150 kVA transformer: max efficiency load

Given: 150 kVA, 11000/230 V, core loss $P_i = 1.4$ kW, full-load copper loss $P_{cu,fl} = 1.6$ kW.

(i) Load for maximum efficiency

At maximum efficiency, copper loss equals core loss:

$$x^2P_{cu,fl} = P_i \Rightarrow x = \sqrt{\frac{1.4}{1.6}} = 0.9354$$ $$\text{kVA at } \eta_{max} = 0.9354\times150 = 140.3\ \text{kVA}$$

The power factor is not given for this part; taking the same 0.8 lagging as in part (ii):

$$\begin{aligned} P_{out} &= 140.3\times0.8 = 112.25\ \text{kW} \\ \text{Losses} &= 2P_i = 2.8\ \text{kW} \\ \eta_{max} &= \frac{112.25}{112.25 + 2.8}\times100 = 97.57\% \end{aligned}$$

(At unity power factor the same load would give $\eta_{max} = \dfrac{140.31}{140.31 + 2.8}\times100 = 98.04\%$.)

(ii) Efficiency at 0.8 pf lagging

Full load ($x = 1$):

$$\eta = \frac{150\times0.8}{150\times0.8 + 1.4 + 1.6}\times100 = \frac{120}{123}\times100 = 97.56\%$$

Half load ($x = 0.5$): output $= 0.5\times150\times0.8 = 60$ kW, copper loss $= 0.25\times1.6 = 0.4$ kW.

$$\eta = \frac{60}{60 + 1.4 + 0.4}\times100 = \frac{60}{61.8}\times100 = 97.09\%$$

Answer: (i) 140.3 kVA, $\eta_{max} = 97.57\%$ (at 0.8 pf); (ii) full load 97.56%, half load 97.09%