Roles of Kirchhoff's laws

Ohm's law alone cannot solve circuits with several loops and sources. Kirchhoff's laws are the basic rules that apply to any lumped circuit, and every method of circuit analysis (mesh, nodal, superposition, Thevenin) is built on them.

Kirchhoff's current law (KCL)

The algebraic sum of currents at a node is zero: ΣI_in = ΣI_out. It follows from conservation of charge (charge cannot pile up at a node).

Kirchhoff's voltage law (KVL)

The algebraic sum of voltages around any closed loop is zero: ΣE = ΣIR. It follows from conservation of energy.

Roles in analysis

• KCL gives node equations: the basis of nodal analysis.
• KVL gives loop equations: the basis of mesh (loop) analysis.
• They find unknown branch currents and voltages in circuits that cannot be reduced by series-parallel rules.
• They hold for DC and AC (in phasor form), for linear and non-linear elements, and for any type of source.
• They are used to check results, for example that the power supplied equals the power absorbed.

Justification with an example

Two batteries E₁ = 10 V and E₂ = 4 V, with R₁ = 2 Ω and R₂ = 1 Ω, feed a load R₃ = 3 Ω. By KCL, I₃ = I₁ + I₂. By KVL on the two loops: 10 = 2I₁ + 3I₃ and 4 = I₂ + 3I₃. These three equations give the three unknowns uniquely: I₁ = 28/11 = 2.55 A, I₂ = −10/11 = −0.91 A (battery E₂ is being charged) and I₃ = 18/11 = 1.64 A. This cannot be done with Ohm's law alone.