Average over a half-cycle: For a sine wave with peak value 40 V, what is the average (mean) value over one half-cycle (i.e., the average-rectified value)?

Introduction / Context:
Two common averages for sinusoids are the full-cycle algebraic average (zero for a centered sine) and the average-rectified value (over a half-cycle, or equivalently the average of the absolute value over a full cycle). Rectifier and power supply calculations often use the half-cycle average.

Given Data / Assumptions:

• Sine wave peak Vp = 40 V.
• No DC offset; pure sinusoid.
• We want the average value over a single half-cycle.

Concept / Approach:

The half-cycle average of a sine is V_avg(half) = (2/π) * Vp ≈ 0.63662 * Vp. This comes from integrating sin(θ) from 0 to π and dividing by π.

Step-by-Step Solution:

Verification / Alternative check:

Compare with RMS: Vrms = Vp/√2 ≈ 28.28 V. It is logical that the RMS exceeds the half-cycle average for a sine. Values are consistent.

Why Other Options Are Wrong:

6.37 V corresponds to 10 V peak's half-cycle average. 14.14 V is the RMS of a 20 V peak, not applicable. 50.96 V is 1.274 * Vp, not a standard sine factor.

Common Pitfalls:

Confusing RMS with average-rectified; forgetting to multiply by 2/π instead of 1/π.

Final Answer:

25.48 V