LC Oscillations

1. Basic Concepts:

• Introduction:

  • An LC circuit consists of an inductor and a capacitor connected in series or parallel.
  • When the capacitor is charged and the circuit is closed, the stored electrical energy is transferred between the inductor and the capacitor, resulting in oscillations.

• Inductance and Capacitance:

  • Inductance (L): Property of a conductor to oppose changes in current flow, measured in henries (H).
  • Capacitance (C): Ability of a conductor to store electrical charge, measured in farads (F).

• Energy Stored:

  • Energy stored in an inductor: (E_L = 1/2 LI^2)
  • Energy stored in a capacitor: (E_C = 1/2 CV^2)

2. Differential Equation of LC Oscillations:

• Derivation:

  Using Kirchhoff’s voltage law and the relationships between current and voltage in an inductor and capacitor, we obtain the differential equation:

  $$L\frac{d^2q}{dt^2}+\frac{q}{C}=0$$

• Solutions:

  The solutions to this equation are sinusoidal functions:

  $$q(t) = Q_{max} \cos(\omega t + \phi)$$

  where (Q_{max}) is the maximum charge, $\omega$ is the angular frequency, and $\phi$ is the phase angle.

• Angular Frequency and Time Period:

  Angular frequency: (\omega = \frac{1}{\sqrt{LC}}) Time period: (T = \frac{2\pi}{\omega} = 2\pi \sqrt{LC} )

3. Energy Conservation in LC Oscillations:

• Principle:

  Total energy in the circuit (sum of electrical and magnetic energies) remains constant during oscillations.

• Proof:

  $$E_{total} = E_L + E_C = \frac{1}{2}LI^2 + \frac{1}{2}CV^2$$

Taking the derivative with respect to time and using the differential equation, we get:

$$\frac{dE_{total}}{dt} = LI\frac{dI}{dt} + CV\frac{dV}{dt} = 0$$

Therefore, the total energy remains constant.

4. Phase Difference and Amplitude:

• Phase Difference:

  Current (I) and voltage (V) differ in phase by 90 degrees in an LC circuit. When current is maximum, voltage is zero, and vice versa.

• Amplitude:

Maximum current amplitude: (I_{max} = \frac{Q_{max}}{\sqrt{L}}) Maximum voltage amplitude: (V_{max} = Q_{max}\sqrt{\frac{1}{C}})

5. Quality Factor:

• Definition:

  Quality factor (Q) represents the energy loss per oscillation. $$Q = \frac{\omega_0L}{R}$$ where $\omega_0$ is the resonant frequency and R is the resistance in the circuit.

• Significance:

  Higher Q indicates lower energy loss and more sustained oscillations.

6. Damped LC Oscillations:

• Causes:

  Energy loss due to resistance in the circuit causes damping of oscillations.

• Differential Equation: $$L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{q}{C}=0$$

• Solutions: ( q(t) = Q_0 e^{-\alpha t} \cos(\omega ’ t + \phi) )

  where (Q_0) is the initial charge, $\alpha$ is the decay constant, $\omega ‘$ is the damped angular frequency, and $\phi$ is the phase angle.

• Decay Constant and Logarithmic Decrement:

  Decay constant: (\alpha = \frac{R}{2L}) Logarithmic decrement: (\delta = \frac{2\pi \alpha}{T})

7. Resonance in LC Circuits:

• Condition:

  Resonance occurs when the angular frequency of the applied voltage matches the natural angular frequency of the LC circuit: (\omega = \omega_0).

• Sharpness:

  Sharpness of resonance is characterized by the quality factor (Q). Higher Q indicates a sharper resonance.

• Bandwidth and Selectivity:

  Bandwidth (BW): Frequency range around the resonant frequency where the amplitude drops to (1/\sqrt{2}) of the maximum amplitude. Selectivity: Ability of the circuit to distinguish between signals of different frequencies. Higher Q implies higher selectivity.

8. Coupled LC Circuits:

• Introduction:

  Two or more LC circuits that interact through mutual inductance are known as coupled LC circuits.

• Coefficient of Coupling:

  Measures the degree of magnetic coupling between the coils. (0\le k\le 1).

• Energy Transfer:

  Energy oscillates between the coupled circuits, with the frequency and amplitude depending on the coupling coefficient.

• Normal Modes of Oscillation:

  Two distinct frequencies at which the coupled circuits oscillate independently.

9. Applications of LC Oscillations:

• LC Oscillators:

  Generate sinusoidal oscillations used in various electronic devices.

• Tuning Circuits:

  Used in radios and television to select a specific frequency from the electromagnetic spectrum.

• Filters:

  Used in electronic circuits to pass or reject certain frequency bands.

• Energy Storage Devices:

  Capacitors and inductors can store electrical energy in LC circuits.