Capacitor Smoothing Circuits & Calculations

- notes, details & calculations for smoothing capacitor circuits used with rectifiers with details of ripple voltage and ripple current.

Rectifiers are normally used in circuits that require a steady voltage to be supplied.

To provide a steady DC output. The raw rectified DC requires a smoothing capacitor circuit to enable the rectified DC to be smoothed so that it can be used to power electronics circuits without large levels of voltage variation.

Capacitor smoothing basics

The raw DC supplied by a 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 supply 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.

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.. This 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.

Smoothing action of a reservoir capacitor
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 backflow through the transformer, etc..

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:

Rload   *   C   >>   1   /   f

Rload = 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 for smoothed diode rectifier circuit
Peak to peak ripple for
smoothed diode rectifier circuit

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

Vripple   =   Iload   /   2   f   C

Half wave rectifier

Vripple   =   Iload   /   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 the smoothing capacitor during discharge
  • 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 large the capacitor, the better it reduces the ripple and the shorter the charge period.

    Peak current supplied by the smoothing capacitor during charge

In view of the large currents involved, care must be taken to ensure that he ripple current does not exceed the rated values for the capacitor.

By Ian Poole

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