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Circuit Surgery Regular clinic by Ian Bell Measuring circuit frequency response using a PC sound card, part 11: anti-aliasing & reconstruction filter circuits O ver the past few months, we have been looking at circuit frequency response, including implementing a measurement system using the ‘sound card’ (audio interface) of a PC or laptop together with free software called REW (Room EQ Wizard). This was motivated by the need to measure the response of a practical implementation of an example digital filter, from our series on digital signal processing (DSP), without using advanced lab test equipment. We have described various aspects of the signal conditioning circuitry that might be needed to interface the circuit under test to the sound card’s line inputs and outputs. As noted, the inspiration for the measurement system was a DSP system, and in general, DSP circuits need analog anti-aliasing and reconstruction filters. The requirements for these filters were discussed in depth in the DSP series, but we did not look at specific circuit designs then. Our simulations used idealised behavioural anti-aliasing and reconstruction filters where needed, as we were focusing on the digital aspects of the system. The signal conditioning circuit for the measurement system did not include a filter, as this was considered part of the circuit under test (the digital filter system). However, an op amp based reconstruction filter was designed specifically for use in the test setup. This month, we will look at analog filter design with this reconstruction filter as an example. First, we will quickly recap anti-aliasing and reconstruction filters, then cover some key basic analog filter circuit concepts using single op amp stages. As with other recent articles, our discussion will have wider applicability than just the frequency response Analog input Antialiasing filter measurement and DSP systems, with the reconstruction filter used as an example. Anti-aliasing and reconstruction filters DSP systems often require analog filters at their inputs and outputs (see Fig.1) to prevent signal degradation due to the fundamental mathematical properties of the DSP system. Most DSP systems take samples of their analog input signal at a fixed rate (the sampling frequency or rate, fs) and convert these samples to digital data for processing. The system may have an adjustable sampling frequency, but generally the processing operations occur at fixed intervals once a specific rate is set. Mathematical analysis of the process of sampling a signal shows that signals with a frequency of less than half the sampling frequency are reliably represented in the sampled data. This is known as the Nyquist sampling theorem. The theory shows that if the input does not contain any frequency components at or beyond half the sampling frequency (fs/2, known as the Nyquist frequency), it is possible to perfectly reproduce the original signal from the sampled data. If the signal contains frequencies at or above fs/2, an effect known as aliasing occurs. These frequencies become indistinguishable from other frequencies in the range up fs/2, which means that the sampled signal is not an accurate representation of the original signal. So, if the input signal contains frequencies at or above fs/2, these must be removed by an anti-aliasing filter before the signal is sampled to avoid affecting the resulting samples. How demanding this filtering is depends on how close the maximum Conversion and digital processing Fig.1: the role of analog filters in a DSP system. 40 Reconstruction filter Analog output wanted signal frequency is to fs/2 and the amplitude of any unwanted signals at or just above fs/2. Demanding conditions require a filter which has a very sharp cut-off close to fs/2, with a gain that rapidly drops to a very low value. The demand on the filter can be reduced by increasing the sampling frequency so that fs/2 is further from the maximum wanted signal frequency. In that case, the cut-off doesn’t have to be as sharp to avoid affecting the wanted signals while heavily attenuating the unwanted ones. The DSP system under test in the frequency response measurement system did not include an anti-aliasing filter because the signal generation system was not configured to produce frequencies above the Nyquist frequency. The spectrum (frequency content) of a sampled signal comprises images of the original signal repeated at multiples of the sampling frequency. At the top of Fig.2 is the spectrum of a signal sampled at fs = 50kHz where the original signal has a frequency content from DC up to a maximum of 20kHz. The images occur centred on 50kHz, 100kHz and other multiples of 50kHz (not shown). When the sampled signal is converted to analog form using a DAC, Sampled signal –2fs –fs –100 –50 xs(f) 0 –50 