analyse the effect of a single capacitor in smoothing, including the effect of the values of capacitance and the load resistance

Rectification and Smoothing – Cambridge IGCSE/A‑Level (9702) – Objective 21.2

Learning Objective

Analyse the effect of a single capacitor in smoothing a rectified voltage and explain how the values of capacitance (C) and load resistance (R_L) influence ripple voltage, ripple factor and the DC output. Include the role of diode forward‑voltage, transformer regulation and a brief comparison with other common filter types.

1. Rectifier Topologies

Key points

• Full‑wave rectifiers give twice the ripple frequency of a half‑wave, so the ripple amplitude for a given filter is roughly halved.
• Diode forward‑voltage drop (typically 0.6 – 0.8 V for silicon) reduces the usable DC level. In a bridge the drop is twice that of a half‑wave.

2. Ripple – Definitions

• Ripple voltage V_r – peak‑to‑peak variation of the filtered output.
• Ripple factor r – ratio of ripple voltage to the average (DC) output:

  \[

  r = \frac{V{r}}{V{DC}}\times 100\%

  \]

• DC output V_DC – average (mean) value of the filtered waveform. For a well‑filtered supply it is close to the capacitor’s peak charge voltage.

3. Single‑Capacitor (Capacitor‑Input) Filter

3.1 How it works

1. During each peak of the rectified waveform the capacitor charges almost instantly to the peak voltage V_p (minus diode drops).
2. When the input voltage falls below the capacitor voltage, the diode becomes reverse‑biased and the capacitor discharges through the load RL until the next peak re‑charges it.

3.2 Derivation of the ripple‑voltage formula

During the discharge interval the capacitor voltage follows the exponential law

\[

V(t)=V{p}\,e^{-\frac{t}{R{L}C}}

\]

For a full‑wave rectifier the time between successive peaks is T = 1/(2f); for a half‑wave it is T = 1/f. The peak‑to‑peak ripple is the difference between the voltage at the start of the interval (V_p) and at the end of the interval (V(T)):

\[

V{r}=V{p}-V{p}e^{-\frac{T}{R{L}C}}

=V{p}\Bigl(1-e^{-\frac{T}{R{L}C}}\Bigr)

\]

If the ripple is small (T ≪ R_LC) the exponential can be linearised (first‑order Taylor expansion):

\[

e^{-\frac{T}{R{L}C}}\approx 1-\frac{T}{R{L}C}

\]

Thus

\[

V{r}\approx V{p}\frac{T}{R_{L}C}

\]

Since the average load current is I_L = V_DC/R_L ≈ V_p/R_L, we obtain the widely‑used linear‑discharge expression

\[

\boxed{V{r}\;\approx\;\frac{I{L}}{f\,C}}

\qquad\text{(full‑wave: }f\rightarrow2f\text{)}

\]

From this, the ripple factor becomes

\[

\boxed{r\;\approx\;\frac{I{L}}{f\,C\,V{DC}}\times100\%}

\]

3.3 Influence of C and R_L

• Capacitance – larger C stores more charge, increasing the time constant τ = R_LC and therefore reducing V_r and r proportionally.
• Load resistance – a higher R_L (lower load current) also raises τ, giving a smaller ripple. For a given C,

  \[

  V{r}\propto\frac{1}{R{L}}\;\;\text{or}\;\;V{r}\propto I{L}

  \]

4. Diode Forward‑Voltage and Transformer Regulation

• Diode drop reduces the peak that the capacitor can charge to:

  • Half‑wave: V_DC ≈ V_p − V_D
  • Full‑wave bridge: V_DC ≈ V_p − 2V_D

• Transformer regulation (percentage) quantifies how much the secondary voltage falls when the load is applied:

  \[

  \text{Regulation} = \frac{V{no‑load}-V{full‑load}}{V_{full‑load}}\times100\%

  \]

  Poor regulation adds extra ripple because the peak voltage supplied to the capacitor drops under load.

5. Other Common Filter Types (Brief Overview)

For the IGCSE/A‑Level syllabus the single‑capacitor filter is the required focus, but awareness of LC/π filters helps to answer extension questions.

6. Design Example – 12 V DC Supply (Full‑Wave Bridge)

1. Specification:

   • Load current I_L = 100 mA
   • Maximum ripple V_r ≤ 0.5 V
   • Supply from 230 V r.m.s. mains, 50 Hz

2. Transformer secondary (r.m.s.) ≈ 12 V → peak

   \[

   V_{p}=12\sqrt{2}=16.97\;\text{V}

   \]

3. Diode losses (bridge, 2 × 0.7 V):

   \[

   V{DC}\approx V{p}-2V_{D}=16.97-1.4=15.6\;\text{V}

   \]

   (enough head‑room for a 12 V regulator).

4. Ripple frequency (full‑wave): \(f=2\times50=100\;\text{Hz}\).
5. Capacitance required using \(V{r}=I{L}/(fC)\):

   \[

   C=\frac{I{L}}{fV{r}}=\frac{0.10}{100\times0.5}=2.0\times10^{-3}\;\text{F}=2000\;\mu\text{F}

   \]

6. Component choice: 2200 µF, 25 V electrolytic (standard value, 25 V ≥ 1.5 × peak).
7. Check ripple factor:

   \[

   r=\frac{0.5}{12}\times100\% \approx 4.2\%

   \]

   Well within typical limits for low‑cost circuits.

7. Practical Considerations

• In‑rush current – at turn‑on the capacitor looks like a short circuit. Use a series resistor, NTC thermistor or a soft‑start circuit to limit the surge.
• Voltage rating – select a capacitor rating ≥ 1.5 × peak rectified voltage (e.g., 25 V for a 16 V peak).
• Polarity – electrolytic capacitors must be connected correctly; reverse‑bias can cause catastrophic failure.
• Temperature & ageing – electrolytic capacitance can fall by up to 20 % after several years; design with a safety margin.
• Physical size & cost – larger capacitance gives smoother output but increases size, cost and surge‑current stress.

8. Summary

• A single capacitor smooths the pulsating DC by charging to the peak of the rectified waveform and discharging through the load between peaks.
• Peak‑to‑peak ripple (for small ripples): \(\displaystyle V{r}\approx\frac{I{L}}{fC}\).
• Ripple factor: \(\displaystyle r\approx\frac{I{L}}{fC\,V{DC}}\times100\%.\)
• Increasing capacitance or increasing load resistance (reducing load current) lowers ripple proportionally.
• Full‑wave rectifiers halve the ripple amplitude compared with half‑wave because the ripple frequency is doubled.
• Diode forward‑voltage and transformer regulation affect the usable DC level and must be accounted for in design.
• Other filter types (inductor, LC/π) provide better smoothing but are beyond the core syllabus requirement.
• Design involves balancing ripple specifications, component size, cost, voltage rating and reliability.