Capacitor Smoothing Circuits & Calculations

Reservoir capacitors are used to smooth the raw rectified waveform in a power supply - it is important to chose the right capacitor with the correct value and ripple current rating.

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In a power supply, whether it is a linear power supply or a switch mode power supply using an AC power source and diode rectifiers, the raw rectified output is normally smoothed using a reservoir capacitor before being applied to any regulators or other similar electronic circuitry.

Aluminium electrolytic capacitors are ideal for for acting as smoothing capacitors as many electrolytics are able to provide a sufficiently high capacitance and withstand the level of ripple current needed to smooth the waveform.

Aluminium electrolytic capacitor of the form used in power supply smoothing — Typical electrolytic capacitor used for smoothing applications

Essentially the smoothing circuit fills in the major dips in the raw rectified waveform so that the linear regulator or switch mode power supply circuitry can operate correctly.

The smoothing capacitor changes the waveform from one that changes from zero to the peak voltage over the course of the cycle of the incoming power waveform to one where the voltage changes are very much less. Essentially they smooth the waveform out, and this gives rise to the name.

As smoothing capacitors are used in both linear regulated power supplies and switch mode power supplies, they form an essential part of many of these electronic circuits.

Capacitor smoothing basics

Capacitor smoothing is used for most types of power supply, whether a linear regulated power supply, a switch mode power supply, or even just a smoothed and non-regulated form of power supply.

The raw DC supplied by a diode rectifier on its own would consist of a series of half sine waves with the voltage varying between zero and √2 times the RMS voltage (ignoring any diode and other losses).

A waveform of this nature would not be of any use for powering circuits because any analogue circuits would have the huge level of ripple superimposed on the output, and any digital circuits would not function because the power would be removed every half cycle.

The capacitor smoothing enables the following stages of the linear regulated power supply, or the switch mode power supply to operate correctly.

To smooth the output of the rectifier a reservoir capacitor is used - placed across the output of the reciter and in parallel with the load.

The smoothing works because the capacitor charges up when the voltage from the rectifier rises above that of the capacitor and then as the rectifier voltage falls, the capacitor provides the required current from its stored charge.

Full wave rectifier with smoothing capacitor

In this way the capacitor is able to provide charge when it is not available from the rectifier, and accordingly the voltage varies considerably less than if the capacitor were not present.

The capacitor smoothing will not provide total voltage stability, there will always be some variation in the voltage. In fact the higher the value of the capacitor, the greater the smoothing, and also the less current that is drawn, the better the smoothing.

Action of a smoothing capacitor on the rectified waveform in a power supply - either switch mode power supply or linear regulated power supply — Smoothing action of a reservoir capacitor

It should be remembered that the only way discharge path for the capacitor, apart from internal leakage is through the load to the rectifier / smoothing system. The diodes prevent back-flow through the transformer, etc..

A further point to remember, is that capacitor smoothing does not give any form of regulation and the voltage will vary according to the load and any input variations.

Voltage regulation can be provided by a linear regulator or a switch mode power supply.

Half wave and full wave rectification

It is possible to use either a half wave, or full wave rectifier system. For the half wave rectification, one half of the waveform is blocked and not used. This means that the waveform consists of a series of half waveforms spaced by half a waveform.

Comparison between the half wave and full wave rectifier operation showing that in a full wave rectifier both halves of the waveform are used. — Comparison between the half wave and full wave rectifier operation

Full wave rectification utilises both halves of the waveform and there are no large gaps where no voltage is being provided.

Using full wave rectification is not only more efficient, but it also enables better smoothing to be achieved because the gaps between peaks of the waveforms is less.

With a shorter gap between peaks the full wave rectifier enables better smoothing — Full wave rectifier enables better smoothing

Where possible, full wave rectification is always preferable because it enables better smoothing to be achieved.

It can also be seen that the basic waveform frequency for the output of the half wave rectifier is the same as the input frequency. For the full wave rectifier it is double this because the second half of the waveform is used and the peak occur at twice the frequency of the input.