fs 2fs 50 100 f/kHz 50 100 f/kHz xc(f) After reconstruction filter –100 Ideal filter response 0 Fig.2: the action of a reconstruction filter in the frequency domain. Practical Electronics | August | 2026 the image frequencies will be present in the DAC output signal. To correctly recover only the wanted signals , it is necessary to remove image frequencies using an analog reconstruction filter. An ideal filter with a cut-off of fs/2 will achieve this (see Fig.2). If the filter is able to perform this task perfectly, its output will only contain the wanted signals (the lower spectrum in Fig.2). In simple terms, you can think of this as the filter smoothing out the stepped output from the DAC. Note that negative frequencies are also shown in Fig.2, which is usual when analysing DSP systems and related to the mathematics involved. In this case, the negative part of the spectrum is just a reflection of the positive part. Also, the output spectrum of a DAC is not exactly the same as that of the set of samples (as shown in Fig.2) due to its frequency response, which follows the sinc mathematical function. These and other details of sampled signals, the requirements of anti-aliasing and reconstruction filters, and many other aspects of DSP were covered in depth in our series on the topic. Our initial testing of the DSP hardware implementation used a simple RC filter, which was sufficient for basic checking but had some impact on the overall frequency response. To prevent this influencing the results with the PC-based measurement system, it was replaced with an active filter that had a flat response in the frequency range used for testing. The filter cut-off was 20kHz and the DSP filter’s Nyquist limit was 24kHz. Bandforms and filter features In simple terms, a filter is a circuit that passes certain frequencies (in the passband) and rejects others (in the stopband). Based on this definition, there are four basic classes of filter, referred to as bandforms, which are shown in idealised form in Fig.3. These are graphs of the filter’s frequency response, plots of gain versus frequency. Low-pass filters let low frequencies through and block high frequencies. High-pass filters let high frequencies through and block low frequencies. Bandpass filters let a specific range of frequencies through. Bandstop filters reject a specific range of frequencies. Some other terms are used for specific cases. A notch filter is a bandstop filter with a very narrow stopband, which can be useful for rejecting a specific unwanted frequency. A comb filter has multiple, regularly spaced stopbands and passbands. In addition to these frequency-­ dependent gain responses, there are Practical Electronics | August | 2026 Gain Low-pass Filter Gain Bandpass Filter Gain High-pass Filter Gain Bandstop Filter Frequency ( f ) Frequency ( f ) Frequency ( f ) Frequency ( f ) Fig.3: basic filter bandforms. Real filters don’t have such steep transitions! also all-pass filters which have frequency-dependent phase shift (delay) but equal gain at all frequencies (see the recent Audio Out articles by Jake Rothman for more on these circuits). Anti-aliasing and reconstruction filters are low-pass filters, so we’ll concentrate on those in this article. However, many of the points we discuss will be relevant to other types, as are the tools which can be used to facilitate their design. The idealised (so-called ‘brick wall’) filters in Fig.3 cannot be implemented. For real filters, the transition from the passband to the stopband occurs over a range of frequencies. Furthermore, the gain in the passband may not be constant (as it is implied by Fig.3) and the stopband gain will not be either zero (complete rejection) or constant at some low level. Fig.4 shows a low-pass response with features that occur in real circuits. Without a perfectly sharp cut-off, we need to specify exactly what we mean by the cut-off frequency. For a flat passband, the cut-off is often defined as the frequency where the filter’s gain is -3dB with respect the passband gain. This is Gain (dB) the point where the output signal has half the power it does in the passband. For filters with passband ripple, the cut-off frequency may be defined as the frequency where the response has dropped by an amount equal to the ripple size relative to the maximum passband gain. However, it may also be defined in terms of the gain relative to that deep in the passband (for example, very low frequencies in the case of a low-pass filter). When looking at filter specifications or design tools, take care to check how the cut-off frequency is defined. To design a filter, we need to specify our requirements with respect to the features shown in Fig.4. By defining the cut-off frequency and gain at cutoff, the stopband frequency and gain at this frequency, and the passband gain and ripple, we effectively define a region of the response graph into which an acceptable response curve must fit (the green area in Fig.4). Filter design tools often require these values to be entered directly, although the ripple and stopband values may be determined by other choices; for Passband Passband gain Passband ripple Transition region Stopband gain Roll-off gradient (dB/decade) Stopband Stopband ripple Cutoff frequency ( fc) Stopband frequency ( fs) Frequency ( f ) (log scale) Fig.4: features of filter frequency response curves. 