Question $4$

$($i$)$ A Forward converter has the following parameters:

$-$ Input voltage, $\backslash ($ V$\_\{\backslash$text $\{$in $\}\}=30\backslash$mathrm$\{$~V$\}\backslash ),$

$-$ Output voltage, $\backslash ($ V$\_\{\backslash$text $\{$out $\}\}=3\mathrm{.}3\backslash$mathrm$\{$~V$\}\backslash ),$

$-$ Output current, $\backslash ($ I$\_\{\backslash$text $\{$out $\}\}=10\backslash$mathrm$\{$~A$\}\backslash ).$

$-$ Switching frequency $\backslash (=500\backslash$mathrm$\{$kHz$\}\backslash ),$

$-$ Transformer turns ratio is $\backslash ($ N $1$: N $2=3$: $1\backslash ),$

$-$ Output $\backslash ($ L $\backslash )$ value $\backslash (=2\mathrm{.}8\backslash$mu H $\backslash ),$

$-$ Output $\backslash ($ C $\backslash )$ is implemented using $4$ parallel $\backslash (2\backslash$mu $\backslash$mathrm$\{$~F$\}\backslash )$ capacitors each with an ESR of $\backslash (50\backslash$mathrm$\{$~m$\}\backslash$Omega $\backslash ).$

Sketch a block diagram of the closed loop converter control, indicating the error amplifier, sampling, PWM generation and converter.

Give the transfer function for the sampling, PWM generation and converter blocks.

Assuming a $5$ V $($peak to peak$)$ triangle wave is used to generate the PWM and the error amplifier uses a reference voltage of $2\mathrm{.}5$ V roughly sketch the Bode plot $($magnitude and phase$)$ for the expected combined open loop response of these converter blocks indicating the main points. You can use the Bode plot template at the end of the paper.

$($ii$)$ What type of error amplifier should be used to ensure stability of this converter if the cross$-$over frequency is to be placed at $100$ kHz $?$ Explain your answer.

Calculate the zero and pole frequencies and gain at cross$-$over frequency for the error amplifier required to ensure stability of the converter with a phase margin of $60$ degrees at a cross$-$over frequency of $100$ kHz $.$

Roughly, sketch the Bode plot $($magnitude and phase$)$ for the error amplifier, indicating the required gain at the cross$-$over frequency, and the zero and the pole frequencies.

$($iii$)$ If instead of the controller designed in part $($ii$)$ a PI controller with

$\backslash ($ k$\_\{$p$\}=0\mathrm{.}0066,$ k$\_\{$i$\}=1384\mathrm{.}7\backslash )$ is used for the converter, estimate by means of a Bode plot sketch or otherwise, the cross$-$over frequency and phase margin for the system with this PI controller.

Comment on the performance of the system with the PI controller compared to the controller designed in part $($ii$).$