Q$4.($a$)$ What are the purposes of doing short$-$circuit and open$-$circuit tests for a transformer?

$(5$ marks$)$

$($b$)$ A $400$kVA,$50Hz,11K\frac{V}{220}V$ single$-$phase transformer has an internal impedance of $5\%$ p$.$u$.$ The open$-$circuit test $($at rated voltage$)$ of the transformer shows it has a power loss of $3$ kW $.$ The short$-$circuit test $($at rated current$)$ of the transformer shows it has a power loss of $6$ kW $.$

When the high voltage side of the transformer is fixed at $11$ kV $,$ estimate:

$($i$)$ The transformer's internal impedance refers to the lower voltage side;

$(5$ marks$)$

$($ii$)$ The transformer voltage regulation at a load power factor of $0\mathrm{.}8$ lagging;

$(10$ marks$)$

$($iii$)$ The transformer efficiency when it is half$-$loaded at a power factor of $0\mathrm{.}8$ lagging.

$(5$ marks$)$