41 example, if a specific response type (see below) is selected. Some tools display graphics similar to Fig.4 to provide feedback to the user as values are entered. On filter response plots, the gain or attenuation is usually plotted in decibels, which is a logarithmic scale. For a voltage gain A, the value in decibels is 20log10(A)dB. The frequency axis of the graph is also usually logarithmic with the scale marked in multiple-often steps, referred to as decades. However, linear scales are also used when appropriate, to emphasise specific features or when the range plotted is small (eg, for narrow band filters). Response types There are various standard filter response types available, named after their related mathematics. Widely used responses include Butterworth, Bessel and Chebyshev, but these are not the only ones. The features shown in Fig.4 vary with different response types, and there is a trade-off in terms of desirable features. Various common filter responses are shown in Fig.5 and Fig.6. These all have a cut-off frequency of 20kHz, defined here as being the point the gain falls to 3dB below the low frequency gain (DC gain), which in this case is 0dB. Butterworth filters have maximally flat passbands with a relatively slow gain change near the cut-off. Chebyshev filters have passband ripple, but their gain decreases more rapidly near the cut-off. The amount of ripple can vary, with higher ripple giving faster cut-off (Fig.5, which shows examples with 3dB and 1.5dB ripple). Bessel filters have maximally flat delay, which is equivalent to a nearly linear increase in phase shift with frequency across the passband (see Fig.6), resulting in minimal distortion. Fig.6 is plotted using a linear frequency axis so that the linearity (or otherwise) of the phase responses is clear. Although we have named certain response types, in practice, a continuum of responses is available. Chebyshev filters with very small ripple are similar to Butterworth filters, and responses in between pure Butterworth and Bessel are available too. The roll-off, or slope, of the frequency response in the transition region/stop band (see Fig.4) indicates how quickly the filter’s gain drops as the frequency moves away from cut-off. Roll-off is usually measured in dB per decade and typically changes with frequency near the cut-off. However, for frequencies well into 42 Fig.5: the magnitude response for various second-order 20kHz low-pass filters. Fig.6: the phase response of various low-pass filters in the 20kHz passband, shown on a linear frequency axis. the transition region/stop band, the response of many filters tends to decrease as a straight line when plotted on a graph of decibels against log frequency, as in Fig.5. The ultimate roll-off is determined by the order of the filter; the higher the order, the faster the falloff. A single RC filter is ‘first order’ and rolls off at 20dB/decade. An Nth-order filter rolls off at 20(N)dB/decade. The order of the filter determines the ultimate roll-off, but the response type (Butterworth, Chebyshev etc.) determines the shape of the response near the cut-off. All the filter responses shown in Fig.5 are second-order types (N = 2) with an ultimate roll-off of 40dB/decade. The response curves run in parallel at higher frequencies, which is more obvious if you plot a larger frequency range than in Fig.5. Resonance A value called Q, or quality factor, of a filter determines the response type (the shape of the frequency response). For the responses shown in Fig.5, the Q values are 0.577 (Bessel), 0.707 = √2 ÷ 2 (Butterworth), 1.04 (Chebyshev 1.5dB ripple) and 1.3 (Chebyshev 3dB ripple). Higher Q values result in larger peaks in the response. Low- and high-pass filters typically have relatively low Q values, while bandpass filters (particularly narrowband types) have higher Q values. The properties of many filters are related to resonance. Resonance is often experienced in the tendency of some systems to respond at greater amplitude when applied oscillations match the system’s natural frequency of vibration (the resonant frequency). Often quoted real-world examples Practical Electronics | August | 2026 Fig.7: an RLC second order low-pass filter. Vin R1 L1 Vout C1 include playground swings (you have to push at the correct rate) or wine glasses shattered by singers hitting just