DC value of a rectified waveform

The DC value of a rectified waveform is the time-average of a rectifier’s output — the constant level you would measure with a DC voltmeter. It is what you ultimately want a Rectifier to deliver, and it is far smaller than the peak because rectified humps spend a lot of time near zero.

Half-wave: $V_p/\pi$

A Half-wave rectifier passes a sine peak $V_p$ for half of each period and outputs zero for the other half. The DC value is the integral of the output over one full period $T$ divided by $T$. Take $v(t)=V_p\sin (\omega t)$, passing the positive half ($0$ to $T/2$) and blocking the rest:

$V_{DC}=\frac{1}{T}{\int }_0^T{v}_{o}dt=\frac{1}{T}{\int }_0^{T/2}{V}_p\sin (\omega t)dt$

With $\omega =2\pi /T$, the integral of $\sin (\omega t)$ from $0$ to $T/2$ is $[-\frac{1}{\omega }\cos (\omega t){]}_0^{T/2}=\frac{1}{\omega }(1-\cos \pi )=\frac{2}{\omega }$. So

$V_{DC}=\frac{V_p}{T}\cdot \frac{2}{\omega }=\frac{V_p}{T}\cdot \frac{2T}{2\pi }=\frac{V_p}{\pi }\approx 0.318V_p$

So the average of half-wave-rectified sine is only about a third of the peak.

Full-wave: $2V_p/\pi$

[The full-wave value and its derivation are filled in from general knowledge; the source PDF gives the half-wave figure explicitly but only states the full-wave benefit qualitatively.] A Full-wave rectifier flips the negative half-cycles up instead of discarding them, so the output is $|V_p\sin (\omega t)|$ — a hump every half-period. The waveform now repeats every $T/2$, and the average over a half-period is

$V_{DC}=\frac{2}{T}{\int }_0^{T/2}{V}_p\sin (\omega t)dt=\frac{2}{T}\cdot V_p\cdot \frac{2}{\omega }=\frac{2V_p}{\pi }\approx 0.637V_p$

Exactly double the half-wave value, which is the quantitative reason full-wave rectification is preferred.

Both $V_p/\pi$ and $2V_p/\pi$ assume an ideal diode, so $V_p$ is the full source peak. A real diode subtracts its forward drop from the conducting peak, so the true averages are lower — $(V_p-V_D)/\pi$ half-wave, $2(V_p-V_D)/\pi$ full-wave (a Bridge rectifier loses $2V_D$). The ideal-diode forms are kept here because they isolate the averaging factor ($1/\pi$, $2/\pi$); the diode-drop correction layers on top of it.

Why a smoothing capacitor is still needed

Even $0.637V_p$ is a poor DC source: the waveform is still pulsating between $V_p$ and 0, and any circuit drawing power from it sees that swing. The “DC value” is just the average — the variation about it is enormous. To get something usable you add a smoothing capacitor, which holds the output near the peak between conduction bursts. The resulting output sits near $V_p$ (not the unsmoothed average) with only a small Ripple voltage on top. Computing the raw DC value matters for understanding why smoothing is necessary and for estimating power, but a real supply runs on the smoothed, near-peak voltage, not on $V_p/\pi$.