Toward better behaved Sallen-Key low pass filters

The most common type of filter is probably the low pass. Among active filters, the Sallen-Key is the most widely used topology (Figure 1).

Figure 1 A first order op-amp-based Sallen-Key low-pass filter with broadband gain 1+Rf0/Rg0.

Wow the engineering world with your unique design: Design Ideas Submission Guide

The transfer function of this filter is H(s) = (Rf0/Rg0) / (1+s·C0·R0). The op-amp-configured gain has no effect on the C0-R0 filter, for which C0·R0 = 1/ω0. The values of C0 and R0 can be varied without modifying the filter characteristics as long as their product remains unchanged.

Figure 2 A second order op-amp-based Sallen-Key low-pass filter with broadband gain 1+Rf/Rg.

The transfer function of the filter in Figure 2 is shown in Equation (1):

H(s) = (Rf/Rg) / [1 + s·(C2·(R1+R2) – Rf·R1·C1/Rg) + s2·R1·R2·C1·C2] = (Rf/Rg) / [1 + s/(Q·ω0) + (s/ω0)2] (1)

From Equation (1), equating like powers of s yields Equations (2) and (3):

ω0 = 1/sqrt(R1·R2·C1·C2) (2)

and

Q = 1/((C2·(R1+R2) – Rf·R1·C1/Rg)·ω0) (3)

If you are unfamiliar with s, Q or ω0, this reference [1] gives a good and brief tutorial.

Complete filters might consist of a first and/or second order section or, of multiple second order sections possibly cascaded with a first. Even for a given cutoff frequency, second order sections can have various values of Q which depend on response types such as Bessel, Butterworth, and Chebyshev, etc. It is somewhat of an art to arrange multiple sections in an order which enhances both noise and clipping headroom. Noise can be reduced while maintaining the same filter characteristics by reducing resistor values, but for those resistors other than Rf and Rg, this unfortunately results in physically larger and generally more expensive capacitors. And for response accuracy, higher Q sections place greater demands on op amp gain-bandwidths. But this Design Idea will not be addressing any of these issues. Instead, its goal will be to specify how to select resistor and capacitor value sets which lessen the effects of their tolerance-associated variations on the responses of second order sections. To accomplish this, use can be made of a filter design tool of the kind available from multiple semiconductor manufacturers [2-4]. Such tools have the advantage of automatically generating component values from a set of performance requirements. Unfortunately, none of these tools addresses the aforementioned goal. But calculations employing these values can evaluate the sections’ Q and ω0 parameters, which can in turn be used to generate new component value sets that realize more stable responses.

Taming response variations

To quote a reference [5], “desensitization is obtained in a dual way by increasing the value of the capacitance ratio ρ while keeping the resistance ratio r equal to unity.” In the case of Figure 2 above:

ρ = C1/C2 (4)

and

r = 1 = R2/R1 so that R1 = R2 = R (5)

Applying Equations (4) and (5) to (3), we obtain Equation (6):

R·(2·C1/ρ – C1·Rf/Rg) = 1/(Q·ω0) (6)

Substituting the value of ω0 from Equation (2) into (6) and again making use of Equations (4) and (5), we find that:

Rf/Rg = 2/ρ – 1/(Q·sqrt(ρ)) (7)

It’s clear that the largest value of ρ that produces a non-negative, and therefore realizable value of Rf/Rg (one where Rf/Rg = 0 and which would be implemented by making Rf a short and removing Rg from the circuit), is ρ = 4·Q2, so that C1 = 4·Q2·C2. For a filter in which R1 = R2 having a given Q, this would yield the response with the least possible sensitivity to component values. Let’s apply some of these findings to an example.

An example

This is a second order section obtained from one manufacturer’s tool:

Figure 3 An example of a second order section from a manufacturer’s tool.

Right away, we notice one thing that’s odd: every component is of a standard value with the exception of C1. To get within 1% of C1, at least two capacitors would have to be used. As we’ll see, a redesign can avoid this extra component. Applying Equations (2) and (3), we find that Q = 3.127 and ω0 = 5048. Let’s keep the value of C2 and choose the next higher standard value for C1 from what is shown, 33n. Solving in Equation (6) for R = R1 = R2, we obtain 11693, the nearest standard value of which is 11.8k. Since ρ = 3.3, from Equation (7), Rf/Rg = .4300. We can therefore let Rg = 2490 and Rf = 1071 ≈ 1070.

Figure 4 shows a 100 sample Monte Carlo run with capacitor tolerances of 5% and resistor tolerances of 1%. Here, the revised design is superimposed on top of the original one from Figure 3. (The broadband filters’ gains due to Rf/Rg have been normalized to unity so that the response variations can be more readily compared.) Note that the revised filter has less variation. This is mostly because R1 and R2 have been made equal and less so because ρ has been (only) slightly increased.

Figure 4 A 100 sample Monte Carlo run of the original Figure 3 filter and a revised version where R1 and R2 have been set to be equal and the ratio C1/C2 only slightly increased. The filters’ Q’s and ω0’s are identical. The broadband gains due to Rf/Rg have been normalized to aid in the comparison of response variations.

