Question

For overdamped systems, the closed loop poles are

• A. real and equal.
• B. real and distinct.
• C. purely imaginary.
• D. complex conjugate.

Hint:

Poles on the real axis $\rightarrow$ non-oscillatory response.
Poles off the real axis $\rightarrow$ oscillatory response.
Overdamped systems do not oscillate because their poles are entirely on the real axis and distinct.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question relates to the transient response characteristics of a second-order control system, specifically focusing on the pole locations for an overdamped system.

Step 2: Key Formula or Approach:
The characteristic equation of a standard second-order system is:
\[ s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \]
The roots (poles) of this equation are:
\[ s_{1,2} = -\zeta\omega_n \pm \omega_n \sqrt{\zeta^2 - 1} \]

Step 3: Detailed Explanation:
We classify the system behavior based on the damping ratio $\zeta$:

• Overdamped ($\zeta > 1$): The term under the square root ($\zeta^2 - 1$) is positive. Thus, the roots $s_{1,2}$ are real, negative, and unequal (distinct).

• Critically Damped ($\zeta = 1$): The term under the square root is zero, so the roots are real and equal ($s_{1,2} = -\omega_n$).

• Underdamped ($0 < \zeta < 1$): The term under the root is negative, leading to complex conjugate roots.

• Undamped ($\zeta = 0$): The roots are purely imaginary ($s_{1,2} = \pm j\omega_n$).

Step 4: Final Answer:
For an overdamped system, the closed-loop poles are real and distinct, which corresponds to Option (B).