Systematic procedure

Kirchhoff’s two laws become truly powerful when applied in an orderly way: following the same steps every time leads to a linear system of equations in the branch currents, which is solved with the methods of algebra. The difficulty lies not in the physics but in keeping track of signs — for this reason it is worth fixing rigid conventions from the outset and not changing them halfway through.

The steps

1. Assign a conventional direction to each branch current. The choice is free: if a value turns out negative at the end, it simply means the current flows in the direction opposite to the one assumed.
2. Write the node equations, in a number equal to the number of nodes minus one (the last equation would be a linear combination of the others and adds no information).
3. Choose a set of independent loops and write the loop equation for each, applying the sign rules.
4. Solve the resulting linear system.

Sign rules around a loop

Travelling around the loop in the chosen direction:

• source crossed from $-$ to $+$: contribution $+\mathcal{E}$;
• source crossed from $+$ to $-$: contribution $-\mathcal{E}$;
• resistor traversed in the direction of the current: contribution $-R\,i$;
• resistor traversed against the current: contribution $+R\,i$.

The sign of the resistors reflects the fact that, following the current, the potential drops (one descends along the drop $R\,i$); going back against the current the potential rises. The sign of the sources instead depends only on the direction in which the poles are crossed, not on the current.

A standard scheme

A two-loop circuit sharing a central branch is the classic test bench for the method: two sources $\mathcal{E}_1$ and $\mathcal{E}_2$, resistors $R_1$ and $R_2$ on the upper path, and $R_3$ on the central branch connecting nodes A and B.

Two-loop scheme: branch currents are assigned, then the node equation at A and the equations for the two independent loops are written.

With the currents oriented, the node equation at A links the three branch currents, while the two loops supply the missing equations; the resulting system, three equations in three unknowns, uniquely determines all the currents.

Links

Topics: Electric circuits Concepts: Kirchhoff’s laws Skills: Applying Kirchhoff’s laws

Related exercises: Problem — two batteries with Kirchhoff · Worked exercise — two sources, two loops · Worked exercise — three-branch circuit with two sources