors across the same two nodes combine reciprocally.
1 / R_total = 1 / R1 + 1 / R2 + 1 / R3 + …
The total is always smaller than the smallest resistor in the group. Voltage is the same across every resistor; current divides in inverse proportion to each resistor’s value.
For only two resistors in parallel there’s a convenient shortcut:
R_parallel = (R1 × R2) / (R1 + R2)
The “product over sum” formula. Two equal resistors in parallel = half the value. Two unequal resistors in parallel = less than the smaller one, but always more than half of it.
Quick mental shortcuts
- N equal resistors in parallel = R / N. Two 1 kΩ in parallel = 500 Ω. Four 10 kΩ in parallel = 2.5 kΩ.
- Very unequal parallel ≈ the smaller value. 1 kΩ ∥ 100 kΩ ≈ 990 Ω. The larger resistor barely affects the total when it’s much bigger than the smaller one.
- Very unequal series ≈ the larger value. 100 kΩ + 1 kΩ ≈ 101 kΩ. The smaller resistor barely matters when the larger one dominates.
E-series standard values (IEC 60063)
Resistors aren’t made in every possible value — they’re made in standardized “preferred” values called E-series. Each series is a geometric progression that repeats every decade:
- E12 — 12 values per decade, ±10% tolerance. The hobbyist and general-purpose standard. 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 (and their decade multiples).
- E24 — 24 values per decade, ±5% tolerance. Adds intermediate values like 11, 13, 20, 30 to E12.
- E48 — 48 values per decade, ±2% tolerance. Three-significant-figure precision.
- E96 — 96 values per decade, ±1% tolerance. Standard for modern precision resistors.
The tighter the tolerance, the more values are made — a 1% resistor needs finer spacing than a 10% one, because you can distinguish nearby values reliably.
Reverse mode — the practical bench workflow
The reverse-search mode solves a real problem: you calculated that you need, say, 750 Ω for a voltage divider, but nobody sells exactly that. What do you use? The calculator checks all pairs from the chosen E-series and reports:
- The closest single value from the series (may be worse than a pair)
- The closest series pair (two resistors added)
- The closest parallel pair (two resistors in parallel)
For 750 Ω from E12, the parallel combination 1.5 kΩ ∥ 1.5 kΩ gives you exactly 750 Ω — better than any single E12 value. The single closest is 680 Ω or 820 Ω, both about 9% off. This kind of match is impossible to spot by inspection.
Power dissipation matters
Every resistor turns some electrical energy into heat. The power dissipated is P = I²R (using the current through it) or P = V²/R (using the voltage across it). If a resistor’s continuous power exceeds its rating, it will overheat — burn, discolor, drift out of tolerance, or fail open.
Common ratings:
- 1/8 W (0.125 W) — small through-hole and most SMD; general purpose signal circuits
- 1/4 W (0.25 W) — the classic hobbyist size
- 1/2 W (0.5 W) — for currents above ~30 mA in typical 12 V circuits
- 1 W and up — power supplies, LED drivers, motor circuits
When you supply a voltage or current in the calculator, it shows the power dissipation for each resistor so you can size them properly. If a value looks close to a rating boundary, go one size up — a 0.24 W circuit on a 0.25 W resistor gives no safety margin and runs uncomfortably hot.
What this calculator doesn’t cover
- Mixed series-parallel networks — this calculator handles all-series or all-parallel arrangements. For mixed networks (e.g. two resistors in parallel, in series with a third), calculate the parallel section first, then treat the result as a single resistor in series with the third.
- Non-resistive impedances — capacitors and inductors also have series/parallel combining rules, but they’re frequency-dependent (impedance, not resistance).
- Tolerance stacking — the calculator uses exact E-series values; real resistors vary within their tolerance band. For precision circuits, use tighter-tolerance parts and consider tolerance stack-up in your design.
- Temperature coefficient effects — resistor values drift with temperature; low-TCR precision resistors matter for stable measurement circuits.