Question

If the system is stable, the response is smooth and non-oscillatory (damping coefficient, $\xi>1$), the response is referred to as

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

Hint:

Visualize the step response of a second-order system for different damping coefficients. Overdamped response creeps slowly to the final value, critically damped reaches it quickly without oscillation, underdamped oscillates before settling, and undamped oscillates indefinitely.

Answer 1

The Correct Option is A

Step 1: Understand the concept of damping in a second-order system.
The damping coefficient (\(\xi\), zeta) is a dimensionless parameter that describes how oscillations in a system decay after a disturbance. It is a crucial factor in determining the nature of the system's transient response. A second-order system's characteristic equation is often written in terms of the damping coefficient and the natural frequency (\(\omega_n\)): $$s^2 + 2\xi\omega_n s + \omega_n^2 = 0$$ The roots of this equation determine the system's response characteristics. 
Step 2: Analyze the different types of damping based on the value of the damping coefficient (\(\xi\)).
Undamped (\(\xi = 0\)): In this ideal case, there is no energy dissipation, and the system oscillates indefinitely with a constant amplitude at its natural frequency. The roots of the characteristic equation are purely imaginary (\(s = \pm j\omega_n\)). Underdamped (\(0<\xi<1\)): Here, there is some energy dissipation, but not enough to completely eliminate oscillations. The system oscillates with a decaying amplitude. The roots are complex conjugates with negative real parts (\(s = -\xi\omega_n \pm j\omega_n\sqrt{1-\xi^2}\)). Critically damped (\(\xi = 1\)): This is the boundary case where the damping is just sufficient to prevent oscillations. The system returns to equilibrium as quickly as possible without overshooting. The roots are real and equal (\(s = -\omega_n, -\omega_n\)). Overdamped (\(\xi>1\)): In this case, the damping is greater than the critical damping. The system returns to equilibrium slowly without any oscillations. The roots are real and distinct (\(s = -\xi\omega_n \pm \omega_n\sqrt{\xi^2-1}\), both negative). 
Step 3: Match the description in the question with the types of damping.
The question describes a system that is stable implies roots in the left half of the s-plane, and its response is smooth and non oscillatory with a damping coefficient \(\xi>1\).  
Step 4: Evaluate the given options.
(1) Overdamped: This matches the description provided in the question.
(2) Critically damped: This corresponds to \(\xi = 1\), not \(\xi>1\).
(3) Underdamped: This corresponds to \(0<\xi<1\) and exhibits oscillations.
(4) Undamped: This corresponds to \(\xi = 0\) and exhibits sustained oscillations.