Classify the following R L C circuits as (i) underdamped,...

Classify the following $R L C$ circuits as (i) underdamped, (ii) critically damped, or (iii) overdamped: (a) $R=1, L=2, C=4$, (b) $R=4, L=3, C=1$,
(c) $R=2, L=3, C=3$, (d) $R=4, L=10, C=2$, (e) $R=1, L=1, C=3$.

Key Concepts

RLC Circuits

RLC circuits consist of a resistor (R), an inductor (L), and a capacitor (C) connected in series or parallel. The interplay between these components creates a system that is modeled by a second?order differential equation, which captures the dynamics of current and voltage in response to a disturbance. This analysis forms the basis for understanding oscillatory and transient behaviors in electrical circuits.

Second Order Differential Equations

The dynamic behavior of an RLC circuit is described by a second order differential equation. The characteristic equation associated with this differential equation provides key insights into the natural response of the circuit by determining the system’s eigenvalues, which indicate whether the transient response oscillates, decays in a non-oscillatory manner, or a mix of both.

Characteristic Equation and Discriminant Analysis

The characteristic equation of an RLC circuit is typically derived from applying Kirchhoff’s voltage law and has the form s² + (R/L)s + 1/(LC) = 0. The discriminant of this quadratic equation, given by (R/L)² - 4/(LC), is central to classifying the damping behavior. Its value determines the nature of the roots: a positive discriminant yields two distinct real roots (overdamped), a zero discriminant yields a repeated real root (critically damped), and a negative discriminant yields complex conjugate roots (underdamped).

Damping Classification

Damping classification in RLC circuits involves determining whether the circuit’s natural response decays via oscillations or a monotonic decrease. An underdamped system (negative discriminant) exhibits oscillatory decay due to complex conjugate roots, a critically damped system (zero discriminant) represents the fastest non-oscillatory response without overshoot, and an overdamped system (positive discriminant) shows a slower, non-oscillatory decay governed by two distinct real roots.

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