RP Sallen and EL Key of the Massachusetts Institute of Technology’s Lincoln Laboratory in 1955 introduced the SallenKey analog filter topology. Engineering literature extensively discusses the secondorder section that creates two filter poles (
Figure 1 and
references 1 to
4). You can also make a thirdorder filter using two op amps (
Figure 2). For filter gains of one or two, you can make a thirdorder filter with one op amp (
Figure 3). Such a configuration has been addressed in a limited manner for op amp gains of 1 and 2 (
references 5 and
7). Unitygain filters have low sensitivities to component values, but they can require large ratios of capacitor values. Gainoftwo filters allow capacitors of similar or identical values, but generally are much more sensitive.
Using the following design procedure, you can convert sets of two or three poles into single opamp filters. The procedure does not place undue restrictions on opamp gains or component values. You can select standardvalue capacitors and resistors and then calculate the remaining resistor values from filter specifications. The procedure produces designs with both low sensitivities and moderate ranges of userspecified capacitor values. A figure of merit compares filter sensitivities.
The tendency of the filters to oscillate can be assessed. The procedure also demonstrates the superiority of thirdorderfilter stopbandleakage characteristics compared with those of secondorder filters. You can perform the associated calculations for this procedure in this
Excel spreadsheet.
Secondorder section design
You first use
Equation 1 to determine the transfer function for the secondorder section to start the design procedure:
You then select standardvalue capacitors. If you use capacitor values that are too large, the capacitors will be expensive or will occupy excessive space. If the values are too small, the PCB’s (printedcircuitboard’s) and op amp’s parasitic capacitances will affect the filter’s response.
In
Equation 1, the denominator is (s−p2)×(s−p3), where p2 and p3 are the realvalued and often complex filter poles. By equating the denominator coefficients in s of
Equation 1 with those in the expression containing p2 and p3, you can write an equivalence for a term you define as B and then solve for A
_{MIN}, the minimum opamp gain for the filter, as
equations 2 and
3 show.
You then choose the values of R
_{F} and R
_{G} that will keep the gain above A
_{MIN}, as
Equation 4 shows.
You then create equivalences for terms you define as D and E, as
equations 5 and
6 show.
This approach lets you conveniently write
equations 7 through
10, which define two sets of the two resistor values.
Resistor values R
_{1A} and R
_{2A} yield one solution, and R
_{1B} and R
_{2B} define another. You now choose the nearest standardvalue resistors for R
_{1} and R
_{2}. The resistor tolerances should be 1% or better, and capacitor tolerances should be 2% or better. When R
_{F}×C
_{2}>R
_{G}×C
_{3}, the second pair of values will be negative and will constitute an unusable solution.
Thirdorder section designClosedform solutions for thirdorder filter sections do not exist. However, you can employ numerical techniques to achieve suitable results. Once again, you start with a derivation of the filtertransfer function, as
Equation 11 shows:
You define the B terms with three equivalences, as
equations 12 through
14 show.
You can express the denominator as (s−p1)×(s−p2)×(s−p3), where p1, p2 and p3 are the poles of the filter you are designing. You can then equate appropriate coefficients of powers of s to b0, b1, and b2. By solving
Equation 14 for r1 and substituting into
equations 12 and
13, you get
equations 15 and
16:
The K terms in these two equations are defined in
equations 17 through
23.
Note that
Equation 15 has a degenerate form when K equals zero. This form allows you to generate four equivalences based on the value of K by solving for r2 in
equations 15 and
16, as
equations 24 and
25 show:
You can now choose values for the capacitors and for resistors R
_{F} and R
_{G} and evaluate the K's in
equations 17 through
23. Equating
equations 24 and
25 and solving for r2 gives sets of values for r2 and r3. You can then easily rearrange
Equation 14 to solve for r1. One straightforward way to find solutions is to iterate r3 from values of 10Ω to 1 MΩ in a spreadsheet, subtracting expressions for r2
_{α} from those for r2
_{β} in ranges in which both have positive real values. If successive subtractions have opposing signs, then a solution lies between them.
In searching for solutions, it is sometimes helpful to graph equations 24 and 25. One, many, or no solutions are possible. If there are no curve intersections, the graphs can show whether a new set of values moves the curves closer to or farther from one. The graph can show cases that find a single solution (Figure 4). If you just make arbitrary selections for the capacitors and for resistors R_{F} and R_{G}, you will generally not make a successful design. The samplefilterdesign section of this article provides guidance for value selections.
Stability of the secondorder sectionAs with any active circuit, oscillation will occur if the zerophaseshift frequency loop gain exceeds unity. Accordingly, you must calculate that gain. First, break the connection between C
_{2} and the opamp output of
Figure 1. Then, connect a voltage source, V
_{I} to C
_{2}. You ground the filter input at R
_{2} because the source driving the filter must have zero impedance if the filter is to function as you design it. Using an opamp output voltage that you define as V
_{O}, you can calculate the transfer function V
_{O}/V
_{I} using
Equation 26.
Because the zerophaseshift frequency occurs when s
^{2}=−1/(R2×R3×C2×C3), you can solve for the loop gain under that condition using
Equation 27.
