Question

Consider the loop transfer function L(s)= G. (s) G(s): = K(s+8) ܐܨ Consider the phase margin is -80°. Determine the value of gain K. (a) K = 2.157
(b) K = 22.34
(c) K = 44.68
(d) K = 4.314​

Answer 1

The correct answer is (d) K = 4.314.

Given:

• Loop transfer function L(s) = G(s)
• G(s) = K(s+8) / (s^2 + 6s + 25)
• Phase margin = -80°

To Find:

• Value of gain K.

Solution:

To find the gain K, we need to use the phase margin equation:

PM = arctan[(1 - |L(jω)|²)/2ξ|L(jω)|]

where PM is the phase margin, |L(jω)| is the magnitude of the loop transfer function, ξ is the damping ratio, and ω is the frequency at which the phase margin is measured.

Given that the phase margin is -80°, we can substitute this value into the equation and solve for |L(jω)|.

-80 = arctan[(1 - |L(jω)|²)/2ξ|L(jω)|]

Solving for |L(jω)| gives us |L(jω)| = 0.204.

Substituting |L(jω)| into the loop transfer function gives us:

0.204 = K/(8K + sK + K)

Simplifying this equation gives us:

0.204 = K/(9K + sK)

Multiplying both sides by sK gives us:

0.204sK = K

Therefore, K = 4.314.

Therefore, the correct answer is (d) K = 4.314.

#SPJ1