Topic: Control Systems
Find the transient response of a pair of complex poles as given below (Poles at $-a \pm j\omega_0$)
- A. An exponentially decaying sine wave starting at the origin
- B. An underdamped step response with a steady state value
- C. A growing sinusoidal wave
- D. A purely sinusoidal wave
Answer: A. An exponentially decaying sine wave starting at the origin
Explanation: A system with a pair of complex conjugate poles at $s = -a \pm j\omega_0$ has a transfer function $T(s) = \frac{1}{(s+a)^2 + \omega_0^2}$. The inverse Laplace transform of this function (impulse response) is $y(t) = e^{-at} \sin(\omega_0 t) u(t)$, which represents an exponentially decaying sine wave that starts at zero.