Solving circuit problems with Kirchhoff's laws

By the end of this lesson

By the end you can solve a mixed series-parallel circuit by reducing it in stages, and check your answer with the junction rule and the loop rule — and you can set up loop equations for a network with more than one loop.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$R_p$	combined resistance of resistors in parallel
$V_s$	p.d. across the series part
$V_p$	p.d. across the parallel part

A method that always works

Most circuit problems fall to the same four steps. The trick is not to rush the arithmetic.

Mark the currents. Label the current in each branch, and draw an arrow for each one. Your arrow is a guess, and a negative answer later just means the real current flows the other way.
Reduce what is clearly in series or parallel. Series resistances add; parallel ones add as reciprocals. Keep going until you have one combined resistance.
Work back out. Use $I = V/R$ for the total current, then $V = IR$ to find p.d.s and branch currents from the outside in.
Check with Kirchhoff. The branch currents must add up at each junction, and the p.d.s must add up round each loop. A check that fails means an arithmetic slip, not a new law.

Worked example: Reducing a mixed circuit

A 12 V supply is connected to a 2.0 Ω resistor in series with a parallel pair of 6.0 Ω and 3.0 Ω. Find the total current, the p.d. across the parallel pair and the current in each parallel branch.

Have a go first?

Quick check

CalculateQuestion 1 · Apply

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Answer the 2 questions above to continue.