A.C. voltage, V= V0 sin ωt As,t = πω = 12.2πω = 12T, therefore, first half cycle (T/2). Hence, average value of AC voltage, Eav = 2V0π.

Average Value of AC Voltage V = V0 Sin ωt

To find the average value of an alternating current (AC) voltage, we need to consider a complete cycle of the waveform. In the case of V = V0 Sin ωt, where V0 is the maximum voltage, ω is the angular frequency, and t is the time, the waveform is a sine function. The average value can be determined by integrating the waveform over one complete cycle and dividing it by the period.

Integration of Sin ωt:
The integral of Sin ωt can be found using basic trigonometric identities. The integral of Sin ωt is equal to -Cos ωt divided by ω.

Average Value Calculation:
To find the average value of V = V0 Sin ωt, we need to integrate the waveform over one complete cycle and divide it by the period.

1. Determine the period (T) of the waveform:
The period is the time taken for one complete cycle. For the waveform V = V0 Sin ωt, the period is T = 2π/ω.

2. Integrate the waveform over one complete cycle:
Integrating Sin ωt over one complete cycle (0 to 2π) gives us:
∫(0 to 2π) V0 Sin ωt dt = -V0/ω [Cos ωt] (0 to 2π)

3. Substitute the limits of integration:
Plugging in the limits of integration, we get:
-V0/ω [Cos ω(2π) - Cos ω(0)]

4. Simplify:
Cos ω(2π) = Cos 2π = 1 (Cosine function repeats after every 2π)
Cos ω(0) = Cos 0 = 1

Simplifying further, we have:
-V0/ω [1 - 1] = 0

5. Divide by the period:
Dividing the integral result by the period (T = 2π/ω) gives us the average value of V.
Average Value = 0 / (2π/ω) = 0

Explanation:
The average value of the AC voltage V = V0 Sin ωt is zero. This means that over a complete cycle of the sine waveform, the positive and negative values cancel each other out, resulting in an average value of zero. This is because the waveform is symmetrical, spending an equal amount of time in the positive and negative regions.

It is important to note that the average value represents the DC component of the AC waveform. In the case of V = V0 Sin ωt, there is no DC component present, resulting in an average value of zero. However, the instantaneous voltage (V0 Sin ωt) still oscillates between positive and negative values.

Summary:
The average value of the AC voltage V = V0 Sin ωt is zero. This is because over a complete cycle, the positive and negative values of the sine waveform cancel each other out. It is important to distinguish between the average value (DC component) and the instantaneous voltage (AC component) in AC waveforms.