Element tolerance sensitivities of single op-amp filter sections

Editor’s word: This DI invitations the reader to reference customized Excel sheets for:

Please refer to those as you evaluate this DI.
—Aalyia Shaukat

A number of producers provide op amp-based filter design instruments [1-3]. Some instruments select off-the-shelf capacitor values, however others choose non-standard ones. The choice to change passive part (resistor and capacitor) values whereas sustaining a given response is commonly restricted, if out there in any respect. Sure instruments appear to contemplate the results of specific passive part combos on undesirable variations in filter responses, however others don’t. Some restrict designs to a selected set of filter traits (Butterworth, Bessel, Chebyshev) when filter design tables of high quality issue (Q) and resonance frequency f₀ (Hz) for different response varieties are available (see part 8.4  in [4] and [5]).

Wow the engineering world along with your distinctive design: Design Concepts Submission Information

05.31.2024

05.30.2024

05.28.2024

This text addresses second-order op amp-based low, band, and excessive move filter sections. A reference for lots of the design equations used within the article will be discovered right here [6]. Excel spreadsheets for every filter kind permit the person to specify three defining traits: passband acquire, Q, and resonance frequency f₀. It requests the tolerances in % of the capacitors and the resistors for use. Every filter has a minimal of 4 passives, and so there may be an infinite variety of combos of values which can fulfill the three traits. As a result of the distinction between successive customary capacitor values is at the very least 10% whereas that for 1% resistors is simply 2%, and since high quality capacitors are typically costlier, the person is given the choice of specifying the 2 capacitors’ values relatively than these of the resistors. This leaves it to the spreadsheet to calculate the latter. If desired, close to precise resistor values will be applied in a bodily filter cheaply through the use of two customary components.

The primary objective of this text is to reveal graphically and numerically how totally different units of passive part values and tolerances contribute to undesirable variations in filter responses. From these, the person can readily choose capacitor values which decrease the mix of a filter’s response sensitivity and part measurement and value.

Earlier than moving into an in depth dialogue about sensitivity, it’s value discussing the sorts of passive elements (see [4] pp. 8.112-8.113) that must be utilized in filters. For SMD and through-hole functions, 1% metallic movie resistors are a very good, cheap alternative as are NPO ceramics (steer clear of the monolithic, excessive dielectric worth ceramics). For floor mount functions solely, there are skinny movie capacitors. For through-hole, polystyrene, polypropylene, and Teflon capacitors can be found. As for lively elements, this text assumes ideally suited op amps (which we all know are troublesome to supply). The reference (see [4] pp. 8.114-8.115) provides a dialogue of what’s required of this part, the largest concern of which is the acquire out there at f₀ Hz. By “rule of thumb” this could exceed 4·Q² for the filter by an element of 10 or extra.

However relatively than coping with a rule of thumb, it’s endorsed to begin by simulating the filter utilizing nominal worth passive elements and an op amp with no excessive frequency roll-off and a resonance frequency acquire of 1000·4·Q² or extra. Then, cut back the acquire and introduce a excessive frequency roll off till a response change is seen. Lastly, an op amp with matching or superior traits can then be chosen and utilized in a simulation for design affirmation.

Typically, a variation in a passive’s worth will lead to some change in filter response traits. If that change is sufficiently small, there will probably be some sensitivity S which is a continuing of proportionality relating the filter parameter y variation to the passive x’s change. To maintain S dimensionless, will probably be helpful to narrate fractional modifications within the passive’s worth to these of the parameter. Mathematically,

Fixing for within the restrict as Δx goes to zero, now we have:

The situations of x which are of concern are the resistor and capacitor values that make up the filter. The situations of y are the defining filter parameters: passband acquire, Q and ω₀ = 2π·f₀. The next is an instance of how the assorted S values are computed for the low move filter in Determine 1.

Determine 1 A pattern lowpass filter used to compute varied S values.

The frequency area (s-plane) switch operate of the above filter is:

For such a piece, this is the same as:

By equating like phrases, the assorted parameters will be computed. However what is basically wanted is a few complete sensitivity of every y parameter to an entire filter design, one which entails all its passive elements. A method to do that is to make use of the next equation:

That is the sq. root of the sum of the squares of the sensitivities of a selected y to every of the i part’s x_i multiplied by the tolerance of x_i in %, pct_tol_xi. This expression is beneficial for comparisons between the general sensitivities of implementations with totally different units of part values.

Confer with Determine 2 which exhibits the spreadsheet LPF.xlsx used for the design and evaluation of low move filters. A lot of its traits are similar to those used within the excessive and bandpass spreadsheets.

Determine 2 A screenshot of the low move filter spreadsheet the place the yellow values are entered by the person, the orange cells are filter part values mechanically calculated by the spreadsheet, the underside parameters are intermediate calculation required by the spreadsheet, and columns F and G include the sensitivity values. There’s additionally a graph which ignores _ρ and shows a variety of attainable part values from which the person might select.

