Using the general equation for sensitivity, the sensitivities of Q and ω_n are:

(22)

(23)

(24)

(25)

(26)

(27)

Note that the natural frequency sensitivities, as with the passive filter above, are either ± 0.5 or zero, whereas Q sensitivities are significantly more complex.

Natural frequency sensitivities to R_a and R_b (as well as to K) are zero. This means that the natural frequency is independent of the values of these resistors (and to the DC gain).

These simpler sensitivities are constants. We can do nothing within this circuit topology to change these sensitivities. As discussed earlier, sensitivities of magnitude 0.5 and less are as good as it gets. So for this circuit, let's concern ourselves with Q sensitivities.

The example in Figure 6 is a 1 kHz low-pass filter with a nominal Q of 1. We'll vary R_b to change the Q without changing the natural frequency.

Let's vary R_b from 1k to 19.9k. At R_b = 20k, the equation for Q goes to infinity and should be avoided.

View the full-size image

Since R_b is a gain-determining component, the DC gain, shown in Figure 7, changes along with the Q. Monitor the feedback voltage at the op amp's inverting terminal, Vfb in the schematic, to see the gain-normalized responses, Figure 8.

View the full-size image

View the full-size image

The results in Figure 8 are much the same as with the passive filter in Figure 7.

Moving forward from here, we know that we can calculate sensitivities for this and other circuits. We can also use this information to determine which components are contributing most significantly to response variation so we can specify more precise (and more expensive) parts in these locations. Sometimes this is all that we can do.

The Multiple Feedback Filter

Another very common filter is the multiple feedback filter (MFB) shown in Figure 9.

View the full-size image

This filter has two feedback paths, thus, its name. It is an inverting circuit, whereas the Sallen-Key utilized a non-inverting amplifier. The transfer function of this circuit is given by:

(28)

The natural frequency and Q are given by:

(29)

(30)

The sensitivity equations are:

(31)

(32)

(33)

(34)

(35)

(36)

Equation 30 tells us that, once again we have a component (R₁) that can be varied to change the Q without affecting the natural frequency.

Equations 31 and 32 indicate that all sensitivities to the capacitors are –0.5 or 0.5.

Equations 34, 35, and 36 are quite complex. Unfortunately, many textbooks leave these equations in this form.¹ However, if we simply plug in Q from Equation 29 to each of these, we greatly simplify these equations:

(37)

(38)

(39)

where "//" means "in parallel with."

By inspection we can tell that these Q sensitivities to the resistors are all less than one. The numerator in Equation 36 must be less than the denominator as R₁ paralleled with any other resistors necessarily is smaller than R₁ itself. Similarly, numerators in Equations 37 and 38 must be smaller than the denominators. All the terms are the same but, while a difference is taken in the numerator, a sum is taken in the denominator. The half term out front means that Q sensitivities to R₂ and R₃ are actually less than 0.5.

As discussed, having all sensitivities with magnitudes of one or less is as good as it usually gets. Does this mean that the MFB filter is always less sensitive than the Sallen-Key? In many instances it does. Digging deeper into the math, however, we may find some surprises.

In Part 2, which will be posted online at www.embedded.com, we will delve deeper into the mathematics of the Sallen-Key filter, and find that we can implement highly insensitive filters with this topology. We will also look at the effects of op amp bandwidth, as well as how to use Monte Carlo analysis in determining how much a filter's transfer function will vary in production.

For the last five years, Mark Fortunato has been an analog field applications manager for Texas Instruments where his staff covers Southwest USA. He has a BS in electrical engineering from CalTech and works primarily on DC to 200 GHz.

Endnotes:

1. Huelsman, L.P. and Allen, P.E. Introduction to the Theory and Design of Active Filters. McGraw-Hill, New York, 1980.

2. Budak, Aram. Passive and Active Network Analysis and Synthesis. Houghton Mifflin Company, Boston, 1974.

3. Ghausi, M.S. and Laker, K.R. Modern filter Design: Active RC and Switched Capacitor. Prentice-Hall, Englewood Cliffs, N.J., 1981.

4. Sallen, R.P. and Key, E.L. "A Practical Method of Designing Active Filters," IRE Transactions on Circuit Theory, vol. CT-2, pp.74-85, March 1955.

Further reading:

Fortunato, M. "Circuit Sensitivity; With Emphasis On Analog Filters," Texas Instruments Developer Conference 2007, March 2007: http://focus.ti.com/general/docs/tidc/general.tsp?templateId=6180&navigationId=12622&path=templatedata/cm/tidcgeneral/data/am_landing/ww_07 presentations.