Which statement is true about the relationship between average and peak value for a sine wave?

Which statement is true about the relationship between average and peak value for a sine wave?

• A. The average value equals 0.636 times the peak value
• B. The average value equals peak value times 1.57
• C. The average value equals the peak value
• D. The average value equals half the peak value

Answer: (A)

When you look at the average value of a sine wave, you have to be clear what "average" means. For the raw sine wave, the positive and negative halves cancel, so the average over a full cycle is zero. But if you’re talking about the average value of the magnitude (the rectified sine, where negative parts are taken as positive), the average is a fixed fraction of the peak. Compute the average of the absolute value of a sine wave v(t) = Vp sin(ωt): Vavg = (1/T) ∫_0^T |Vp sin(ωt)| dt, where T = 2π/ω. Let φ = ωt. This becomes Vavg = (Vp/2π) ∫_0^{2π} |sin φ| dφ. The integral of |sin φ| over one full cycle is 4, so Vavg = (Vp/2π)·4 = (2/π) Vp ≈ 0.637 Vp. So the true relationship is that the average value is about 0.636 times the peak value for the magnitude of a sine wave (the rectified case). The other options either imply the average is larger than the peak, or equal to the peak, or equal to half the peak, none of which match the actual average magnitude of a sine.

When you look at the average value of a sine wave, you have to be clear what "average" means. For the raw sine wave, the positive and negative halves cancel, so the average over a full cycle is zero. But if you’re talking about the average value of the magnitude (the rectified sine, where negative parts are taken as positive), the average is a fixed fraction of the peak.

Compute the average of the absolute value of a sine wave v(t) = Vp sin(ωt):

Vavg = (1/T) ∫_0^T |Vp sin(ωt)| dt, where T = 2π/ω.

Let φ = ωt. This becomes Vavg = (Vp/2π) ∫_0^{2π} |sin φ| dφ. The integral of |sin φ| over one full cycle is 4, so Vavg = (Vp/2π)·4 = (2/π) Vp ≈ 0.637 Vp.

So the true relationship is that the average value is about 0.636 times the peak value for the magnitude of a sine wave (the rectified case). The other options either imply the average is larger than the peak, or equal to the peak, or equal to half the peak, none of which match the actual average magnitude of a sine.