In one line
The average value of a sine wave over a half cycle is \( 2V_m/\pi \approx 0.637V_m \); form factor is RMS ÷ average (1.11) and peak factor is peak ÷ RMS (1.414).
Explained simply
If you average a full sine wave, the positive half cancels the negative half and you get zero, which tells you nothing useful. So for symmetrical AC we average only one half-cycle. It is like measuring how much water flows one way in a tide, ignoring the return flow.
Technically, the average value is the arithmetic mean of all instantaneous values over a half-cycle. It matters in rectifiers and battery charging, where the charge transferred depends on the average current. Form factor and peak factor describe the shape of a waveform.
How it works
- Take \( v = V_m\sin\theta \) over \( 0 \) to \( \pi \).
- \( V_{avg} = \dfrac{1}{\pi}\int_0^{\pi} V_m\sin\theta\,d\theta = \dfrac{V_m}{\pi}[-\cos\theta]_0^{\pi} = \dfrac{2V_m}{\pi} \).
- So \( V_{avg} = 0.637V_m \).
- Form factor \( K_f = V_{rms}/V_{avg} = 0.707/0.637 = 1.11 \).
- Peak (crest) factor \( K_p = V_m/V_{rms} = 1.414 \).
| Waveform | Average (half cycle) | RMS | Form factor | Peak factor |
|---|---|---|---|---|
| Sine | 0.637\( V_m \) | 0.707\( V_m \) | 1.11 | 1.414 |
| Square | \( V_m \) | \( V_m \) | 1 | 1 |
Key formula
\( V_{avg} = \dfrac{2V_m}{\pi}, \quad K_f = \dfrac{V_{rms}}{V_{avg}}, \quad K_p = \dfrac{V_m}{V_{rms}} \)
\( V_m \) = peak voltage (V), \( V_{avg} \) = average value over a half-cycle (V), \( V_{rms} \) = RMS value (V). \( K_f \) and \( K_p \) have no units.
Quick example
For \( V_m = 100 \) V: \( V_{avg} = 63.7 \) V, \( V_{rms} = 70.7 \) V, \( K_f = 70.7/63.7 = 1.11 \), \( K_p = 100/70.7 = 1.414 \). Average-responding meters are calibrated using the 1.11 form factor, so they read correctly only for pure sine waves.
Common mistakes
- Writing the full-cycle average of a sine wave as \( 0.637V_m \); over a full cycle it is zero.
- Swapping form factor and peak factor definitions.
- Using \( 2V_m/\pi \) for half-wave rectified output; that is \( V_m/\pi \) because the second half is missing.
How MU asks it
Typically 5 marks: "Derive the average value of a sinusoidal voltage and define form factor and peak factor", or a numerical on a given waveform (half-wave rectified, triangular, square). For full marks: state that the average is taken over a half-cycle for symmetrical waves, show the integral, give the numeric results 0.637, 1.11, 1.414, and keep units.