The nominal responses for these two circuits, Figure 4, are the same except for the expected gain difference of two (6dB).

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Now that we have established the component values, let's calculate the sensitivities associated with both circuits (Table 1).

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All the natural frequency sensitivities match between the two methods but, as expected, Q sensitivities to each of the components are better with the new method. Only the sensitivity of Q to K is worse with the new method but, as discussed earlier, K does not vary as long as we are using an adequate op amp.

But how much better in practice will this new method be over the cookbook method? An easy way to evaluate the two circuits is to run AC Monte Carlo analyses on both circuits. A Monte Carlo analysis runs the same analysis multiple times (AC analysis in our case), but randomizes each component value within it's tolerance for each run. The result is similar to a production run.

Let's use one percent resistors and five percent capacitors, and a flat (uniform) distribution.

Figure 5 shows that the cookbook version of the filter has about 1.5dB variation near the natural frequency, while the unity-gain version has only 0.5dB of variation.

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As one might expect, we don't get "something for nothing" with this low-sensitivity version of the Sallen-Key filter. We cannot get gain out of this circuit. Secondly, there is a practical limit to the magnitude of Q. To get high Qs, the capacitor value ratio gets very high making it hard to implement with practical capacitor values. We can determine this relationship very easily from the equation for Q, which has been greatly simplified to:

(31a)

or:

(31b)

Thus, the capacitors are related as follows:

(32a)

or:

(32b)