The average value of one half-cycle is what fraction of the peak value?

The average value of one half-cycle is what fraction of the peak value?

• A. 0.636 x peak
• B. 0.5 x peak
• C. 1 x peak
• D. 0.318 x peak

Answer: (A)

For a sine wave, the average value over one half-cycle is 2/π times the peak amplitude. If the peak is Vp, the half-cycle runs from 0 to π with the waveform v = Vp sin θ. The average over that interval is (1/π) ∫0^π Vp sin θ dθ = (Vp/π)[-cos θ]0^π = Vp(2/π) ≈ 0.636 Vp. So the average value is about 0.636 times the peak. That 0.636 value is specific to the half-cycle of a sine wave. Other options don’t match this integral result: 0.5 would be half the peak, not the true average over the half-cycle; 1 would be the peak itself; 0.318 would be 1/π times the peak, which isn’t the average over the half-cycle.

For a sine wave, the average value over one half-cycle is 2/π times the peak amplitude. If the peak is Vp, the half-cycle runs from 0 to π with the waveform v = Vp sin θ. The average over that interval is (1/π) ∫0^π Vp sin θ dθ = (Vp/π)[-cos θ]0^π = Vp(2/π) ≈ 0.636 Vp. So the average value is about 0.636 times the peak.

That 0.636 value is specific to the half-cycle of a sine wave. Other options don’t match this integral result: 0.5 would be half the peak, not the true average over the half-cycle; 1 would be the peak itself; 0.318 would be 1/π times the peak, which isn’t the average over the half-cycle.