Sensitivity of ωo in Tow-Thomas LPF

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 ($V_3 = -V_2$) built with two equal resistors $r$ (the gain stays $-1$ when both change together).

        +------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).