Analyzing circuit sensitivity for analog circuit design

Mark Fortunato, Texas Instruments

April 16, 2008

Mark Fortunato, Texas Instruments

These sensitivity equations are simply the exponents of the components in Equations 8 and 9. They are all constants, which means there is nothing we can do within this circuit topology to improve sensitivities. Also, note that the natural frequency sensitivity to R₁ is zero. This confirms our earlier statement that R₁ has no effect on the natural frequency.

These results are useful in understanding what sensitivity equations mean. For example, if C₁ is increased by a factor of four, the natural frequency decreases by a factor of two (Equation 14):

(14)

Q, on the other hand, increases by a factor of two when C₁ increases by four.

Transfer function sensitivity
As discussed earlier, we can also calculate the sensitivity of the transfer function to each of the component values. Using Equation 1, which defines sensitivity, the sensitivity of the transfer function for this circuit to C₁ is:

(15)

Without showing all the intermediate steps (I had to pull out my old calculus text book to get this right), the end result is:

(16)

Similarly, the transfer function sensitivities to L₁and R₁ are:

(17)

and:

(18)

These results are far more complex than the Q and natural frequency sensitivities, and thus difficult to put to use, especially since they are functions of both the components and frequency.

We can plot these sensitivity functions using the nominal component values from the schematic. The plot is shown in Figure 4.

A lot of good information is in these plots, but we get all we need for most situations with the far simpler Q and ω_n sensitivities.

Passive filters, like the RLC lowpass filter in Figure 3 always have Q and ω_n sensitivities in the –1 to +1 range, with ±0.5 being most common. Sensitivities in this range generally are considered as good as it gets. Active filters have more flexibility in choices with increased complexity of sensitivities.

Active filter example: The Sallen-Key Filter
We'll use the venerable and ubiquitous Sallen-Key for our first active filter. The Sallen-Key lowpass filter shown in Figure 5 first described over 50 years ago⁴ is one of the most common filter topologies.