Analog frequency-selective filters are useful for noise reduction, antialiasing before digitizing a signal, frequency response correction, and more. In this article, Circuit Cellar columnist Robert Lacoste explains the differences between filters and how to design them with computer-aided tools.
Welcome back to the Darker Side. I spoke about operational amplifiers (op-amps) in my last few columns. Op-amps shine in plenty of applications—in particular, to build active filters. This month, I’ll focus on filters—more precisely, analog frequency-selective filters, which are used in audio devices, as well as for noise reduction, antialiasing before digitizing a signal, separation of frequency-multiplexed signals, frequency response correction, and so on.
So analog filters must be in the bag of tricks of any designer. Unfortunately, filter design, or even their use, is often perceived as a difficult task close to black magic. This is, well, unfortunate. Filters are definitively useful, simple, and even fun. I bet a textbook about filters full of math would bore you, right? Well, relax. My goal for this article is more pragmatic. I will try to help you to specify a filter, understand the main filter variants, and efficiently use some great computer-aided design tools. I promise, no Laplace transforms or poles or zeros, just electronics.
Let’s start with some vocabulary. By definition, a filter is a circuit that attenuates some signals more than others, depending on their frequency. Figure 1 depicts the most classic filter types. A low-pass filter lets the low frequencies pass through, but attenuates high-frequency signals. It is perfect for removing high-frequency noise on a signal coming from a sensor.
FIGURE 1: Four classic types of frequency filters. Each one attenuates a specific frequency range. Click image to enlarge.
Conversely, a high-pass filter attenuates the low frequencies, and could in particular remove any DC component of a signal. Band-pass filters are a combination of both, and they attenuate all frequencies below or above a given range. For example, any radio frequency receiver is a band-pass filter, providing attenuation of all signals except for frequencies close to its preset frequency. Lastly, a band stop filter, often called a notch filter, does the opposite, and it attenuates a selected range of frequencies. For example, a 50- or 60-Hz notch filter is included in virtually every weight scale to remove EMC perturbations from the surrounding power lines.
Want to specify a filter? Figure 2 illustrates this on a low-pass filter. The first parameter is the filter cut-off frequency, of course. By definition, this is the frequency at which the filter attenuates the power of the signal by 50%. This means that the losses of the filter will be 3 dB at that frequency. Aren’t you fluent with decibels? A decibel is one tenth of a Bel, and a Bel is the base-10 logarithm of the ratio of two powers. Take your calculator and enter 10 × log(0.5), you will get –3.01, which everybody rounds to –3 dB.
FIGURE 2: A filter (low-pass in this case) is specified by its cutoff frequency f3dB, its ripple in the pass-band, and its rejection in the stopband. Click image to enlarge.
But perhaps an attenuation of 3 dB is already too much for your application. The maximum tolerated variation of signal power in the pass-band (here from DC to fPB) is called the ripple of the filter. Lastly, you will very probably want to specify that the filter must provide a given minimum attenuation, called rejection, above some frequency fSB. Of course, these specifications must be established with care. If you decide that you need a filter with 0.01 dB of ripple up to 10 kHz and 100 dB of rejection from 11 kHz upward, you will probably need plenty of time and cash for the design.
I propose to start with the most basic designs: RC filters. The basic low-pass filter is built with one series resistor and one capacitor to ground (see Figure 3). The capacitor impedance gets lower when the frequency increases, and the signal power is attenuated. This filter is called a first-order filter, and it provides an attenuation of 6 dB per octave or 20 dB per decade. (. Simply because 23.33 = 10, and 3.33 × 6 = 20.) That means that, above its cut-off frequency, its attenuation is increased by 6 dB each time the frequency is doubled, or by 20 dB each time it is multiplied by 10. I did the simulation for you with Labcenter Electronics Proteus. Figure 3 shows the result. You can do the same with any Spice-based simulator like the free LT-Spice. The attenuation of this RC filter is –20 dB at 100 kHz, and 20 dB more, meaning –40 dB, at 10 × 100 kHz = 1 MHz as expected.