the right note. Resonance involves the exchange of energy between two different storage mechanisms. For mechanical examples, like the glass and swing, these are potential and kinetic energy. At the resonant frequency, the energy exchange is at its most efficient, which means that minimal energy is lost in the transfer. This means that any energy input by a driving force (the sound waves from the singer or the person pushing the swing) produces the greatest amplitude of oscillation when it is at the resonant frequency, because relatively little energy is dissipated in each cycle. Q is a measure of how efficient the energy exchange at resonance is. It is proportional to the energy stored divided by the energy dissipated in a cycle. A higher Q means lower dissipation at resonance and hence a higher amplitude peak for a given driving force or input. RLC circuits (for example, Fig.7) are a well-known way to implement filters. They are resonant, with the energy exchange being between that stored in an electric field (in the capacitor) and a magnetic field (in the inductor). The resistance affects the Q – higher resistance means more energy is dissipated in each cycle, so a lower Q. RC and RL circuits can also be used as filters, but are not resonant, as they only have one storage mechanism. It is possible to find a Q value using an energy stored to dissipated ratio, but it is low (0.5 or less) and, unlike for an (ideal) RLC circuit, varies with frequency. This means that we cannot make sharp cut-off filters with just RC or RL circuits. Active filters built from op amps resistors and capacitors (for example, Fig.8) lack the magnetic field storage provided by inductors in RLC circuits, but they are able to behave in ways that are mathematically equivalent to RLC resonant circuits (the circuits in Fig.7 and Fig.8 can provide equal responses). They are therefore able to achieve high Q values if required. The name ‘operational amplifier’ (abbreviated to ‘op amp’) comes from their ability to implement mathematical operations – they were widely used in analog computers to solve mathematical problems before the advent of digital computers. For bandpass filters, Q is the ratio of the bandwidth relative to the centre frequency. If a band-pass filter has a centre frequency of f0 and a -3dB bandwidth of fb then Q = f0 ÷ fb. For low-pass filters, we typically require relatively low Q values because higher Q values result in a peak around the cut-off frequency (for example, the Chebyshev responses in Fig.5). Usually, we would not want this peak to be very high. The situation is the same for high-pass filters. For these filters, the damping factor (ζ = 1 ÷ 2Q) is often used instead of Q, as it is perhaps more intuitive. Higher damping results in a slower, smoother transition from the passband to the stopband. Resonant and cutoff frequencies The frequency of the peaks seen in Fig.5 is related to the resonant frequency of the filter, but filters using resonant circuits (or mathematical equivalents) without peaks in their responses still have a resonant frequency. Low Q (high damping) produces responses without peaks, but the resonant frequency and Q still determine the response shape. The resonant frequency is a fundamental property of the filter, related to the mathematics of the filter’s function. The cutoff frequency is effectively arbitrarily defined, so they are not the same thing. Fig.9 shows plots of the same filter response types as in Fig.5 (with the same scales and trace colours), but with each having the same resonant frequency (20kHz), rather than the same -3dB cut-off. The cut-off frequencies for the responses in Fig.9 are not the same as those in Fig.5, except for the Butterworth response, where both have cut-off frequencies at 20kHz (equal to the resonant frequency). The responses in Fig.5 are the same at frequencies a long way from cut-off (both high and low), demonstrating the close mathematical relationship between filters with the same resonant frequency but different Q values. These responses differ around the cut-off point. The filters with higher Q have more peaking and a faster roll-off near their cut-off frequency. For a particular response type, order and definition of cut-off frequency (fc), there will be a fixed relationship between the resonant frequency (f0) and cut-off frequency that can be expressed as a frequency scaling factor (Fs ), such that f0 = Fs fc. The scaling factors are typically, but not always, in the range of 0.2 to 2. For Butterworth filters with the cutoff defined as -3dB, the frequency scaling factor is 1; hence the equal responses in Fig.5 and Fig.9. Analog filter circuit design There are many different ways to implement analog filters; the best approach will depend on your requirements. As is usually the case in C1 Input R1 R2 + C2 Output – 0V Fig.8: an active second-order low-pass filter (unity-gain Sallen-Key type). Practical Electronics | August | 2026 Fig.9: the magnitude response for various second-order low-pass filters with a 20kHz resonant frequency. 