Question

A closed-loop system has a loop transfer function

L(s) = G (s)G(s) = K / s(s +8)(s + 12) =
Determine the gain K so that the phase margin is P.M. = 50°.

Answer 1

To determine the gain K for a phase margin of 50° in a closed-loop system, find the gain crossover frequency, ω, where the phase is -130°. Substitute s = jω into the loop transfer function and solve for K using the magnitude of the loop transfer function at that frequency.

Step-by-step explanation:

The phase margin, denoted as PM, is the difference between the phase of the open-loop system at the gain crossover frequency and -180 degrees. To determine the gain K for a phase margin of 50°, we can use the Bode plot or Nyquist diagram to find the gain crossover frequency. At this frequency, the phase is -180 + PM, which is -130° in this case.

Next, we substitute s = jω into the loop transfer function to find the gain crossover frequency, ω, where the phase is -130°. Then, we can solve for K using the magnitude of the loop transfer function at that frequency.