Combined Series-Parallel Circuit Calculations

Many real circuits combine series and parallel sections. To solve these, break the circuit into simpler parts.

Strategy

1. Identify which resistors are in parallel and calculate their combined resistance.
2. Replace the parallel combination with a single equivalent resistor.
3. Add any series resistors to find the total circuit resistance.
4. Calculate the total current using $I = V/R_{\text{total}}$.
5. Work backward to find individual voltages and currents.

Worked example

A 12 V battery is connected to a 4.0 $\Omega$ resistor in series with a parallel combination of 6.0 $\Omega$ and 12 $\Omega$.

Step 1: find the parallel combination.

$\frac{1}{R_p} = \frac{1}{6.0} + \frac{1}{12} = \frac{3}{12} \quad \Rightarrow \quad R_p = 4.0\text{ }\Omega$

Step 2: find total resistance.

$R_{\text{total}} = 4.0 + 4.0 = 8.0\text{ }\Omega$

Step 3: find total current.

$I = \frac{12}{8.0} = 1.5\text{ A}$

Step 4: find p.d. across the series resistor.

$V_1 = 1.5 \times 4.0 = 6.0\text{ V}$

Step 5: p.d. across the parallel section.

$V_p = 12 - 6.0 = 6.0\text{ V}$

Step 6: current in each parallel branch.

$I_{6\Omega} = \frac{6.0}{6.0} = 1.0\text{ A} \qquad I_{12\Omega} = \frac{6.0}{12} = 0.50\text{ A}$

Check: $1.0 + 0.50 = 1.5$ A (matches total current).

Common errors and how to correct them

Calculating parallel resistance for series components or vice versa. Always identify the configuration first: components in series share the same current; components in parallel share the same p.d.

Not checking answers for consistency. Branch currents should add up to the total current. P.d.s across series components should add up to the supply voltage.

How to apply this in an exam

Draw or annotate the circuit to identify series and parallel sections. Work step by step: find parallel equivalents first, then total resistance, then current, then work backward for individual values. Show all steps.