Question

For an ideal 2-winding transformer, an impedance${\mathbf{Z}}_{\mathbf{2}}$connected acrosswinding 2 (secondary) is referred to winding 1 (primary) by multiplying by

(a) The turns ratio$\left(\frac{N_1}{N_2}\right)$
(b) The square of the turn’s ratiorole="math" localid="1655113801735" ${\mathbf{\left(}\frac{{\mathbf{N}}_{\mathbf{1}}}{{\mathbf{N}}_{\mathbf{2}}}\mathbf{\right)}}^{\mathbf{2}}$
(c) The cubed turn’s ratio${\mathbf{\left(}\frac{{\mathbf{N}}_{\mathbf{1}}}{{\mathbf{N}}_{\mathbf{2}}}\mathbf{\right)}}^{\mathbf{3}}$

Short answer

Therefore, the correct option is (b): The square of the turn’s ratio${\left(\frac{N_1}{N_2}\right)}^2$

Step-by-step solution

Use the basic relationship for ideal transformer.

The basic relationship for the ideal transformer

$$\begin{gathered} \frac{{\mathbf{E}}_{\mathbf{1}}}{{\mathbf{E}}_{\mathbf{2}}}\mathbf{=}\frac{{\mathbf{N}}_{\mathbf{1}}}{{\mathbf{N}}_{\mathbf{2}}} \\ {\mathbf{E}}_{\mathbf{1}}\mathbf{=}\mathbf{\left(}\frac{{\mathbf{N}}_{\mathbf{1}}}{{\mathbf{N}}_{\mathbf{2}}}\mathbf{\right)}{\mathbf{E}}_{\mathbf{2}} \\ {\mathbf{E}}_{\mathbf{1}}\mathbf{=}{\mathbf{aE}}_{\mathbf{2}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}.....\left(1\right) \end{gathered}$$

Similarly, the relationship of turns and the current.

$$\begin{gathered} \frac{I_2}{I_1}=\left(\frac{N_1}{N_2}\right) \\ {I}_1=\left(\frac{N_1}{N_2}\right)I_2 \\ {I}_1=\frac{I_2}{a}......\left(2\right) \end{gathered}$$

When impedance Z2 is connected across winding 2.

The impedance $Z_2$can be calculated as

$Z_2=\frac{E_2}{I_2}$

Consider the impedance measured from the winding 1.

$Z_1=\frac{E_1}{I_1}$

Substitute the values from equations (1) and (2)

$$\begin{gathered} Z_1=\frac{a{E}_2}{{\displaystyle \frac{I_2}{a}}} \\ {Z}_1=a^2\frac{E_2}{I_2} \\ {Z}_1=a^2{Z}_2 \end{gathered}$$

Hence impedance $Z_2$is referred to winding 1 by$Z_2$ multiplied by$a^2$ that is square of the turn ratio.