Gustav Kirchhoff (1824–1887) devised two rules known as Kirchhoff's rules to analyze complex circuits, which cannot be analyzed with series-parallel techniques. These rules can be used to analyze any circuit, simple or complex.

Kirchhoff's first rule is called the junction rule. A junction, also known as a node, is a connection of three or more wires. The rule states that the sum of all currents entering a junction must equal the sum of all currents leaving the junction.

This law is based on the charge conservation principle. Since the current is the flow of charge and the charge is conserved, whatever charge flows into the junction must flow out.

Kirchhoff's second rule—the loop rule—applies to the potential differences and, hence, is also referred to as the voltage rule. It states that the algebraic sum of the potential differences in any loop, including the voltage supplied by the voltage sources and resistive elements, must be equal to zero.

This rule is based on the conservation of energy. However, the loop rule is stated in terms of potential (V) rather than potential energy (U), but the two terms are related, since U = qV. In a closed loop, the energy supplied by the battery (voltage source) must be transferred into other forms by the devices in the loop, as there is no other way to transfer energy into or out of the circuit.