Question

Maximum efficiency of a transformer depends on:
(A) The working condition
(B) Whether copper loss = 1/2x iron loss
(C) Whether copper loss = iron loss
(D) Whether copper loss = 2x iron loss

Answer 1

In general efficiency is defined as the ratio of output vs input and greater be the efficiency the more productive is the machine is. The efficiency of the transformer is defined as the ratio of useful output parts to the input power. The input and the output are measured in the same unit.

Complete step by step solution:
$r=\dfrac{output\,\,power}{input\,\,power}=\dfrac{output\,\,power}{output\,\,power\,\,+\,\,loss}$
$r=\dfrac{output\,\,power}{input\,\,power\,\,+\,\,iron\,\,loss\,\,+\,\,copper\,\,loss}$
$r=\dfrac{{{V}_{2}}{{I}_{2}}\cos {{\phi }_{2}}}{{{v}_{2}}{{I}_{2}}\,\,ocs{{\phi }_{2}}+{{P}_{i}}+{{P}_{c}}}$ …(1)
Where
${{V}_{2}}\to$ Secondary terminal voltage.
${{I}_{2}}\to$ Full load secondary current
$\operatorname{Cos}{{\phi }_{2}}\to$ Iron losses = hysteresis loss + eddy current loss
${{P}_{C}}=$ Full load copper
From equation (1)
$M=\dfrac{{{V}_{2}}\,\,{{I}_{2}}\,\,\cos {{\phi }_{2}}}{{{v}_{2}}\,\,{{I}_{2}}\,\,\cos {{\phi }_{2}}+Pi+I_{2}^{2}\,\,\operatorname{Re}s}$
$M=\dfrac{{{V}_{2}}\,\cos {{\phi }_{2}}}{{{v}_{2}}\,\,\cos {{\phi }_{2}}+\dfrac{Pi}{{{I}_{2}}}+{{I}_{2}}\operatorname{Re}s}$ …(2)
The value of terminal voltage ${{V}_{2}}$is approximately constant. Thus, for a given power factor i.e. transformer efficiency depends upon the load current ${{I}_{2}}$. In eq. (2), the numerator is constant and the transformer efficiency will be maximum if the denominator with respect to variable ${{I}_{2}}$ is equated to zero.
$\dfrac{d}{d{{I}_{2}}}\left( {{V}_{2}}\cos {{\phi }_{2}}+\dfrac{Pi}{{{I}_{2}}}{{I}_{2}}\operatorname{Re}s \right)=0$
$0-\dfrac{pi}{I_{2}^{2}}+\operatorname{Re}s=0$ $$$$ …(3)
${{m}_{\max }}=\dfrac{{{V}_{2}}{{I}_{2}}\cos {{\phi }_{2}}}{{{V}_{2}}{{I}_{2}}\cos {{\phi }_{2}}+2Pi} as\,\,\left( P_c=P_i \right)$
Hence, while seeing all the above equations we got the final result that the transformati
Thus, the transformer will give the maximum efficiency when their copper loss is equal to iron loss.

Hence the correct option is (c)

Note: Efficiency of a transformer depends upon many factors: its build quality, the material used, the flux loss etc.
The efficiency of the transformer is defined as the ratio of useful output parts to the input power.