Kirchhoff's Laws — KCL and KVL with Worked Examples

Introductory College⏱ 11 min readElectricity & Magnetism

Kirchhoff's current and voltage laws: conservation of charge and energy in circuits. Rules, sign conventions, and a worked multi-loop example.

Quick answer

KCL: at any junction, current in equals current out (charge is conserved). KVL: around any closed loop, voltage rises equal voltage drops (energy is conserved). Together they solve any circuit.

Example: At a junction with 5 A entering and branches of 2 A and I leaving: 5 = 2 + I, so I = 3 A — KCL in one line.

What you will learn

State Kirchhoff's current law (KCL) and voltage law (KVL)
Apply consistent sign conventions
Set up equations for a two-loop circuit
Solve for unknown currents in a simple network

Why we need them

Series and parallel rules handle simple circuits. But real circuits — two batteries, three branches, a bridge — need something more general. Kirchhoff's laws (1845) are that something: two conservation statements that generate exactly enough equations to solve any circuit.

KCL (current law): conservation of charge at junctions.
KVL (voltage law): conservation of energy around loops.

KCL: current in equals current out

Kirchhoff's current law: the algebraic sum of currents entering any junction is zero.

$\sum I_{\text{in}} = \sum I_{\text{out}}$

Charge can't pile up at a node (in steady circuits), so whatever flows in must flow out. At a junction with 5 A arriving and two branches leaving carrying 2 A and $I$ : $5 = 2 + I$ , so $I = 3\ \mathrm{A}$ . One line, no fuss.

\sum I = 0 \quad \text{(at any junction)}

Currents entering positive, leaving negative (or vice versa — be consistent)

KVL: loop voltages sum to zero

Kirchhoff's voltage law: the algebraic sum of voltages around any closed loop is zero.

$\sum V = 0 \quad \text{(around any loop)}$

Going around a loop, every voltage rise (battery, + to −... careful: traversing a battery from − to + is a rise) must be balanced by voltage drops (resistors, traversed with the current). It's energy bookkeeping: charge gains energy at sources and spends it all in loads — none left over after a full lap.

Sign conventions that save you

Sign errors kill more Kirchhoff solutions than any conceptual gap. Use this ritual:

Mark assumed current directions on every branch (guesses are fine — a negative answer just flips them).
Traverse each loop in a chosen direction.
Resistor: voltage drop $IR$ if traversed with the current, rise if against.
Battery: rise if traversed from − to +, drop from + to −.
Set the sum to zero and solve the system.

Warning

A negative current is not a mistake — it means the actual direction is opposite your arrow. Flip the arrow and continue; the magnitudes are correct.

Worked mini-example: two loops

Two loops share a middle branch. Battery 12 V (left loop) and 6 V (right loop) oppose through a shared $2\ \Omega$ resistor; outer resistors are $4\ \Omega$ each. Let left-loop current be $I_1$ (clockwise), right-loop $I_2$ (counterclockwise); middle branch carries $I_1 + I_2$ downward.

Left loop (KVL): $12 - 4I_1 - 2(I_1 + I_2) = 0$ → $6I_1 + 2I_2 = 12$ .

Right loop (KVL): $6 - 4I_2 - 2(I_1 + I_2) = 0$ → $2I_1 + 6I_2 = 6$ .

Solving: multiply the second by 3 → $6I_1 + 18I_2 = 18$ ; subtract the first: $16I_2 = 6$ → $I_2 = 0.375\ \mathrm{A}$ . Then $I_1 = (12 - 0.75)/6 \approx 1.875\ \mathrm{A}$ . Middle branch: $2.25\ \mathrm{A}$ downward. KCL check at the top node: $1.875 + 0.375 = 2.25$ ✓.

Variables and units

Symbol	Name	Unit	Meaning
$I$	Current	A	Current through a branch
$V$	Voltage	V	Voltage rise or drop across a component

Common mistakes

✗ KCL means current is never lost anywhere.

KCL is about junctions: the algebraic sum of currents entering a node is zero. Current can absolutely be 'lost' from one branch — it just appears in another.

✗ KVL works even with changing magnetic fields through the loop.

KVL assumes no changing magnetic flux. With induction (transformers, wireless charging), you need Faraday's law — KVL's 'sum to zero' breaks down.

✗ The signs don't matter as long as the magnitudes are right.

Signs encode direction, and direction is the physics. A wrong sign is a wrong equation — though a negative answer just means your assumed direction was backward, which is fine.

Real-world applications

Circuit simulators (SPICE and its descendants) solve millions of KCL/KVL equations to predict chip behavior before manufacturing.
Electricians troubleshoot by checking KVL around loops — a voltage that doesn't add up pinpoints the faulty component.
PCB designers verify that every node satisfies KCL so no trace carries unexpected current.

Frequently asked questions

What do KCL and KVL actually conserve?

KCL conserves charge (what flows in must flow out); KVL conserves energy (gains and drops around any closed loop must balance). Both are conservation laws wearing circuit clothes.

Which sign convention should I use?

Pick one and stay consistent: either sum of currents entering = sum leaving, or algebraic sum = 0 with entering positive. Mixing conventions mid-problem is the top error source.

Do Kirchhoff's laws work for AC circuits?

Yes, with phasors — but that's beyond this article. The laws themselves hold; only the arithmetic gets complex-valued (literally).

How are these different from Ohm's law?

Ohm's law relates V, I, R in a single component. Kirchhoff's laws relate currents at junctions and voltages around loops — they're the network-level rules that let you use Ohm's law everywhere at once.