SSC JE Electrical Engineering 24 Jan 2018 Evening Paper

Efficiency of transformer (η)
$\left(\mathit{\eta }\right)=\frac{{\mathit{V}}_2{\mathit{I}}_2\cos \mathit{\varphi }}{{\mathit{V}}_2{\mathit{I}}_2\cos \mathit{\varphi }+{\mathit{P}}_{\mathit{i}}+{\mathit{I}}_2^2\mathit{R}}$
The value of terminal voltage ${\mathit{V}}_2$ is approximately constant. Thus for a given power factor the transformer efficiency
$\frac{\mathit{d}\left(\mathit{\eta }\right)}{\mathit{d}{\mathit{I}}_2}=\left({\mathit{V}}_2\cos {\mathit{\varphi }}_2+\frac{{\mathit{P}}_{\mathit{i}}}{{\mathit{I}}_2}+{\mathit{I}}_2\mathit{R}\right)=0$
$-\frac{{\mathit{P}}_{\mathit{i}}}{{\mathit{I}}_2}+\mathit{R}=0$
So, for maximum efficiency core loss (${\mathit{P}}_{\mathit{i}}$ ) is equal to copper loss (${\mathit{P}}_{\mathrm{cu}}$ )