Differentiation is the mathematical opposite of integration, detecting the instantaneous slope of a function.
We took a look at op amp integrators in the previous article, Op amps do integration, so it makes sense to round out the picture by covering differentiator circuits. Of course, differentiation is the mathematical opposite of integration, detecting the instantaneous slope of a function.
In the earlier article, we showed that the transfer function for the circuit in Figure 1 is:
Figure 1 This is the generic op amp inverter circuit.
An inverting differentiator (Figure 2) can be created by substituting a feedback resistor R2 for Z2 and using an input capacitor C1 for Z1, giving this transfer function:
The operator s represents differentiation, so taking the inverse Laplace transform provides this time domain function for the circuit:
The output voltage of the circuit is equal to the derivative of the input voltage scaled by −R2C1.
Figure 2 Here is the classic operational amplifier differentiator circuit, with inverting output.
The time domain operation of the differentiator is shown in Figure 3. The bottom waveform is a square wave that is applied to the input of the circuit. The rise time of the square wave is set at 10 ns to avoid sending the differentiator output to infinity. (The output may still hit the power supply rails, depending on the specific component values used.)
The upper waveform in Figure 3 is the output of the differentiator, which has a positive or negative spike when the input waveform transitions. Note that the differentiator circuit is inverting, so positive square wave transitions cause a negative output spike.
Figure 3 In this example of the time domain operation of the differentiator, the bottom waveform is a square wave input to the circuit and the top waveform is the resulting output voltage.
In the frequency domain, the amplitude of the transfer function is a straight line, increasing with frequency (Figure 4). The differentiator produces high gain at high frequencies, often creating high-frequency noise or instability.
Figure 4 The frequency response of the differentiator circuit (amplitude only) is a straight line, increasing with frequency.
The practical differentiator
One way to deal with the excessive gain and noise at high frequencies is to add an input resistor, R1 to the circuit (Figure 5). As the frequency increases, C1 begins to look like a short circuit and the amplifier gain reverts to being the ratio of R2 and R1. Figure 6 shows the gain flattening out at high frequencies. (This plot is for a circuit with R2/R1 = 1, resulting in a high frequency gain of 0 dB.) The idea is that the circuit has the straight-line differentiator response at low frequencies, but the gain is limited at high frequencies.
Figure 5 The practical differentiator circuit offers one way to deal with excessive gain and noise at high frequencies.
Figure 6 Adding input resistor R1 causes the frequency response of the circuit to flatten out at high frequencies.
Another design approach is to also add the feedback capacitor C2 to the circuit (Figure 5). This capacitor causes the frequency response to not just flatten out but to decrease at high frequencies. Again, the idea is to maintain the straight-line response at low frequency so that the circuit behaves like a differentiator while reducing the high frequency response.
Figure 7 Adding a feedback capacitor C2 provides additional roll off at high frequencies.
Analyzing the circuit, we find that the transfer function is:
The numerator of the function is sR2C1, the same as the basic differentiator circuit. There are two poles in the denominator, determined by R1C1 and R2C2. As before, the design goal is to have the circuit act like a differentiator at low to mid frequencies and then roll off the gain at higher frequencies. The choice of component values will depend on the specific application and the frequency range of interest.
Just a flexible inverting amplifier
Another look at this circuit reveals that it can also be thought of as an inverting amplifier that rolls off for low frequencies and for high frequencies. Put another way, it’s an AC-coupled amplifier with a lowpass filter applied. This is all going to depend on the component values chosen.
Figure 8 This is the frequency response of the circuit shown in Figure 5 with different component values.
Figure 8 shows the amplitude of the transfer function with a different set of component values: R1=R2= 1 kΩ and C1 = 10 μF and C2= 1 nF. These components set the frequency response to be flat from 100 Hz to 30 kHz, rolling off both the low-end and high-end responses.
The circuit shown in Figure 5 is quite versatile. Removing the capacitors causes the circuit to degenerate to the inverting amplifier configuration (see Common op-amp circuits). Leaving the feedback capacitor, C2, in place creates an integrator or a lowpass filter. As we’ve just seen, C1 can implement a differentiator or provide low frequency rejection for the inverting amplifier.
This article was originally published on EDN.
Bob Witte is President of Signal Blue LLC, a technology consulting company.