Kirchhoff's Laws: KCL and KVL Made Simple

Kirchhoff’s Laws are two core rules used to analyze electrical circuits.

1) Kirchhoff’s Current Law (KCL)

At any node (junction), total current entering equals total current leaving.

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

Quick Example

If 5 A enters a node and one branch carries 2 A out, the other branch must carry 3 A out.

$$5 = 2 + 3$$

2) Kirchhoff’s Voltage Law (KVL)

Around any closed loop, the algebraic sum of voltages is zero.

$$\sum V = 0$$

Quick Example

For a loop with a 12 V source and two resistor drops of 7 V and $V_x$:

$$12 - 7 - V_x = 0 \Rightarrow V_x = 5\text{ V}$$

How to Use KCL + KVL Together

1. Label current directions and voltage polarities.
2. Apply KCL at key nodes to relate branch currents.
3. Apply KVL on independent loops to relate voltages.
4. Solve the resulting equations.

Common Mistakes

• Mixing current direction signs.
• Forgetting that voltage rises and drops need consistent sign convention.
• Writing dependent equations from the same loop/node multiple times.

Kirchhoff’s Laws are the foundation for nodal and mesh analysis, and once your sign conventions are consistent, circuit problems become much easier to solve.