Step 1: Concept Definition:
Peak Inverse Voltage (PIV) is the maximum reverse voltage a diode in a rectifier must withstand without breakdown. For a half-wave rectifier, during the negative half-cycle, the diode is reverse-biased. The maximum voltage experienced is the secondary AC waveform's peak voltage.
Step 2: Methodology:
1. Determine the secondary RMS voltage (\(V_{s,rms}\)).
2. Calculate the secondary peak voltage (\(V_{s,peak}\)).
3. For a half-wave rectifier, PIV equals \(V_{s,peak}\).
Relevant formulas:
\[ \frac{V_s}{V_p} = \frac{N_s}{N_p} \implies V_{s,rms} = V_{p,rms} \times \frac{N_s}{N_p} \]
\[ V_{peak} = V_{rms} \times \sqrt{2} \]
Step 3: Detailed Calculation:
1. Secondary RMS Voltage (\(V_{s,rms}\)):
Given: Primary voltage \(V_{p,rms} = 220\) V. Turns ratio \( \frac{N_p}{N_s} = \frac{20}{1} \).
Calculation:
\[ V_{s,rms} = V_{p,rms} \times \frac{N_s}{N_p} = 220 \, V \times \frac{1}{20} = 11 \, V \]
2. Secondary Peak Voltage (\(V_{s,peak}\)):
Relationship for sinusoidal AC:
\[ V_{s,peak} = V_{s,rms} \times \sqrt{2} \]
Calculation:
\[ V_{s,peak} = 11 \, V \times \sqrt{2} \approx 11 \times 1.414 \]
\[ V_{s,peak} \approx 15.554 \, V \]
3. Peak Inverse Voltage (PIV):
In a half-wave rectifier, the full secondary peak voltage appears across the diode during the negative cycle.
\[ \text{PIV} = V_{s,peak} \approx 15.554 \, V \]
Step 4: Conclusion:
The peak inverse voltage for the diode is approximately 15.554 V, corresponding to option (D).