43 engineering, the better you understand and define what is needed, the more likely the outcome is to be successful. The bandform and cut-off frequency are usually straightforward to determine from the application. The order, response type (Butterworth, Chebyshev etc) and circuit implementation choices will be based on the frequency and performance requirements. For example, how close are unwanted signals to wanted ones, how much do they need to be attenuated, and how important is flat passband gain? One key decision is whether to use an active filter (op-amp-based) or a passive one (using just resistors, capacitors and inductors). For undemanding requirements, passive RC filters may suffice. RLC filters can deliver high performance, but at low frequencies the inductors required may be bulky and expensive. Active filters are therefore often the best choice at lower frequencies (eg, audio), whereas at very high frequencies (RF), RLC filters may be a better choice because the inductors will be small and it could be difficult to find op amps with sufficient bandwidth. For active filters, there are a variety of circuits that can be configured to implement a desired response; for example, Sallen-Key, multiple feedback (MFB) and state-variable filters. These have different non-ideal characteristics, which might need to be taken into consideration and balanced against factors such as cost and complexity. Filters can be designed from first principles, based on mathematical analysis of the chosen circuit and the filter response required. This can be time-consuming and challenging if you are not already familiar with all the theory. Fortunately, this is usually not necessary because standard filter circuits are so widely used that design formulae are readily available. There are even design tools that will do all the calculations for you. Even before the wide availability of software tools, filter design was eased by printed tables or graphs from which relevant values (typically Q and Fs ) could be looked up and used, along with the use of straightforward formulae. A good example of this is the Active Filter Cookbook by Don Lancaster, which was first published in 1975 and is described as the best-selling book on active filter design. As well as providing design tables, the book also discussed the theory behind the circuits in detail. I have a well-used copy from the 1980s. Another useful source is the “Active 44 Fig.12: Sallen-Key and MFB filter responses (2nd-order, 20kHz, Butterworth). Low-Pass Filter Design” application note by Jim Karki from Texas Instruments (SLOA049, see www.ti.com/lit/ pdf/sloa049). Several semiconductor manufacturers provide online tools for active filter design. There is the Analog Filter Wizard from Analog Devices (https:// tools.analog.com/en/filterwizard/) and FilterLab Active Filter Designer from Microchip (https://pemag.au/link/acc3), both of which provide low-pass, highpass and band-pass designs. The Filter Design Tool from Texas Instruments (https://pemag.au/link/ acc4) also provides bandstop and allpass designs, but unlike the other two, requires an account login. These tools allow you to input the filter specifications (as discussed above) from which they will produce schematics with all component values calculated for you. Of course, they will use the manufacturers’ own op amps, but other similar devices can be used instead. Free online tools tend to provide a limited range of implementation and response options; still, they are very useful for many applications. Circuit choices Two widely used active filters are the Sallen-Key and MFB circuits. These are both second-order filters equivalent to an RLC filter (eg, Fig.7) using a single op amp. They are popular because they provide reasonable performance at relatively low complexity and component count. We will show some low-pass versions here, but other bandforms can be obtained by switching the positions of resistors and capacitors in the circuit. Fig.8 shows a unity-gain Sallen-Key filter, which is a simplification of the Sallen-Key filter shown in Fig.10. The frequency-dependent circuit is similar to a two-stage RC filter, but with C1 connected to positive feedback from the amplifier, rather than to ground. This allows the circuit to have much higher Q values than can be obtained with the RC filter, which, as already noted, has a maximum Q of 0.5; it is often lower. The version in Fig.10 is more complex but can provide gains other than unity. This circuit has an advantage for higher-­ Q filters, where (due to the way the maths works), the unity gain circuit can require impractical ratios between the capacitor values (this means you