RLC

RLC refers to a circuit or system containing a resistor (R), inductor (L), and capacitor (C) arranged in series or parallel. It is a fundamental second-order electrical circuit and a standard model for many physical systems. In an RLC circuit, energy continuously exchanges between the magnetic field of the inductor and the electric field of the capacitor, with the resistor providing damping.

In series form, the circuit is described by the equation L d^2q/dt^2 + R dq/dt + q/C = V(t), where q is the charge on the capacitor and i = dq/dt is the current. The total impedance as a function of angular frequency ω is Z(ω) = R + j(ωL − 1/(ωC)). In parallel form, the total admittance is Y(ω) = 1/R + 1/(jωL) + jωC. The natural frequency is ω0 = 1/√(LC). The damping ratio for the series form is ζ = R/2 · √(C/L), with α = R/(2L) and ω0 as above. The system can be underdamped (ζ < 1), critically damped (ζ = 1), or overdamped (ζ > 1). In underdamped cases, oscillations occur at ωd = ω0√(1 − ζ^2) and decay over time. A forced response near ω0 exhibits a resonance peak when damping is moderate, characterized by a quality factor Q, with Q = ω0L/R for series or Q = R√(C/L) for parallel configurations.

Applications include filters and tuning circuits in radio and signal processing, where they select or reject specific frequencies, as well as impedance matching and energy storage in power electronics. Energy storage in L is (1/2)LI^2, and in C is (1/2)CV^2, reflecting the dynamic exchange that underpins the circuit’s behavior.