For a sinusoidal waveform with peak value Vp = 12 V, what is the average (rectified) value over a half cycle (i.e., the mean of |v(t)| for a sine wave with Vp = 12 V)?

Introduction:This problem tests understanding of average (also called rectified average) for a sinusoidal waveform. For AC power and measurements, three common measures are used: peak value Vp, root-mean-square (rms) value Vrms, and average of the rectified waveform Vavg. Knowing how these relate is essential for converting between specification formats and instrument readings.

Given Data / Assumptions:

• A pure sinusoidal voltage with peak value Vp = 12 V.
• The requested average is the mean of the absolute value over one full period (equivalently, the average over one half cycle of the positive sine).
• No DC offset is present.

Concept / Approach:For a sine wave v(t) = Vp * sin(ωt), the average of |v(t)| over a period is Vavg = (2/π) * Vp. This is different from Vrms = Vp / √2 and the algebraic average over a full cycle, which is zero for a centered sine. The key constant for the rectified average is 2/π ≈ 0.637.

Step-by-Step Solution:Use Vavg = (2/π) * VpCompute numerical factor: 2/π ≈ 0.637Multiply by Vp: Vavg = 0.637 * 12 VVavg ≈ 7.644 VRounded to two decimals: 7.64 V

Verification / Alternative check:Compare with Vrms: Vrms = 12 / √2 ≈ 8.49 V, which is higher than the rectified average as expected. The full-cycle algebraic average is 0 V, reinforcing that the problem specifically asks for rectified average, not algebraic average.

Why Other Options Are Wrong:

• 3.82 V or 4.24 V: Do not correspond to standard constants times 12 V.
• 9.42 V: Too high; would exceed Vrms which contradicts the known relationships.
• 12.0 V: The peak value, not the average of the rectified sine.

Common Pitfalls:

• Confusing average-rectified value with rms value.
• Using the algebraic average over the full period (which is zero) for a centered sine wave.
• Rounding constants too early; keep precision until the end.

Final Answer:7.64 V