FIGURE 3: A first-order RC filter (top) provides an attenuation of 20 dB/decade (green curve), whereas a second-order filter provides 40 dB/decade (red). Click image to enlarge.
Such a RC filter can be designed for any cutoff frequency. Just select the proper values for R and C. You might wonder how to calculate the values of the R and C. For a single RC cell, it is really easy. The cutoff frequency is 1/(2pRC).
If you want to increase the steepness of the attenuation, you can chain several RCs. For example, I simulated a second-order RC filter, with two RC cells in series (see Figure 3). As expected, the attenuation is now 12 dB (i.e., 2 × 6) per octave, or 40 dB (i.e., 2 × 20) per decade. Nothing magic. The 3-dB cutoff frequency is pushed downward as compared to a single RC cell, simply because at the 3-dB cutoff of each cell the attenuation is now 6 dB. However, you can see in the graph that even if the falloff in high frequencies is two times better, the drop around the cutoff frequency isn’t improved: it is still “soft.” That’s a limitation of cascaded RC cells. I will present you with a better solution.
Maybe a low-pass filter isn’t what you need. If you prefer a high-pass filter, then just exchange capacitors and resistors. A series capacitor and a resistor to ground would make it. Do you want a band-pass? Just put a low-pass cell in series with a high-pass cell with the appropriate cutoff frequencies. For example, a 10-to-50-kHz band-pass can be built with a 10-kHz high pass and a 50-kHz low pass. And for a notch filter? Do the same with the two filters in parallel. With the same example, a 10-to-50-kHz band stop may be implemented with a 10-kHz low pass and 50-kHz high pass in parallel. Easy.
For a given filter performance, is it possible to build a passive analog filter with fewer parts than a multicell RC filter and with improved performance? Yes, you can use LC filters. Here the filter is made with capacitors and inductors, as illustrated in Figure 4. How steep can be their attenuation profile? Very easy. It is the same in the case of RC filters. Count the number of capacitors, add the number of inductors, and you’ll get the order or the filter. Then multiply by 6 dB to get the attenuation per octave, or by 20 dB for an attenuation per decade! For example, the first filter simulated in Figure 4 has one inductor and one capacitor. Two parts, so it is a second-order filter, with the same 40 dB/octave attenuation as the dual RC example in Figure 3. The bottom example has three capacitors and two inductors; therefore, its attenuation is 100 dB (i.e., 5 × 5) per decade or 30 dB (i.e., 5 × 6) per octave. I promise. It’s simple!
FIGURE 4: This simulation shows the frequency response of three LC filters, respectively, of order two, three, and five from top to bottom. Their outputs are open-ended, which imply some overshoot. Click image to enlarge.
Well, nearly. Let’s now see the small details. If you refer back to Figure 4, you’ll see that such LC filters have a weird response around their cutoff frequencies. There is an overshoot, which means they have a positive gain at some frequencies. Of course, such passive filters can’t “create energy.” This positive gain is due to the fact that their output is open-circuited so no energy actually flows anywhere. Don’t be confused. This is not an artifact of the simulation. This would be exactly the same on an actual circuit. The amplitude of the overshoot is directly linked to the so-called quality factor of the L and C parts, and in particular their series resistance. If the capacitor and inductors are ideal, then the overshoot will be infinite at the frequency where the L and C oscillate. That’s why I added a small 47-Ω series resistor on the simulations. If you change the value of this series resistor, then the shape of the gain curve changes. I illustrated it in Figure 5 (top graph) which shows a series resistor ranging from 5 to 100 Ω.
FIGURE 5: The top simulation shows the frequency response of an open-ended LC filter with varying serial resistor value. The bottom simulation shows that the overshoot disappears when the filter is connected to a matched load. X varies from 5 to 100 Ω. Click image to enlarge.
