Question

Consider an ideal single-phase winding transformer of turns ratio ${\mathbf{N}}_{\mathbf{1}}\mathbf{/}{\mathbf{N}}_{\mathbf{2}}\mathbf{=}\mathbf{a}$. If it is converted to an autotransformer arrangement with a transformation ratio of ${\mathbf{V}}_{\mathbf{H}}\mathbf{/}{\mathbf{V}}_{\mathbf{X}}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{a}$, (the autotransformer rating/two-winding transformer rating) would then be

(a)$\mathbf{1}\mathbf{+}\mathbf{a}$

(b)$\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{\mathbf{a}}$

(c)$\mathbf{a}$

Short answer

Therefore the correct option is (b): $1+\frac{1}{a}$

Step-by-step solution

Two winding transformer transformation ratio.

The KVA relation between two-winding transformer and auto transformer is given by,

${\left(\mathbf{KVA}\right)}_{\mathbf{Auto}\mathbf{transformer}}\mathbf{=}{\left(\mathbf{KVA}\right)}_{\mathbf{2}\mathbf{-}\mathbf{winding}\mathbf{transformer}}\left(\frac{1}{1-K}\right)$

Rearrange the above expression.

$\frac{{\left(\mathrm{KVA}\right)}_{\mathrm{Auto}\mathrm{transformer}}}{{\left(\mathrm{KVA}\right)}_{2-\mathrm{winding}\mathrm{transformer}}}=\left(\frac{1}{1-\mathrm{K}}\right)$ …… (1)

Here, the value of k is,

$\mathbf{K}\mathbf{=}\frac{\mathbf{Low}\mathbf{voltage}\mathbf{side}\left({\mathbf{V}}_X\right)}{\mathbf{High}\mathbf{voltage}\mathbf{side}\left({\mathbf{V}}_H\right)}$ …… (2)

Consider the expression for the transformation ratio.

$\frac{{\mathrm{V}}_{\mathrm{H}}}{{\mathrm{V}}_{\mathrm{X}}}=1+\mathrm{a}$ ……. (3)

Using the equation (2) and (3), deduce the following expression.

$\mathbf{K}\mathbf{=}\frac{1}{1+a}$

Substitute$\frac{1}{1+a}$ for$K$ in equation (1).

role="math" localid="1655981535739" ….. (1)

Explanation for option (a)

From equation (1) as the autotransformer rating to two-winding transformer is$1+\frac{1}{a}$ and not $1+a$, the option (a) is incorrect.

Thus the (a) is incorrect option.

Explanation for option (b)

Rearrange the expression in equation (1).

${\left(\mathrm{KVA}\right)}_{\mathrm{Auto}\mathrm{transformer}}=\left(\frac{1}{1+\mathrm{a}}\right){\left(\mathrm{KVA}\right)}_{2-\mathrm{winding}\mathrm{transformer}}$.

Hence, the KVA rating of autotransformer is $\left(\frac{1}{1+a}\right)$times the KVA rating of two-winding transformer.

Thus the correct option is (b).

Explanation for option (c)

From equation (1) as the autotransformer rating to two-winding transformer is$1+\frac{1}{a}$ and not $a$, the option (a) is incorrect.