1. Why Resistors Are Combined
In real circuits, we rarely use just one resistor. Multiple resistors are combined to get the desired overall resistance. Depending on how they are connected, the total resistance can become larger or smaller.
The two most common ways of connecting resistors are series and parallel combinations.
2. Resistors in Series
Resistors are connected in series when they are placed one after another, so the same current flows through each of them. The potential difference gets divided across the resistors.
2.1. Definition of Series Combination
In a series combination, resistors are connected end-to-end, and there is only one path for current to flow.
2.2. Equivalent Resistance in Series
The total resistance is simply the sum of individual resistances:
\( R_{\text{eq}} = R_1 + R_2 + R_3 + \cdots \)
This means series connections always increase the net resistance.
2.2.1. Small Example
If three resistors of 2 Ω, 3 Ω and 5 Ω are connected in series, the total resistance is:
\( R_{\text{eq}} = 2 + 3 + 5 = 10\,\Omega \)
3. Resistors in Parallel
In a parallel connection, resistors are connected across the same two points. The potential difference across each resistor is the same, but the current divides among them.
3.1. Definition of Parallel Combination
In a parallel combination, each resistor is connected directly across the supply, forming multiple current paths.
3.2. Equivalent Resistance in Parallel
The reciprocal of the total resistance is the sum of the reciprocals of individual resistances:
\( \dfrac{1}{R_{\text{eq}}} = \dfrac{1}{R_1} + \dfrac{1}{R_2} + \dfrac{1}{R_3} + \cdots \)
Parallel combinations always reduce the net resistance.
3.2.1. Small Example
For two resistors of 6 Ω and 3 Ω in parallel:
\( \dfrac{1}{R_{\text{eq}}} = \dfrac{1}{6} + \dfrac{1}{3} = \dfrac{1}{6} + \dfrac{2}{6} = \dfrac{3}{6} \)
\( R_{\text{eq}} = 2\,\Omega \)
4. Current and Voltage Distribution
In series, the current remains the same through all resistors while voltage divides. In parallel, voltage remains the same across each branch while current divides according to resistance.
4.1. Series Behaviour
- Current: same through each resistor
- Voltage: distributed according to resistance
4.2. Parallel Behaviour
- Voltage: same across each resistor
- Current: larger current flows through smaller resistance
5. Quick Comparison
| Series | Parallel |
|---|---|
| One path for current | Multiple paths for current |
| Current is same | Voltage is same |
| Resistance adds | Reciprocal adds |
| Net resistance increases | Net resistance decreases |