Problem $1$:

$1-$kW$,3,380-$V$,\backslash$Delta $-$connection, $2\mathrm{.}5-$A$,0\mathrm{.}83$ as power factor, $2780$ rpm : induction machine has

the following parameters obtained by dc test, free$-$shaft test, blocked rotor test:

r$\_($s$)=8\mathrm{.}125\backslash$Omega $,$r$\_($r$)=4\mathrm{.}85\backslash$Omega $,$l$\_($m$)=0\mathrm{.}489$H$,$l$\_($ls$)=0\mathrm{.}015$H$,$l$\_($lr$)=0\mathrm{.}015$H$,$

J$=0\mathrm{.}0001$kgm$^(2),$F$=0\mathrm{.}001$Fms

The state$-$variable model is obtained when realizing PARK transformation.

$($di$\_($ss$))/($dt$)=(1)/(\backslash$sigma l$\_($s$))$v$\_($su$)-(($r$\_($s$))/(\backslash$sigma l$\_($s$))+($M$^(2)$r$\_($r$))/(\backslash$sigma l$\_($s$)$l$\_($r$)^(2)))$i$\_($sd$)+\backslash$omega $\_($e$)$i$\_($sq$)+($r$\_($r$)$M$)/(\backslash$sigma l$\_($s$)$l$\_($r$)^(2))\backslash$phi $\_($r$)$

$($di$\_($sq$))/($dt$)=(1)/(\backslash$sigma l$\_($x$))$v$\_($sq$)-(($r$\_($s$))/(\backslash$sigma l$\_($x$))+($M$^(2)$r$\_($r$))/(\backslash$sigma l$\_($s$)$l$\_($r$)^(2)))$i$\_($sq$)-\backslash$omega $\_($ds$)$i$\_($su$)-($M$)/(\backslash$sigma l$\_($s$)$l$\_($r$))\backslash$omega $\_($m$)\backslash$phi $\_($r$)$

$($d$\backslash$phi $\_($r$))/($dt$)=($l$\_($m$)$r$\_($r$))/($l$\_($r$))$i$\_($sd$)-($r$\_($s$))/($l$\_($r$))\backslash$phi $\_($r$)$

J$($d$\backslash$Omega $)/($dt$)=($PM$)/($l$\_($r$))(\backslash$phi $\_($r$)$i$\_($xq$))-$F$\backslash$Omega $-$C$\_($c$)$

Where

$\{($:$[$l$\_($s$)=$l$\_($ts$)+$M$]),($l$\_($r$)=$l$\_($tr$)+$M$),($M$=3$l$\_($m$))$:$\},\backslash$sigma $=1-($M$^(2))/($l$\_($s$)$l$\_($r$))$

$\backslash$phi $\_($r$)$v$\_($sd$)^($c$),$v$\_($sq$)^($c$)($i$\_($sd$),$i$\_($sq$))$ to $($v$\_($sd$)^($c$),$v$\_($sq$)^($c$))$ : calculate the static

gain and the constant time of this transfer function.

Determine the transfer function lying the rotor flux $\backslash$phi $\_($r$)$ to i$\_($sd$)$ calculate the static gain and

the constant time this transfer function.

Determine the transfer function lying the rotor flux $\backslash$omega to i$\_($sq$)$ calculate the static gain and

the constant time this transfer function.

Draw the bloc diagram of dq$-$model of induction machine using the previous transfer

function.