Question

Give the RMS value, average value, form factor, and peak factor of sinusoidal waveform.

Answer 1

For a sinusoidal waveform, the following values can be calculated:

1. **RMS Value (Root Mean Square)**:
   \[
   V_{rms} = \frac{V_{m}}{\sqrt{2}}
   \]
   where \( V_{m} \) is the peak value of the sinusoidal waveform.

2. **Average Value**:
   \[
   V_{avg} = \frac{2V_{m}}{\pi}
   \]
   for one complete cycle.

3. **Form Factor**:
   \[
   \text{Form Factor} = \frac{V_{rms}}{V_{avg}} = \frac{\frac{V_{m}}{\sqrt{2}}}{\frac{2V_{m}}{\pi}} = \frac{\pi}{2\sqrt{2}} \approx 1.1107
   \]

4. **Peak Factor (or Crest Factor)**:
   \[
   \text{Peak Factor} = \frac{V_{m}}{V_{rms}} = \frac{V_{m}}{\frac{V_{m}}{\sqrt{2}}} = \sqrt{2} \approx 1.4142
   \]

In summary:
- RMS Value: \( \frac{V_{m}}{\sqrt{2}} \)
- Average Value: \( \frac{2V_{m}}{\pi} \)
- Form Factor: \( \frac{\pi}{2\sqrt{2}} \approx 1.1107 \)
- Peak Factor: \( \sqrt{2} \approx 1.4142 \)

Answer 2

For a sinusoidal waveform, the following values can be calculated:

### 1. RMS Value (Root Mean Square)
The RMS value of a sinusoidal waveform is given by:

\[ \text{RMS} = \frac{V_{\text{peak}}}{\sqrt{2}} \]

where \( V_{\text{peak}} \) is the peak value of the waveform.

### 2. Average Value
The average value of a sinusoidal waveform over one complete cycle is zero. This is because the positive and negative halves of the waveform cancel each other out:

\[ \text{Average Value} = 0 \]

However, if you're looking for the average value of the absolute value of the waveform (which is sometimes used), it would be:

\[ \text{Average Value (absolute)} = \frac{2 \cdot V_{\text{peak}}}{\pi} \]

### 3. Form Factor
The form factor is the ratio of the RMS value to the average value (absolute) of the waveform:

\[ \text{Form Factor} = \frac{\text{RMS}}{\text{Average Value (absolute)}} \]

Substituting the values:

\[ \text{Form Factor} = \frac{\frac{V_{\text{peak}}}{\sqrt{2}}}{\frac{2 \cdot V_{\text{peak}}}{\pi}} = \frac{\pi}{2 \sqrt{2}} \approx 1.11 \]

### 4. Peak Factor (or Crest Factor)
The peak factor is the ratio of the peak value to the RMS value:

\[ \text{Peak Factor} = \frac{V_{\text{peak}}}{\text{RMS}} \]

Substituting the values:

\[ \text{Peak Factor} = \frac{V_{\text{peak}}}{\frac{V_{\text{peak}}}{\sqrt{2}}} = \sqrt{2} \approx 1.414 \]

### Summary
- **RMS Value**: \( \frac{V_{\text{peak}}}{\sqrt{2}} \)
- **Average Value**: \( 0 \) (or \( \frac{2 \cdot V_{\text{peak}}}{\pi} \) if considering the absolute value)
- **Form Factor**: \( \frac{\pi}{2 \sqrt{2}} \approx 1.11 \)
- **Peak Factor**: \( \sqrt{2} \approx 1.414 \)