Question

The open loop transfer function of a unity feedback system is: $G(s) = \frac{1 + s}{s(1 + 0.5s)}$. The corner frequencies are

• A. 0 and 2
• B. 1 and 1
• C. 0 and -1
• D. 1 and 2

Hint:

Always convert terms to the $(1 + s/ \omega_c)$ format. For example, $(1 + 0.5s)$ is the same as $(1 + s/2)$, making it easy to see that 2 is the corner frequency.

Answer 1

The Correct Option is D

Corner frequencies (or break frequencies) in a Bode plot are the frequencies at which the slope of the magnitude curve changes. These frequencies are derived from the time constants ($\tau$) of the zeros and poles. 1. Standard Form: We rewrite the transfer function in the time-constant form $(1 + s\tau)$: $$G(s) = \frac{1 + s}{s(1 + 0.5s)}$$

2. Identifying Time Constants:

• From the numerator zero $(1 + s\tau_z)$: $\tau_z = 1$.

• From the denominator pole $(1 + s\tau_p)$: $\tau_p = 0.5$.

• Note: The $s$ in the denominator represents a pole at the origin, which provides a constant slope but does not have a "corner" frequency in the same sense.

3. Calculating Frequencies ($\omega = 1/\tau$):

• Corner frequency from zero: $\omega_{c1} = 1/\tau_z = 1/1 = \mathbf{1 \text{ rad/s}}$.

• Corner frequency from pole: $\omega_{c2} = 1/\tau_p = 1/0.5 = \mathbf{2 \text{ rad/s}}$.
Therefore, the corner frequencies where the Bode plot changes its slope are 1 and 2.