The yellow values in column C rows 5 via 14 are the one values entered by the person. These embrace the filter characterization parameters Q, Acquire, and f₀; in addition to the ratio _ρ = C1/C2 (be aware of the remark related to cell C10); values for C1 and RG (reference designators for the elements within the schematic seen in columns B via D and rows 26 via 37); and the % tolerances of the resistors (r_tol) and the capacitors (c_tol) supposed for use within the filter.

The orange cells, columns B and C, rows 20 to 24, are filter part values calculated by the spreadsheet from these person entries. Columns C and D, rows 43 to 48 include among the intermediate calculations required by the spreadsheet.

Columns F and G include the , , and sensitivities related to every part x. Solely these which have non-zero results on the entire sensitivity parameters S^Q, S^Acquire, and S^ω0 (additionally proven in these columns) are listed. Discover that the equation for each parameter calculated by the spreadsheet seems to the suitable of the parameter worth. There’s additionally a graph which ignores _ρ and shows a variety of attainable part values from which the person might select.

Now let’s check out the curves on the graph for parameters _ρ = _C1 /_C2 and sensitivities S^Q and S^ω0 that are parameterized by _r = _R2/(1/_R1a + 1/_R1b) for values from .01 to 100. These rely solely on Q, Acquire, and _r. all these are dimensionless.

The _ρ curve exhibits that for this specific filter, there are not any options for values lower than 4·Q² = 4. (If you happen to had entered such a worth for _ρ, Excel would return the #NUM! error for a lot of spreadsheet calculations.) The curve for Sensitivity of Acquire, S^Acquire, can’t be proven on a logarithmic scale—cell G25 exhibits it to be equal to zero. Why? The move band (low frequency) Acquire is 1, RF is zero, R1b is infinite (the spreadsheet exhibits it to be ridiculously massive), and no passive elements have any impact on Acquire. (In a bodily filter, there may be nonetheless a sensitivity to the unity gain-configured op amp’s acquire, which is definitely lower than unity as a consequence of its finite acquire bandwidth product. Therefore one of many causes to simulate filter designs with the supposed op amp.) Curiously, the part sensitivities to S^ω0 are unbiased of Q, _r, _ρ, and Acquire for features higher than or equal to unity, being depending on tolerances r_tol and c_tol solely. If Acquire is unity, the one general sensitivity that may be influenced is S^Q, which is minimized on this case for _ρ = 4·Q² = 4.

When 12.0E-9 is entered for _C1, the expression = 12/2.7 ≈ 4.44 for _ρ is near 4 to permit using customary worth capacitors. Will probably be seen that for high and low move filters, the least delicate alternative is for a Acquire of unity. Determine 3 exhibits what occurs when the Acquire requirement is elevated by even a small quantity to 1.5.

Determine 3 The low move filter design of Determine 2 with the Acquire parameter elevated from 1 to 1.5.

S^ω0 is unchanged as anticipated, however the most effective S^Q has now greater than doubled and S^Acquire has made a displaying, though it’s not a lot of a priority. The one excellent news is that _ρ = _C1/_C2 may very well be lowered to 2.2/1 and _C1 to 2.2E-9 (not proven in Determine 3) with no important impact on S^Q. A big enhance in Acquire is unquestionably not really useful, because it causes a big bounce in S^Q, as will be seen in Determine 4.

Determine 4 Low Cross Filter screenshot with Acquire leaping from from a worth of 1 to five, leading to a big bounce in S^Q.

Such massive acquire values enhance the most effective obtainable worth of S^Q by an element of 6 compared to the Determine 3 design. The issue is compounded for larger values of Q and for part tolerances higher than 1%.

Low move filter design abstract

It’s no shock that the most effective outcomes will probably be obtained with the bottom tolerance passive elements. There’s little that may be completed to affect the worth of S^ω0 which is fixed for Acquire values higher than or equal to unity, and which falls by small quantities just for smaller features. Fortuitously, its worth is comparatively small. For given values of Q and f₀, the least delicate low move filter designs general have a Acquire of unity. For such a case, S^Acquire is zero and S^Q is at its minimal. Beneficial properties of unity or much less depart S^Q  unchanged, however could cause S^Acquire to rise a small quantity above the very steady S^ω0. The true downside comes with Acquire values higher than unity: Even barely larger values trigger S^Q to extend considerably and overwhelm the contributions of S^Acquire and S^ω0, however they’ll cut back the minimal usable worth of _ρ, which can be an appropriate tradeoff towards elevated S^Q for some excessive Q instances. Typically, although, it’s clever to keep away from Acquire values a lot higher than unity, you’ll be able to confirm that the generally really useful case of Acquire = 2 to permit _ρ = 1 for equal capacitor values can produce a horrendous enhance in S^Q.

