Combining the two laws to solve a circuit

Why combine the two laws?

A circuit with several loops has several unknown currents. The node law alone, or the loop law alone, is generally not enough to determine them all: a system of equations combining both laws must be written.

General method

1. Identify the nodes and name one current per branch (I1, I2, I3...).
2. Apply the node law at each independent node (the last node does not provide a new equation; it is already contained in the previous ones).
3. Apply the loop law to each independent loop of the circuit.
4. Solve the resulting system of equations.

Example: two-loop circuit

A generator E supplies two resistors R1 and R2 in parallel. Let I be the total current, I1 in R1, I2 in R2.

• Node law: I = I1 + I2
• Loop law (loop 1): E = R1 x I1
• Loop law (loop 2): E = R2 x I2

With E = 10 V, R1 = 5 ohm, R2 = 10 ohm: I1 = 2 A, I2 = 1 A, so I = 3 A.

Common pitfall

A circuit with n nodes provides only (n - 1) independent equations from the node law: the equation for the last node is a consequence of the others (redundant). Likewise, independent loops must be chosen (do not write the same loop twice in another form), otherwise you get an inconsistent or insufficient system.

Key takeaway

The rigorous combination of Kirchhoff's two laws allows any linear circuit, however complex, to be fully solved.