How to choose analog-signal-chain components: part 3

Use op amps, resistors, and capacitors to build high-pass, low-pass, and bandpass filters.

In part 1 of this series, we looked at a noisy signal and discussed why it might be better to use an analog filter instead of a digital filter. Then, in part 2, we looked at the operational amplifier (op amp) and described it as the basic building block of the analog signal chain.

Q: Can we use an op amp to build a filter?
A: Yes. Figure 1 shows how we convert our voltage follower from part 2 into a simple low-pass filter (LPF) by adding a resistor and capacitor.

Figure 1. An RC network changes a voltage follower into a low-pass filter.

Q: How do we set the filter cutoff frequency?
A: Note that the resistor and capacitor form a voltage divider, and our filter cutoff frequency f_C occurs when V₁ is half of V_IN. That occurs at the frequency at which R equals the absolute value of the capacitor’s impedance Z_C, where

If we equate R with Z_C, we can calculate f_C:

Let’s say we want a cutoff frequency of 20 kHz. We can choose 1 kW for R, and we can calculate C:

Q: What’s the transfer function of the Figure 1 circuit?
A: To plot the frequency response of our filter, we first need to calculate V₁ = V_OUT as a function of frequency. We know that I_IN equals V_IN divided by the series combination of R and Z_C. Because the total series impedance has orthogonal real and reactive components, we calculate it as follows:

I_IN is simply V_IN divided by the total series impedance:

Then, we can calculate V_OUT, which equals V₁ because of our unity-gain voltage follower configuration:

Now we can derive our transfer function:

We can also express the gain in dB:

We can plot the transfer function in dB as a function of f and get the Bode plot shown by the blue trace in Figure 2, where the gain is down 3 dB at our 20-kHz cutoff frequency.

Figure 2. You can use an op amp, resistor, and capacitor to build a low-pass filter (blue), high-pass filter (red), or bandpass filter (dashed curve).

Q: How do we build a high-pass filter?
A: We simply switched the locations of the resistor and capacitor in Figure 1, as shown in Figure 3. Note that the DC gain of the high-pass filter is 0, and as frequency increases, the gain approaches 1. If we keep the numerical values for R and C the same as for the Figure 1 example, we will retain the same 20-kHz cutoff frequency, as shown by the red trace in Figure 2.

Figure 3. Switching the locations of the resistor and capacitor in Figure 1 creates a high-pass filter.

Q: Can we create a notch filter using these building blocks?
A: We can put a low-pass filter in series with a high-pass filter to create a notch, or bandpass filter. If we put our Figure 1 and Figure 3 circuits in series but increase the cutoff frequency of our low-pass filter to 100 kHz, we get the response shown by the dashed trace in Figure 2.

Q: Why haven’t we used inductors in any of these filters?
A: We could! We could use inductor-resistor, inductor-capacitor, or inductor-resistor-capacitor combinations. But generally, modern designers try to avoid inductors because they tend to be larger than the alternatives, especially at lower frequencies, and they tend to exhibit the worst aspects of the alternatives — they have parasitic winding resistance and can have parasitic interwinding capacitance. A good resource on this topic is the Handbook of Operational Amplifier Active RC Networks [1], written in 1966 and revised in 2001! So, 59 years after being written and 24 years after the last revision, it’s still relevant.

Q: What else should we know about op-amp filters?
A: The simple examples presented here are first-order filters. If you look at the right-hand portion of the blue trace in Figure 2, you’ll see that it approaches a roll-off, or linear decrease after the cutoff frequency, of 20 dB per decade — that is, for every tenfold increase in frequency, the gain decreases 20 dB. The inverse holds for the left-hand tail of the high-pass filter response. Often, we want a sharper roll-off, which we can achieve with higher-order filters. We’ll take a look next time. Also, we have been looking at amplifier circuits with voltage inputs and outputs, so we will conclude with a brief look at current-input or output devices, called transimpedance and transconductance amplifiers.