A series–parallel circuit contains both series and parallel connections. It is analysed by repeatedly replacing parallel groups with their equivalent resistance and adding series resistances, then working back to find branch voltages and currents.
Learning objectives
- Identify series and parallel sections in a mixed circuit.
- Reduce a series–parallel network to a single equivalent resistance.
- Work back to find branch currents and voltages.
- Verify results with Kirchhoff’s current and voltage laws.
Progressive explanation
Start with the plain-language idea, then move into the engineering detail. Each level stands on its own, so you can stop at the depth you need.
Beginner The core idea
Once you have the total resistance, find the total current from the source voltage. Then work backwards through the circuit: the total current flows through the series elements, and it splits between parallel branches according to their resistances. Checking that the voltages and currents add up correctly confirms the result.
Intermediate How it behaves in a circuit
Kirchhoff’s laws provide a check. The sum of the branch currents equals the current entering the parallel group, and the sum of all voltage drops around any loop equals the source voltage. A common mistake is to reduce the network correctly but then misapply the current when working back, so label each intermediate value clearly.
Advanced Limits, assumptions, engineering detail
In real circuits, component tolerances and temperature coefficients mean the calculated division is nominal. Non-linear elements require iterative solution or load-line methods. At high frequency, parasitic capacitance and inductance make the simple resistive model inadequate, and the physical layout can create unintended coupling between branches. The reduction method remains a valuable first estimate and a sanity check for simulation results.
Formulas and variables
Units of measurement
| Quantity | Symbol | Unit |
|---|---|---|
| Resistance | R | ohm (Ω) |
| Voltage | V | volt (V) |
| Current | I | ampere (A) |
| Power | P | watt (W) |
Worked examples
Series resistor with a parallel pair
Problem. A 10 Ω resistor is in series with a parallel pair of two 20 Ω resistors, across a 40 V source. Find the total current, the voltage across the parallel pair and the current in each 20 Ω branch.
Solution. Parallel pair: 20 || 20 = 10 Ω. R_total = 10 + 10 = 20 Ω. I_total = 40/20 = 2.0 A. Voltage across the series resistor = 2.0 × 10 = 20 V, so the parallel pair has 40 − 20 = 20 V. Each 20 Ω branch current = 20/20 = 1.0 A, and 1.0 + 1.0 = 2.0 A checks with KCL.
Unequal parallel branches
Problem. A 12 Ω resistor is in series with a parallel pair of 24 Ω and 8 Ω, across a 36 V source. Find the total current and the current in each parallel branch.
Solution. Parallel pair: 24 || 8 = (24 × 8)/(24 + 8) = 192/32 = 6.0 Ω. R_total = 12 + 6 = 18 Ω. I_total = 36/18 = 2.0 A. Voltage across 12 Ω = 2.0 × 12 = 24 V; parallel pair voltage = 36 − 24 = 12 V. Branch currents: 12/24 = 0.50 A and 12/8 = 1.5 A; sum = 2.0 A.
Practical applications
- Automotive lighting and accessory circuits
- Sensor bridges and signal conditioning
- Industrial control and indicator circuits
- Ladder networks and attenuators
Automotive applications
Vehicle circuits commonly combine series and parallel sections: a switch and fuse are in series with a lamp, while several lamps may be in parallel across the supply. A parallel branch that loses its ground can force current through another path, changing voltages in ways that simple series or parallel reasoning does not predict. Reducing the network section by section helps explain measured voltages and locate the fault.
Industrial applications
Industrial control panels often contain mixed networks of indicator lamps, relay coils and resistors. A resistor in series with an LED indicator, with several indicators in parallel, is a typical series–parallel arrangement. Understanding the reduction helps select resistor power ratings and diagnose why one indicator is dim or another does not light.
Measurement procedure
- Measure the source voltage and total current first to check the overall equivalent resistance.
- Measure the voltage across each parallel group; branch currents then follow from the branch resistances.
- De-energise and isolate a branch before measuring its resistance, because the rest of the network forms parallel paths.
- Compare measured branch currents with the calculated division and investigate significant discrepancies as possible faults.
Common mistakes
- Reducing a parallel group but then applying the total current to a single branch.
- Treating a bridge network as a simple series–parallel combination when it is not reducible that way.
- Forgetting that the voltage across a parallel group is the source voltage minus the series drops.
- Ignoring component tolerances when comparing calculated and measured values.
Safety considerations
- Mixed networks can keep a branch live through an unexpected path; isolate and prove dead before work.
- Check the power rating of every resistor in a divider, not only the total.
- Do not assume a low-voltage circuit is harmless; batteries can deliver high fault currents through parallel paths.