Sensitivity of 0.5; single and multi-parameter sensitivity

Meaning of $S^y_x = 0.5$

$$\frac{\Delta y}{y} \approx 0.5\frac{\Delta x}{x}$$

A 1% increase in $x$ causes about a 0.5% increase in $y$; the error is halved, so $y$ has low sensitivity to $x$. This happens, for example, when $y \propto \sqrt{x}$ ($S^{\sqrt{x}}_x = 1/2$), as with $\omega_o$ of an LC circuit ($\omega_o \propto 1/\sqrt{LC}$, magnitude 1/2).

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 $\sqrt{4 \times 0.25}\times 1\% = 1\%$.

Other single-parameter measures:

• Function sensitivity $S^{T(j\omega)}_x$: change of the whole response with $x$; its real part gives the magnitude change, $S^{|T|}_x = \text{Re}\,S^{T}_x$.
• Root (pole/zero) sensitivity $S^{p}_x = x\,\dfrac{\partial p}{\partial x}$ (unnormalized form): movement of a pole in the s-plane.

Multi-parameter use in design: single-parameter values show which element matters most, while the multi-parameter sum shows the total expected error from all elements together, which decides the component tolerances needed to meet the specification.