would need to use either an excessively large or excessively small value for one of the components). The ability to implement non-­unity gain for the Sallen-Key circuit in Fig.10 may be directly useful in some applications (eg, it can save another op amp that would be required for a separate gain stage). On the other hand, the unity-gain version has the highest gain accuracy because it does not depend on the accuracy of any gain-setting resistors (R3 and R4 in Fig.10). Fig.11 shows an MFB filter – its name comes from the fact that there are two feedback paths in the frequency-setting C1 Input R1 R2 + C2 Output – 0V R3 R4 Fig.10: a 0V Sallen-Key second-order low-pass filter. Practical Electronics | August | 2026 R4 f 0= network (R3 and C2 in Fig.11), whereas the Sallen-Key filter only has one (C1 in Fig.8 and Fig.10). Sallen-Key filters tend to be easier to design (manually) than MFB filters, which accounts for their popularity. The MFB circuit has better performance at high frequencies, is better suited to higher-Q circuits, and is overall less sensitive to component variations. However, the unity-­gain Sallen-Key is the least sensitive to gain variations. If the component values in a filter circuit do not exactly match the designed values, it will deviate from the desired response. This is inevitable because components are available in limited sets of values (E12, E24 series etc), which usually do not match the required values, and individual components vary within their tolerance range and with temperature. The degree to which a circuit deviates from its designed performance in response to a given amount of component variation is termed sensitivity and can be subjected to detailed mathematical analysis. It can also be investigated in simulation by varying components within their tolerance range and obtaining an envelope of possible responses. Sensitivity varies with filter implementation; for higher-order and higher-Q circuits, the deviations may be very significant. Single op amp filters such as the Sallen-Key and MFB circuits do not provide the lowest possible sensitivity. If they cannot meet requirements, filters using several op amps to implement a second-order response (eg, state-variable filters) need to be used instead. Sallen-Key circuits are non-inverting, whereas MFB filters are inverting. Sallen-Key low-pass circuits have a high input impedance at DC because the input is fed to the non-inverting input, whereas the input feeds the virtual ground at the inverting input in the MFB filter, so the input impedance is less than R1 + R2 (due to C1’s finite impedance and feedback via R3). Both filters have non-ideal performance at high frequencies, where their gain does not continue to decrease as frequency increases in line with an ideal second-order response. R3 Input R1 C1 R2 C2 + – 0V Fig.11: a multiple-feedback (MFB) second-order low-pass filter. Practical Electronics | August | 2026 Output However, the Sallen-Key filter is significantly worse in this respect, as shown in Fig.12. This shows the responses of a 20kHz, 2nd order, low-pass Butterworth filter implemented with both Sallen-Key (green trace) and MFB (blue trace) circuits. The gain of the Sallen-Key filter in Fig.12 increases above about 1MHz, whereas the MFB circuit’s gain continues to decrease before levelling off around 5MHz. The problem is caused by the increase in output impedance as frequency increases. As the frequency increases, the op amp’s open-loop gain reduces, decreasing the ability of the negative feedback to reduce the effective output impedance. The op amp then struggles to drive the output towards 0V (which is what the output should be with high attenuation in the stop band) against the input signal feeding through to the output via the resistors and capacitors. The MFB filter has a less direct, higher impedance path from input to output, and C1 helps to shunt more input signal at higher frequencies, so it suffers less from this effect. A large number of op amps are suitable for use in Sallen-Key and MFB circuits, but not all op amps will suit every filter. In general, the op amp should have a low input bias current to prevent DC errors, sufficient slew rate to handle the maximum amplitude required at the highest pass-band frequency required, and a low output resistance to limit the non-ideal high-frequency behaviour just described. A key op amp characteristic is the gain-­bandwidth product (GBP), which must be much larger than the filter cut-off. A rule of thumb is that GBP > 100Ap⸳fC, or GBP > 100Ap⸳Q 3⸳fC for Sallen-Key filters with Q > 1, where Ap is the passband gain. Design tools will often assist with op amp selection. A Sallen-Key low-pass filter As an example of the manual approach to filter design, we will look at the Sallen-Key low-pass filter shown in Fig.10. It comprises a frequency-­ dependent circuit