Editor’s note: This DI invites the reader to reference custom Excel sheets for:

• Low pass filter,
• High pass filter, and
• Bandpass filter design.

Please refer to these as you review this DI.
—Aalyia Shaukat

Several manufacturers offer op amp-based filter design tools [1-3]. Some tools choose off-the-shelf capacitor values, but others select non-standard ones. The option to alter passive component (resistor and capacitor) values while maintaining a given response is often limited, if available at all. Certain tools seem to consider the effects of particular passive component combinations on unwanted variations in filter responses, but others do not. Some limit designs to a specific set of filter characteristics (Butterworth, Bessel, Chebyshev) when filter design tables of quality factor (Q) and resonance frequency f₀ (Hz) for other response types are readily available (see section 8.4  in [4] and [5]).

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

This article addresses second-order op amp-based low, band, and high pass filter sections. A reference for many of the design equations used in the article can be found here [6]. Excel spreadsheets for each filter type allow the user to specify three defining characteristics: passband gain, Q, and resonance frequency f₀. It requests the tolerances in percent of the capacitors and the resistors to be used. Each filter has a minimum of four passives, and so there is an infinite number of combinations of values which will satisfy the three characteristics. Because the difference between successive standard capacitor values is at least 10% while that for 1% resistors is only 2%, and because quality capacitors are generally more expensive, the user is given the option of specifying the two capacitors’ values rather than those of the resistors. This leaves it to the spreadsheet to calculate the latter. If desired, near exact resistor values can be implemented in a physical filter cheaply by using two standard parts.

The main purpose of this article is to demonstrate graphically and numerically how different sets of passive component values and tolerances contribute to unwanted variations in filter responses. From these, the user can readily select capacitor values which minimize the combination of a filter’s response sensitivity and component size and cost.

Types of passives used in filters

Before getting into a detailed discussion about sensitivity, it’s worth discussing the types of passive components (see [4] pp. 8.112-8.113) that should be used in filters. For SMD and through-hole applications, 1% metal film resistors are a good, inexpensive choice as are NPO ceramics (stay away from the monolithic, high dielectric value ceramics). For surface mount applications only, there are thin film capacitors. For through-hole, polystyrene, polypropylene, and Teflon capacitors are available. As for active components, this article assumes ideal op amps (which we know are difficult to source). The reference (see [4] pp. 8.114-8.115) gives a discussion of what is required of this component, the biggest concern of which is the gain available at f₀ Hz. By “rule of thumb” this should exceed 4·Q² for the filter by a factor of 10 or more.

But rather than dealing with a rule of thumb, it is recommended to start by simulating the filter using nominal value passive components and an op amp with no high frequency roll-off and a resonance frequency gain of 1000·4·Q² or more. Then, reduce the gain and introduce a high frequency roll off until a response change is seen. Finally, an op amp with matching or superior characteristics can then be selected and used in a simulation for design confirmation.

Quantizing filter response variations due to component tolerances

Generally, a variation in a passive’s value will result in some change in filter response characteristics. If that change is small enough, there will be some sensitivity S which is a constant of proportionality relating the filter parameter y variation to the passive x’s change. To keep S dimensionless, it will be useful to relate fractional changes in the passive’s value to those of the parameter. Mathematically,

Solving for in the limit as Δx goes to zero, we have:

The instances of x that are of concern are the resistor and capacitor values that make up the filter. The instances of y are the defining filter parameters: passband gain, Q and ω₀ = 2π·f₀. The following is an example of how the various S values are computed for the low pass filter in Figure 1.

Figure 1 A sample lowpass filter used to compute various S values.

The frequency domain (s-plane) transfer function of the above filter is:

For such a section, this is equal to:

By equating like terms, the various parameters can be computed. But what is really needed is some total sensitivity of each y parameter to a complete filter design, one which involves all its passive components. One way to do this is to use the following equation:

This is the square root of the sum of the squares of the sensitivities of a specific y to each of the i component’s x_i multiplied by the tolerance of x_i in percent, pct_tol_xi. This expression is useful for comparisons between the overall sensitivities of implementations with different sets of component values.

The general filter design approach

Refer to Figure 2 which shows the spreadsheet LPF.xlsx used for the design and analysis of low pass filters. Many of its characteristics are identical to the ones used in the high and bandpass spreadsheets.

Figure 2 A screenshot of the low pass filter spreadsheet where the yellow values are entered by the user, the orange cells are filter component values automatically calculated by the spreadsheet, the bottom parameters are intermediate calculation required by the spreadsheet, and columns F and G contain the sensitivity values. There is also a graph which ignores _ρ and displays a wide range of possible component values from which the user may choose.