How do you avoid such oscillations? Simply connect the filter’s output to a proper load. If you are a regular reader of my columns, you won’t be surprised: this load must provide an impedance matching with the source impedance. Look at the second example in Figure 5. I added a load resistor R3 of the same value as the source resistor R2 (denoted X). I then asked the simulator to show the resulting gain versus frequency graph with different values for these resistors X, ranging from 5 to 100 Ω again. The shapes are varying, but there are no overshoots. Moreover a precise resistor value provides a very clean and flat response, linked of course with the values of the L and C parts. This value, here 50 ohm, is the characteristic impedance of the LC filter.
So LC filters must be calculated to get the required frequency response but also taking into account the impedance of the load. For second order filters, using just one inductor and one capacitor, the calculation are straightforward. The cutoff frequency is f3dB = 1/[2p√(LC)], and the characteristic impedance is Z = √(L/C). If you know the required cutoff frequency and designed impedance, then you can easily calculate L and C from these two formulas.
The calculation is not so straightforward for higher-order filters, especially as the design choices are numerous. More on that below. Our ancestors used the abacus; now we can use web-based design tools. (Refer to the Resources section of this article for some links to free LC filter calculators.) There is even a great design tool from Coilcraft that allows you to directly order the samples of the required inductors with a mouse click. Easy, I promised.
FROM LC TO ACTIVE FILTERS
Using inductors often isn’t pleasant. They can be heavy and large, and they’re always significantly more expensive than capacitors and resistors. Moreover, inductors are often quite far from ideal components. They can have a high series resistance as well as parasitic capacitance, nasty electromagnetic compatibility behavior, and a couple of other issues. How can you keep the performance of an LC filter without using inductors? With an active filter, usually built around our dear friend the op-amp.
There are basically three ways to build an active filter. The first is to simply add an amplifier to the RC filters I’ve already talked about. For example, you can add a voltage follower after an RC cell in order to reduce its output impedance or to provide some gain. You can also wire an op-amp as a differentiator or integrator, which are first-order filters.
The second solution is to build a switched-capacitor filter circuit. (I devoted my Circuit Cellar 277 column to the subject.) So let’s talk about the third option, which is based on so-called gyrators. What is that? A gyrator is a circuit that mimics the behavior of an inductor, using an op-amp and only resistors and capacitors. You will find plenty of literature on the subject. Of course, this is explained in the bible, Paul Horowitz and Winfield Hill’s The Art of Electronics, but Rod Elliott provides a clear presentation on the subject in “Active Filters Using Gyrators – Characteristics, and Examples,” (Elliott Sound Products, 2014).
Look at Figure 6 where I have illustrated the basic concept. The top part of the schematic is a classic second-order LC high-pass filter with matched source and load impedances. I used a 390-nF capacitor and a 1-mH inductor, resulting in a cutoff frequency of 8 kHz and a characteristic impedance of 50 Ω—here roughly matched with source and load 56-Ω resistors.
FIGURE 6: This simulation shows the transformation of an LC high-pass filter (top) into a gyrator-based circuit (middle), which is really close to the common Sallen Key filter (bottom). Click image to enlarge.
The response curve shows noting surprising with a 40 dB/decade (i.e., 2 × 20) attenuation in the stopband. Its gain is –6 dB in the passband, as the voltage is divided by two due to the source and load resistors. (The power is divided by 22 = 4, giving –6 dB.) Now look at the middle section of the schematic in Figure 6. The circuit is exactly the same, but I replaced the inductor with an op-amp, a capacitor, and two resistors. That’s a gyrator. If you look now at the resulting graph, you will see that its frequency response is exactly the same as the LC version, at least up to 1 MHz where the characteristics of the op-amp start to be limiting.
Now another magic trick. Compare the gyrator-based schematic with the schematic at the bottom of Figure 6. If you move the parts and the wires around, you will see that they are exactly identical, except the output is now directly connected to the op-amp output. Do you recognize the new schematic? It is a Sallen-Key second-order active high-pass filter. I modified the part values to a more reasonable range, but you can see that the output frequency response is still the same. More precisely, it doesn’t suffer from the 6-dB losses as the signal is taken directly at the output of the op-amp. So Sallen-Key filters, gyrator-based filters, and LC filters are more than cousins.
