Kirchhoff’s laws turn two conservation principles into circuit equations. The current law expresses conservation of charge at a node; the voltage law expresses conservation of energy around a closed loop. They hold for linear and non-linear circuits alike, which is why they are the foundation of network analysis.
Kirchhoff’s current law (KCL)
Equivalently, the algebraic sum of all currents at a node is zero when entering currents are given one sign and leaving currents the other. The choice of sign is free, but it must stay the same throughout the calculation.
Current at a junction
Problem. A node receives 5 A and supplies two branches carrying 2 A and 1.5 A. What current leaves through the third branch?
Solution. By KCL, 5 = 2 + 1.5 + I3, so I3 = 5 − 3.5 = 1.5 A.
Kirchhoff’s voltage law (KVL)
Voltages around a loop
Problem. A 12 V source feeds three series elements. Two drop 4 V and 5 V. What is the third drop?
Solution. The drops must sum to the source voltage: 12 = 4 + 5 + V3, so V3 = 3 V.
- Choose a current direction and a loop direction before writing equations.
- A voltage rise in the loop direction is written with the opposite sign to a drop.
- A negative result is valid: it means the real direction is opposite to the assumed one.
- KCL and KVL apply to every node and every loop of a network, not only to simple circuits.
- KCL: currents into a node equal currents out.
- KVL: voltage changes around a closed loop sum to zero.
- A consistent sign convention is essential.
- A negative answer indicates the opposite of the assumed direction.
Worked example
A node has 3 A and 2 A entering and 4 A leaving through one branch. KCL gives 3 + 2 = 4 + I4, so I4 = 1 A leaves through the remaining branch.