However, things can get better. If we retain 10n for C2 and set ρ to be slightly less than 4·Q2 so as to use the standard value of 390n for C1, the nearest standard R value becomes 3160. Rf/Rg falls almost to 0, so we replace Rf with a short and remove Rg. Figure 5 shows the result.

Figure 5 A 100 sample Monte Carlo run of the original Figure 3 filter and a newly revised version. R1 and R2 have been set to be equal, and the ratio C1/C2 increased to a near optimal value slightly less than the realizable maximum of 4·Q2. The broadband gains due to Rf/Rg have been normalized to aid in the comparison of response variations.

The Rf/Rg ratio in the manufacturer’s Figure 3 design comes from a requirement for a total gain of 10dB in a four-section filter of which this section is a part. The manufacturer decided to require each section to have a gain of 2.5 dB = 20·log10(1 + Rf/Rg). From a sensitivity point of view, this is obviously not the best choice. Ideally, second order sections’ op-amps should be configured for unity gain (Rf/Rg = 0). We know this from the previously quoted statement from a reference [5] to minimize sensitivity by setting r to 1 and maximizing ρ, and from applying that maximized value to Equation (7).

You don’t need a manufacturer’s tool

You can design filter sections from tables of filter characteristics [6]. These tables list the Q’s and the ω0’s (shown in the reference as F0’s) for filters of multiple response types and orders from 2 through 10. Even number E orders require E/2 second order sections, while odd number O orders demand (O – 1)/2 second order sections and one first order section. The F0’s (which are the same as the ω0’s in this Design Idea) are shown for a 3 dB attenuation frequency in radians per second listed in the column labeled -3 dB FREQUENCY. Simply multiply all F0’s by 2·π·F to change the 3 dB attenuation frequency from 1 radian per second to F Hz. The Q’s are unchanged. The Q’s and resulting ω0’s are required for deriving the component values for each section. Working from the tables is actually more accurate than working from the tools. This is because some or all of the tools’ component values have been approximated with standard values.

Summary

This Design Idea shows how to create filters whose amplitude responses are minimally affected by tolerance-associated variations in components’ values. First order filters’ ω0’s and second order filters’ ω0’s and Q’s can be obtained from semiconductor manufacturers’ tools or from filter design tables. Using these values, one can proceed by first choosing a standard capacitor value C. For a first order filter, set:

C0 = C and R0 = 1/(C0· ω0)

and choose the nearest standard value for R0.

For each second order filter, choose a value of ρ which is the ratio of two standard value capacitors. ρ should be greater than unity and large enough to attain the desired reduction in response sensitivity, but no larger than 4·Q2 for the section. Then set:

C2 = C

C1 = C2·ρ

R1# = R2# = 1/(ω0· sqrt(C1·C2) )

Rf/Rg = 2/ρ – 1/(Q·sqrt(ρ))

Choose the nearest standard value for all components with the # superscript. Approximate Rf/Rg by choosing standard values for two resistors, considering that smaller values minimize noise contributions but could overload the op amp’s output stage and/or exceed AC power consumption requirements.

Consider employing aggregate filters of only odd orders and placing all the requirement’s gain in the first order section where it will generally have the least effect on the aggregate frequency response. To maximize both noise and headroom, connect the output of the lowest Q second order section to the input of the next higher Q section and so on so that the last stage is the first order one. Reversing the connection order minimizes noise and headroom. Some compromise between the two would likely be the best choice.

One more note: because in high pass Sallen-Key filters the placements of resistors R1 and R2 are swapped with those of capacitors C1 and C2, response sensitivities for this topology are minimized when the capacitor values are equalized and the resistor ratio is maximized! Perhaps this is a topic for a future Design Idea. (Oops, I just spilled the beans! Not much more to it than that.)

The design procedure presented in this Design Idea provides the opportunity to minimize filter response sensitivities to variations due to the tolerances of resistor and capacitor values. You may wish to consider this for your next design.

References

https://www.ti.com/lit/an/sloa049d/sloa049d.pdf?ts=1695449683656, see especially sections 3 and 6.
https://webench.ti.com/filter-design-tool/filter-response
https://www.microchip.com/en-us/development-tool/filterlabdesignsoftware
https://tools.analog.com/en/filterwizard/
https://hrcak.srce.hr/file/78626
https://www.analog.com/media/en/training-seminars/design-handbooks/basic-linear-design/chapter8.pdf, specifically Figures 8.26 through 8.36. This reference does a great job of describing the differences between the filter response types and filter realization in general.

Christopher Paul has worked in various engineering positions in the communications industry for over 40 years.

<!–
googletag.cmd.push(function() { googletag.display(‘div-gpt-ad-native’); });
–>

The post Toward better behaved Sallen-Key low pass filters appeared first on EDN.

Become a member

Become a member

Become a subscriber

Become a sponsor