Question

An ideal transformer has \(1: 2.5\) turns ratio. The peak value of the ac is \(28 \mathrm{~V}\). The rms secondary voltage is nearest to \(\ldots \ldots\) (a) \(50 \mathrm{~V}\) (b) \(70 \mathrm{~V}\) (c) \(100 \mathrm{~V}\) (d) \(40 \mathrm{~V}\)

Short answer

The rms secondary voltage can be calculated using the turns ratio and the peak primary voltage. First, find the peak secondary voltage (V2) using the turns ratio: V2 = V1 * (N2/N1) = 28 V * (2.5) = 70 V. Then, calculate the rms voltage from the peak voltage: V_rms = V_peak / sqrt(2) = 70 V / sqrt(2) ≈ 49.5 V. The closest answer choice to the calculated rms secondary voltage is (a) 50 V.

Step-by-step solution

Understanding the turns ratio and its relationship to voltages

The turns ratio is the ratio of the number of turns in the primary coil to the number of turns in the secondary coil (N1:N2). In this problem, the turns ratio is 1:2.5. The voltage ratio across an ideal transformer is directly proportional to the turns ratio; as such, we can write the equation V1/V2 = N1/N2, where V1 is the primary voltage and V2 is the secondary voltage.

Find the peak secondary voltage

Given the turns ratio (N1/N2 = 1/2.5) and the peak primary voltage (V1 = 28 V), we can solve for the peak secondary voltage (V2) using the equation V1/V2 = N1/N2. Rearranging the equation, we get V2 = V1 * (N2/N1). V2 = 28 V * (2.5) V2 = 70 V So, the peak secondary voltage is 70 V.

Calculate the rms voltage from the peak voltage

The rms (root mean square) voltage can be found by dividing the peak voltage by the square root of 2. V_rms = V_peak / sqrt(2) In our case, the peak secondary voltage V2 (70 V) is used to find the rms secondary voltage: V_rms = 70 V / sqrt(2) V_rms ≈ 49.5 V

Determine the closest answer choice

Now we compare our calculated rms secondary voltage (49.5 V) to the given answer choices: (a) 50 V (b) 70 V (c) 100 V (d) 40 V The closest answer to our calculated rms secondary voltage is (a) 50 V.

Key concepts

Turns Ratio

The turns ratio in a transformer is an essential concept that relates to how transformers step up or step down voltages. It is expressed as the ratio of the number of turns in the primary coil (N1) to the number of turns in the secondary coil (N2).
For example, a turns ratio of 1:2.5, as provided in our exercise, indicates that for every single turn in the primary coil, there are 2.5 turns in the secondary coil.
This relationship is crucial because it determines how the voltages are transformed across the transformer. The voltage ratio between primary and secondary is directly tied to this turns ratio.

Primary and Secondary Voltages

Primary and secondary voltages in an ideal transformer refer to the electrical potential difference at the input and output of the transformer, respectively. They are related through the turns ratio according to the equation:

• \( \frac{V_1}{V_2} = \frac{N_1}{N_2} \)

Here, \( V_1 \) and \( V_2 \) are the primary and secondary voltages.
In our example, given a primary voltage (peak value) of 28 V and a turns ratio of 1:2.5, the secondary voltage can be calculated by rearranging the formula to \( V_2 = V_1 \times \frac{N_2}{N_1} \).
This means the secondary voltage is reached by multiplying the primary voltage by the turns ratio, yielding a peak secondary voltage.

Peak and rms Voltage

Understanding peak and rms (root mean square) voltages is crucial in AC voltage discussions.
The peak voltage is the maximum voltage reached by the waveform, whereas the rms voltage is a measure of the equivalent DC value that delivers the same power.
To convert from peak to rms voltage, use:

• \( V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \)

In the current problem, the secondary peak voltage is found to be 70 V. Applying our formula, the rms voltage becomes approximately 49.5 V, rounding close to the nearest answer choice of 50 V.

Voltage Transformation Equations

Voltage transformation equations are pivotal in defining how voltage is altered in transformers.
From the ideal transformer perspective, the ratio of the primary and secondary voltages is equivalent to the turns ratio:

• \( \frac{V_1}{V_2} = \frac{N_1}{N_2} \)

Furthermore, the process of converting peak to rms voltage is achieved using the transformation:

• \( V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \)

Together, these equations allow us to predict how an input voltage will change in magnitude and form at the transformer's secondary side, essential for both theoretical understanding and practical applications in electrical engineering.