Full Wave Rectifier Average Output Voltage Calculation

This solution explains how to calculate the average output voltage for a full wave rectifier when the input AC supply voltage is given. We will analyze the relationship between the input AC voltage and the output DC voltage of the rectifier.

Understanding Full Wave Rectifiers

A full wave rectifier is an electronic circuit that converts an alternating current (AC) input into a direct current (DC) output. It utilizes both the positive and negative halves of the AC input waveform. For a sinusoidal AC supply, a full wave rectifier essentially takes the entire waveform, including the negative parts, and makes them positive, resulting in a pulsating DC output.

Input AC Supply Voltage

The problem states that the full wave rectifier is supplied from a 20 V AC supply. In electronics, when an AC voltage value is given without specifying whether it's peak or RMS, it is conventionally assumed to be the RMS (Root Mean Square) voltage. Therefore:

• Input RMS Voltage, $V_{rms} = 20$ V

Calculating Peak Voltage ($V_m$)

The relationship between the RMS voltage ($V_{rms}$) and the peak voltage ($V_m$) for a sinusoidal AC waveform is given by the formula:

$$ V_{rms} = \frac{V_m}{\sqrt{2}} $$

To find the peak voltage ($V_m$), we rearrange the formula:

$$ V_m = V_{rms} \times \sqrt{2} $$

Substituting the given $V_{rms}$ value:

$$ V_m = 20 \text{ V} \times \sqrt{2} $$

$$ V_m \approx 20 \text{ V} \times 1.4142 $$

$$ V_m \approx 28.284 \text{ V} $$

Average Output Voltage ($V_{avg}$) for Full Wave Rectifier

For an ideal full wave rectifier, the average value of the output voltage ($V_{avg}$) is related to the peak voltage ($V_m$) by the following formula:

$$ V_{avg} = \frac{2 V_m}{\pi} $$

This formula arises from integrating the rectified waveform over one full cycle and dividing by the period.

Performing the Calculation

Now, we substitute the calculated peak voltage ($V_m \approx 28.284$ V) into the formula for $V_{avg}$:

$$ V_{avg} = \frac{2 \times (20 \sqrt{2})}{\pi} \text{ V} $$

$$ V_{avg} = \frac{40 \sqrt{2}}{\pi} \text{ V} $$

Let's calculate the numerical value:

$$ V_{avg} \approx \frac{40 \times 1.4142}{3.14159} $$

$$ V_{avg} \approx \frac{56.568}{3.14159} $$

$$ V_{avg} \approx 18.005 \text{ V} $$

Comparing with Options

The calculated average output voltage is approximately $18.005$ V. Let's compare this with the given options:

• Option 1: $12.73$ V
• Option 2: $20.00$ V
• Option 3: $18.01$ V
• Option 4: $9.00$ V

Our calculated value of $18.005$ V is very close to $18.01$ V.

Conclusion

Based on the standard formulas for AC RMS voltage and the average output of a full wave rectifier, the average output voltage for a 20 V RMS AC supply is approximately $18.01$ V.