Chapter 6, Problem 16PEB

A step-down transformer connected to a 120 V electric generator has 30 loops on the primary for each loop in the secondary. (a) What is the voltage of the secondary? (b) If the transformer has a 90.0 A current in the primary, what is the current in the secondary? (c) What are the power input and output?

To determine

The secondary voltage of a step-down transformer.

Answer to Problem 16PEB

Solution:

The secondary voltage is $\text{4 V}$.

Explanation of Solution

Given data:

A step-downtransformer with 30 loops for every loop at the primary side. The primary coil is supplied with $120\text{ V}$ and $90\text{ A}$.

Formula used:

The expression that relates the number of loops and the voltage is written as,

$\frac{V_p}{N_p}=\frac{V_s}{N_s}$

Here, $V_p$ is the primary voltage, $N_p$ are the number of loops at the primary side, $V_s$ is the secondary voltage and $N_s$ are the number of loops at the secondary side.

Explanation:

Consider the expression.

$\frac{V_p}{N_p}=\frac{V_s}{N_s}$

Rearrange to get the secondary voltage $V_s$.

$V_s=N_s\frac{V_p}{N_p}$

The number of loops at the primary and the secondary sides are dependent on each other.

Consider theloops at secondary side be $x$, then loops at the primary end will be equal to $30x$.

Substitute $x$ for $N_s$, $120\text{ V}$ for $V_p$ and $30x$ for $N_p$.

$\begin{array}{c}{V}_s=N_s\frac{V_p}{N_p}\\ =x\times \frac{120\text{ V}}{30x}\\ =4\text{ V}\end{array}$

Conclusion:

Hence, the secondary voltage is $4\text{ V}$.

To determine

The secondary current in the step-down transformer.

Answer to Problem 16PEB

Solution:

The secondary current is $2700\text{ A}$.

Explanation of Solution

Given data:

A step-down transformer with 30 loops for every loop at the primary side. The primary coil is supplied with $120\text{ V}$ and $90\text{ A}$.

Formula used:

The expression for power in a transformer is written as,

$V_p{I}_p=V_s{I}_s$

Here, $I_p\text{ and }{I}_s$ are current at primary and secondary side, respectively.

Explanation:

Consider the expression for power in the transformer.

$V_p{I}_p=V_s{I}_s$

Rearrange the power expression to get the secondary current $I_s$.

$I_s=\frac{V_p{I}_p}{V_s}$

Substitute $120\text{ V}$ for $V_p$, $90\text{ A}$ for $I_p$ and $4\text{ V}$ for $V_s$.

$\begin{array}{c}{I}_s=\frac{V_p{I}_p}{V_s}\\ =\frac{120\times 90}{4}\\ =2700\text{ A}\end{array}$

Conclusion:

Hence, the current at the secondary side is $2700\text{ A}$.

To determine

The power of the transformer at the input and the output sides.

Answer to Problem 16PEB

Solution:

The power at the input and the output side is same and is equal to $1.08\times {10}^4\text{ W}$.

Explanation of Solution

Given data:

A step-down transformer with 30 loops for every loop at the primary side. The primary coil is supplied with $120\text{ V}$ and $90\text{ A}$.

Formula used:

The expression for power at the input side is written as,

$P{i}_{}=V_p{I}_p$

Here, $P_i$ is the input power, $V_p$ is the primary voltage and $I_p$ is the primary current.

The expression for power at the output side is written as,

$P{o}_{}=V_s{I}_s$

Here, $P_o$ is the output power, $V_s$ is the secondary voltage and $I_s$ is the secondary current.

Explanation:

Consider the expression for the input power.

$P{i}_{}=V_p{I}_p$

Substitute $120\text{ V}$ for $V_p$ and $90\text{ A}$ for $I_p$.

$\begin{array}{c}{P}_{in}=V_p{I}_p\\ =120\text{ V}\times 90\text{ A}\\ =10800\text{ W}\end{array}$

Similarly, consider the expression for the output power.

$P{o}_{}=V_s{I}_s$

Substitute $4\text{ V}$ for $V_s$ and $2700\text{ A}$ for $I_s$.

$\begin{array}{c}{P}_{out}=V_s{I}_s\\ =4\text{ V}\times 2700\text{ A}\\ =10800\text{ W}\end{array}$

Conclusion:

Hence, the power at the input and the output side is same and is equal to $10800\text{ W}$.