# Op Amp Low Pass Filter: active filter circuit design

### Operational amplifiers or op amps provide an easy and effective method or creating active low pass filters with a minimum of components.

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Operational amplifiers or op-amps provide a very effective means of creating low pass filters without the need for inductors.

By incorporating the filter elements into the feedback loop of an op amp, high performance low pass filters are easily created with a minimum number of components and without the need for inductors.

Op amp low pass filters can be used in many areas power supplies to the outputs of digital to analogue converters to remove alias signals and many more applications.

## What is a low pass filter

As the name implies, a low pass filter is a filter that passes the lower frequencies and rejects those at higher frequencies.

The shape of the curve is of importance with features like the cut-off frequency and roll off being key to the operation.

The cut-off frequency is normally taken as the point where the response has fallen by 3dB as shown.

Another important feature is the final slope of the roll off. This is generally governed by the number of 'poles' in the filter. Normally there is one pole for each capacitor inductor in a filter.

When plotted on a logarithmic scale the ultimate roll-off becomes a straight line, with the response falling at the ultimate roll off rate. This is 6dB per pole within the filter.

## Single pole active low pass filter circuit

The simplest circuit low pass filter circuit using an operational amplifier simply places a capacitor across the feedback resistor. This has the effect as the frequency rises of increasing the level of feedback as the reactive impedance of the capacitor falls.

The break point for this simple type of filter can be calculated very easily by working out the frequency at which the reactance of the capacitor equals the resistance of the resistor. This can be achieved using the formula:

**Where:**

**Xc** is the capacitive reactance in ohms

**Π** is the greek letter and equal to 3.142

**f** is the frequency in Hertz

**C** is the capacitance in Farads

The in band gain for these circuits is calculated in the normal way ignoring the effect of the capacitor.

While these operational amplifier circuits are useful to provide a reduction in gain at high frequencies, they only provide an ultimate rate of roll off of 6 dB per octave, i.e. the output voltage halves for every doubling in frequency. This type of filter is known as a one pole filter. Often a much greater rate of rejection is required, and to achieve this it is possible to incorporate a higher performance filter into the feedback circuitry.

## Two pole low pass filter op-amp circuit

Although it is possible to design a wide variety of filters with different levels of gain and different roll off patterns using operational amplifiers, the filter described on this page will give a good sure-fire solution. It offers unity gain and a Butterworth response (the flattest response in band, but not the fastest to achieve ultimate roll off out of band).

The calculations for the circuit values are very straightforward for the Butterworth response and unity gain scenario. Critical damping is required for the circuit and the ratio of the resistor and capacitor values determines this.

$C}_{1}=2{C}_{2$

$f=\frac{\sqrt{2}}{4\pi R{c}_{2}}$

When choosing the values, ensure that the resistor values fall in the region between 10 kΩ and 100 kΩ. This is advisable because the output impedance of the circuit rises with increasing frequency and values outside this region may affect the performance.

Op amp low pass filters are easy to design, especially when a Butterworth filter type is used as above. More sophisticated designs using different types of filter can also be developed, although the mathematics does become more complicated and decisions need to be made about the optimum type of filter to be used. For most applications, the basic Butterworth provides excellent filter performance.

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