Kirchhoff’s Current and Voltage Laws

Kirchhoff’s current law (KCL) and voltage law (KVL) stated plainly, with sign conventions and three worked circuits you can check by hand.

The two laws

Example 1: a node (KCL)

1. Currents into the node: 5 A and 3 A. Out through one branch: 2 A.
2. 5 + 3 = 2 + I, so I = 6 A out through the last branch.

Example 2: a series loop (KVL)

A 12 V source drives 100 Ω and 220 Ω in series.

1. I = V / (R1 + R2) = 12 / 320 = 0.0375 A
2. V1 = I × R1 = 3.75 V and V2 = I × R2 = 8.25 V
3. KVL check: 3.75 + 8.25 = 12 V, the source voltage

Example 3: a parallel pair (KCL at the top node)

The same 12 V source across 100 Ω and 220 Ω in parallel.

1. I1 = 12 / 100 = 0.12 A and I2 = 12 / 220 = 0.05455 A
2. The source supplies I1 + I2 = 0.1745 A
3. Equivalent resistance = 12 / 0.1745 = 68.75 Ω, which equals (R1 × R2) / (R1 + R2)

When the laws are used

Every circuit method is built on them: series and parallel reduction, the voltage and current dividers, mesh and node analysis. They apply to any circuit made of lumped components, linear or not.

Frequently asked questions

What is Kirchhoff’s current law?

The currents into a node add up to the currents out of it. Charge does not collect at a junction. If 5 A and 3 A flow in and 2 A flows out through one wire, the other wire carries 6 A.

What is Kirchhoff’s voltage law?

Around any closed loop the voltage rises and drops add up to zero. In a series circuit the drops across the resistors equal the source voltage.

How do I handle signs?

Choose a direction for each current and a direction around the loop. A drop across a resistor in the direction of the current is positive; going through a source from − to + is a rise. If an answer comes out negative, the real direction is opposite to the one you chose.

Do the laws work for AC?

Yes, for the instantaneous values, and for phasors with complex impedances. They assume lumped components, so very high frequencies and long lines need transmission line theory.

Related

Ohm’s Law FormulasCoulomb’s LawVoltage Divider CalculatorConversion Centre