What is the relation between average current and rms current over half cycle.

Answer

To find the relation between average current (I_avg) and root mean square current (I_rms) over a half cycle, we can follow these steps:

### Step 1: Define the Peak Current
Let \( I_0 \) be the peak current. This is the maximum value of the current during the half cycle.

### Step 2: Calculate RMS Current
The RMS current for a half cycle is given by the formula:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]
This formula arises because RMS is defined as the square root of the average of the squares of the current over one complete cycle.

### Step 3: Calculate Average Current
The average current over a half cycle can be calculated using the formula:
\[ I_{avg} = \frac{2 I_0}{\pi} \]
This formula is derived from integrating the current over the half cycle and normalizing it over the time period.

### Step 4: Relate Average Current to RMS Current
Now, we can express the average current in terms of the RMS current. We know:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \quad \Rightarrow \quad I_0 = I_{rms} \sqrt{2} \]
Substituting this into the average current formula:
\[ I_{avg} = \frac{2 (I_{rms} \sqrt{2})}{\pi} = \frac{2 \sqrt{2}}{\pi} I_{rms} \]

### Conclusion
Thus, the relationship between average current and RMS current over a half cycle is:
\[ I_{avg} = \frac{2 \sqrt{2}}{\pi} I_{rms} \]

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