Topic: Control Systems
Consider the system with transfer function $H(s)=(s+1)/(s^{2}+2s+2)$. Which of the following best describes the nature of its poles?
- A. Real and equal
- B. Real and distinct
- C. Complex conjugates with negative real parts
- D. Purely imaginary
Answer: C. Complex conjugates with negative real parts
Explanation: The poles of a transfer function are found by setting the denominator polynomial to zero: $s^2 + 2s + 2 = 0$.
Using the quadratic formula to solve for $s$: $s = \frac{-2 \pm \sqrt{2^2 - 4(1)(2)}}{2} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = \frac{-2 \pm j2}{2} = -1 \pm j1$.
These roots are a complex conjugate pair with a negative real part (-1), meaning the system is underdamped and stable.