14.5 Oscillations in an LC Circuit

LC circuits are like energy ping-pong matches. Energy bounces back and forth between the capacitor and inductor, creating electrical oscillations. This dance of charge and current forms the basis for many electronic applications, from radio tuners to signal processing.

Understanding LC circuits is key to grasping alternating current (AC) behavior. The frequency of these oscillations depends on the circuit's components, following a simple formula. This relationship between energy, charge, and frequency is fundamental to electrical engineering and physics.

Oscillations in an LC Circuit

Energy transfer in LC circuits

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• Energy oscillates between the capacitor and inductor in an LC circuit
  • Initially, the capacitor is charged, storing electrical potential energy ($\frac{1}{2}CV^2$)
  • As the capacitor discharges, current flows through the inductor
    • Inductor builds up a magnetic field, storing magnetic potential energy ($\frac{1}{2}LI^2$)
  • When the capacitor is fully discharged, the inductor's magnetic field collapses
    • Induces a current in the opposite direction, charging the capacitor again (electromotive force, emf)
  • Process repeats, with energy continuously transferring between the capacitor and inductor (oscillations)
• Total energy in the system remains constant, assuming no energy losses due to resistance
  • Sum of the capacitor's electrical potential energy and the inductor's magnetic potential energy remains constant ($\frac{1}{2}CV^2 + \frac{1}{2}LI^2 = \text{constant}$)
  • In reality, some energy is lost as heat due to the circuit's resistance (damped oscillations)

Oscillation frequency of LC circuits

• Frequency of oscillations in an LC circuit depends on the capacitance and inductance values
  • Frequency is given by the equation: $f = \frac{1}{2\pi\sqrt{LC}}$
    • $f$ is the frequency of oscillations (Hz)
    • $L$ is the inductance (henries, H)
    • $C$ is the capacitance (farads, F)
  • Example: LC circuit with $L = 10 \text{ mH}$ and $C = 100 \text{ nF}$ has a frequency of $f = \frac{1}{2\pi\sqrt{(10 \times 10^{-3})(100 \times 10^{-9})}} \approx 15.9 \text{ kHz}$
• Angular frequency of oscillations is given by: $\omega = 2\pi f = \frac{1}{\sqrt{LC}}$
  • $\omega$ is the angular frequency (radians per second, rad/s)
  • Related to the frequency by a factor of $2\pi$ ($\omega = 2\pi f$)
• Period of oscillations, $T$, is the reciprocal of the frequency: $T = \frac{1}{f} = 2\pi\sqrt{LC}$
  • $T$ is the time for one complete oscillation (seconds, s)
  • Example: LC circuit with $f = 15.9 \text{ kHz}$ has a period of $T = \frac{1}{15.9 \times 10^3} \approx 62.9 \text{ µs}$
• The frequency at which the circuit's response is maximized is called resonance

Charge and current expressions

• Charge on the capacitor and current in the circuit vary sinusoidally with time
• To derive the expressions, start with the differential equation for an LC circuit: $L\frac{d^2q}{dt^2} + \frac{1}{C}q = 0$
  • $q$ is the charge on the capacitor (coulombs, C)
  • $t$ is time (seconds, s)
• Solution to this differential equation is: $q(t) = Q_0 \cos(\omega t + \phi)$
  • $Q_0$ is the maximum charge on the capacitor (coulombs, C)
  • $\phi$ is the initial phase angle (radians)
• Current in the circuit is the derivative of the charge with respect to time: $I(t) = \frac{dq}{dt} = -\omega Q_0 \sin(\omega t + \phi)$
  • $I(t)$ is the current in the circuit (amperes, A)
  • Current leads the charge by 90° ($\frac{\pi}{2}$ radians)
• Initial conditions determine the values of $Q_0$ and $\phi$
  1. Capacitor initially charged to a voltage $V_0$ and initial current is zero: $Q_0 = CV_0$ and $\phi = 0$
  2. Initial charge is zero and initial current is $I_0$: $Q_0 = \frac{I_0}{\omega}$ and $\phi = -\frac{\pi}{2}$
• Example: LC circuit with $L = 10 \text{ mH}$, $C = 100 \text{ nF}$, and initial capacitor voltage $V_0 = 5 \text{ V}$ has charge and current expressions:
  • $q(t) = (100 \times 10^{-9})(5) \cos(10^5 t) = 500 \text{ nC} \cos(10^5 t)$
  • $I(t) = -(10^5)(500 \times 10^{-9}) \sin(10^5 t) = -50 \text{ mA} \sin(10^5 t)$
• Phasor diagrams can be used to visualize the relationship between charge and current in the circuit

Circuit characteristics

• Impedance in an LC circuit is the total opposition to current flow, combining the effects of inductance and capacitance
• Quality factor (Q factor) is a measure of the circuit's efficiency in storing energy, related to the sharpness of the resonance peak
• Bandwidth refers to the range of frequencies around the resonance frequency where the circuit's response remains significant