Match the following lists : Correct codes are :List - I List - II a. Negative real and simple roots i. Sustained oscillatory b. Negative real and equal roots ii. Overdamped c. Complex conjugate roots iii. Critically damped d. Imaginary conjugate roots iv. Underdamped
a-ii, b-iii, c-iv, d-i
The question requires matching terms related to roots of a characteristic equation in control systems to their corresponding system behaviors.
- Negative real and simple roots: When a system has distinct negative real roots, it is referred to as an overdamped system. This means the system returns to equilibrium without oscillating, and the response is slower. Therefore,
a-ii. - Negative real and equal roots: This condition results in a critically damped system. The system returns to equilibrium as quickly as possible without oscillating. Thus,
b-iii. - Complex conjugate roots: These roots indicate an underdamped system. The system exhibits oscillations that gradually decay over time. Hence,
c-iv. - Imaginary conjugate roots: These roots suggest a sustained oscillatory system with no damping (purely oscillatory behavior). So,
d-i.
Thus, the correct matching of List-I to List-II is:
- a-ii: Negative real and simple roots - Overdamped
- b-iii: Negative real and equal roots - Critically damped
- c-iv: Complex conjugate roots - Underdamped
- d-i: Imaginary conjugate roots - Sustained oscillatory
Hence, the correct answer is: a-ii, b-iii, c-iv, d-i.
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