Aside from a number of variations associated to interchanging the remedies of R1 and R2 with these of C1 and C2, excessive move filter design and the excessive move filter design spreadsheet proven in Determine 5 are very similar to these for the low move filter. The largest variations are first, that parameterization of the graph’s curves is by _ρ = _C1/_C2 (assuming values from .01 to 100) relatively than by _r = _R2/_R1. For the low move, any worth of _r produces a realizable consequence, whereas that is true for _ρ for the excessive move. Second, there is no such thing as a C1b/C1a voltage divider akin to the low move filter’s R1b/R1a—there may be solely _C1. The introduction of a capacitive voltage divider would require a previous stage to drive a capacitive load, courting oscillation. And so, though the excessive move filter can’t help Acquire values lower than unity, the excessive and low move designs present important similarities. A comparability between Determine 4 and Determine 5 graphs, which make use of the identical Q, Acquire, and f₀, present just about similar outcomes (with _ρ and _r switched).

Determine 5 Excessive Cross Filter screenshot with the identical Q, Acquire, and f₀ necessities as these of Determine 4.

Excessive Cross filter design abstract

The feedback discovered within the “Low move filter design abstract” part apply right here too, besides that there is no such thing as a choice for Acquire values lower than unity.

Though the least delicate topology for part tolerances in excessive and low move filters is the Sallen-Key, for the bandpass it’s the Delyannis-Buddy (aka the a number of suggestions configuration). A screenshot of the bandpass filter spreadsheet will be seen in Determine 6.

Consumer knowledge entry with the bandpass is very similar to that for the high and low move instances, besides that there is no such thing as a _RG (and subsequently no _RF). As soon as once more, please concentrate on the feedback within the notes in columns D and E. If the background of cell C6 (filter Acquire at resonance) is pink, there are not any realizable filters, calculations in columns C via G must be ignored, and the graph will probably be clean.

In some instances, the cell C6 background shade would be the regular white, however filters will probably be realizable for sure smaller values of _ρ solely, and the graph’s curves will probably be displayed accordingly. The curves may be absent, or partially or totally current, whatever the worth of _ρ in cell C10. But when C10’s background shade is pink, the _ρ-dependent calculations in columns C via G must be ignored. Determine 6 is an instance the place the filter Acquire at resonance is shut sufficient to the utmost attainable worth of two·Q² to render excessive values of _ρ (higher than 30) unrealizable.

Determine 6 A bandpass filter screenshot the place person entry knowledge (yellow) is just like the high and low move filter excel sheets.

Bandpass filter design abstract

It’s stunning that the passive sensitivity curves will be proven to be nearly fully unbiased of the user-specified filter Acquire at resonance. It is because for a given Q and f₀, the filter Acquire is ready by the ratio of R1a to R1b. The parallel mixture of those elements is unbiased of filter Acquire, and the rest of the filter sees no distinction in aside from sign degree. (Designers must be conscious that the op amp can simply clip at or close to resonance with too excessive a acquire.) Surprisingly, sensitivities are unbiased of Q. Nonetheless, the upper the Q, the upper the op amp open loop acquire should be to offer sufficient margin to precisely implement the required op amp closed loop acquire. Simulation of the filter design utilizing the op amp supposed for it, or one with related acquire traits, is strongly really useful.

Wanting on the sensitivity curves solely, it may very well be concluded that the only option can be for a _ρ of 1 or much less. _ρ = 1 has the benefit of the smallest ratio _r = R2 / (R1a || R1b). However contemplate the Acquire of op amp at resonance: Much less acquire is required at larger values of _ρ, placing much less of a burden on op amp open loop acquire necessities to offer sufficient margin to fulfill the closed loop acquire requirement.

Greater values of _ρ enhance the overriding S^Acquire by solely a small quantity. Clearly, there’s a relatively massive drawback to values of _ρ lower than unity when the demand on op amp closed loop acquire is taken into account. Maybe the only option is _ρ = 1. The matched capacitors will be any customary worth, S^Acquire is close to its smallest worth, _r is at its smallest worth, and there may be solely a modest enhance within the op amp closed loop (and subsequently open loop) acquire requirement.

This text and its attendant spreadsheets present an understanding of the sensitivities of move band features, Q’s, and resonance frequencies to the almost infinite combos of passive elements that may make up low, band, and excessive move, single op amp filters. The flexibility to implement designs utilizing capacitors of available values is offered. It’s hoped that filter designers will discover these to be a helpful set of instruments whose options will not be discovered elsewhere.

Christopher Paul has labored in varied engineering positions within the communications trade for over 40 years.

Associated Content material

References

1. Texas Devices. WEBENCH® Filter Design Device. https://webench.ti.com/filter-design-tool/design/8
2. Analog Gadgets. Analog Filter Wizard. https://instruments.analog.com/en/filterwizard/
3. FilterLab Lively Filter Designer. https://www.microchip.com/en-us/development-tool/filterlabdesignsoftware
4. Zumbahlen, Hank. “Chapter 8: Analog Filters.” Linear Circuit Design Handbook. Elsevier, 2008, https://www.analog.com/en/assets/technical-books/linear-circuit-design-handbook.html.
5. Williams, Arthur Bernard. Analog Filter and Circuit Design Handbook. McGraw-Hill, 2014.
6. Jurišić, D., Moschytz, G. S., & Mijat, N. (2010). Low-Sensitivity Lively-RC Allpole Filters Utilizing Optimized Biquads. Automatika, 51(1), 55–70. https://doi.org/10.1080/00051144.2010.11828355