# Op Amp Differentiator Circuit

## It is easy to design a differentiator using an op amp circuit that provides an accurate analogue implementation of this function.

The op amp circuit for a differentiator is one that has been used within analogue computing for many years. Although analogue differentiator circuits using differential amplifiers made with discrete electronic components have been used for many years, the introduction of the op amp integrated circuit has revolutionised the electronic circuit design process.

The very high level of gain of the operational amplifier means that it can provide a very high level of performance - much better than that which could be obtained using discrete electronic components.

One of the applications for, analogue differentiator circuits is for transforming different types of waveform as shown below.

## Op amp differentiator basics

A differentiator circuit is one in which the voltage output is directly proportional to the rate of change of the input voltage with respect to time.

This means that a fast change to the input voltage signal, the greater the output voltage change in response.

As a differentiator circuit has an output that is proportional to the input change, some of the standard waveforms such as sine waves, square waves and triangular waves give very different waveforms at the output of the differentiator circuit.

For these waveforms it can be seen that the greater the rate of change of the waveform at the input, the higher the output voltage at that point. In fact for the square wave input, only very short spikes should be seen. The spikes will be limited by the slope of the edges of the input waveform and also the maximum output of the circuit and its slew rate and bandwidth. The spikes should also decay swiftly. Again this may be limited by the circuit and on the diagram, the decay is not shown to be infinitely fas, representing better what a real life waveform may look like.

The triangular wave input transforms to a square wave in line with the rising and falling levels of the input waveform.

The sine wave is converted to a cosine waveform - giving 90° of phase shift of the signal. This can be useful in some circumstances.

## Op amp differentiator circuit

It can be seen that the op amp circuit for an integrator is very similar to that of the differentiator. The difference is that the positions of the capacitor and inductor are changed.

In its basic form the centre of the circuit is based around the operational amplifier itself. In addition to this a couple of other electronics components are required: a capacitor is connected from the input of the whole circuit to the inverting input of the operational amplifier. A feedback resistor is then used to provide the negative feedback around the op amp chip - this is connected from the output of the operational amplifier to its inverting input. The non-inverting input is connected to ground.

Unlike the integrator circuit, the operational amplifier differentiator has a resistor in the feedback from the output to the inverting input. This gives it DC stability - an important factor in many applications.

## Electronic circuit design equations

In order to develop the electronic component values for the differentiator circuit, it is necessary to determine the performance that is required.

The voltage output for the operational amplifier differentiator can be determined from the relationship below:

Where:
Vout = output voltage from op amp differentiator
Vin = input voltage
t = time in seconds
R = resistor value in the differentiator in Ω
C = capacitance of differentiator capacitor in Farads
dVin/dt = rate of change of voltage with time.

As mentioned, differentiators have issues with noise and sometimes instabilities at high frequencies as a result of the gain and also the internal phase shifts within the operational amplifier.

These issues can be overcome by adding some HF roll off. Only two additional electronic components are required to achieve this.

The choice of the electronic components: the capacitor C2 and resistor R2 depends very much upon the conditions - the level of noise and the differentiator bandwidth needed. The larger values of the electronic components provide increased stability and noise reduction at the cost of bandwidth.

The value of R2 can be calculated from the equation:

${R}_{1}=0.5\sqrt{\frac{{R}_{2}}{{C}_{1}{f}_{t}}}$

Although not always included, the capacitor C2 can be added for further noise reduction. A suitable starting value for this can be estimated from the equation below.

$f>>\frac{1}{2\pi {R}_{1}{C}_{1}}$

With the additional electronic components,, C2 and R2, the circuit starts to become an integrator at high frequencies (f » 1 / 2 π R1 C1 ). This occurs as a result of the feedback flatness and the overall compensation within the operational amplifier itself.

## Op amp differentiator design considerations

There are a number of electronic circuit design considerations that need to be taken into account when using an op amp differentiator circuit.

• Remember output rises with frequency:   One of the key facets of having a series capacitor is that it has an increased frequency response at higher frequencies. The differentiator output rises linearly with frequency, although at some stage the limitations of the op amp will mean this does not hold good.

Accordingly precautions may need to be made to account for this during the electronic circuit design and build process. The circuit, for example will be very susceptible to high frequency noise, stray pick-up, etc. The circuit, and its input in particular must be protected from stray pick up, otherwise this may disrupt its operation.

• Electronic component value limits:   It is always best to keep the values of the electronic components, i.e. the capacitor and particularly the resistor within sensible limits. Often values of less than 100kΩ for the resistor are best. In this way the input impedance of the op amp should have no effect on the operation of the circuit.

## Applications

The differentiator circuit has many applications in a number of areas of electronic design. The op amp differentiator is particularly easy to use and therefore is possibly one of the most widely used versions.

Obviously the circuit is used in analogue computers where it is able to provide a differentiation manipulation on the input analogue voltage.

Possibly the differentiator circuit is used most widely in process instrumentation. Here it can be used to monitor the rate of change of various points. If the measurement device returns a rate of change greater than a certain value, this will give an output voltage above a certain threshold and this can be measured using a comparator and used to set an alarm or warning indication.

In fact there are many signal conditioning applications where a differentiator may be required. Of the various options open to the electronic circuit designer, often the op amp solution is often the most attractive, requiring few components while still giving an excellent level of performance.

The op amp circuit for a differentiator has been used in many analogue computer applications, however it is also used in waveform transformations whee signals need to be processed. The gain of the op amp circuit means that the transformation is almsot perfect, although noise can be an issue and for this reason, these circuits may not be as widely used as they otherwise might.

Using just a few electronic components, and some simple electronic circuit design equations, these op amp circuits are easy to implement.

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