28 Jun 2025
Rectifier Output, Regulation, Ripple, Efficiency, and Filters
Average and RMS output, ripple factor, efficiency, voltage regulation, and basic smoothing filters.
For a periodic output of period $T_o$, the dc and RMS values are
\[V_{dc}=\frac1{T_o}\int_0^{T_o}v_o(t)\,dt, \qquad V_{rms}=\sqrt{\frac1{T_o}\int_0^{T_o}v_o^2(t)\,dt}.\]For an ideal diode, no filter, and a purely resistive load $R_L$, direct integration gives:
| Rectifier | $V_{dc}$ | $V_{rms}$ | $I_{dc}$ | $I_{rms}$ |
|---|---|---|---|---|
| Half wave | $V_m/\pi$ | $V_m/2$ | $V_m/(\pi R_L)$ | $V_m/(2R_L)$ |
| Full wave | $2V_m/\pi$ | $V_m/\sqrt2$ | $2V_m/(\pi R_L)$ | $V_m/(\sqrt2R_L)$ |
The full-wave $V_m$ is the peak presented to the load: one half-secondary peak for a center-tapped circuit and the complete-secondary peak for a bridge. Constant diode drops reduce all output values and change the conduction interval, so the table is not exact for a real diode.
Ripple factor and rectification efficiency
The RMS value contains dc and ac components:
\[V_{rms}^2=V_{dc}^2+V_{ac,rms}^2.\]Hence the ripple factor is
\[\boxed{r=\frac{V_{ac,rms}}{V_{dc}} =\sqrt{\left(\frac{V_{rms}}{V_{dc}}\right)^2-1}}.\]For the unfiltered ideal waveforms,
\[r_{\rm half}=\sqrt{\frac{\pi^2}{4}-1}=1.211, \qquad r_{\rm full}=\sqrt{\frac{\pi^2}{8}-1}=0.483.\]Rectification efficiency is the dc load power divided by the ac power entering the rectifying path:
\[\eta_r=\frac{P_{dc}}{P_{ac}}.\]If diode resistance and transformer losses are neglected,
\[\boxed{\eta_{r,\rm half}=\frac4{\pi^2}=0.406}, \qquad \boxed{\eta_{r,\rm full}=\frac8{\pi^2}=0.812}.\]These are limiting values, not percentages of voltage converted.
The exact integrals and identities are checked in rectifier-check.mac; every displayed residual is zero.
Voltage regulation
Rectifier output falls when load current increases because of transformer, diode, and source resistances. A standard measure is
\[\boxed{\%\text{ regulation} =\frac{V_{NL}-V_{FL}}{V_{FL}}\times100},\]where $V_{NL}$ and $V_{FL}$ are the no-load and specified full-load dc voltages. Smaller positive regulation is better. This quantity describes load dependence; it is distinct from the operation of a voltage-regulator circuit.
Filters
A shunt capacitor charges near each rectified peak and discharges through $R_L$ between peaks. If the ripple is small, load current is approximately constant, and diode conduction time is short,
\[V_{r,pp}\simeq\frac{I_L}{f_rC}, \qquad V_{r,rms}\simeq\frac{V_{r,pp}}{2\sqrt3},\]where $f_r=f$ for half-wave and $f_r=2f$ for full-wave rectification. With $I_L\simeq V_{dc}/R_L$,
\[\boxed{r\simeq\frac1{2\sqrt3\,f_rR_LC}}.\]$R_LC$ has units seconds, so the expression is dimensionless. The approximation fails for large ripple, rapidly changing load, or significant source resistance.
A series inductor has reactance $X_L=2\pi f_rL$: it passes dc but opposes ripple current. A shunt capacitor has $X_C=1/(2\pi f_rC)$ and bypasses ripple across the load. LC and $\pi$ filters combine these actions; their useful smoothing requires $X_L$ large and $X_C$ small at the ripple frequency while dc voltage and component current ratings remain adequate.
Discussion