Electronics
Kirchhoff's laws explained: KCL and KVL
Kirchhoff's laws are the two rules every circuit obeys. The current law (KCL) says the current flowing into a junction equals the current flowing out. The voltage law (KVL) says that round any closed loop, the voltage rises add up to the voltage drops. Gustav Kirchhoff published them in 1845.
The current law: in equals out
A junction is a point where three or more wires meet. Charge can't pile up there or vanish, so:
Σ means "add them all up". If 30 mA flows in and 12 mA leaves on one wire, the other wire carries 30 − 12 = 18 mA.
Live experiment
Balance the junction
Four wires meet at a junction. Three currents are given, flowing in or out; slide the fourth until what flows in equals what flows out. Unbalanced, charge piles up or runs dry, which real charge can't do. Three junctions to solve.
“Kirchhoff's current law” is lesson 4 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 10 before it first.
How current splits
When current splits between branches, the easier branch takes more. For two branches, the current divider rule gives each share, with the other branch's resistance on top:
Example: 40 mA into 1 kΩ and 3 kΩ in parallel. The 1 kΩ takes 40 × 3 over 4 = 30 mA and the 3 kΩ the other 10 mA. KCL checks out: 30 + 10 = 40 mA.
The voltage law: round the loop
The battery is a rise; each part is a drop. Go once round any closed loop and they balance:
Example: a 9 V battery, a resistor and a red LED in one loop. The LED drops 2.0 V, so 9 = 2.0 + VR, and the resistor drops 7 V. That's the familiar "the resistor gets what's left", with its proper name.
KVL also explains the open switch. No current flows, so every resistor drops I × R = 0 V, and the whole supply appears across the gap.
Why they're true
KCL is conservation of charge; KVL is conservation of energy. A charge gains energy crossing the battery and gives exactly that much away in the drops.
Together with each part's own rule, like Ohm's law, they solve any circuit. Write KCL at every node, with the node voltages as unknowns, and you get nodal analysis. That's how circuit simulators work, including the one in the lessons.
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.
“Kirchhoff's current law” is lesson 4 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 10 before it first.
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Practice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: add up so that the rises equal the drops. That's Kirchhoff's voltage law. The energy charge gains in the battery is all given up in the drops, so you end where you started.
- Why not “are the full battery voltage across each part”? Only a part wired straight across the battery gets its full voltage. Parts in series share it, and their drops add up to the battery's voltage.
- Why not “depend on how long the wires are”? Kirchhoff's voltage law holds round any loop, whatever the wires' length. Ordinary hookup wire drops almost nothing, so it barely shows up in the sum.
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”? Parts in series share the same current too, and so do equal resistors in parallel. Equal currents alone don't tell you how they're wired.
- Why not “they're drawn next to each other”? A drawing can put parts side by side without joining them that way. What counts is the connections: follow the wires from each end.
Show the answer
Answer: Goes round again. The charge goes round and round. The battery hands it energy, the LED uses that energy to make light, and the charge carries on back to the battery.
- Why not “Is used up”? Charge isn't consumed. What the LED uses is energy, and the same charge carries on round the loop, which is why the current in equals the current out.
- Why not “Leaks out of the wire”? In a closed loop, charge stays inside the conductors. If it leaked out, the current into and out of each part wouldn't match, and it always does.
- Why not “Turns into light”? The light comes from energy the charge gives up in the LED, not from the charge itself. The charge carries on round to the battery.
“Kirchhoff's current law” is lesson 4 of 7 in Junction City, Electronics World 2. Free with an account. Lessons unlock in order, so you’ll do the 10 before it first.
Questions people ask
What is Kirchhoff's current law?
Kirchhoff's current law (KCL) says the total current flowing into a junction equals the total current flowing out. If 30 mA flows in and 12 mA leaves on one wire, the other wire must carry 18 mA. It holds because charge can't pile up or vanish.
What is Kirchhoff's voltage law?
Kirchhoff's voltage law (KVL) says that around any closed loop, the voltage rises add up to the voltage drops. With 9 V from a battery and a red LED dropping 2.0 V, the resistor in the same loop drops the other 7 V. It's conservation of energy.
What are Kirchhoff's laws used for?
They're the two rules behind all circuit analysis. Writing KCL at every node, with the node voltages as unknowns, is called nodal analysis, and it's how circuit simulators, including the one in the lessons, solve circuits.
Why is there voltage across an open switch?
An open switch carries no current, so every resistor in the loop drops I × R = 0 V. KVL still holds, so the whole supply appears across the switch's gap. That's why a switched-off mains switch is still live inside.
Related topics
- Series vs parallel circuitsOne path or many: what stays the same, what adds up, and how to combine resistors.
- Voltage divider explainedTwo resistors that split a voltage: the formula, potentiometers, and why a load drags the output down.
- What is electric current?How much charge flows past a point each second, in amps, and why it's the same all round a loop.