Circuit Sensitivity Analysis--An Important Tool for Analog Circuit Design: Part 2
The Sallen-Key filter can result in very sensitive circuits. This is the second part of a two-part article on how to analyze circuit sensitivity.
Figures 10 through 12 show the variations in AC Frequency response that we get using one percent resistors and five percent capacitors for all three filter circuits by running Monte Carlo analyses with 300 runs.
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The low-sensitivity Sallen-Key and MFB versions exhibit about 3dB of gain variation around the natural frequency, or about ± 15 percent. The cookbook version has over 14dB gain variation meaning the gain of any two units in production would could differ by as much as a factor of five!
Op amp open loop gain considerations
We made several statements like "...as long as we have chosen op amps with sufficient open-loop gain over our frequency range...." What happens if you use an op amp with insufficient loop gain? What loop gain is sufficient? Even a follower's gain will decrease with insufficient loop gain.
Let's model an otherwise ideal op amp with 120dB low frequency open-loop gain and a single dominant pole to role off the open-loop gain. We make the model, Figure 13 such that the GBWP can be varied for our experiment.
Implementing the high-Q stage of the low-sensitivity Sallen-Key filter with this simplified op amp model, the frequency response changes with the different GBWP versions (Figure 14).
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Both 10M and 100M GBWP versions seem to be identical to the ideal op amp, while the 1M op amp seems close and the 100k GBWP is clearly inadequate. Let's take a closer look in Figure 15.
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This close-up shows that the 10M GBWP op amp deviates slightly from the ideal, while 100M still seems identical to the ideal.
As with passive component variations, we can use Monte Carlo analysis to vary the GBWP. This model allows us to readily modify the GBWP simply by varying R3. Real op amp GBWPs tend to vary ±30 percent. With the tolerance of R3 set to ±30 percent, we run Monte Carlo analyses on the 1M, 10M and 100M GBWP op amps.
The 100MHz GBWP op amp shows no perceivable difference from the ideal op amp. The 10MHz op amp deviates from the ideal by about 0.25dB with about 0.05dB variation. The 1MHz op amp peak deviates in frequency from the ideal by about six percent, and about 3dB in amplitude with about 0.7dB variation in amplitude.
