The required gain-bandwidth product (GBP) design margin in active filter designs regularly comes up. It is inevitably a judgement call, but engineers rarely detail the decisions going into that assessment. At the low end of the design-margin targets, we ask “what minimum GBP will deliver the desired response shape with acceptable response spread over GBP variation?” At higher GBP margins, we ask “what minimum loop gain (LG) is necessary to hit the desired in band harmonic distortion?” Let’s consider the minimum GBP required for response accuracy–going up from there will usually improve distortion.

Engineers that set out to provide an active-filter design tool need an estimate of minimum required GBP for the implementation op-amp. The simplest approaches (page 24, Ref. 1) simply target a significant design margin. While functional, excess GBP adds power and cost. Ideally, “just enough” GBP margin will yield a production ready design with little excess quiescent current and cost. There is certainly no single best way to approach this, but here are issues to consider using a low-pass, second-order, multiple feedback (MFB) design.

- The nominal response fit to target can be very good, even with very low GBP margin, using an RC adjustment solution accounting for the op amp’s GBP. (Ref. 2)
- Starting with a close nominal fit, production variation will have both an RC tolerance contribution and a GBP variation impacting the deviation from nominal fit.
- GBP variation for part-to-part and over temperature using modern voltage feedback amplifiers (VFAs) can be conservatively estimated at ±30% for devices not using a supply current trim and ±20% for those that do.
- Considering the nominal and worst-case minimum loop gain (LG) is one approach to account for the op amps GBP impact on response variation. The minimum LG is a function of the op amps’ GBP and the over frequency noise gain (NG) response for the filter’s RC implementation.
- The RC solutions’ LG over frequency is a gain-error term inside the filter. While you can nominally design it out using a GBP adjusted flow for the RC solutions, GBP variation will change that minimum LG from nominal, changing the maximum gain error over GBP variation.
- The NG for the low pass MFB has second-order poles identical to the desired filter shape and second-order real zeroes that depend partially, but not completely, on the RC solution for the desired filter response.
- Setting a GBP margin via the minimum LG can only be done in the context of an RC solution flow that is considering the peak Noise Gain (NG).
- It is possible to tilt those solutions towards zeroes that move the lower zero up in frequency. That will reduce the peak noise gain for the same filter response allowing lower GBP solutions versus design flows that further peak the NG.
- Whichever RC solution flow you use, those will produce a peak noise gain – adding a desired margin above that at the peak frequency will set a lower bound on the required GBP for a solution op amp.
- That desired margin (minimum LG) at the peak NG is subjective – the more margin you ask for, the higher the minimum GBP op amps required.

**Figure 1** shows an example design using the OPA134.

This solution has adjusted for the 9.7 MHz GBP for the OPA134 and then picked the best fit E24 capacitors and E96 standard resistor values. It is imperative to use the “true” GBP in the cubic coefficients for the GBP adjusted RC flow (Ref. 2). Using the OPA134 reported 8 MHz GBP would have given less accurate results. (note, those cubic coefficient equations in Reference 2 were corrected from the older Ref. 3).

Hiding within this design is a NG response given by Eq. 1 (Eq. 19, Ref. 3) using the RC numbering of Fig. 1.

This NG has been recast as much as possible in terms of the desired filter terms. This shows an added 1/(R_{2}C_{2}) degree of freedom in the NG zeroes. These will always be two real zeroes where the solutions can be adjusted to move the lower zero up in frequency (equivalent to saying an increase for the Q of the numerator – always <0.5). Once you choose RC values, you can then place them into Eq. 1 and sweep them over frequency to generate the NG magnitude over frequency. You can then locate the NG peak and a desired minimum LG target by adding that margin to the peak NG. Then that target A_{OL} magnitude at the peak NG frequency will set a minimum GBP required. **Figure 2** shows this plot for the design of Fig. 1.

Here, the minimum GBP calculation on the right has targeted a minimum LG of 20 dB. Adding that to the peak NG of 20.9 dB works out to a required A_{OL} at 91.1 kHz of 111 V/V. From that, you can calculate the minimum required GBP as 91.1 kHz×111 V/V = 10.11 MHz. This is slightly higher than the 9.7 MHz GBP available in this plot for the OPA134 solution – resulting in the slightly <20 dB minimum LG at the NG peak frequency. This approach strongly depends on using the “true” GBP for the candidate op amps. This minimum LG approach is happening at a frequency far below the A_{OL} = 0 dB frequency where the true single pole roll-off projection to A_{OL}=0 dB is needed for correct analysis. While the peak NG was used here to set the margin, an alternate approach would be to use the minimum LG at the F_{O }frequency.

