Single/multi-parameter sensitivity; ωo of Sallen-Key

Sensitivity of a filter

Sensitivity is the per-unit (fractional) change in a filter parameter $y$ (such as $\omega_o$, $Q$ or gain) caused by a per-unit change in an element value $x$:

$$S^y_x = \frac{\partial y / y}{\partial x / x} = \frac{x}{y}\frac{\partial y}{\partial x} = \frac{\partial \ln y}{\partial \ln x}$$

So $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$. It measures how much a filter parameter ($\omega_o$, $Q$, gain, or $|T(j\omega)|$) shifts when components deviate from their nominal values because of tolerance, temperature or ageing.

Single-parameter and multi-parameter sensitivity

Single-parameter sensitivity measures the effect of a change in one element while all others stay fixed:

$$S^y_x = \frac{x}{y}\frac{\partial y}{\partial x}, \qquad \frac{\Delta y}{y} \approx S^y_x\frac{\Delta x}{x}$$

Examples are $S^{\omega_o}_{R_1}$, $S^Q_K$, or the pole sensitivity $S^{p}_x$. Properties: $S^{x^n}_x = n$, $S^{1/y}_x = -S^y_x$, $S^{y_1y_2}_x = S^{y_1}_x + S^{y_2}_x$, $S^{ky}_x = S^y_x$.

Multi-parameter sensitivity gives the change in $y$ when all elements $x_1, x_2, \dots, x_n$ change together, as they do in practice. From the total differential:

$$\frac{\Delta y}{y} \approx \sum_{i=1}^{n} S^{y}_{x_i}\frac{\Delta x_i}{x_i}$$

• Worst case: if every element can be off by up to $\varepsilon_i$, the largest error is $\left|\dfrac{\Delta y}{y}\right|_{max} = \sum_i |S^y_{x_i}|\,\varepsilon_i$.
• Statistical (random, independent tolerances): the standard deviation is $\sigma_{y}/y = \sqrt{\sum_i (S^y_{x_i})^2\sigma_{x_i}^2}$, which is usually much smaller than the worst case.
• The Schoeffler sensitivity $\sum_i (S^y_{x_i})^2$ is a single figure of merit for comparing circuits.

Example: for the Sallen-Key $\omega_o$, all four sensitivities are −1/2. With ±1% elements the worst-case error is $4 \times 0.5 \times 1\% = 2\%$, while the statistical spread is $\sqrt{4 \times 0.25}\times 1\% = 1\%$.

Sensitivity analysis of $\omega_o$ of the Sallen-Key low-pass 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.)