Question

If \( L = 1\text{ H} \), \( C = 1\text{ F} \) and \( R = 1\ \Omega \) in a series RLC circuit, then the circuit is:

• A. Pure oscillating
• B. Critically damped
• C. Overdamped
• D. Underdamped

Hint:

An alternative condition for a series RLC circuit can be evaluated using the damping ratio \(\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}\): - If \(\zeta > 1\): Overdamped - If \(\zeta = 1\): Critically Damped - If \(\zeta < 1\): Underdamped Here, \(\zeta = \frac{1}{2}\sqrt{\frac{1}{1}} = 0.5 < 1\), confirming it is underdamped.

Answer 1

The Correct Option is D

Concept: The transient characteristic response of a second-order series RLC network is fundamentally dictated by its characteristic equation, derived from the governing differential equation. For a series arrangement, this equation takes the form: \[ s^2 + \frac{R}{L}s + \frac{1}{LC} = 0 \] This standard equation is compared with the canonical form of a second-order system equation: \[ s^2 + 2\alpha s + \omega_0^2 = 0 \] where:
• \(\alpha\) is the attenuation factor or damping coefficient, defined for a series layout as \(\alpha = \frac{R}{2L}\).
• \(\omega_0\) is the undamped natural resonant angular frequency, defined as \(\omega_0 = \frac{1}{\sqrt{LC}}\). The relative comparison between \(\alpha\) and \(\omega_0\) determines the nature of the damping:
• If \(\alpha > \omega_0\), the roots are real and distinct; the system is Overdamped.
• If \(\alpha = \omega_0\), the roots are real and identical; the system is Critically damped.
• If \(\alpha < \omega_0\), the roots are complex conjugates; the system is Underdamped.
• If \(\alpha = 0\) (i.e., \(R=0\)), the roots are purely imaginary; the system is Undamped or Pure oscillating.

Step 1: Computing the damping factor \(\alpha\).
From the values given in the problem statement (\(R = 1\ \Omega\), \(L = 1\text{ H}\), \(C = 1\text{ F}\)), we evaluate \(\alpha\): \[ \alpha = \frac{R}{2L} = \frac{1}{2 \times 1} = 0.5\text{ rad/s} \]

Step 2: Computing the natural resonant frequency \(\omega_0\).
Next, we calculate the value of \(\omega_0\) using the given component parameters: \[ \omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{1 \times 1}} = 1\text{ rad/s} \]

Step 3: Comparing the values to characterize the system.
We compare the calculated values of \(\alpha\) and \(\omega_0\): \[ 0.5 < 1 \quad \implies \quad \alpha < \omega_0 \] Because the damping factor is strictly less than the natural frequency of oscillation, the roots of the system are complex numbers with a negative real part, which forces an exponentially decaying sinusoidal output response. Therefore, the circuit is underdamped.