The yellow values in column C rows 5 through 14 are the only values entered by the user. These include the filter characterization parameters Q, Gain, and f₀; as well as the ratio _ρ = C1/C2 (take note of the comment associated with cell C10); values for C1 and RG (reference designators for the components in the schematic seen in columns B through D and rows 26 through 37); and the percent tolerances of the resistors (r_tol) and the capacitors (c_tol) intended to be used in the filter.

The orange cells, columns B and C, rows 20 to 24, are filter component values calculated by the spreadsheet from these user entries. Columns C and D, rows 43 to 48 contain some of the intermediate calculations required by the spreadsheet.

Columns F and G contain the , , and sensitivities associated with each component x. Only those which have non-zero effects on the total sensitivity parameters S^Q, S^Gain, and S^ω0 (also shown in these columns) are listed. Notice that the equation for every parameter calculated by the spreadsheet appears to the right of the parameter value. There is also a graph which ignores _ρ and displays a wide range of possible component values from which the user may choose.

Low Pass filter design

Now let’s take a look at the curves on the graph for parameters _ρ = _C1 /_C2 and sensitivities S^Q and S^ω0 which are parameterized by _r = _R2/(1/_R1a + 1/_R1b) for values from .01 to 100. These depend only on Q, Gain, and _r. all these are dimensionless.

The _ρ curve shows that for this particular filter, there are no solutions for values less than 4·Q² = 4. (If you had entered such a value for _ρ, Excel would return the #NUM! error for many spreadsheet calculations.) The curve for Sensitivity of Gain, S^Gain, can’t be shown on a logarithmic scale—cell G25 shows it to be equal to zero. Why? The pass band (low frequency) Gain is 1, RF is zero, R1b is infinite (the spreadsheet shows it to be ridiculously large), and no passive components have any effect on Gain. (In a physical filter, there is still a sensitivity to the unity gain-configured op amp’s gain, which is actually less than unity due to its finite gain bandwidth product. Hence one of the reasons to simulate filter designs with the intended op amp.) Interestingly, the component sensitivities to S^ω0 are independent of Q, _r, _ρ, and Gain for gains greater than or equal to unity, being dependent on tolerances r_tol and c_tol only. If Gain is unity, the only overall sensitivity that can be influenced is S^Q, which is minimized in 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 close to 4 to allow the use of standard value capacitors. It will be seen that for low and high pass filters, the least sensitive choice is for a Gain of unity. Figure 3 shows what happens when the Gain requirement is increased by even a small amount to 1.5.

Figure 3 The low pass filter design of Figure 2 with the Gain parameter increased from 1 to 1.5.

S^ω0 is unchanged as expected, but the best S^Q has now more than doubled and S^Gain has made a showing, although it’s not much of a concern. The only good news is that _ρ = _C1/_C2 could be reduced to 2.2/1 and _C1 to 2.2E-9 (not shown in Figure 3) with no significant effect on S^Q. A significant increase in Gain is definitely not recommended, as it causes a large jump in S^Q, as can be seen in Figure 4.

Figure 4 Low Pass Filter screenshot with Gain jumping from from a value of 1 to 5, resulting in a large jump in S^Q.

Such large gain values increase the best obtainable value of S^Q by a factor of 6 in comparison to the Figure 3 design. The problem is compounded for higher values of Q and for component tolerances greater than 1%.

Low pass filter design summary

It’s no surprise that the best results will be obtained with the lowest tolerance passive components. There is little that can be done to influence the value of S^ω0 which is constant for Gain values greater than or equal to unity, and which falls by small amounts only for smaller gains. Fortunately, its value is relatively small. For given values of Q and f₀, the least sensitive low pass filter designs overall have a Gain of unity. For such a case, S^Gain is zero and S^Q is at its minimum. Gains of unity or less leave S^Q  unchanged, but can cause S^Gain to rise a small amount above the very stable S^ω0. The real problem comes with Gain values greater than unity: Even slightly higher values cause S^Q to increase significantly and overwhelm the contributions of S^Gain and S^ω0, but they will reduce the minimum usable value of _ρ, which may be an acceptable tradeoff against increased S^Q for some high Q cases. Generally, though, it’s wise to avoid Gain values much greater than unity, you can verify that the commonly recommended case of Gain = 2 to allow _ρ = 1 for equal capacitor values can produce a horrendous increase in S^Q.

