Chapter 23, Problem 6TP

To determine

The output power and the input current of an ideal step-up transformer.

Answer to Problem 6TP

  $17.14\text{A}$

  $120\text{W}$

Explanation of Solution

Given information:

An ideal step-up transformer with turn ratio $\text{1:30}$ is supplied with an input power of $\text{120}$ W.

The output voltage is $\text{210}$ V.

Formula used:

In transformer, the voltages across the primary and secondary coils are given by,

  $\frac{{\text{V}}_{\text{S}}}{{\text{V}}_{\text{P}}}\text{=}\frac{{\text{N}}_{\text{S}}}{{\text{N}}_{\text{P}}}$

Where ${\text{V}}_{\text{P}}$ and ${\text{V}}_{\text{S}}$ are the voltages across primary and secondary coils having ${\text{N}}_{\text{P}}$ and ${\text{N}}_{\text{S}}$ turns.

And the currents ${\text{I}}_{\text{P}}$ and ${\text{I}}_{\text{S}}$ in the primary and secondary coils are given by,

  $\frac{{\text{I}}_{\text{S}}}{{\text{I}}_{\text{P}}}\text{=}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S}}}$

Power is calculated as,

  $\text{P=VI}$

Where $\text{I}$ is current and $\text{V}$ is voltage.

Given:

  $\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S}}}=\frac{1}{30}$

  ${\text{P}}_{\text{P}}\text{=120W}$

  ${\text{V}}_{\text{s}}\text{=210}$ V

It is known that in transformer, the voltages across the primary and secondary coils are given by,

  $\frac{{\text{V}}_{\text{S}}}{{\text{V}}_{\text{P}}}\text{=}\frac{{\text{N}}_{\text{S}}}{{\text{N}}_{\text{P}}}\mathrm{...}\text{eq}\left(1\right)$

Where ${\text{V}}_{\text{P}}$ and ${\text{V}}_{\text{S}}$ are the voltages across primary and secondary coils having ${\text{N}}_{\text{P}}$ and ${\text{N}}_{\text{S}}$ turns.

And the currents ${\text{I}}_{\text{P}}$ and ${\text{I}}_{\text{S}}$ in the primary and secondary coils are given by,

  $\frac{{\text{I}}_{\text{S}}}{{\text{I}}_{\text{P}}}\text{=}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S}}}\mathrm{...}\text{eq}\left(2\right)$

Input power is given as,

  ${\text{P}}_{\text{p}}{\text{=V}}_{\text{p}}{\text{I}}_{\text{p}}$

Substitute all the values in ${\text{P}}_{\text{p}}{\text{=V}}_{\text{p}}{\text{I}}_{\text{p}}$.

  $\begin{array}{l}\text{From eq}\left(1\right){\text{V}}_{\text{p}}=\frac{{\text{V}}_{\text{S}}{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S}}}\\ {\text{P}}_{\text{p}}\text{=}\frac{{\text{V}}_{\text{S}}{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S}}}{\text{I}}_{\text{p}}\\ \Rightarrow 120\text{=}\frac{\left(210\right)\left(1\right)}{30}{\text{I}}_{\text{p}}\\ \Rightarrow \frac{120\left(30\right)}{210}{\text{=I}}_{\text{p}}\\ \Rightarrow \frac{120}{7}{\text{=I}}_{\text{p}}\\ \Rightarrow {\text{I}}_{\text{p}}=\frac{120}{7}\text{=}17.14\text{A}\end{array}$

Output power is given as,

  ${\text{P}}_{\text{s}}{\text{=V}}_{\text{s}}{\text{I}}_{\text{s}}$

Substitute all the values in ${\text{P}}_{\text{s}}{\text{=V}}_{\text{s}}{\text{I}}_{\text{s}}$.

  $\begin{array}{l}\text{From eq}\left(2\right){\text{I}}_{\text{S}}\text{=}\frac{{\text{N}}_{\text{P}}{\text{I}}_{\text{P}}}{{\text{N}}_{\text{S}}}\\ \text{So }\\ {\text{P}}_{\text{s}}\text{=}\frac{{\text{N}}_{\text{P}}{\text{I}}_{\text{P}}}{{\text{N}}_{\text{S}}}{\text{ V}}_{\text{s}}\\ \Rightarrow {\text{P}}_{\text{s}}\text{=}\frac{\left(120\right)\left(1\right)}{\left(\text{30}\right)\left(7\right)}\text{ 210}\\ \Rightarrow {\text{P}}_{\text{s}}\text{=120}\\ \Rightarrow {\text{P}}_{\text{s}}=120\text{W}\end{array}$

Conclusion:

Thus, the output power of an ideal step-up transformer is $17.14\text{A}$ voltage and the input current of an ideal step-up transformer is $120\text{W}\text{.}$