A parallel circuit is a circuit in which branches are connected across the same two nodes, so every branch experiences the same voltage. The total current is the sum of the branch currents, and the equivalent resistance is smaller than the smallest branch resistance.
Learning objectives
- Calculate the equivalent resistance of parallel branches.
- Explain why each parallel branch has the same voltage.
- Calculate branch currents and the total current.
- Describe the effect of adding or losing a parallel branch.
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
Household wiring is in parallel: every lamp receives the same supply voltage, and switching one off does not affect the others. The same is true in a vehicle, where headlights, horn and control modules are all connected across the battery through their own circuits. Adding more parallel branches increases the total current drawn from the source.
Intermediate How it behaves in a circuit
The equivalent resistance is always smaller than the smallest branch. Adding another parallel branch provides another path for current, so total current rises for the same voltage. This is why adding loads to an already loaded supply increases the current and can overload it. In real circuits, branch resistances may be non-linear, but the same voltage-sharing principle applies.
Advanced Limits, assumptions, engineering detail
At high frequency, parallel branches interact through mutual inductance and stray capacitance, and current sharing can become uneven because the impedance of each path differs with frequency. On circuit boards, parallel traces carrying high current must be balanced to avoid thermal hotspots. In power systems, parallel generators must be synchronised in voltage, frequency and phase before being connected together.
Formulas and variables
Units of measurement
| Quantity | Symbol | Unit |
|---|---|---|
| Resistance | R | ohm (Ω) |
| Voltage | V | volt (V) |
| Current | I | ampere (A) |
| Conductance | G | siemens (S) |
Worked examples
Two parallel resistors
Problem. A 6 Ω and a 3 Ω resistor are connected in parallel across 12 V. Find the equivalent resistance, branch currents and total current.
Solution. R_eq = (6 × 3)/(6 + 3) = 2 Ω. I₆ = 12/6 = 2.0 A, I₃ = 12/3 = 4.0 A. I_total = 2.0 + 4.0 = 6.0 A. Check: I_total = 12/2 = 6.0 A.
Three parallel branches
Problem. Resistances of 10 Ω, 15 Ω and 30 Ω are in parallel across 30 V. Find the equivalent resistance and the total current.
Solution. 1/R_eq = 1/10 + 1/15 + 1/30 = 3/30 + 2/30 + 1/30 = 6/30 = 1/5, so R_eq = 5 Ω. I_total = 30/5 = 6.0 A. Branch currents: 3.0 A, 2.0 A and 1.0 A, which sum to 6.0 A.
Practical applications
- Household and vehicle load wiring
- Redundant power supplies and battery banks
- Distributing high current across conductors
- Adding loads to a common bus
Automotive applications
Vehicle loads are wired in parallel across the battery through individual fuses. Each load receives essentially the same system voltage, and one failed lamp does not switch off the others. Adding accessories in parallel increases the total current from the alternator and battery, which is why the charging capacity and fuse ratings must be checked when equipment is added.
Industrial applications
Industrial distribution uses parallel feeders and parallel-connected loads on a busbar. Parallel cables can share large currents, but they must be matched in length and termination to share current evenly. Parallel generators require synchronisation, and parallel sources must be designed so that a fault in one does not back-feed the others.
Measurement procedure
- Measure the voltage across each branch; in an ideal parallel circuit all branches have the same voltage.
- Measure branch currents individually and add them to obtain the total current.
- To measure a branch resistance, isolate it from the other branches; otherwise the parallel path lowers the reading.
- For current sharing between parallel conductors, use a clamp meter on each conductor separately.
Common mistakes
- Adding parallel resistances directly instead of using reciprocals.
- Expecting the branch currents to be equal regardless of resistance.
- Forgetting that the equivalent resistance is smaller than any single branch.
- Leaving a parallel path connected while trying to measure one branch’s resistance.
Safety considerations
- A parallel branch remains live even when another branch is switched off; isolate the correct circuit before working.
- Adding parallel loads increases total current; check fuse, cable and supply ratings before adding equipment.
- Parallel battery connections require matched voltages and appropriate protection to avoid circulating currents.