Series and parallel
Resistors can be connected end to end (in series) or side by side (in parallel). The arrangement changes the total, or equivalent, resistance that the rest of the circuit sees.
- Series: the same current flows through each resistor and the resistances add: R = R1 + R2 + R3 + …. The total is always larger than the largest resistor.
- Parallel: each resistor sees the same voltage and the currents add. The conductances (1 ÷ R) add: 1 ÷ R = 1 ÷ R1 + 1 ÷ R2 + 1 ÷ R3 + …. The total is always smaller than the smallest resistor.
Worked example: 100 Ω, 220 Ω and 470 Ω
- In series: 100 + 220 + 470 = 790 Ω.
- In parallel: 1 ÷ 100 + 1 ÷ 220 + 1 ÷ 470 = 0.01 + 0.004545 + 0.002128 = 0.016673 S.
- R = 1 ÷ 0.016673 = 59.98 Ω, below the smallest resistor of 100 Ω.
Shortcuts for parallel resistors
- Two resistors: R = (R1 × R2) ÷ (R1 + R2), the “product over sum”. 100 Ω and 220 Ω give 22,000 ÷ 320 = 68.75 Ω.
- n identical resistors: R ÷ n. Four 1 kΩ resistors in parallel make 250 Ω.
- One very large resistor in parallel barely changes the total, while a small one dominates it.
Effect of adding a fourth resistor
| R4 | Series total | Parallel total |
|---|---|---|
| none | 790 Ω | 59.98 Ω |
| 10 Ω | 800 Ω | 8.57 Ω |
| 100 Ω | 890 Ω | 37.49 Ω |
| 1 kΩ | 1,790 Ω | 56.58 Ω |
| 10 kΩ | 10,790 Ω | 59.62 Ω |
Mixed circuits
Real circuits often combine both. Solve them step by step: replace each group of resistors in parallel by its equivalent, then add up the series parts, and repeat until a single value remains. For example, a 100 Ω resistor in series with two 220 Ω resistors in parallel gives 100 + 110 = 210 Ω.
Practical uses
- Getting a value you don’t have: two 1 kΩ resistors in series make 2 kΩ; in parallel, 500 Ω.
- Sharing power: several resistors in parallel spread the heat. Two 1 W resistors of 200 Ω in parallel act as a 100 Ω resistor rated 2 W.
- LED strings: LEDs with their resistor in series share the same current; branches in parallel each need their own resistor.
- Speakers: two 8 Ω speakers in parallel present 4 Ω to the amplifier; in series, 16 Ω. Check that your amplifier supports the result.
Capacitors and inductors
Inductors follow the same rules as resistors. Capacitors do the opposite: capacitances add in parallel, and their reciprocals add in series.