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

Rectifier type	Configuration	Peak inverse voltage (PIV)	Typical applications
Half‑wave	Single diode in series with load	$V_{peak}$	Low‑power signal detection
Full‑wave centre‑tapped	Two diodes with centre‑tapped transformer	$2V_{peak}$	Audio power supplies
Full‑wave bridge (Graetz)	Four diodes in bridge configuration	$2V_{peak}$	General DC power supplies

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:

$$V_r \approx \frac{I_{load}}{f C}$$

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: $V_r = r \times V_{DC} \approx 0.02 \times 12\;\text{V}=0.24\;\text{V}$.
4. Ripple frequency: $f = 2 \times 50\;\text{Hz}=100\;\text{Hz}$.
5. Required capacitance: $$C \ge \frac{I_{load}}{f V_r}= \frac{0.5}{100 \times 0.24}=0.0208\;\text{F}\approx 22\,000\;\mu\text{F}$$

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

Concept	Quantum origin	Practical implication
Diode forward conduction	Band‑gap reduction under forward bias; carrier diffusion	Defines $V_F$ and maximum forward current
Reverse blocking	Depletion region widens, creating a potential barrier	Determines PI \cdot and leakage current
Ripple reduction	Capacitor stores charge during peaks, releases during troughs	Ripple voltage $V_r$ inversely proportional to $C$ and $f$