Question

The response of a control system having damping factor unity will be :

• A. Oscillatory
• B. Under damped
• C. Critically damped
• D. Cannot be predicted

Hint:

For second-order systems: \[ \zeta=1 \] implies a critically damped response, which is the fastest response without oscillations.

Answer 1

The Correct Option is C

Concept: The damping ratio (or damping factor) of a second-order control system determines the nature of the system response. For a standard second-order system: \[ \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \] where:
• \(\omega_n\) is natural frequency,
• \(\zeta\) is damping ratio. Different values of \(\zeta\) produce different responses.

Step 1: Understanding different damping conditions.
• If \[ \zeta=0 \] system is undamped and oscillatory.
• If \[ 0<\zeta<1 \] system is underdamped.
• If \[ \zeta=1 \] system is critically damped.
• If \[ \zeta>1 \] system is overdamped.

Step 2: Using the given damping factor. The question states: \[ \zeta=1 \] Hence the system corresponds to the critically damped condition.

Step 3: Understanding critically damped response. A critically damped system:
• reaches steady state quickly,
• does not oscillate,
• avoids overshoot,
• provides fastest response without oscillation.

Step 4: Checking each option carefully.
• Option \(1\): Oscillatory \(\rightarrow\) Incorrect
• Option \(2\): Under damped \(\rightarrow\) Incorrect
• Option \(3\): Critically damped \(\rightarrow\) Correct
• Option \(4\): Cannot be predicted \(\rightarrow\) Incorrect Thus the correct answer is: \[ \boxed{(3)\; \text{Critically damped}} \]