Quantum physics

Rectification and Smoothing – A‑Level Physics 9702 (Paper 4, Topic 21.1)

Learning objectives

• Recall the basic sinusoidal description of an AC source (period, frequency, peak value).
• Explain the quantum‑mechanical origin of a p‑n junction and the resulting diode I‑V characteristic.
• Identify the three common rectifier configurations and calculate the peak‑inverse voltage (PIV) for each.
• Derive the expressions for ripple voltage, ripple factor and the required filter capacitance.
• Apply the formulas to a design problem and evaluate the choice of diode and capacitor.

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1. Sinusoidal AC fundamentals

An ideal alternating voltage can be written as

\(v(t)=V_{\!0}\sin(\omega t)\)

• Peak value \(V_{\!0}\) – maximum instantaneous voltage.
• Angular frequency \(\omega =2\pi f\) (rad s\(^{-1}\)).
• Frequency \(f\) – number of cycles per second (Hz).
• Period \(T = 1/f\) – time for one complete cycle.
• Root‑mean‑square (rms) value: \(\displaystyle V_{\rm rms}= \frac{V_{\!0}}{\sqrt2}\), \(\displaystyle I_{\rm rms}= \frac{I_{\!0}}{\sqrt2}\).

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.

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2. Quantum‑mechanical basis of a diode

• At 0 K the valence band is full and the conduction band empty. Thermal energy creates electron‑hole pairs; the occupation probability follows the Fermi‑Dirac distribution.
• In a p‑n junction a depletion region forms, producing an internal electric field that creates a potential barrier (≈ 0.7 V for silicon).
• Forward bias lowers the barrier, allowing majority carriers to diffuse across the junction. Reverse bias widens the depletion region, preventing carrier flow.

The ideal diode current–voltage relation (Shockley equation) is

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

• \(I_{\!S}\) – saturation (reverse‑leakage) current.
• \(q\) – elementary charge (1.60 × 10\(^{-19}\) C).
• \(k\) – Boltzmann’s constant (1.38 × 10\(^{-23}\) J K\(^{-1}\)).
• \(T\) – absolute temperature (K).

Key practical parameters for a rectifier diode:

Parameter	Symbol	Typical value (Si)	Effect on a rectifier
Forward voltage drop	\(V_F\)	0.6–0.7 V	Reduces the peak output by \(V_F\) per conducting diode.
Reverse‑leakage current	\(I_R\)	\(<10^{-6}\) A (small‑signal)	Ideally negligible; large \(I_R\) 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.

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3. Rectifier configurations

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

• Half‑wave: one pulse per mains period → ripple frequency = \(f\).
• Full‑wave (centre‑tapped or bridge): two pulses per period → ripple frequency = \(2f\). The higher frequency shortens the discharge time of the filter capacitor, giving a smaller ripple for the same capacitance.

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4. Smoothing with a filter capacitor

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

\(V_r \;\approx\; \dfrac{I_{\text{load}}}{f_{\text{ripple}}\,C}\)

• \(I_{\text{load}}\) – average load current (A).
• \(f_{\text{ripple}}\) – ripple frequency (Hz) = \(2f\) for full‑wave.
• \(C\) – filter capacitance (F).

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

\(V_{r(\text{rms})}= \dfrac{V_r}{\sqrt3}\)

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.

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5. Worked design example

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 \(r \le 0.02\).

1. Peak voltage of the transformer secondary \[ V_{\text{peak}} = \sqrt2 \times V_{\text{rms}} = \sqrt2 \times 12\;\text{V}=16.97\;\text{V} \]
2. Peak‑inverse voltage for each diode \[ V_{PIV}=2V_{\text{peak}} = 33.9\;\text{V} \] Choose diodes with a rating of at least 40 V (e.g. 1N4007, \(V_{BR}=1000\) V gives ample margin).
3. Allowable rms ripple voltage \[ V_{r(\text{rms})}= r\,V_{\text{DC}} = 0.02 \times 12 = 0.24\;\text{V} \] Peak‑to‑peak ripple (triangular approximation) \[ V_r \approx \sqrt3\,V_{r(\text{rms})}=0.42\;\text{V} \]
4. Ripple frequency \[ f_{\text{ripple}} = 2f = 2 \times 50 = 100\;\text{Hz} \]
5. Required capacitance \[ C \ge \frac{I_{\text{load}}}{f_{\text{ripple}}\,V_r} = \frac{0.5}{100 \times 0.42} = 1.19\times10^{-2}\;\text{F} \approx 12\,000\;\mu\text{F} \] A standard electrolytic capacitor of 15 000 µF, rated ≥ 35 V, satisfies the requirement and provides a safety margin.
6. Check the DC output 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.

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6. Summary of key concepts

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.	\(V_r \propto \dfrac{I_{\text{load}}}{f_{\text{ripple}}C}\); 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.

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7. Quick reference formulas

• Sinusoid: \(v(t)=V_{\!0}\sin(2\pi ft)\)
• RMS values: \(V_{\rm rms}=V_{\!0}/\sqrt2\), \(I_{\rm rms}=I_{\!0}/\sqrt2\)
• Mean power in a resistor: \(\overline P = \frac12 V_{\!0}I_{\!0}=V_{\rm rms}I_{\rm rms}\)
• Diode I‑V: \(I = I_S\!\left(e^{qV/kT}-1\right)\)
• Ripple voltage (full‑wave): \(V_r \approx \dfrac{I_{\text{load}}}{2fC}\)
• Ripple factor: \(r \approx \dfrac{I_{\text{load}}}{\sqrt3\,2fC\,V_{\text{DC}}}\)
• PIV for bridge: \(V_{PIV}=2V_{\text{peak}}\); centre‑tapped: \(V_{PIV}=V_{\text{peak}}\); half‑wave: \(V_{PIV}=V_{\text{peak}}\).

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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.