Average value of alternating current for one positive half cycle is ………. Of the peak value .

Answer

To find the average value of alternating current for one positive half cycle in terms of the peak value, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Alternating Current (AC) Function**:
The alternating current can be represented mathematically as:
\[ I(t) = I_0 \sin(\omega t + \phi) \]
where \( I_0 \) is the peak current, \( \omega \) is the angular frequency, and \( \phi \) is the phase angle.

2. **Identifying the Positive Half Cycle**:
The positive half cycle of the sine wave occurs from \( t = 0 \) to \( t = \frac{T}{2} \), where \( T \) is the time period of the AC signal.

3. **Calculating the Average Value**:
The average value of the current over one positive half cycle is given by the formula:
\[ I_{\text{avg}} = \frac{1}{T/2} \int_0^{T/2} I(t) \, dt \]
Substituting \( I(t) \):
\[ I_{\text{avg}} = \frac{2}{T} \int_0^{T/2} I_0 \sin(\omega t + \phi) \, dt \]

4. **Evaluating the Integral**:
The integral can be evaluated as follows:
\[ I_{\text{avg}} = \frac{2I_0}{T} \int_0^{T/2} \sin(\omega t + \phi) \, dt \]
The integral of \( \sin \) function is:
\[ \int \sin(x) \, dx = -\cos(x) \]
Therefore,
\[ \int_0^{T/2} \sin(\omega t + \phi) \, dt = \left[-\frac{1}{\omega} \cos(\omega t + \phi)\right]_0^{T/2} \]

5. **Substituting the Limits**:
Evaluating the limits gives:
\[ = -\frac{1}{\omega} \left[ \cos\left(\frac{\omega T}{2} + \phi\right) - \cos(\phi) \right] \]
Since \( \omega T = 2\pi \), we have:
\[ = -\frac{1}{\omega} \left[ \cos(\pi + \phi) - \cos(\phi) \right] \]
Using the property \( \cos(\pi + x) = -\cos(x) \):
\[ = -\frac{1}{\omega} \left[ -\cos(\phi) - \cos(\phi) \right] = \frac{2\cos(\phi)}{\omega} \]

6. **Final Calculation of Average Current**:
Substituting back into the average current formula:
\[ I_{\text{avg}} = \frac{2I_0}{T} \cdot \frac{2\cos(\phi)}{\omega} \]
Since \( \frac{T}{\omega} = 2\pi \):
\[ I_{\text{avg}} = \frac{2I_0}{2\pi} \cdot 2\cos(\phi) = \frac{2I_0 \cos(\phi)}{\pi} \]
For a standard sine wave where \( \phi = 0 \):
\[ I_{\text{avg}} = \frac{2I_0}{\pi} \]

7. **Expressing in Terms of Peak Value**:
Therefore, the average value of the alternating current for one positive half cycle is:
\[ I_{\text{avg}} = \frac{2}{\pi} I_0 \approx 0.637 I_0 \]

### Conclusion:
The average value of alternating current for one positive half cycle is approximately \( 0.637 \) times the peak value \( I_0 \).