Learning objectives
An ideal alternating voltage can be written as
\(v(t)=V_{\!0}\sin(\omega t)\)
For a purely resistive load the instantaneous power is \(p(t)=v(t)i(t)\). The mean (average) power over one cycle is
\(\displaystyle \overline P = \frac{1}{2}P_{\max}\)
where \(P{\max}=V{\!0}I_{\!0}\) is the product of the peak voltage and peak current.
The ideal diode current–voltage relation (Shockley equation) is
\(I = I_{\!S}\!\left(e^{\frac{qV}{kT}}-1\right)\)
Key practical parameters for a rectifier diode:
| Parameter | Symbol | Typical value (Si) | Effect on a rectifier |
|---|---|---|---|
| Forward voltage drop | \(VF\) | 0.6–0.7 V | Reduces the peak output by \(VF\) per conducting diode. |
| Reverse‑leakage current | \(IR\) | \(<10^{-6}\) A (small‑signal) | Ideally negligible; large \(IR\) adds a DC offset. |
| Breakdown (reverse‑voltage) rating | \(V_{BR}\) | ≥ 50 V (common rectifier) | Must exceed the peak‑inverse voltage (PIV) the diode experiences. |
| Maximum forward current | \(I_{F(\max)}\) | 1–5 A (typical) | Limits the load current to avoid overheating. |
| Configuration | Diodes used | Peak‑inverse voltage (PIV) | Ripple frequency | Typical applications |
|---|---|---|---|---|
| Half‑wave | 1 diode | \(V{PIV}=V{\text{peak}}\) | \(f_{\text{ripple}} = f\) | Low‑power signalling, simple hobby supplies |
| Full‑wave centre‑tapped | 2 diodes + centre‑tapped transformer | \(V{PIV}=V{\text{peak}}\) | \(f_{\text{ripple}} = 2f\) | Audio amplifiers where a transformer is already required |
| Full‑wave bridge (Graetz) | 4 diodes (bridge) | \(V{PIV}=2V{\text{peak}}\) | \(f_{\text{ripple}} = 2f\) | General DC supplies, portable equipment |
Waveform comparison
A capacitor placed across the load charges to the peak rectified voltage during each conduction interval and discharges through the load when the input falls, thereby reducing the AC component.
For a full‑wave rectifier feeding a resistive load, the approximate peak‑to‑peak ripple voltage is
\(Vr \;\approx\; \dfrac{I{\text{load}}}{f_{\text{ripple}}\,C}\)
The ripple factor, a convenient measure of the remaining AC component, is defined as
\(r = \dfrac{V{r(\text{rms})}}{V{\text{DC}}}\)
For a roughly triangular ripple (the usual case with a large capacitor) the rms value is
\(V{r(\text{rms})}= \dfrac{Vr}{\sqrt3}\)
Combining the two expressions gives
\(r \;\approx\; \dfrac{I{\text{load}}}{\sqrt3\,f{\text{ripple}}\,C\,V_{\text{DC}}}\)
Typical design targets are \(r<0.05\) for general electronics and \(r<0.02\) for precision circuits.
Requirement: Obtain a regulated 12 V DC output from a 50 Hz mains supply using a full‑wave bridge rectifier. Load current \(I_{\text{load}} = 0.5\) A and ripple factor \(r \le 0.02\).
\[
V{\text{peak}} = \sqrt2 \times V{\text{rms}} = \sqrt2 \times 12\;\text{V}=16.97\;\text{V}
\]
\[
V{PIV}=2V{\text{peak}} = 33.9\;\text{V}
\]
Choose diodes with a rating of at least 40 V (e.g. 1N4007, \(V_{BR}=1000\) V gives ample margin).
\[
V{r(\text{rms})}= r\,V{\text{DC}} = 0.02 \times 12 = 0.24\;\text{V}
\]
Peak‑to‑peak ripple (triangular approximation)
\[
Vr \approx \sqrt3\,V{r(\text{rms})}=0.42\;\text{V}
\]
\[
f_{\text{ripple}} = 2f = 2 \times 50 = 100\;\text{Hz}
\]
\[
C \ge \frac{I{\text{load}}}{f{\text{ripple}}\,V_r}
= \frac{0.5}{100 \times 0.42}
= 1.19\times10^{-2}\;\text{F}
\approx 12\,000\;\mu\text{F}
\]
A standard electrolytic capacitor of 15 000 µF, rated ≥ 35 V, satisfies the requirement and provides a safety margin.
The filtered DC voltage is approximately
\[
V{\text{DC}} \approx V{\text{peak}} - 2V_F
= 16.97\;\text{V} - 2(0.7\;\text{V}) \approx 15.6\;\text{V}
\]
A simple linear regulator (e.g. 7812) can then drop this to a stable 12 V with the ripple already within the allowed limit.
| Concept | Quantum origin (brief) | Practical implication for rectifiers |
|---|---|---|
| Forward conduction | Band‑gap narrowing under forward bias; diffusion of majority carriers across the depletion region. | Defines the forward voltage drop \(V_F\); two drops in a bridge reduce the peak output by ≈ 2 V. |
| Reverse blocking | Widening of the depletion region creates a potential barrier that prevents carrier flow. | Determines the peak‑inverse voltage rating; must exceed the maximum reverse voltage each diode sees. |
| Ripple voltage | Capacitor stores charge during voltage peaks and releases it during troughs. | \(Vr \propto \dfrac{I{\text{load}}}{f_{\text{ripple}}C}\); higher ripple frequency (full‑wave) and larger \(C\) give smoother DC. |
| Ripple factor | Statistical measure of the residual AC component after filtering. | Guides selection of \(C\) and load current; low \(r\) (<0.05) required for most electronic circuits. |
These notes now cover every sub‑point of Cambridge International AS & A Level Physics syllabus 21.1, present the quantum‑mechanical foundation at an appropriate depth, and provide the practical formulas and examples needed for exam success.