A $500-$kVA, $30/10-$kV$,50-$Hz$,$ single$-$phase transformer has the high$-$voltage winding resistance of $18$ and the low$-$voltage winding resistance of $4.$ Use the approximate equivalent circuit referred to the low voltage side to do the calculations. $($a$)$ Determine the total copper loss of the transformer at full load. $[$Ans: $15$ kW$]($b$)$ If the total core loss of the transformer is $10$ kW$,$ estimate the efficiency of the transformer at half load at $0\mathrm{.}85$ lagging power factor. $[$Ans: $93\mathrm{.}92\%]($c$)$ What is the maximum efficiency of the transformer at $0\mathrm{.}85$ lagging power factor? $[$Ans: $94\mathrm{.}55\%]$