High pass filter design

Other than a few differences related to interchanging the treatments of R1 and R2 with those of C1 and C2, high pass filter design and the high pass filter design spreadsheet shown in Figure 5 are much like those for the low pass filter. The biggest differences are first, that parameterization of the graph’s curves is by _ρ = _C1/_C2 (assuming values from .01 to 100) rather than by _r = _R2/_R1. For the low pass, any value of _r produces a realizable result, while this is true for _ρ for the high pass. Second, there is no C1b/C1a voltage divider corresponding to the low pass filter’s R1b/R1a—there is only _C1. The introduction of a capacitive voltage divider would require a prior stage to drive a capacitive load, courting oscillation. And so, although the high pass filter cannot support Gain values less than unity, the high and low pass designs show significant similarities. A comparison between Figure 4 and Figure 5 graphs, which employ the same Q, Gain, and f₀, show virtually identical results (with _ρ and _r switched).

Figure 5 High Pass Filter screenshot with the same Q, Gain, and f₀ requirements as those of Figure 4.

High Pass filter design summary

The comments found in the “Low pass filter design summary” section apply here too, except that there is no option for Gain values less than unity.

Bandpass filter design

Although the least sensitive topology for component tolerances in high and low pass filters is the Sallen-Key, for the bandpass it’s the Delyannis-Friend (aka the multiple feedback configuration). A screenshot of the bandpass filter spreadsheet can be seen in Figure 6.

User data entry with the bandpass is much like that for the low and high pass cases, except that there is no _RG (and therefore no _RF). Once again, please be aware of the comments in the notes in columns D and E. If the background of cell C6 (filter Gain at resonance) is red, there are no realizable filters, calculations in columns C through G should be ignored, and the graph will be blank.

In some cases, the cell C6 background color will be the normal white, but filters will be realizable for certain smaller values of _ρ only, and the graph’s curves will be displayed accordingly. The curves might be absent, or partially or fully present, regardless of the value of _ρ in cell C10. But if C10’s background color is red, the _ρ-dependent calculations in columns C through G should be ignored. Figure 6 is an example where the filter Gain at resonance is close enough to the maximum possible value of 2·Q² to render high values of _ρ (greater than 30) unrealizable.

Figure 6 A bandpass filter screenshot where user entry data (yellow) is similar to the low and high pass filter excel sheets.

Bandpass filter design summary

It’s surprising that the passive sensitivity curves can be shown to be almost completely independent of the user-specified filter Gain at resonance. This is because for a given Q and f₀, the filter Gain is set by the ratio of R1a to R1b. The parallel combination of these components is independent of filter Gain, and the remainder of the filter sees no difference in other than signal level. (Designers should be aware that the op amp can easily clip at or near resonance with too high a gain.) Surprisingly, sensitivities are independent of Q. However, the higher the Q, the higher the op amp open loop gain must be to provide enough margin to accurately implement the required op amp closed loop gain. Simulation of the filter design using the op amp intended for it, or one with similar gain characteristics, is strongly recommended.

Looking at the sensitivity curves only, it could be concluded that the best choice would be for a _ρ of 1 or less. _ρ = 1 has the advantage of the smallest ratio _r = R2 / (R1a || R1b). But consider the Gain of op amp at resonance: Less gain is required at higher values of _ρ, putting less of a burden on op amp open loop gain requirements to provide enough margin to meet the closed loop gain requirement.

Higher values of _ρ increase the overriding S^Gain by only a small amount. Clearly, there is a rather large disadvantage to values of _ρ less than unity when the demand on op amp closed loop gain is considered. Perhaps the best choice is _ρ = 1. The matched capacitors can be any standard value, S^Gain is near its smallest value, _r is at its smallest value, and there is only a modest increase in the op amp closed loop (and therefore open loop) gain requirement.

Flexible passive component values

This article and its attendant spreadsheets provide an understanding of the sensitivities of pass band gains, Q’s, and resonance frequencies to the nearly infinite combinations of passive components that can make up low, band, and high pass, single op amp filters. The ability to implement designs using capacitors of readily available values is provided. It is hoped that filter designers will find these to be a useful set of tools whose features are not found elsewhere.

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

Related Content

• A Sallen-Key low-pass filter design toolkit
• Optimizing a simple analog filter for any PWM
• Double up on and ease the filtering requirements for PWMs
• Toward better behaved Sallen-Key low pass filters
• Parsing PWM (DAC) performance: Part 1—Mitigating errors

References

1. Texas Instruments. WEBENCH® Filter Design Tool. https://webench.ti.com/filter-design-tool/design/8
2. Analog Devices. Analog Filter Wizard. https://tools.analog.com/en/filterwizard/
3. FilterLab Active 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/resources/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 Active-RC Allpole Filters Using Optimized Biquads. Automatika, 51(1), 55–70. https://doi.org/10.1080/00051144.2010.11828355

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