To solve a series-parallel DC circuit, identify which resistors share one path and which connect across the same two nodes, simplify those groups one at a time, then use Ohm’s law and work backward to find individual currents and voltages. This guide explains the method and gives you checks to catch common worksheet errors.
Identify series and parallel connections first
Look at how components are connected, not just how they are drawn. Resistors are in series when they lie along one current path with no branching between them. Resistors are in parallel when both ends of each resistor connect to the same two nodes. A mixed circuit combines these arrangements, so one part can often be reduced before the next.
Mark each simple series group and parallel group on the diagram. Start with a group whose connections are clear, often the innermost parallel pair or series chain. After replacing it with an equivalent resistor, redraw the circuit so the remaining connections are easier to see. OpenStax explains the same staged approach in its series-and-parallel resistor examples.
Use the right rule for each group
Resistors in series
In a series path, the same current passes through every resistor. Add the resistances to find the equivalent resistance:
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Rs = R1 + R2 + …
The voltage drop across each resistor depends on its resistance and the current through it, using V = IR. The drops across all resistors in an ideal series path account for the source voltage.
Resistors in parallel
In parallel, each resistor connects across the same two nodes, so each branch has the same voltage. Current divides among the branches, and the branch currents add to the current entering the parallel group. Find equivalent resistance with the reciprocal rule:
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1/Rp = 1/R1 + 1/R2 + …
Add the reciprocals first, then take the reciprocal of the result. The equivalent resistance is less than the smallest individual branch resistance. For the definition and worked examples, see OpenStax’s parallel-circuits section.
Solve a mixed circuit step by step
- Label the topology. Mark each set of resistors in series and each set connected in parallel across the same two nodes.
- Reduce one simple group. Add resistances for a series group. For a parallel group, add reciprocals and invert the sum.
- Redraw the circuit. Replace the reduced group with its equivalent resistor while preserving the connections to the rest of the circuit.
- Repeat until one equivalent resistance remains. Continue reducing groups until the source sees a single total resistance.
- Find the source current. Use Ohm’s law with the source voltage and total resistance: Itotal = Vsource/Rtotal.
- Work backward through the reductions. For a series group, use the group current to calculate each drop with V = IR. For a parallel group, use the shared branch voltage to find each branch current with I = V/R.
- Check the values. Branch currents in each parallel group should sum to the current entering it. Voltage drops along a series path should account for the voltage across that path.
This sequence follows the method in OpenStax’s mixed series-parallel worked example: reduce a parallel group, combine its equivalent with a series resistor, then use the simplified circuit to recover the original quantities.
Keep current and voltage relationships straight
Current is not used up by a resistor. In a series path it is the same through each component; at a parallel junction it divides among branches and recombines, so the branch currents sum to the total. Voltage behaves differently: parallel branches share the same voltage, while voltage drops across series resistors depend on their resistance and the path current.
OpenStax’s key equations for circuits include Ohm’s law and the series and parallel equivalent-resistance rules. Apply one relationship at a time and keep track of which value belongs to the whole circuit, a group, or an individual resistor.
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Common worksheet mistakes to avoid
- Calling components parallel because they look side by side. Confirm that both ends connect to the same two nodes.
- Adding parallel resistances directly. Use the reciprocal formula; direct addition is the series rule.
- Assuming current is equal in every branch. Parallel branches share voltage, not necessarily current. Their currents depend on their resistances.
- Using total current for every parallel resistor. First find the voltage across the parallel group, then calculate each branch current.
- Skipping the redraw after a reduction. A new sketch helps prevent treating a simplified group as though its original internal connections still affect the remaining circuit.
- Mixing up total and component values. Label quantities as total, branch, or resistor-specific, with units, as you calculate.
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