Capacitive Circuits - Alternating Currents

1. Capacitors and Capacitance:

• Concept: A capacitor is a passive electronic component that stores electrical energy in an electric field.
• Units: The unit of capacitance is farad (F), named after the English physicist Michael Faraday.
• Factors Affecting Capacitance:
  • Plate Area (A): The larger the plate area, the greater the capacitance.
  • Distance Between Plates (d): The smaller the distance between the plates, the greater the capacitance.
  • Dielectric Material: The type of insulating material (dielectric) between the plates affects the capacitance. Common dielectrics include ceramic, paper, plastic, and electrolytes.

2. Capacitors in AC Circuits:

• Alternating Currents (AC): AC is an electrical current that reverses direction periodically.

• Capacitive Reactance (X_C): The opposition offered by a capacitor to the flow of AC is called capacitive reactance. It depends on the capacitance (C) and the angular frequency (ω) of the AC:

  • X_C = 1/(2πfC), where f is the frequency in hertz (Hz).

• Phase Difference: In a capacitive circuit, the current leads the voltage by 90 degrees.

• Vector Diagrams: Vector diagrams are used to represent the phase differences between voltage and current in AC circuits.

3. RC Circuits:

• Components: An RC circuit consists of a resistor (R) and a capacitor (C) connected in series or parallel.
• Time Constant (τ): The time constant of an RC circuit is the time it takes for the voltage or current to reach 63.2% or 1 - 1/e of its final value.
  • τ = RC, where R is in ohms (Ω) and C is in farads (F).
• Transient Response: The transient response of an RC circuit is the initial, non-steady state behavior of the circuit before it reaches its steady state.
• Steady-State Response: The steady-state response of an RC circuit is the constant, long-term behavior of the circuit after it has reached its equilibrium.

4. Capacitors in Series and Parallel:

• Series Combination: Capacitors in series have a combined capacitance (C_eq) given by:
  • 1/C_eq = 1/C₁ + 1/C₂ + …
• Parallel Combination: Capacitors in parallel have a combined capacitance (C_eq) given by:
  • C_eq = C₁ + C₂ + …
• Voltage Division: In a series combination, the voltage across each capacitor is proportional to its capacitance.

5. Resonance in Capacitive Circuits:

• Resonance: Resonance occurs in a series RLC circuit when the inductive reactance (X_L) equals the capacitive reactance (X_C).
• Resonance Frequency (f₀): The resonant frequency is the frequency at which resonance occurs.
  • f₀ = 1/(2π√(LC)), where L is the inductance in henries (H) and C is the capacitance in farads (F).
• Quality Factor (Q): Quality factor represents the sharpness of the resonance peak. A higher Q-factor indicates a more selective circuit.

6. Applications of Capacitors in AC Circuits:

• Energy Storage: Capacitors can store electrical energy and release it when needed.
• Capacitor-Start Induction Motors: Capacitors are used to provide the initial torque to start induction motors.
• Power Factor Correction: Capacitors can be used to improve the power factor of an AC circuit, reducing energy losses.
• Filter Circuits: Capacitors are used in filter circuits to remove unwanted frequencies from signals.

7. Phasor Diagrams:

• Phasors: Phasors are complex numbers that represent the amplitude and phase of AC quantities.
• Phasor Diagrams: Phasor diagrams graphically represent the phase relationships between AC voltages and currents.

8. Problems and Numerical Analysis:

• Practice solving numerical problems involving capacitive circuits.
• Analyze circuit behavior for different values of capacitance, frequency, and other circuit elements.
• Interpret phasor diagrams to understand phase relationships and circuit behavior.

References:

• NCERT Physics, Class 11, Chapter 4 - Alternating Current and Electromagnetic Induction.
• NCERT Physics, Class 12, Chapter 7 - Alternating Current.