Depending on the value of relative to and , the RLC equation has three qualitative behaviours. The discriminant of the characteristic equation changes sign at the threshold , and with it the character of the solution.
Three regimes, one equation
- Underdamped (small ): the system oscillates with exponentially decreasing amplitude, like a pendulum in a thin syrup.
- Critical (): the system returns to equilibrium in the shortest possible time without oscillating.
- Overdamped (large ): the system returns to equilibrium slowly, with no oscillations.
The critical regime is not just a mathematical curiosity: it is the one to which car shock absorbers are designed. After a pothole, the suspension must return to position quickly but without bouncing (which would give a car that jolts) and without being too slow (which would give a soft, wallowing car). The optimal compromise is exactly critical damping, and the mathematics is identical to that of this circuit.
Collegamenti
Argomenti: Electromagnetic induction Concetti: LC and RLC oscillations · Damped oscillations
Esercizi collegati: Problem — Natural frequency of an LC circuit · Problem — Capacitor of an AM radio receiver · Worked exercise — Forced RLC, find the generator’s EMF