The RC values in Fig. 1 have been nominally adjusted to account for this relatively low GBP margin. To test response variation over GBP tolerancing for this non-supply current trimmed op amp (±30% GBP variation), modify the TINA (Ref. 4) OPA134 model to produce those GBP endpoints, rerun the response curve, and extract F_{O} and Q as shown in **Table 1**.

The response Q’s really don’t change much with GBP variation. The response F_{O} drops 2% for a 30% low GBP and increases 1% for a 30% high GBP. The RC tolerances will then expand the response range around these endpoints. Recent price checks indicate that 1% C0G MLCC capacitors are relatively inexpensive – as are 0.5% resistors. Those tighter RC tolerances expand the production range still more than this GBP induced spread using the nominal GBP RC adjustment flow. **Figure 3** shows a 500-case Monte Carlo simulation on the RC’s showing about 0.44 dB gain variation at 100 kHz using the nominal 9.7 MHz GBP for the OPA134.

One way to think about this GBP induced fit error is the change in the LG/(LG+1) gain error term. Much like the DC gain error introduced by the DC LG/(LG+1) idea, this same error term is in the response shape across frequency. Adjusting RC’s for the nominal GBP takes care of this at the minimum LG point. Using a 20 dB margin target is accounting for a 10/11 = 0.909 gain error term at the peak NG. Letting the GBP vary ±30% from that is equivalent to a 7/8 = 0.875 max LG error (down 4% from nominal) and 13/14 = 0.928 minimum LG error terms (up 2% from nominal). These are relatively low percentage changes from an RC adjusted design for nominal GBP.

If the worst-case GBP induced F_{O} variation in Table 1 is unacceptable, simply increase your min LG target from 20 dB (and/or use a supply current trimmed op amp with ±20% GBP variation). For a GBP adjusted RC solution, probably no more than 28 dB minimum LG would be required to drop the GBP induced spread well below the RC tolerancing spread. Where no RC adjustments are being made (Ref. 5) probably more like 34 dB minimum LG would be required to get the nominal fit within 1%. This explains some of the much higher minimum GBP requirements coming out of the older tools (page 24, Ref. 1).

So, if someone asks what minimum GBP is needed for an active filter design, the answer is “it depends.”

- Are you using a reduced NG peaking RC solution flow? If not, you’ll need higher GBP than a reduced noise gain peaking solution.
- Are you using a nominal GBP adjusted RC solution flow? If not, you will need a much higher GBP to get a good nominal fit and tolerance.
- Are you using a supply current trimmed device? If not, you will need to account for more variation due to min/max GBP.
- How much do you care about GBP induced design spread? The tighter spread you want, the higher that minimum LG target will need to be and hence the higher the minimum GBP will become.

All of this starts with the “true” op amp GBP. Confirm that first (Ref. 6, Fig. 1) to render the subsequent analysis more accurate. This discussion used the second-order low pass MFB filter for illustration. These ideas apply as well to the other implementations (Sallen-Key) and types (bandpass/high-pass) of active filters.

Next up, understanding slew rate in modern high-speed devices and the transition from non-slew-limited step responses to slew limited edges.

**References**

- TI application note, “Filterpro
^{TM}Users Guide”, 2011, http://www.ti.com/lit/an/sbfa001c/sbfa001c.pdf - Planet Analog article, Michael Steffes, Feb. 8, 2018,”Include the op amp gain bandwidth product in the rauch low pass active filter performance equations”, https://www.planetanalog.com/include-the-op-amp-gain-bandwidth-product-in-the-rauch-low-pass-active-filter-performance-equations/
- TI Application note, “Design Methodology for MFB Filters in ADC Interface Applications”, Michael Steffes, Feb. 2006, http://www.ti.com/lit/an/sboa114/sboa114.pdf
- TINA simulator available from DesignSoft for <$350 for the Basic Plus edition. Includes a wide range of vendor op amp models and is the standard platform for TI op amp models.
- TI online active filter tool, https://webench.ti.com/filter-design-tool/filter-type
- Planet Analog article, Michael Steffes, Sept 23, 2019, ”Why is Gain Bandwidth Product (GBP) so Confusing, Insight #12”. https://www.planetanalog.com/why-is-amplifier-gbp-so-confusing-insight-12/

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