Question

The second order system with the transfer function \(1/(s^2 + 2s + 1)\) is a

• A. Underdamped system
• B. Overdamped system
• C. Undamped system
• D. Critically damped system

Hint:

Alternatively, you can find the roots of the denominator polynomial (poles of the transfer function):
\( s^2 + 2s + 1 = (s+1)^2 = 0 \implies s = -1, -1 \).
Since the poles are real, negative, and repeated, the system is critically damped.
Real, distinct poles mean overdamped; complex conjugate poles mean underdamped; purely imaginary poles mean undamped.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question requires us to classify the damping behavior of a second-order system represented by a given transfer function.
A second-order system's dynamic response is characterized by its damping ratio ($\zeta$), which indicates whether the system oscillates or decays without oscillation.

Step 2: Key Formula or Approach:
The standard form of a second-order system transfer function is given by: \[ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \] where:
$\omega_n$ is the natural frequency of oscillation.
$\zeta$ is the damping ratio.
The system characteristics based on the value of $\zeta$ are defined as follows:
1. Undamped system: $\zeta = 0$
2. Underdamped system: $0 \lt \zeta \lt 1$
3. Critically damped system: $\zeta = 1$
4. Overdamped system: $\zeta \gt 1$

Step 3: Detailed Explanation:
We are given the transfer function: \[ G(s) = \frac{1}{s^2 + 2s + 1} \] By comparing the denominator of the given transfer function with the standard characteristic equation: \[ s^2 + 2\zeta\omega_n s + \omega_n^2 = s^2 + 2s + 1 \] Comparing the constant term: \[ \omega_n^2 = 1 \quad \implies \quad \omega_n = 1 \, \text{rad/s} \] Comparing the coefficient of $s$: \[ 2\zeta\omega_n = 2 \] Substituting the value of $\omega_n = 1$: \[ 2\zeta(1) = 2 \quad \implies \quad \zeta = 1 \] Since the damping ratio is exactly $\zeta = 1$, the system is critically damped.
A critically damped system returns to its steady-state value in the fastest possible time without any overshoot or oscillation.

Step 4: Final Answer
Hence, the system is critically damped, which corresponds to option (D).