formed by R1, R2, C1 and C2, plus a non-inverting op amp configuration. The equations needed for the design are widely published. The amplifier gain (A) is set by R3 and R4 and is given by: R A=1+ R 33 A=1+ R 4 R4 The resonant frequency, f0, is: 1 1 f 0= f 0= 2 π √ R1 R2 C 1 C 1 2 π √ R1 R2 C 1 C 1 C √√ RR11 RR22 C C 11 C 11 Q= Q= R 1 C 1 + R 2 C 1 + R 1 C 2 ( 1 − A ) R1 C 1 + R2 C 1 + R1 C 2 ( 1 − A ) 1 The Q is: 1 2 π √ R1 R2 C 1 C 1 √ R1 R 2RRC31 C 1 3 A=1+ R 1 C 1 +A=1+ R 2 C 1 +RR 1 C 2 ( 1− A ) R4 These equations are 4quite cumber1 1can be made some, butFsimplifications s f c =f 0 =1 f = f 00 =constraints2 on by placing the design, R 2πRC C C 2 √ RRgain 1 C 11unity C 11 (as in 2π πthe 1 R2 such as fixing at 1 Fig.8). Another Sallen-Key equaQ= R C √ way R 113R R−22 C C tions can be simplified A 11 Cis11 by using Q= Q= = ))R2 equal component R1 A R 11 + R 11 C C 11 + +R R 22 C Cvalues, +R R 11 C CR22 ((=1− 1− A Q= and C = C1 = C2, which gives: 11 F F ss ff cc =f =f 00 = = 2 πRC 2 πRC 1 1 Q= Q= 3 − A 3− A This provides very simple design equations but does not allow the overall gain of the filter to be set independently of the Q value. Still, the overall gain can be adjusted by adding a suitable gain stage. The equal-component Sallen-Key filter circuit can be designed by choosing a required response, which sets Q and Fs (these can be looked up in filter design tables). The Q value sets the gain (A), which is set by choosing R3 and R4. The frequency-setting R and C values are found by first finding f0 from the required cut-off frequency (fC) and Fs. Then we can either choose R and calculate C, or vice versa. The SLOA049 application note from Texas Instruments provides four simplified approaches to Sallen-Key design and two for MFB filters, along with tables of values for Q and Fs. The Active Filter Cookbook provides a similar approach for a wider range of filters. Multi-stage filters Multiple second-order low-pass or high-pass Sallen-Key or MFB stages can be cascaded to form higher (even) order filters (4th, 6th etc) for faster cutoff and greater stopband attenuation. If an odd order is required (3rd, 5th etc), a first-order op amp filter can be used as one of the stages. [Editor’s note – some active filter circuits can realise a third-order response with a single op amp, but this is not possible for all topologies and may impose other restrictions.] Fig.13 shows a unity-gain typical firstorder stage, which is simply an RC filter buffered by an op amp. The circuit can be given gain using feedback resistors in the same way as an inverting amplifier. Input R1 + C1 Output – 0V Fig.13: a unity-gain first-order stage. 45 The design of multi-stage filters is more complex than simply joining multiple copies of the same filter. For example, you cannot make a 4th-order 20kHz Butterworth filter by cascading two 2nd-order 20kHz Butterworth filters. With a bit of thought, this might be obvious – if both filters attenuate by 3dB at 20kHz, the two-stage filter’s response will be 6dB down at 20kHz. Unfortunately, the answer is not simply to shift the cut-off frequencies of the two stages up a bit so that the two-stage filter has a -3dB cut-off at the correct frequency. This can be done, but the response will not be a Butterworth shape – the response will have a softer cut-off and less attenuation at frequencies above cut-off. These problems are illustrated in Fig.14. The green trace is a correctly designed 4 th -order Butterworth response using two unity-gain SallenKey stages. The orange trace is the response from cascading two 20kHz 2nd order Butterworth stages (these both individually have the same response as shown in Fig.3 and Fig.5). This response has -3dB cut-off at around 15.8kHz. The magenta plot is for two cascaded 24.5kHz, 2nd order Butterworth stages (24.5kHz = 20kHz × (20kHz ÷ 15.77kHz)). This has a cut-off around 20kHz, but as noted above, it has a softer cut-off than the Butterworth response. Fig.15 shows the two individual 5-year collections 2019-2023 responses (brown and grey traces) of the two stages that comprise the 4th order, 20kHz, low-pass Butterworth filter, along with the overall response (green trace), which is the same as the one as in Fig.12. One of the individual responses shown in Fig.13 obviously has a higher Q than a Butterworth filter – there is a clear peak in the stage 2 (grey trace) response. Intuitively, we can think of this as compensating for the soft response obtained by simply cascading two Butterworth stages (as shown in Fig.12). Actually obtaining the correct response requires getting the mathematical details of each stage correct which, as mentioned, may require significant effort. Fortunately, filter tables provide the numbers you need for each stage for common responses. Design tools will provide you with the entire circuit and all component values. Mathematically, it