Smoothing capacitor value

The choice of the capacitor value needs to fulfil a number of requirements. In the first case the value must be chosen so that its time constant is very much longer than the time interval between the successive peaks of the rectified waveform:

R l o a d \cdot C > > \frac{1}{f}

Where:
R_load = the overall resistance of the load for the supply
C = value of capacitor in Farads
f = the ripple frequency - this will be twice the line frequency a full wave rectifier is used.

Smoothing capacitor ripple voltage

As there will always be some ripple on the output of a rectifier using a smoothing capacitor circuit, it is necessary to be able to estimate the approximate value. Over-specifying a capacitor too much will add extra cost, size and weight - under-specifying it will lead to poor performance.

Peak to peak ripple on smoothed power supply from rectifier — Peak to peak ripple for output from smoothing capacitor on a power supply (full wave)

The diagram above shows the ripple for a full wave rectifier with capacitor smoothing. If a half wave rectifier was used, then half the peaks would be missing and the ripple would be approximately twice the voltage.

For cases where the ripple is small compared to the supply voltage - which is almost always the case - it is possible to calculate the ripple from a knowledge of the circuit conditions:

Full wave rectifier

V_{ripple} = \frac{I_{load}}{2 f C}

Half wave rectifier

V_{ripple} = \frac{I_{load}}{f C}

These equations provide more than sufficient accuracy. Although the capacitor discharge for a purely resistive load is exponential, the inaccuracy introduced by the linear approximation is very small for low values of ripple.

It is also worth remembering that the input to a voltage regulator is not a purely resistive load but a constant current load. Finally, the tolerances of electrolytic capacitors used for rectifier smoothing circuits are large - ±20% at the very best, and this will mask any inaccuracies introduced by the assumptions in the equations.

Ripple current

Two of the major specifications of a capacitor are its capacitance and working voltage. However for applications where large levels of current may flow, as in the case of a rectifier smoothing capacitor, a third parameter is of importance - its maximum ripple current.

The ripple current is not just equal to the supply current. There are two scenarios:

Capacitor discharge current: On the discharge cycle, the maximum current supplied by the capacitor occurs as the output from the rectifier circuit falls to zero. At this point all the current from the circuit is supplied by the capacitor. This is equal to the full current of the circuit.

Peak current supplied by capacitor in discharge phase
Capacitor charging current: On the charge cycle of the smoothing capacitor, the capacitor needs to replace all the lost charge, but it can only achieve this when the voltage from the rectifier exceeds that from the smoothing capacitor. This only occurs over a short period of the cycle. Consequently the current during this period is much higher. The larger the capacitor, the better it reduces the ripple and the shorter the charge period.

The shorter charging time gives rise to very large peak current levels as the smoothing capacitor needs to absorb sufficient charge for the discharge period in a very short time.

Period over which power supply capacitor charges

Pi section smoothing networks

In some applications a linear voltage regulator would not be used, an improved form of smooth could be required. This could be provided by using two capacitors and a series inductor or resistor.

The smoothed power supply approach is used in some high voltage systems and in some other specialist areas, but it is not nearly as common as linear regulated power supplies and switch mode power supplies which provide much better regulation and smoothing.

This approach can also be seen in many vintage wireless sets where the use of a linear regulated power supply was not feasible.

Pi smoothing filter using two capacitors and an inductor or resistor — Pi section smoothing filter

There are two options for a π section smoothing system. With two capacitors between the the line and ground, the series element was either and inductor or a resistor. The inductor cost much more and gave better performance, but the resistor was a much cheaper option although it did dissipate more power.

Smoothing capacitors are essential elements of both linear power supplies and switch mode power supplies and as such they are widely used.

When selecting a reservoir capacitor for smoothing applications in a power supply, not only is the value in terms of capacitance important to give the required ripple voltage reduction, but it is also very important to ensure that the capacitor ripple current rating is not exceeded. If too much current is drawn, the capacitor will heat up and its life expectancy reduced, or in extreme cases it could fail, sometimes catastrophically.