If you want to design a single-cell filter, either a first-order RC filter, a second-order LC, or an active filter, then you will not have a lot of design choices. You can select the desired filter type, cutoff frequency, and impedance, but nothing more. However, for higher-order filters, the choices are wider. The filter is made of several cells, and you can tune each cell separately. Therefore, you will have a better attenuation curve thanks to the higher order (remember, 6 dB per octave multiplied by the order of the filter), as well as more control on the shape of the filter.
Nothing prevents you from designing your own filter, tweaking each cell however you want. However, mathematicians have already calculated several “optimal” filters for certain applications. Do you want to have a response curve as flat as possible in the passband? Stephen Butterworth calculated it for you in 1930. It’s now called the Butterworth filter, of course. Do you prefer to attenuate as quickly as possible the stop-band even if it implies a higher level of ripple in the pass-band? Use a Chebyshev filter, derived from the Chebyshev polynomials. More precisely, this is a family of filters based on the acceptable ripple (e.g., 0.5 dB). The so-called elliptic filters are close.
The last common variant, the Bessel filter, is a little more complex. A Bessel filter is not a great option both in terms of flatness and attenuation; however, it has a key advantage in the time domain. Its so-called group delay is nearly flat. That brings us a little too far here, but these characteristics preserve the shape of the filtered signals in the time domain. I will tackle that subject in another article.
Of course, each variant has drawbacks. For the same filter complexity, a higher ripple in the passband must be accepted to get a higher attenuation in the stop-band. Similarly, a better phase flatness implies a worse frequency response. Life is difficult, but you are the designer, so you have the control. Figure 7 shows the characteristic responses of each filter variant. For more information, I strongly encourage you to have a look at the “Analog Filters” chapter in Hank Zumbahlen’s Linear Circuit Design Handbook (Analog Devices, 2008).
FIGURE 7: These plots show the typical frequency and time (step and impluse) response of the three most common filter variants. Click image to enlarge. (Source: Linear Circuit Design Handbook, Analog Devices)
So you have plenty of options when designing a filter. Fortunately, there are great computer-based design tools made for the design engineer. Some are expensive, but plenty are free. In particular, several op-amp suppliers offer filter design tools for their products. I like Analog Devices’s Analog Filter Wizard (www.analog.com/designtools/en/filterwizard/). It’s powerful and doesn’t require a PC installation. Other solutions include Texas Instruments’s Webench Filter Designer, Microchip Technology’s FilterLab, Linear Technology’s FilterCAD, and some others.
FIGURE 8: With a tool like the Analog Filter Wizard (Analog Devices), life is easy. Click image to enlarge.
As an example, Figure 8 shows a typical session with Analog Devices’s Analog Filter Design. Basically, you start by selecting the filter type (here a low-pass), the required gain in the pass-band, the cutoff frequency, and the attenuation you want at a given stop-band frequency. A slider enables you to browse through several designs—namely, Chebyshev, Butterworth, and others. The next window enables you select the desired tolerance for the capacitors and resistors and actually draw the filter’s full schematic (of course using an op-amp from the supplier who offered the tool). Lastly, the resulting frequency, phase, and time plots are generated, taking into account the tolerance of the parts. Other options enable you to calculate the power consumption of the design or its noise figure. Of course, the beauty of such a tool is that you can try tens of designs in minutes and select the most adequate for your specifications and budget.
Here we are. As always, I have only scratched the subject’s surface. Anyway, I hope you grasped the key concepts. Go through the content listed in the Resources section of this article, and don’t forget to practice on your own. Maybe you should stop reading this magazine now (don’t forget to come back to the issue later), download one of the filter design tools, and play with the settings. It would be the best way to really understand the difference between a fourth-order Butterworth filter and a third-order Chebyshev filter. Have fun and don’t be afraid of filters.
Robert Lacoste lives in France, near Paris. He has 25 years of experience in embedded systems, analog designs, and wireless telecommunications. A prize winner in more than 15 international design contests, in 2003 he started his consulting company, ALCIOM, to share his passion for innovative mixed-signal designs. His book (Robert Lacoste’s The Darker Side) was published by Elsevier/Newnes in 2009.