does not matter what order the stages are in; the overall response is the same. However, there are practical considerations for real circuits. The most common approach is to arrange the stages in order from lowest to highest Q because this reduces the likelihood of clipping. This optimises the filter’s voltage handling, but other orders may help optimise other aspects such as noise or power consumption. Bandpass and bandstop filters with wide differences between the cut-off frequencies can be realised using a combination of high-pass and lowpass stages. For narrower bands, where the Q is not too large, a single op amp filter can be used. F o r h i g h e r- Q n a r r o w e r b a n d Fig.14: a comparison of 4th-order responses. £49.95 2018-2022 £49.95 2017-2021 £49.95 2016-2020 £44.95 Purchase and download at: www.electronpublishing.com Fig.15: individual stage responses in a 4th-order 20kHz low-pass Butterworth filter. 46 Practical Electronics | August | 2026 Simulation files Most Circuit Surgery columns feature the use of the free circuit simulation software LTSpice. It is used to support descriptions and analysis in Circuit Surgery. The examples and files for this issue are available for download: https://pemag.au/link/acby stages (possibly above 5, certainly above 20), multi-op amp circuits such as state-variable and bi-quad filters should be used. Using design tools The reconstruction filter used in the DSP system-under-test by the PC sound-card-based measurement system was designed using Analog Devices’ Analog Filter Wizard. Fig.16 shows some of the on-screen dialogs encountered while using this tool, which are stepped through from left to right as the design progresses. Other tools provide similar functionality. After selecting the filter type (bandform), low-pass in this case, the frequency and gain requirements can be entered via the Specifications dialog. Unfortunately, the user cannot directly choose the filter order as in some other tools, but if a specific order is required, the stopband can be adjusted to achieve this. The Filter Response slider allows the response type (Chebyshev, Butterworth, Bessel) to be selected. The frequency response and requirement limits (similar to Fig.4) are displayed to the right of the dialog (not shown here). Fig.16: a selection of Analog Filter Wizard dialog boxes, for setting up the Specifications (left), Components (middle) and Tolerances (right). The next step is the Components dialog, which controls the op amp selection and calculation of RC values. The supply voltage you will be using will influence op amp selection. If you choose “Pick for me”, component selection is fully automated, but it can be optimised for specific requirements like the supply voltage range, low power consumption, or low noise. Make this choice before clicking “I want to choose” if you would like more control (as shown in Fig.16). This allows you to pick the op amp, change the relative size of the capacitors and resistors (via the slider) and select the implementation (Sallen-Key or MFB) of each stage separately. In this case, the A and B buttons select the stage being set up by the selection controls. The order of the sections can also be changed by dragging these buttons. The Tolerances dialog is used to choose the E-series and tolerance of the resistors and capacitors. The tool can display a range of information about the circuit, including the frequency response, step response, noise and power, the configuration and response of each stage, the schematic with all the component values and a bill of materials. The information displayed is selected via a drop-down box. As well as drawing the schematic diagram on-screen, the SPICE and other design files can be created and downloaded. Fig.17 shows an LTspice schematic of the reconstruction filter used in our DSP system. This uses the OPAx134 op amps that were used throughout the system, rather than those suggested by the tool. The simulation results in Fig.15 were obtained from this circuit. The actual implementation used E24 resistors with values of 6.2kΩ, 13kΩ and 47kΩ. Usually it’s best to pick from the large range of E24- or E96-series resistor values, as they are common and inexpensive, while capacitors can become pricey if you need unusual values. Conclusion Filters are a large topic, so we have only looked at a small part of it. We have covered the basic concepts that are needed to understand the inputs required by design tools and discussed some of the design trade-offs related to single-IC second-order analog filter stages. These circuits are popular; although they do not provide the best possible performance, they require relatively few components and are suited to a wide range of applications. PE Fig.17: an LTspice schematic of a 4th-order 20kHz low-pass Butterworth filter. Practical Electronics | August | 2026 47