The loop transfer function of a system is given by G(s)H(s) = $\frac{10e^{-Ls}}{s}$. The phase crossover frequency is 5 rad/s. The value of the dead time L is ______.

Concept:

Phase crossover frequency (ω_pc):

It is the frequency at which the phase angle of the open loop transfer function is equal to -180° or -π rad/sec.

Calculation:

Given, G(s)H(s) = $\frac{10e^{-Ls}}{s}$

∠G(s)H(s) = $-\frac{\pi }{2}-L\omega$

At ω = ω_pc = $5rad/s$

∠G(ω_pc)H(ω_pc) = $-\frac{\pi }{2}-L(5)=-\pi$

L = $\frac{\pi }{10}$

Additional Information Gain crossover frequency (ω_gc) 

It is the frequency at which the magnitude of the open loop transfer function is equal to 1.

For a stable system, the value of the phase crossover frequency is greater than the gain crossover frequency.