--- title: "Sensitivity of Q in Tow-Thomas LPF" subject: "Filter Design" chapter: 7 key: "semester_5/filter_design:sensitivity-q-tow-thomas" source: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-q-tow-thomas site: BCT Engineering --- # Sensitivity of Q in Tow-Thomas LPF Filter Design · Chapter 7: Sensitivity - Asked in: 2082 Baisakh, 1+1+4 marks; 2081 Bhadra, 1+5 marks; 2082 Chaitra (new course), 3 marks; 2073 Shrawan, 6 marks; 2071 Shrawan, 1+1+2 marks; 2080 Baisakh, 4 marks (asked 6 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-q-tow-thomas - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-7 ## Question What is sensitivity? Describe its importance in filter design. Perform sensitivity analysis of quality factor (Q) in Tow-Thomas low pass filter. ## Short answer Sensitivity S_x^y=(x/y)(∂y/∂x) measures how component variations shift a parameter, guiding tolerance allocation and circuit choice, especially in high-Q designs. In the Tow-Thomas LP biquad, Q=R2√(C1/(R1R4C2)), so S_{R2}^Q=1, S_{C1}^Q=1/2, S_{R1}^Q=S_{R4}^Q=S_{C2}^Q=−1/2; all are small and Q-independent. ## Answer #### 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 - 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 ($V_3 = -V_2$). ```text +------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. | Element | $S^Q$ | |---|---| | $R_1$ | −1/2 | | $R_2$ | 1 | | $R_3$ | 0 | | $R_4$ | −1/2 | | $C_1$ | 1/2 | | $C_2$ | −1/2 | 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%. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does S^y_x = (x/y)(dy/dx) express? A. Absolute change in y per unit x B. Fractional change in y per fractional change in x C. Ratio of y to x D. Change in x per change in y Answer: B. Sensitivity is per-unit change in y per per-unit change in x, so Dy/y = S Dx/x. (Key point: Definition of sensitivity) 2. Why is sensitivity analysis important in filter design? A. It removes the need for components B. It shows how tolerance and drift of elements shift wo, Q and gain, and lets low-sensitivity designs be chosen C. It increases filter order D. It sets the supply voltage Answer: B. Real elements have tolerance and drift; sensitivity tells which circuit and which elements need tight tolerance. (Key point: Importance) 3. What is Q of the Tow-Thomas low-pass biquad? A. R2 sqrt(C1/(R1 R4 C2)) B. R3 sqrt(C1/(R1 R4 C2)) C. R2 sqrt(C2/(R1 R4 C1)) D. sqrt(R1 R4 C1 C2)/R2 Answer: A. Q = wo R2 C1 with wo = 1/sqrt(R1R4C1C2), giving R2 sqrt(C1/(R1R4C2)). (Key point: Q expression) 4. Which set is correct for the Q sensitivities? A. S to R2 = 1/2, to R1 = -1, to C1 = 1/2 B. S to R2 = 1, to C1 = 1/2, to R1, R4, C2 = -1/2, to R3 = 0 C. S to R2 = 1, to C1 = -1/2, to R1 = 1/2 D. S to R3 = 1, to R2 = 0 Answer: B. Exponents in Q = R2 C1^(1/2) R1^(-1/2) R4^(-1/2) C2^(-1/2) are the sensitivities; R3 does not appear. 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