Sensitivity

What is sensitivity

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 $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$.

Importance in filter design

Importance in filter design:

• Real components have tolerances (e.g. ±5% resistors, ±10% capacitors) and drift with temperature and ageing. Sensitivity tells how much these errors shift $\omega_o$, $Q$ and gain.
• It lets the designer compare circuits that realize the same $T(s)$ and choose the one least affected by element changes (low-sensitivity design).
• It shows which elements need tight tolerance or trimming, which controls cost.
• High-$Q$ filters are especially sensitive, so sensitivity decides whether a design will work after manufacture.

Sensitivity of Q in the Tow-Thomas low-pass filter

Tow-Thomas low-pass biquad: OA1 is a lossy integrator ($R_2 \parallel C_1$ in feedback, inputs $V_{in}$ through $R_3$ and $V_3$ through $R_1$), OA2 is an integrator ($R_4$, $C_2$), and OA3 is a unity-gain inverter ($V_3 = -V_2$).

        +------R2------+
        +------||------+
        |      C1      |        +--||--+
Vin-R3--+--|-\         |        |  C2  |
        |  |  >--------+--R4----+-|-\  |
        |  +|/  V1     (OA1)    | |  >-+-- V2 (LP)
        | GND                GND+|/    |
        |                     (OA2)    |
        +--R1-- V3 --[inverter -1]-----+
                         (OA3)

Node equations (ideal op-amps):

$$\begin{aligned} V_1\left(\frac{1}{R_2} + sC_1\right) &= -\frac{V_{in}}{R_3} - \frac{V_3}{R_1} \\ V_2 &= -\frac{V_1}{sR_4C_2}, \qquad V_3 = -V_2 \end{aligned}$$

Solving:

$$T(s) = \frac{V_2}{V_{in}} = \frac{\dfrac{1}{R_3R_4C_1C_2}}{s^2 + \dfrac{s}{R_2C_1} + \dfrac{1}{R_1R_4C_1C_2}}$$

So

$$\omega_o = \frac{1}{\sqrt{R_1R_4C_1C_2}} = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}, \qquad Q = \omega_o R_2C_1 = R_2\sqrt{\frac{C_1}{R_1R_4C_2}}$$

Sensitivity of $Q$: $Q = R_2\,C_1^{1/2}\,R_1^{-1/2}R_4^{-1/2}C_2^{-1/2}$, so

$$S^{Q}_{R_2} = 1, \quad S^{Q}_{C_1} = \frac12, \quad S^{Q}_{R_1} = S^{Q}_{R_4} = S^{Q}_{C_2} = -\frac12, \quad S^Q_{R_3} = S^Q_r = 0$$

For example, $S^Q_{R_2} = \dfrac{R_2}{Q}\cdot\dfrac{\partial Q}{\partial R_2} = \dfrac{R_2}{Q}\cdot\dfrac{Q}{R_2} = 1$.

Check: sum over resistors $= 1 - \frac12 - \frac12 = 0$, sum over capacitors $= \frac12 - \frac12 = 0$, as expected for a dimensionless quantity.

All $Q$-sensitivities are small (magnitude ≤ 1) and independent of $Q$, so the Tow-Thomas biquad can realize high-$Q$ poles reliably. A 1% rise in $R_2$ raises $Q$ by 1%.

Definition

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 $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$.

Importance of sensitivity analysis

Importance in filter design:

• Real components have tolerances (e.g. ±5% resistors, ±10% capacitors) and drift with temperature and ageing. Sensitivity tells how much these errors shift $\omega_o$, $Q$ and gain.
• It lets the designer compare circuits that realize the same $T(s)$ and choose the one least affected by element changes (low-sensitivity design).
• It shows which elements need tight tolerance or trimming, which controls cost.
• High-$Q$ filters are especially sensitive, so sensitivity decides whether a design will work after manufacture.

Sensitivity analysis of the Tow-Thomas low-pass biquad

Tow-Thomas low-pass biquad: OA1 is a lossy integrator ($R_2 \parallel C_1$ in feedback, inputs $V_{in}$ through $R_3$ and $V_3$ through $R_1$), OA2 is an integrator ($R_4$, $C_2$), and OA3 is a unity-gain inverter ($V_3 = -V_2$).

