Electronics
How to calculate equivalent resistance
Equivalent resistance is the one resistor that could replace a whole network: from its two terminals, it draws the same current from the same voltage. To find it, keep replacing groups that are purely in series (add them) or purely in parallel (product over sum) with a single resistor, until only one is left.
The two rules
In series, resistances add:
In parallel, two resistors give product over sum, and n equal ones give R ÷ n:
A series combination is always bigger than its biggest part. A parallel one is always smaller than its smallest, because each branch adds another path.
Reduce it in steps
- Find a group that's purely series or purely parallel.
- Replace it with one resistor.
- Redraw, and repeat until one resistor is left.
Example: 1 kΩ in series with two 10 kΩ resistors in parallel, on 12 V. The pair gives 10 ÷ 2 = 5 kΩ, then 1 + 5 = 6 kΩ.
Now work back out: the 2 mA flows through the 1 kΩ, dropping 2 V, and splits equally between the two 10 kΩ, 1 mA each, with 10 V across them. The drops add up to 12 V, as they must.
Live experiment
Snap resistors together
One 1 kΩ resistor sits in a loop with a 9 V battery and Pip, a red LED. Add more, after the network or beside it, and watch the one resistor it acts like. Series dims Pip, parallel brightens him, and five side by side is too much.
“Combining resistors” is lesson 6 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 12 before it first.
Networks that share the heat
Equal resistors combine neatly. Two series pairs, put in parallel, make (R + R) ∥ (R + R) = R: the same value as one resistor, but each part carries half the current and takes only a quarter of the power. On 9 V, a single 100 Ω resistor takes 0.81 W, too much for a 0.25 W part; four in that 2 × 2 network take about 0.2 W each.
That's how engineers build odd values and bigger power ratings from standard parts: two 15 kΩ in parallel give 7.5 kΩ. It only works when the current shares evenly, so use equal values.
When steps aren't enough
Some networks never reduce to plain series and parallel steps. The classic is the bridge: two series pairs side by side, with a fifth resistor joining their middles. Solve it with Kirchhoff's current and voltage laws instead. When a bridge is balanced, no current flows through the middle at all, which is how strain-gauge scales sense tiny changes.
Free lesson
Now build it yourself
Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.
“Combining resistors” is lesson 6 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 12 before it first.
Or start from the very first lesson, no account neededWork it out
Practice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: both their ends connect to the same two points. Parallel means both ends share the same two nodes, however the drawing looks. Series parts are on one path, so they always share the same current. Equal currents alone don't prove either.
- Why not “the same current flows through them”? Series parts always share one current, and equal parallel resistors share one too. Equal current alone doesn't tell you which.
- Why not “they're drawn next to each other”? How the parts are drawn doesn't matter. Follow the wires: parallel means both ends join the same two points.
Show the answer
Answer: 4. P = I² × R. Doubling I multiplies I² by 4.
- Why not “It doesn't change”? The resistor's value stays the same, but P = I² × R still grows with the current.
- Why not “½”? More current through the same resistor means more heat, not less.
- Why not “2”? Power goes with current squared, P = I² × R, so doubling the current more than doubles it.
Show the answer
Answer: Quadruples. Squared: twice the volts, four times the watts.
- Why not “Stays”? Twice the voltage across the same resistance always means more power: four times as much.
- Why not “Doubles”? Power goes with voltage squared, P = V² ÷ R, so twice the voltage gives more than twice the power.
- Why not “Halves”? More voltage across the same resistor drives more current, so the power rises, never halves.
“Combining resistors” is lesson 6 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 12 before it first.
Questions people ask
What is equivalent resistance?
The single resistor that would draw the same current from the same voltage as a whole network. Seen from its two terminals, any network of resistors acts exactly like that one value.
How do you find the equivalent resistance of a combination circuit?
Find a group that's purely in series or purely in parallel, replace it with one resistor, redraw, and repeat. Series resistances add; two in parallel give R1 × R2 ÷ (R1 + R2). Work from the far end of the network back towards the terminals.
How can I tell if two resistors are in parallel?
Follow the wires from each end. If both ends reach the same two points without passing through another part, they're in parallel, however they're drawn. Two resistors carrying the same current aren't necessarily in series either: check the nodes.
What if a network won't reduce with series and parallel steps?
Some don't, like the bridge: two series pairs side by side with a fifth resistor joining their middles. Solve those with Kirchhoff's laws, or a delta-to-star conversion. If the bridge is balanced, no current flows in the middle resistor and you can ignore it.
Related topics
- Series vs parallel circuitsOne path or many: what stays the same, what adds up, and how to combine resistors.
- Current divider rule explainedHow a current shares itself between parallel branches, and why the smaller resistor takes the bigger share.
- Kirchhoff's laws explained: KCL and KVLWhat flows into a junction flows out, and the drops round a loop add up to the rise.
- Ohm's law explainedV = I × R: set any two of voltage, current and resistance and the third follows.