Kirchhoff's Laws

Why It Matters

While Ohm's law describes the relationship between voltage, current, and resistance for a single device, most practical electrical systems are networks of multiple interconnected devices. To analyze these complete circuits, engineers use Kirchhoff's laws, which form the most important mathematical foundation for all circuit analysis.

Concept

Kirchhoff's laws are simply the electrical expression of fundamental physics conservation laws. They ensure that neither electrical charge nor energy can mysteriously appear or disappear within a circuit.

Theory & Formulas

Kirchhoff's Current Law (KCL)

KCL is based on the conservation of electric charge. It states that the algebraic sum of currents entering a node (junction) must equal zero. In other words, whatever current flows into a node must flow out in equal quantity.

$\sum I = 0$

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

Kirchhoff's Voltage Law (KVL)

KVL is based on the conservation of energy. It states that the algebraic sum of all voltages around any closed loop in a circuit must equal zero. All the energy gained (e.g., from a battery) must equal all the energy lost (e.g., dropped across resistors).

$\sum V = 0$

Alternatively: $\sum V_{rises} = \sum V_{drops}$

Basic Sign Convention

To apply these laws mathematically, you must strictly follow a sign convention.

• For KCL: Assign currents entering as positive (+) and currents leaving as negative (-), or vice versa. The result is the same as long as you remain consistent.
• For KVL: As you trace around a loop, if you enter a device at the positive (+) terminal and exit at the negative (-), count it as a voltage drop. If you enter at the negative (-) and exit at the positive (+), count it as a voltage rise.