📘 Chapter 28: Combination of Capacitors – Series and Parallel (Class XII)

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🔷 1. Introduction

In practical electrical and electronic circuits, capacitors are rarely used alone. Instead, they are connected in different ways to achieve desired values of capacitance. These arrangements are called combinations of capacitors.

The two most important types are:

• Series combination
• Parallel combination

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🔷 2. Why Combine Capacitors?

• To obtain desired capacitance
• To control voltage distribution
• To store more or less energy
• To suit circuit design requirements

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🔷 3. Capacitors in Series

In a series combination, capacitors are connected end-to-end such that the same charge flows through each capacitor.

🔹 Key Characteristics:

• Same charge on all capacitors
• Voltage divides across capacitors
• Equivalent capacitance decreases

🔹 Formula:

1/C = 1/C₁ + 1/C₂ + 1/C₃ + ...

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📦 4. Important Results – Series Combination

• Charge: Same on all capacitors
• Voltage: Divides
• Equivalent capacitance: Less than smallest capacitor
• Energy: Distributed among capacitors

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🔷 5. Derivation (Series)

Total voltage:

V = V₁ + V₂ + V₃

Using C = Q/V:

V = Q/C₁ + Q/C₂ + Q/C₃

So:

1/C = 1/C₁ + 1/C₂ + 1/C₃

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🔷 6. Capacitors in Parallel

In a parallel combination, capacitors are connected across the same potential difference.

🔹 Key Characteristics:

• Same voltage across each capacitor
• Charge divides among capacitors
• Equivalent capacitance increases

🔹 Formula:

C = C₁ + C₂ + C₃ + ...

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📦 7. Important Results – Parallel Combination

• Voltage: Same across all capacitors
• Charge: Divides
• Equivalent capacitance: Greater than largest capacitor
• Energy storage: Maximum

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🔷 8. Derivation (Parallel)

Total charge:

Q = Q₁ + Q₂ + Q₃

Using Q = CV:

Q = C₁V + C₂V + C₃V

So:

C = C₁ + C₂ + C₃

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🔷 9. Comparison: Series vs Parallel

Property	Series	Parallel
Charge	Same	Different
Voltage	Different	Same
Capacitance	Decreases	Increases
Energy Storage	Less	More

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🔷 10. Energy Considerations

Energy stored:

U = ½ C V²

• Parallel → higher energy storage
• Series → lower energy storage

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🔷 11. Practical Insights

• Series combination used for high voltage tolerance
• Parallel combination used for higher capacitance
• Real circuits use mixed combinations

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🧠 12. Solved Numerical Problems

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🔹 Q1

Find equivalent capacitance of 2F and 3F in series.

Solution:

1/C = 1/2 + 1/3 = (3+2)/6 = 5/6

C = 6/5 = 1.2 F

Answer: 1.2 Farad

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🔹 Q2

Find equivalent capacitance in parallel (2F, 3F).

Solution:

C = 2 + 3 = 5 F

Answer: 5 Farad

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🔹 Q3

Which combination gives maximum capacitance?

Answer:

Parallel

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🔹 Q4

Which combination is used for high voltage?

Answer:

Series combination

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🔹 Q5

Why does capacitance decrease in series?

Answer:

Effective separation between plates increases.

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🔷 13. Advanced Conceptual Insight

Understanding capacitor combinations is essential for circuit design, power electronics, and signal processing. Engineers use combinations to optimize energy storage, voltage handling, and system efficiency.

Complex circuits often involve both series and parallel arrangements simultaneously.

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🔷 14. Applications

• Power supply systems
• Electronic circuits
• Energy storage devices
• Communication systems

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🔷 15. Summary

Capacitors can be combined in series and parallel to achieve desired capacitance. Series reduces capacitance and divides voltage, while parallel increases capacitance and shares voltage equally.

✨ End of Chapter 28: Capacitor Combinations ✨