This article offers a strategy for minimizing filter response sensitivity.
Editor’s note: This article is a follow-up to the 2011 article, ‘Design second- and third-order Sallen Key filters with one op amp’ on EDN.
Figure 1 provides a schematic of the filter section which is the subject of this following article.
Figure 1 A schematic of a Sallen Key 2nd order low pass filter.
There are a number of different filter design packages available on the web. Some ask the user to specify filter order and response type, such as Butterworth or Bessel. Others ask for passband ripple and width, and stopband attenuation, and then select the response type automatically. These may offer optimizations for noise, voltage range, low power operation, or other parameters.
Other design packages provide more flexibility in that rather than dealing with response type, they allow the user to specify the quality factor Q and resonance frequency ω0 radians/sec of each 2nd order section. They constrain the problem by asking the user to specify a combination of additional parameters. These parameters might include DC gain and/or certain component values. With this information, calculations are performed to determine the remaining values. The issue of the sensitivities of Q and ω0 to component tolerances is sometimes mentioned. It is generally known that the filters’ amplitude response at ω0 exhibits the greatest variation, even for modest values of Q, but that this variation is minimized in the case of unity DC gain (Rf/Rg = 0.) Unfortunately, such sections require that C1 = 4·Q2·C2, which can be problematic for large values of Q. This paper offers a strategy for minimizing filter response sensitivity at frequency ω0 when C1 is selected to be less than 4·Q2·C2.
The response of this filter configuration is given by
Figure 2 Filter response in terms of component values, Q and ω0.
It is convenient to define some terms before solving for component values. The range of capacitor values suitable for a design is constrained by size, price and tolerance. As such, we want to exercise control over the ratio of the capacitances. We also wish to control DC gain, which is equal to 1 + Rf/Rg. And so we define
Figure 3 Definitions of some terms that are used in subsequent equations.
We see from  that only ratio Rr of resistors Rf and Rg matters, not their absolute values. So, once we specify Rr, we can choose Rf and Rg to be large to minimize power dissipation, or small to minimize noise. If we also specify C1 and C2, then Cr is determined. From these values, we can find R1 and R2.
Figure 4 Calculations of the values of R1 and R2 in terms of C2, Rr, Cr, Q and ω0. There are two sets of values for the resistors.
Realizability places some constraints on Rr and Cr. To ensure that the square root term, and therefore resistors R1 and R2 are real, it is required that
Figure 5 A realizability constraint for all filters.
Additionally, the expression for R2, in which the square root term is subtracted from 1/(2·Q), must be positive. We’ll call this the subtractive root and the other the additive one. And so, for subtractive root, the total constraint is
Figure 6 A realizability constraint for filters constructed from the subtractive root for R2.
Equation  limits R1 and R2 to a small range of values for the subtractive root. We’ll see later that we’ll want to avoid using this root to implement filters because we can always make less sensitive filters from the positive root for a given Q, ω0, Cr and Rr.
It is traditional to calculate the sensitivity of Q to variations in each of the values of the resistors and capacitors in the design. However, the approach taken here is to calculate the sensitivity of the magnitude of the filter response at frequency ω0, the frequency of greatest variation. This sensitivity to each of the six components is given as follows, where Rr12 = R2/R1:
Figure 7 The sensitivities of the magnitude of the filter response at resonance to each component value.
[Continue reading on EDN US: Find the sum of the squares of these sensitivities]
Christopher Paul has worked in various engineering positions in the communications industry for over 40 years.