Question

The following plot shows

• A. Critical damped system
• B. Under damped system
• C. Over damped system
• D. Undamped system

Hint:

Pole locations for second-order systems: \[ \zeta=0 \Rightarrow s=\pm j\omega_n \] \[ 0<\zeta<1 \Rightarrow \text{Underdamped} \] \[ \zeta=1 \Rightarrow \text{Critically damped} \] \[ \zeta>1 \Rightarrow \text{Overdamped} \] Purely imaginary poles always indicate an undamped oscillatory system.

Answer 1

The Correct Option is D

Concept:& nbsp;
For a standard second-order system, \[ s^2+2\zeta\omega_n s+\omega_n^2=0, \] the location of poles in the \(s\)-plane determines the nature of damping. The damping ratio \(\zeta\) classifies the system as underdamped, critically damped, overdamped, or undamped.& nbsp;

Step 1: Recall the pole locations.
The roots of the characteristic equation are \[ s=-\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}. \] The nature of the system depends on \(\zeta\): \[ \begin{array}{|c|c|} \hline \zeta & \text{Nature of System} \\ \hline 0 & \text{Undamped} \\ 0<\zeta<1 & \text{Underdamped} \\ 1 & \text{Critically Damped} \\ \zeta>1 & \text{Overdamped} \\ \hline \end{array} \]& nbsp;

Step 2: Interpret the given pole location.
The figure shows a circle of radius \[ \omega_n \] centered at the origin of the \(s\)-plane. The marked poles lie on the imaginary axis at \[ s=\pm j\omega_n. \] Hence, \[ \operatorname{Re}(s)=0. \] Comparing with \[ s=-\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}, \] we obtain \[ -\zeta\omega_n=0. \] Since \(\omega_n \neq 0\), \[ \boxed{\zeta=0}. \]& nbsp;

Step 3: Determine the system type.
For \[ \zeta=0, \] the poles become \[ s=\pm j\omega_n. \] These poles are purely imaginary and produce sustained oscillations of constant amplitude. There is no exponential decay term, which means no damping is present. Therefore, the system is \[ \boxed{\text{Undamped}}. \]& nbsp;

Step 4: Verify using the time response.
For an undamped system, \[ c(t)=A\sin(\omega_n t)+B\cos(\omega_n t). \] The oscillations continue indefinitely without reduction in amplitude, confirming that \[ \boxed{\zeta=0}. \]& nbsp;

Final Answer: \[ \boxed{\zeta=0} \] and therefore \[ \boxed{\text{Undamped System}} \] Hence, the correct option is \[ \boxed{\text{(D)}} \]