        +------R2------+
        +------||------+
        |      C1      |        +--||--+
Vin-R3--+--|-\         |        |  C2  |
        |  |  >--------+--R4----+-|-\  |
        |  +|/  V1     (OA1)    | |  >-+-- V2 (LP)
        | GND                GND+|/    |
        |                     (OA2)    |
        +--R1-- V3 --[inverter -1]-----+
                         (OA3)

Node equations (ideal op-amps):

$$\begin{aligned} V_1\left(\frac{1}{R_2} + sC_1\right) &= -\frac{V_{in}}{R_3} - \frac{V_3}{R_1} \\ V_2 &= -\frac{V_1}{sR_4C_2}, \qquad V_3 = -V_2 \end{aligned}$$

Solving:

$$T(s) = \frac{V_2}{V_{in}} = \frac{\dfrac{1}{R_3R_4C_1C_2}}{s^2 + \dfrac{s}{R_2C_1} + \dfrac{1}{R_1R_4C_1C_2}}$$

So

$$\omega_o = \frac{1}{\sqrt{R_1R_4C_1C_2}} = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}, \qquad Q = \omega_o R_2C_1 = R_2\sqrt{\frac{C_1}{R_1R_4C_2}}$$

Sensitivity of $\omega_o$: $\omega_o = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}$, so using $S^{x^n}_x = n$:

$$S^{\omega_o}_{R_1} = S^{\omega_o}_{R_4} = S^{\omega_o}_{C_1} = S^{\omega_o}_{C_2} = -\frac12, \qquad S^{\omega_o}_{R_2} = S^{\omega_o}_{R_3} = S^{\omega_o}_{r} = 0$$

Example of one step: $S^{\omega_o}_{R_1} = \dfrac{R_1}{\omega_o}\dfrac{\partial\omega_o}{\partial R_1} = \dfrac{R_1}{\omega_o}\left(-\dfrac{1}{2}\dfrac{\omega_o}{R_1}\right) = -\dfrac12$.

Check: sum over resistors $= -1$, sum over capacitors $= -1$, as expected for a frequency.

Sensitivity of $Q$: $Q = R_2\,C_1^{1/2}\,R_1^{-1/2}R_4^{-1/2}C_2^{-1/2}$, so

$$S^{Q}_{R_2} = 1, \quad S^{Q}_{C_1} = \frac12, \quad S^{Q}_{R_1} = S^{Q}_{R_4} = S^{Q}_{C_2} = -\frac12, \quad S^Q_{R_3} = S^Q_r = 0$$

For example, $S^Q_{R_2} = \dfrac{R_2}{Q}\cdot\dfrac{\partial Q}{\partial R_2} = \dfrac{R_2}{Q}\cdot\dfrac{Q}{R_2} = 1$.

Check: sum over resistors $= 1 - \frac12 - \frac12 = 0$, sum over capacitors $= \frac12 - \frac12 = 0$, as expected for a dimensionless quantity.

Sensitivity of DC gain $H = \dfrac{R_1}{R_3}$ (from $T(0)$): $S^H_{R_1} = 1$, $S^H_{R_3} = -1$, all others 0.

All sensitivities are at most 1 in magnitude and do not depend on $Q$, so the Tow-Thomas biquad is a low-sensitivity circuit; $R_2$ tunes $Q$ alone, and $R_3$ sets the gain alone.

Sensitivity

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 $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$.

For example, $S^{\omega_o}_R = -0.5$ means a 1% increase in $R$ gives about a 0.5% decrease in $\omega_o$.

Importance of sensitivity analysis

Importance in filter design:

• Real components have tolerances (e.g. ±5% resistors, ±10% capacitors) and drift with temperature and ageing. Sensitivity tells how much these errors shift $\omega_o$, $Q$ and gain.
• It lets the designer compare circuits that realize the same $T(s)$ and choose the one least affected by element changes (low-sensitivity design).
• It shows which elements need tight tolerance or trimming, which controls cost.
• High-$Q$ filters are especially sensitive, so sensitivity decides whether a design will work after manufacture.

Sensitivity of $\omega_o$ in the Sallen-Key low-pass filter

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.)

Importance of sensitivity analysis

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 $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$.

Importance in filter design:

• Real components have tolerances (e.g. ±5% resistors, ±10% capacitors) and drift with temperature and ageing. Sensitivity tells how much these errors shift $\omega_o$, $Q$ and gain.
• It lets the designer compare circuits that realize the same $T(s)$ and choose the one least affected by element changes (low-sensitivity design).
• It shows which elements need tight tolerance or trimming, which controls cost.
• High-$Q$ filters are especially sensitive, so sensitivity decides whether a design will work after manufacture.

Useful properties: $S^{kx}_x = 1$, $S^{x^n}_x = n$, $S^{1/y}_x = -S^y_x$, $S^{y_1y_2}_x = S^{y_1}_x + S^{y_2}_x$, $S^{ky}_x = S^y_x$ ($k$ constant).

Sensitivity of $\omega_o$ of the Sallen-Key low-pass filter

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.)

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.)

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, $\dfrac{\Delta\omega_o}{\omega_o} \approx S^{\omega_o}_x\dfrac{\Delta x}{x}$. 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.)

Meaning of $S^y_x = 10$

$$S^y_x = \frac{\Delta y/y}{\Delta x/x} = 10 \;\Rightarrow\; \frac{\Delta y}{y} \approx 10\frac{\Delta x}{x}$$

A 1% increase in $x$ causes about a 10% increase in $y$ (same direction, since the sign is positive). $y$ is very sensitive to $x$: the element $x$ must have a tight tolerance, or the circuit should be changed to a lower-sensitivity design.

