1. Concept Overview
Often, we need more or less capacitance than what a single capacitor provides. By connecting capacitors together, we can fine-tune the total capacitance to suit a circuit. The two basic ways to combine capacitors are series and parallel. Each arrangement affects charge, voltage and total capacitance differently.
2. Understanding How Capacitors Combine
When capacitors are connected, the distribution of charge and potential between them depends on how they are arranged. In a series connection, the same charge flows through each capacitor. In a parallel connection, all capacitors experience the same voltage.
3. Capacitors in Series
Capacitors are said to be in series when they are connected end-to-end so that the same charge appears on each capacitor. The potential difference divides among them.
3.1. Key Points
- Charge on each capacitor is the same (Q).
- Total voltage is the sum of individual voltages.
- Equivalent capacitance is smaller than the smallest capacitor.
3.2. Formula
The reciprocal of the equivalent capacitance is the sum of reciprocals:
\( \dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + \cdots \)
3.3. Reason for Decrease in Capacitance
Connecting in series effectively increases the separation between plates, reducing the ability to store charge.
3.4. Example
Two capacitors: \(C_1 = 6\,\mu F\) and \(C_2 = 3\,\mu F\).
\( \dfrac{1}{C_{eq}} = \dfrac{1}{6} + \dfrac{1}{3} = \dfrac{1}{6} + \dfrac{2}{6} = \dfrac{3}{6} \)
\( C_{eq} = 2\,\mu F \).
4. Capacitors in Parallel
Capacitors are in parallel when their corresponding terminals are connected together. They share the same voltage across them, but the charge divides among the capacitors.
4.1. Key Points
- Voltage across each capacitor is the same.
- Total charge is the sum of charges on each capacitor.
- Equivalent capacitance is larger than the largest individual capacitor.
4.2. Formula
The equivalent capacitance is simply the sum of the individual capacitances:
\( C_{eq} = C_1 + C_2 + C_3 + \cdots \)
4.3. Reason for Increase in Capacitance
Connecting in parallel effectively increases the plate area, allowing more charge to be stored at the same voltage.
4.4. Example
Two capacitors: \(C_1 = 4\,\mu F\) and \(C_2 = 5\,\mu F\).
\( C_{eq} = 4 + 5 = 9\,\mu F \).
5. Mixed Combinations
In real circuits, capacitors may be arranged in a mixture of series and parallel. To find equivalent capacitance, simplify the combination step-by-step by reducing obvious series or parallel groups first.
5.1. General Strategy
- Identify simple series or parallel pairs.
- Simplify one step at a time.
- Redraw the circuit after each reduction.
- Continue until only one equivalent capacitor remains.
5.2. Example (Simplified)
If \(C_1\) and \(C_2\) are in series, and this combination is in parallel with \(C_3\):
First find series equivalent:
\( C_s = \dfrac{C_1 C_2}{C_1 + C_2} \)
Then total capacitance:
\( C_{eq} = C_s + C_3 \)
6. Energy Distribution in Combined Capacitors
Each capacitor stores energy according to the formula:
\( U = \dfrac{1}{2} C V^2 \)
6.1. Series Connection
Voltage divides across capacitors, so each stores different amounts of energy.
6.2. Parallel Connection
Voltage is the same across all capacitors, so energy stored is directly proportional to their capacitance.
7. Comparison Table
| Feature | Series | Parallel |
|---|---|---|
| Charge | Same on all capacitors | Divided among capacitors |
| Voltage | Divided | Same across each |
| Equivalent Capacitance | Smaller than smallest | Larger than largest |
| Formula | 1/Ceq = Σ (1/C) | Ceq = Σ C |
8. Physical Interpretation
Connecting capacitors in different ways gives precise control over total capacitance. Series connections reduce the ability to store charge, while parallel connections increase it. This flexibility is essential in timing circuits, filters, energy storage modules and virtually all electronics.