You then use
Equation 28 to convert the expression to negative decibels to obtain the gain margin.
If the result is positive, the circuit will be stable, whereas negative results predict instability. You should evaluate
Equation 28 with its components at the tolerance extremes that would lead to the highest possible gains, meaning that you should use the largest possible values for C
_{2}, R
_{2}, and R
_{F} and the smallest for C
_{3}, R
_{3}, and R
_{G}.
Stability of the thirdorder section
You use the same procedure to evaluate the stability of a thirdorder loop. You break the connection between C
_{2} and the opamp output and then ground the filter input at R
_{1}. You then use
Equation 29 to determine the transfer function between the disconnected side of C
_{2} and the op amp’s output.
This form of the
equation tells you that you should choose R
_{1}, R
_{2}, C
_{2}, and R
_{F} at the highest points within their tolerance ranges to maximize gain. Similarly, you should choose C
_{1}, R
_{3}, C
_{3}, and R
_{G} at the lowest points. This rule applies to
equations 29 through
38.
Equation 30 rearranges
Equation 29 in the standard form to find the gain at the zerophaseshift frequency, s
_{0}.
This equation uses the equivalences you define for the n and d terms in
equations 31 through
35.
When the phase shift is zero, the arctangent of the ratio of the imaginary to the real parts of the numerator of
Equation 30 must equal that of the denominator, which means that the ratios themselves must be equal:
Imaginary solutions to
Equation 36 exist at ±s0:
You can use
Equation 38 to convert to negative decibels to get the gain margin from
Equation 29.
As with the secondorder stability analysis, positive values indicate stability, and negative values predict instability.
Sensitivity
A lowsensitivityfilter design is immune to component variations due to manufacturing tolerances. Filter parameters such as gain and phase shift are sensitive to component tolerances, and so productionline filters will have somewhat differing characteristics. You use sensitivity analysis to prevent these differences from becoming unacceptable. You can define the sensitivity of some filter function F(x) to a component value, x, using
Equation 39.
The partial derivative appears because F(x) is also a function of variables other than x. The other terms effectively normalize the sensitivity parameter. It is more intuitive to replace the differentials with small differences and then rewrite the equation (
Equation 40).
You can now see that, when you multiply the sensitivity parameter by a small relative change in the value of component x, you get an associated relative change in F(x). Rather than directly differentiate
Equation 39, it is simpler to use the definition of a derivative and evaluate using a small value of ε, such as 10
^{−6} (
Equation 41).
One important filter function is the absolute amplitude response, H(s). Monte Carlo evaluations reveal significant variations of F(x)=H(s
_{c}) in the vicinity of the filter’s cutoff frequency s
_{c}.
Equations defining H(s) for third and secondorder sections were derived earlier. Those for the firstorder sections to form a composite filter with a secondorder section are trivial. It is better to create a single figure of merit that you can use to compare all filters. One approach might be to take the root/mean/square of the sensitivities of H(s
_{c}, x
_{i})for each of a filter’s i constituent components. You can define parameter S as an aggregate filter sensitivity (
Equation 42), taking into account that components may have different tolerances.
TOL
_{XI} is the tolerance of component x
_{I}, so that a component with a 1% tolerance would have a TOL of 1, one with a 2% tolerance would have a TOL of 2, and so forth.
Selecting samplefilter designsIt is valuable to design filter sections that implement complex pole pairs α±jβ over a range of quality factors Q (
Equation 43).
A ninthorder, 0.1dBripple Chebyschev lowpass filter offers a good selection of quality factors. You can implement second order sections (
Figure 1), composite first and secondorder sections (
Figure 2) and thirdorder sections (
Figure 3) using each of the filter’s four complex pole pairs. Each third and firstorder section of the composite filters will reflect the single real pole. You can implement secondorder sections of any Q with equal values of C
_{2} and C
_{3} if the op amp gain is two (
Reference 2). You can also do it with a unity gain circuit if C
_{2} is four times the product of Q
^{2} and C
_{3}. The sensitivities of the filters’ Q and resonance frequencies are much greater if the op amp gain is two, but the ratio of capacitor values for high Q sections can be too high if the gain is one. By investigating filter sections between gains of one and two, you will be able to see that some gains allow the use of standardvalue capacitors whose ratio is considerably less than 4×Q
^{2} and yet yield S values almost as low as those of the unitygain filters.
You can use the same approach to get similar results for thirdorder filter sections. You can find solutions for opamp gains of two where C
_{1}=C
_{2}=C
_{3}. You can make unitygainfilter sections with values of C
_{1} equal to C
_{2} and greater than 8×Q
^{2}×C
_{3}. Solutions also exist for gains slightly greater than unity if C
_{1}=C
_{2} and C
_{2}/C
_{3}≈R
_{G}/R
_{F}, where C
_{2}/C
_{3} can be much less than 8×Q
^{2}. In this last case, values of S can approach those of unity gain designs. Try small value variations in ranges that satisfy these conditions to uncover the minimum values of S.
