In a half wave rectifier, the r.m.s value of the A.C. component of the wave is

Options

A. equal to D.C. value
B. more than D.C. value.
C. less than D.C. value.
D. zero

Answer

To determine the r.m.s (root mean square) value of the A.C. component of the wave in a half wave rectifier, we can follow these steps:

### Step 1: Understand the Half Wave Rectifier
A half wave rectifier allows only one half (positive or negative) of the AC waveform to pass through, effectively blocking the other half. This results in a waveform that consists of the positive half cycles of the AC input.

### Step 2: Identify the DC and AC Components
In a half wave rectifier, the output consists of a DC component (the average value of the rectified output) and an AC component (the ripple). The DC component can be calculated as:
\[ V_{dc} = \frac{V_m}{\pi} \]
where \( V_m \) is the peak voltage of the AC input.

### Step 3: Calculate the RMS Value of the Output
The RMS value of the output voltage for a half wave rectifier can be calculated using the formula:
\[ V_{rms} = \frac{V_m}{2} \]
This is the RMS value of the rectified output.

### Step 4: Determine the RMS Value of the AC Component
The AC component can be considered as the ripple voltage superimposed on the DC value. The RMS value of the AC component can be calculated using the following relationship:
\[ V_{ac\_rms} = \sqrt{V_{rms}^2 - V_{dc}^2} \]

### Step 5: Substitute the Values
Substituting the values we have:
1. \( V_{rms} = \frac{V_m}{2} \)
2. \( V_{dc} = \frac{V_m}{\pi} \)

Now, substituting these into the equation for the AC component:
\[ V_{ac\_rms} = \sqrt{\left(\frac{V_m}{2}\right)^2 - \left(\frac{V_m}{\pi}\right)^2} \]

### Step 6: Simplify the Expression
Simplifying the expression gives:
\[ V_{ac\_rms} = \sqrt{\frac{V_m^2}{4} - \frac{V_m^2}{\pi^2}} \]
\[ V_{ac\_rms} = V_m \sqrt{\frac{1}{4} - \frac{1}{\pi^2}} \]

### Step 7: Conclusion
From the calculations, we can conclude that the RMS value of the AC component of the wave in a half wave rectifier is less than the DC value. Therefore, the correct answer is that the r.m.s value of the A.C. component of the wave is less than the DC value.

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