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

Rectification and Smoothing

Learning Objective

Analyse the effect of a single capacitor in smoothing a rectified voltage, including how the values of capacitance (C) and load resistance (R_L) influence the ripple voltage and the DC output.

1. Introduction to Rectification

Rectification converts an alternating‑current (AC) waveform into a pulsating direct‑current (DC) waveform. The most common rectifier for A‑Level studies is the single‑phase half‑wave and full‑wave bridge rectifier.

• Half‑wave rectifier: conducts during one half‑cycle.
• Full‑wave bridge rectifier: conducts during both half‑cycles, giving a frequency of the ripple that is twice the mains frequency.

2. Need for Smoothing

The output of a rectifier is not a steady DC level; it contains a periodic variation called ripple. Smoothing reduces this ripple to produce a voltage that is close to a constant DC value.

3. Single Capacitor Filter (Capacitor‑Input Filter)

Placing a capacitor directly across the rectifier output forms a simple low‑pass filter. The capacitor charges to the peak of the rectified waveform and discharges through the load when the input voltage falls below the capacitor voltage.

4. Key Relationships

The performance of the filter is governed by the time constant τ = R_L C. Two important quantities are:

1. Peak voltage (\$V_{\text{p}}\$): the maximum voltage the capacitor charges to, approximately the peak of the rectified waveform minus diode drops.
2. Ripple voltage (\$V_{\text{r}}\$): the peak‑to‑peak variation of the filtered output.

For a full‑wave rectifier supplying a resistive load, the ripple voltage can be approximated by:

\$\$

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

\$\$

where:

• \$I{\text{L}} = \dfrac{V{\text{DC}}}{R_{\text{L}}}\$ is the load current,
• \$f\$ is the ripple frequency (twice the mains frequency for a full‑wave bridge, e.g., 100 Hz in a 50 Hz system),
• \$C\$ is the capacitance.

Alternatively, using the time constant:

\$\$

V{\text{r}} \approx V{\text{p}}\left(1 - e^{-\frac{T}{R{\text{L}}C}}\right) \approx \frac{V{\text{p}}\,T}{R_{\text{L}}C}

\$\$

with \$T = \dfrac{1}{f}\$ being the period of the ripple.

5. Effect of Capacitance (C)

Increasing the capacitance reduces the ripple voltage because the capacitor can store more charge and therefore supplies the load for a longer portion of each cycle.

From the table it is clear that a larger C yields a smaller ripple. However, very large capacitors are bulky, expensive, and may increase the inrush current at turn‑on.

6. Effect of Load Resistance (R_L)

The load resistance determines the discharge rate of the capacitor. A lower resistance (heavier load) draws more current, causing the capacitor voltage to fall more quickly, which increases ripple.

Thus, for a given capacitance, the ripple voltage is inversely proportional to the load resistance.

7. Design Example

Design a 12 V DC supply from a 230 V r.m.s. mains (50 Hz) using a full‑wave bridge and a single smoothing capacitor. Required ripple voltage < 0.5 V when the load draws 100 mA.

1. Calculate the peak secondary voltage (assume a 12 V r.m.s. transformer):

   \$V{\text{p}} = V{\text{rms}}\sqrt{2} \approx 12\sqrt{2} = 16.97\text{ V}\$

2. Subtract diode drops (≈2 × 0.7 V):

   \$V_{\text{DC}} \approx 16.97 - 1.4 = 15.6\text{ V}\$

3. Ripple frequency for full‑wave: \$f = 2 \times 50 = 100\text{ Hz}\$.
4. Required ripple voltage \$V_{\text{r}} = 0.5\text{ V}\$.
5. Using \$V{\text{r}} \approx \dfrac{I{\text{L}}}{fC}\$, solve for \$C\$:

   \$C \approx \frac{I{\text{L}}}{f V{\text{r}}} = \frac{0.1}{100 \times 0.5} = 2\times10^{-3}\text{ F} = 2000\ \mu\text{F}\$

6. Select a standard value, e.g., 2200 µF, 25 V electrolytic.

8. Practical Considerations

• Inrush current: When the supply is switched on, the capacitor appears as a short circuit, causing a large surge. A series resistor or a soft‑start circuit can limit this.
• Voltage rating: The capacitor must tolerate the peak rectified voltage plus a safety margin (typically 1.5× the expected peak).
• Temperature and ageing: Electrolytic capacitors lose capacitance over time; design with a margin of at least 20 %.

9. Summary

A single capacitor filter smooths the pulsating DC from a rectifier by charging to the peak voltage and discharging through the load. The ripple voltage is approximately \$V{\text{r}} \approx I{\text{L}}/(fC)\$, showing that:

• Increasing capacitance reduces ripple proportionally.
• Increasing load resistance (reducing load current) also reduces ripple.
• Design involves balancing ripple specifications, component size, cost, and reliability.