Quantum physics

Rectification and Smoothing

Objective: Understand the quantum‑mechanical basis of rectification and how smoothing converts a pulsating DC into a near‑steady DC.

1. Quantum basis of rectification

Rectification relies on the asymmetric conduction of charge carriers across a p‑n junction. The behaviour is explained by the band theory of semiconductors:

• The valence band is fully occupied at absolute zero, while the conduction band is empty.
• Thermal excitation creates electron‑hole pairs; the probability of occupation follows the Fermi‑Dirac distribution.
• When a forward bias \$V\$ is applied, the potential barrier is reduced, allowing carriers to tunnel or diffuse across the junction.

The current–voltage relationship of an ideal diode is given by the Shockley equation:

\$I = I_S\!\left(e^{\frac{qV}{kT}} - 1\right)\$

where \$I_S\$ is the saturation current, \$q\$ the elementary charge, \$k\$ Boltzmann’s constant and \$T\$ the absolute temperature.

2. Diode characteristics relevant to rectification

Key parameters that affect rectifier performance:

1. Forward voltage drop \$V_F\$ (typically 0.6–0.7 V for silicon).
2. Reverse leakage current \$I_R\$ (ideally negligible).
3. Breakdown voltage \$V_{BR}\$ – the maximum reverse voltage the diode can withstand.
4. Maximum forward current \$I_{F(max)}\$ – limited by device heating.

3. Types of rectifiers

4. Smoothing the rectified output

After rectification the output is a pulsating DC. Smoothing reduces the ripple to produce a near‑constant voltage. The most common method uses a filter capacitor placed across the load.

The ripple voltage \$V_r\$ for a full‑wave rectifier feeding a capacitor \$C\$ is approximated by:

\$Vr \approx \frac{I{load}}{f C}\$

where \$I_{load}\$ is the load current and \$f\$ the ripple frequency (twice the mains frequency for full‑wave).

5. Ripple factor and its significance

The ripple factor \$r\$ quantifies the quality of smoothing:

\$r = \frac{V{r(rms)}}{V{DC}}\$

A lower \$r\$ indicates a smoother DC. Typical design targets are \$r < 0.05\$ for precision electronics.

6. Example calculation

Design a smoothing capacitor for a 12 V RMS, 50 Hz mains supply using a full‑wave bridge. The load draws \$I_{load}=0.5\,\$A and a ripple factor \$r\le0.02\$ is required.

1. Convert RMS mains voltage to peak: \$V_{peak}= \sqrt{2}\times12\;\text{V}=16.97\;\text{V}\$.
2. Peak inverse voltage for a bridge: \$PIV = 2V{peak}=33.9\;\text{V}\$ (choose a diode with \$V{BR}>40\,\$V).
3. Ripple voltage limit: \$Vr = r \times V{DC} \approx 0.02 \times 12\;\text{V}=0.24\;\text{V}\$.
4. Ripple frequency: \$f = 2 \times 50\;\text{Hz}=100\;\text{Hz}\$.
5. Required capacitance:

   \$C \ge \frac{I{load}}{f Vr}= \frac{0.5}{100 \times 0.24}=0.0208\;\text{F}\approx 22\,000\;\mu\text{F}\$

Thus a standard electrolytic capacitor of \$22\,000\;\mu\text{F}\$ (or larger) rated at ≥ 35 V will meet the specification.

7. Summary of key points