Question

If the negative feedback system is marginally stable, then the location of closed loop poles will be

• A. in the right half of s-plane.
• B. in the left half of s-plane.
• C. on the Imaginary axis.
• D. on the real axis.

Hint:

Poles in LHP $\rightarrow$ Transient decays (Stable).
Poles on Imaginary axis $\rightarrow$ Transient oscillates at constant amplitude (Marginally Stable).
Poles in RHP $\rightarrow$ Transient blows up (Unstable).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question relates system stability to the location of the closed-loop poles in the s-plane.

Step 2: Key Formula or Approach:
The stability of a linear time-invariant system is defined by the location of its closed-loop poles on the complex s-plane ($s = \sigma + j\omega$).

Step 3: Detailed Explanation:

• Stable System: All closed-loop poles lie strictly in the left-half of the s-plane ($\text{LHP}$, i.e., $\text{Re}(s) < 0$). The transient response decays to zero.

• Unstable System: At least one closed-loop pole lies in the right-half of the s-plane ($\text{RHP}$, i.e., $\text{Re}(s) > 0$), or there are multiple repeating poles on the imaginary axis. The transient response grows without bound.

• Marginally Stable System: Non-repeated closed-loop poles lie on the imaginary axis ($\text{Re}(s) = 0$, i.e., $s = \pm j\omega_d$). This results in sustained, constant-amplitude oscillations in the transient response.

Step 4: Final Answer:
For a marginally stable system, the closed-loop poles are on the imaginary axis, which corresponds to Option (C).