Kirchhoff's circuit laws are a set of fundamental laws dealing with the conservation of energy, namely voltage and current. They are tools used in circuit analysis to determine currents and voltages in a circuit. Between Kirchhoff's current law and voltage law and Ohm's law, any circuit can be analyzed and reduced to its Thévenin or Norton equivalent.

Kirchhoff's current law[]

Kirchhoff's current law (KCL) deals with the current flow in and out of nodes. All current that flows into a node must come out, so the sum of all currents in and out of a node must equal zero. In the pictured example (a current source and three resistors in parallel), there are four currents leaving node 0. This can be expressed mathematically as . If, for example, is actually flowing in the opposite direction, then its value becomes negative. By initially assuming all currents are leaving the node, sign errors can be avoided while working the algebra. Then negative current that is present after analysis is known to flow in the opposite direction as the arrow drawn in the schematic.

To analyze this circuit, first notice that the voltage across each element is the same. Using Ohm's law, the current through each resistor is which, when applied to the current sum equation, yields . Then, given the source current and resistance values, the voltage at can be found and the current through each resistor.

Kirchhoff's voltage law[]

Kirchhoff's voltage law (KVL) deals with the voltage changes around a loop. All the voltage changes around any loop in a circuit must sum to zero. In the pictured example (a voltage source and three resistors in series), there are four elements that will cause changes in the voltage. Mathematically, this is . As with KCL, it is useful to assign the voltage directions before forming any equations.

To analyze this circuit, first it must be realized that the current through each element is the same. According to Ohm's law, the voltage drop across each resistor is equal to the product of the current, I, and the respective resistance, R. Substituting this into the first equation gives us from which the loop current can be found given the voltage source value and the resistance of each of the resistors.