Sensitivity of $\omega_o$ of the Sallen-Key low-pass filter

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.)

Meaning of $S^x_y = -3$

$$S^x_y = \frac{\Delta x/x}{\Delta y/y} = -3 \;\Rightarrow\; \frac{\Delta x}{x} \approx -3\frac{\Delta y}{y}$$

A 1% increase in $y$ causes about a 3% decrease in $x$; the minus sign means $x$ moves opposite to $y$. Since $|S| > 1$, the change is amplified three times, so $x$ is fairly sensitive to $y$.

Sensitivity of $\omega_o$ of the Sallen-Key low-pass filter

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.)

Meaning of $S^x_y = -5$

$$S^x_y = \frac{\Delta x/x}{\Delta y/y} = -5 \;\Rightarrow\; \frac{\Delta x}{x} \approx -5\frac{\Delta y}{y}$$

A 1% increase in $y$ causes about a 5% decrease in $x$ (opposite direction, five times larger). $x$ is highly sensitive to $y$, so $y$ needs a tight tolerance.

Sensitivity of $\omega_o$ of the Sallen-Key low-pass filter

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.)

Meaning of $S^y_x = 2$

$$S^y_x = \frac{\Delta y/y}{\Delta x/x} = 2 \;\Rightarrow\; \frac{\Delta y}{y} \approx 2\frac{\Delta x}{x}$$

A 1% increase in $x$ causes about a 2% increase in $y$ (same direction, twice as large). For example, if $y \propto x^2$ then $S^y_x = 2$.

Sensitivity of the centre frequency of the Sallen-Key low-pass filter

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.)

Why sensitivity analysis is important

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 $\dfrac{\Delta y}{y} \approx S^y_x \dfrac{\Delta x}{x}$.

Importance in filter design:

• Real components have tolerances (e.g. ±5% resistors, ±10% capacitors) and drift with temperature and ageing. Sensitivity tells how much these errors shift $\omega_o$, $Q$ and gain.
• It lets the designer compare circuits that realize the same $T(s)$ and choose the one least affected by element changes (low-sensitivity design).
• It shows which elements need tight tolerance or trimming, which controls cost.
• High-$Q$ filters are especially sensitive, so sensitivity decides whether a design will work after manufacture.

Sensitivity of $\omega_o$ in the Tow-Thomas low-pass filter

Tow-Thomas low-pass biquad: OA1 is a lossy integrator ($R_2 \parallel C_1$ in feedback, inputs $V_{in}$ through $R_3$ and $V_3$ through $R_1$), OA2 is an integrator ($R_4$, $C_2$), and OA3 is a unity-gain inverter ($V_3 = -V_2$).

        +------R2------+
        +------||------+
        |      C1      |        +--||--+
Vin-R3--+--|-\         |        |  C2  |
        |  |  >--------+--R4----+-|-\  |
        |  +|/  V1     (OA1)    | |  >-+-- V2 (LP)
        | GND                GND+|/    |
        |                     (OA2)    |
        +--R1-- V3 --[inverter -1]-----+
                         (OA3)

Node equations (ideal op-amps):

$$\begin{aligned} V_1\left(\frac{1}{R_2} + sC_1\right) &= -\frac{V_{in}}{R_3} - \frac{V_3}{R_1} \\ V_2 &= -\frac{V_1}{sR_4C_2}, \qquad V_3 = -V_2 \end{aligned}$$

Solving:

$$T(s) = \frac{V_2}{V_{in}} = \frac{\dfrac{1}{R_3R_4C_1C_2}}{s^2 + \dfrac{s}{R_2C_1} + \dfrac{1}{R_1R_4C_1C_2}}$$

So

$$\omega_o = \frac{1}{\sqrt{R_1R_4C_1C_2}} = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}, \qquad Q = \omega_o R_2C_1 = R_2\sqrt{\frac{C_1}{R_1R_4C_2}}$$

Sensitivity of $\omega_o$: $\omega_o = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}$, so using $S^{x^n}_x = n$:

$$S^{\omega_o}_{R_1} = S^{\omega_o}_{R_4} = S^{\omega_o}_{C_1} = S^{\omega_o}_{C_2} = -\frac12, \qquad S^{\omega_o}_{R_2} = S^{\omega_o}_{R_3} = S^{\omega_o}_{r} = 0$$

Example of one step: $S^{\omega_o}_{R_1} = \dfrac{R_1}{\omega_o}\dfrac{\partial\omega_o}{\partial R_1} = \dfrac{R_1}{\omega_o}\left(-\dfrac{1}{2}\dfrac{\omega_o}{R_1}\right) = -\dfrac12$.

Check: sum over resistors $= -1$, sum over capacitors $= -1$, as expected for a frequency.

A 1% increase in $R_1$, $R_4$, $C_1$ or $C_2$ lowers $\omega_o$ by about 0.5%; $\omega_o$ is unaffected by $R_2$ (which sets $Q$) and $R_3$ (which sets gain).