Question

A plant is controlled by a proportional controller. If a time delay element is introduced in the loop then

• A. gain margin remains the same
• B. phase margin remains the same
• C. gain margin decreases
• D. phase margin decreases

Answer 1

The correct option is D phase margin decreases

Let a system is given by
$G\left(s\right)=\frac{1}{(s+1)}$

$\therefore$ $\varphi =-ta{n}^{-1}\omega$

If the system has a dead time of $t_{d}sec,$ then

$G_s^{{}^{\prime }}=\frac{e^{-t_{d}s}}{(s+1)}$

${\varphi }^{\prime }=-{\tan }^{-1}\omega -(57.3\times t_d)$

$PM={180}^o+\varphi$
$andP{M}^{\prime }={180}^o+{\varphi }^{\prime }$
$clearly{\varphi }^{\prime }<\varphi$

$\therefore P{M}^{\prime }<PM$

So, if a time delay element is introduced inthe loop then the phase margin of the system is decreased.