Comparing samplefilter designs
You can generate a set of sample filter designs using the Excel spreadsheet to do the mathematics (
Table 1). The first column assigns a trial number to each design. The subsequent two columns yield pole values that are multiplied by a 2π1000 frequencyscale factor to achieve a 1kHz cutoff frequency. The next two columns give the values of Q and 4×Q
^{2} for each complex pole pair. The next column describes the orders of the implemented filter sections. Columns with headings of C
_{1}, C
_{2}, C
_{3}, R
_{G}, and R
_{F} contain the values you have selected, whereas those with headings of R
_{1}, R
_{2}, and R
_{3} list calculated values that the nearest standard 1%tolerance values approximate. The subsequent column lists gain margin, using worstcase values within tolerances of 1% for resistors and 2% for capacitors. The following column presents the aggregate sensitivity, S, of the thirdorder composite (
Figure 2) or thirdorder single section (
Figure 3). The next column shows S only for the secondorder section in the composite filter. The final column indicates whether other design procedures have confirmed the filtercomponent values in the
table.
Click to enlarge
Perhaps the most notable finding is that the gain margins of trials 3, 6, and 9 (in red) are negative, meaning that these designs will be unstable at componentvaluetolerance extremes. All of these designs have opamp gains of two and relatively high quality factors. You should be cautious when using such designs. Among the composite designs, virtually all the sensitivity lies in the secondorder sections, and little exists in the first.
You can also compare the S values for the thirdorder single sections and their companion composite filters, those having similar or the same opamp gains and implementing the same poles. If you ignore the unstable designs, there is little difference in the aggregate sensitivities. A composite design, which requires an additional op amp, has about the same sensitivity as a thirdorder single section, meaning that you can convert a thirdorder composite design into a thirdorder single section by removing an op amp and adjusting component values. This result exhibits little or no penalty in componenttolerance sensitivity. If your design requires a specific secondorder response, you can add a real pole high enough above the secondorder section’s cutoff frequency to get additional stopband attenuation.
Building and measuring filtersIt is a good idea to validate the solutions to these equations with actual physical filters. Alternatively, for composite and singlesection thirdorder filters with gains of two, you can gain confidence in the design by ensuring that other design procedures give component values identical to these. Because few if any alternative procedures exist for thirdorder singlesection designs with gains other than one or two, you must evaluate those designs by building the circuit. The designs of trials 8 and 11 in
Table 1 were built and tested using highgainbandwidth op amps. The measured dc gains were normalized to unity. Spot checks of each filter were made against the continuous graph of their theoretical responses (
Figure 5).
Stopband leakageThere is one additional benefit to a thirdorder design. The secondorder designs suffer from a highfrequency leakage current through R
_{2} and C
_{2} from the filter’s input to the op amp’s output (
Reference 8). Because the output impedance of an op amp rises with falling openloop gain at higher frequencies, it and the current from the input combine to yield stopband leakage, an unexpected signal at the output. In a thirdorder filter, C
_{1} shunts much of this current to ground. Although the highfrequency current in a secondorder section is simply V
_{IN}/R
_{2}, the current for a thirdorder single section decreases to (V
_{IN}/(R
_{1}+R
_{2}))/(1+sC
_{1}R
_{1}R
_{2}/(R
_{1}+R
_{2})).
Figure 6 shows the measured and theoretical results of the
Table 1 filter designs of trials 11 (secondorder section only) and 8 (thirdorder single section). To exacerbate stopband leakage, they both use a lowgainbandwidth op amp and with components having onetenth of the impedances the
table lists. The stopband leakage of the secondorder Trial 11 section is evident, whereas it is absent from the thirdorder Trial 8 section. Because the measurement noise floor was about −70 dBV, it is not possible to determine at what point leakage appeared in the thirdorder section. The inputs of both filters were driven at 10V rms at frequencies greater than 1500 Hz where filter attenuation ensured that output saturation of the ±12Vpowered op amp was not a concern.
References

 Williams, Arthur B, and Fred Taylor, Electronic Filter Design Handbook, McGrawHill, 1981.
 Van Valkenburg, ME, Analog Filter Design, Van Valkenburg, M.E., Holt, Reinhart and Winston, 1982.
 Cano, Martin, “A new set of SallenKey filter equations,” EDN, Oct 1, 2009.
 Texas Instruments FilterPro, 20012006, Texas Instruments Corp.
 Williams, Arthur B, Active Filter Design, Artech House Inc, 1975.
 Parker, Glenn A, “Reducing Active Filter Costs Using Symbolic ThreePole Synthesis,” RF Design, March 1996.
 Beis, Uwe, “Design and Dimensioning of Active Filters."
 Cano, Martin, “Eliminate SallenKey stopband leakage with a voltage follower,” EDN, May 14, 2009, pg 17.

Author’s biographyChristopher Paul is a director of energy systems at Motorola. He holds a bachelor’s degree in engineering from Brown University (Providence, RI) and master’s degrees in electrical engineering and in electrical engineering with a wireless concentration from California State Polytechnic University (Pomona, CA). He enjoys walking, performing and listening to music, writing for audio hobbyists, and reading on a broad range of topics.
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