Question $3$

The open$-$loop frequency response of the equipment used in a temperature control system was measured to produce the results given in Table $2.$

Table $2$

$($a$)$ Plot the data of Table $2$ onto a Nichols Chart and specify the gain margin $($ dB $)$ and phase margin $($degrees$).$

marks$]$

$($b$)$ A proportional control solution is to be applied with a gain factor of $\backslash (\backslash$mathrm$\{$K$\}=8\mathrm{.}9\backslash$mathrm$\{$~dB$\}\backslash ).$ Plot the new values of open$-$loop frequency response on a Nichol's Chart with this value of gain included. Note: start plotting your new points at $\backslash (\backslash$mathrm$\{$f$\}=0\mathrm{.}25\backslash$mathrm$\{$~Hz$\}\backslash ).$

$($c$)$ For the gain factor of part $($b$),$ given that $\backslash (\backslash$mathrm$\{$M$\}\_\{\backslash$mathrm$\{$o$\}\}=0\backslash$mathrm$\{$~dB$\}\backslash ),$ estimate the percentage overshoot and the frequency of dominant oscillations, $\backslash (\backslash$omega$\_\{\backslash$mathrm$\{$d$\}\}\backslash ),$ produced at the output for a unit step input.

marks$]$