Design second and thirdorder SallenKey filters with one op amp
Christopher Paul, Motorola January 31, 2011
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:
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.
Thirdorder section design
Closedform 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:
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.
As 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.
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.
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:
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.
Selecting samplefilter designs
It is valuable to design filter sections that implement complex pole pairs α±jβ over a range of quality factors Q (Equation 43).
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
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 filters
It 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).
There 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 

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