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.