Signal parameter sensitivity; ωo of Sallen-Key

Signal parameter sensitivity

Signal parameter sensitivity is the sensitivity of a filter's performance parameter, such as the centre (pole) frequency $\omega_o$, quality factor $Q$ or gain $H$, to a change in an element value $x$:

$$S^{\omega_o}_x = \frac{x}{\omega_o}\frac{\partial\omega_o}{\partial x}, \qquad S^{Q}_x = \frac{x}{Q}\frac{\partial Q}{\partial x}$$

It is the per-unit change of the parameter per unit change of the element, $\dfrac{\Delta\omega_o}{\omega_o} \approx S^{\omega_o}_x\dfrac{\Delta x}{x}$. This is the single-parameter sensitivity: one element is varied at a time while all the others stay fixed. These parameters fix the pole positions, so they show directly how the response shifts.

Sensitivity of $\omega_o$ of the Sallen-Key biquad

Sallen-Key low-pass biquad (gain $K = 1 + R_B/R_A$):

              C1
        +-----||-------------+
        |                    |
Vin -R1-+-(A)- R2 -+-(B)--|+\ |
                   |      |  >+---- Vo
                  === C2 +|-/ |
                   |     |    |
                  GND    +-RB-+
                         RA
                         |
                        GND

Its transfer function is

$$T(s) = \frac{K/(R_1R_2C_1C_2)}{s^2 + s\left(\frac{1}{R_1C_1} + \frac{1}{R_2C_1} + \frac{1-K}{R_2C_2}\right) + \frac{1}{R_1R_2C_1C_2}}$$

so

$$\omega_o = \frac{1}{\sqrt{R_1R_2C_1C_2}} = R_1^{-1/2}R_2^{-1/2}C_1^{-1/2}C_2^{-1/2}$$

Sensitivity with respect to $R_1$:

$$\begin{aligned} \frac{\partial\omega_o}{\partial R_1} &= -\frac12 R_1^{-3/2}(R_2C_1C_2)^{-1/2} = -\frac{\omega_o}{2R_1} \\ S^{\omega_o}_{R_1} &= \frac{R_1}{\omega_o}\cdot\left(-\frac{\omega_o}{2R_1}\right) = -\frac12 \end{aligned}$$

In the same way:

$$S^{\omega_o}_{R_1} = S^{\omega_o}_{R_2} = S^{\omega_o}_{C_1} = S^{\omega_o}_{C_2} = -\frac12$$

$\omega_o$ does not contain $K$, $R_A$ or $R_B$, so

$$S^{\omega_o}_{K} = S^{\omega_o}_{R_A} = S^{\omega_o}_{R_B} = 0$$

So a 1% increase in any one R or C lowers $\omega_o$ by about 0.5%; if all four increase by 1%, $\omega_o$ falls by about 2%. (The Q of the Sallen-Key circuit is more sensitive: for the equal-element design $S^Q_K = 3Q - 1$, which grows with Q.)