31.9: RLC Series Circuits
An RLC series circuit comprises an inductor, a resistor, and a charged capacitor connected in series. When the circuit is closed, the capacitor begins to discharge through the resistor and inductor by transferring energy from the electric field to the magnetic field. Here, the resistor connected to the circuit causes energy losses; therefore, on the complete discharge of the capacitor, the magnetic field energy acquired by the inductor is less than the original electric field energy of the capacitor. Similarly, the energy acquired by the capacitor when the magnetic field has decreased to zero is smaller than the original magnetic field energy of the inductor. The energy oscillations between the capacitor and inductor decrease in magnitude with time. The total decrease in electromagnetic energy is equal to the energy dissipated in the resistor. Such energy oscillations are called damped oscillations, and the circuit is said to be an RLC damped circuit. This behavior is analogous to the mass-spring damped harmonic oscillator. Similar to a damped harmonic oscillator, the oscillations in a damped RLC circuit can be of three forms. If the resistance is relatively small, the circuit oscillates but with damped harmonic motion, and the circuit is said to be underdamped. If the resistance in the circuit is increased, the oscillations die out more rapidly. When it reaches a certain value, the circuit no longer oscillates and is termed critically damped. For still larger values of resistance, the circuit is overdamped, and the capacitor charge approaches zero even more slowly.
In a damped RLC circuit, the variation of charge and current with time is given by the following differential equation:
This equation is analogous to the equation of motion for a damped mass-spring system. The form of the solution is different for the underdamped, critically damped, and overdamped cases. In the case of underdamping, the solution of the differential equation is given by
where the angular frequency of the oscillations is given by