--- title: "Sensitivity analysis of Tow-Thomas LPF" subject: "Filter Design" chapter: 7 key: "semester_5/filter_design:sensitivity-tow-thomas-lowpass" source: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-tow-thomas-lowpass site: BCT Engineering --- # Sensitivity analysis of Tow-Thomas LPF Filter Design · Chapter 7: Sensitivity - Asked in: 2078 Bhadra, 1+4 marks; 2074 Chaitra, 1+4 marks; 2074 Asoj, 2+4 marks; 2081 Bhadra, 6 marks; 2078 Bhadra, 5 marks; 2076 Asoj, 1+5 marks (asked 5 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sensitivity-tow-thomas-lowpass - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-7 ## Question Define sensitivity. What is the importance of sensitivity analysis in filter design? Perform the sensitivity analysis of Tow-Thomas low pass biquad filter (with respect to all the resistors and capacitors present in the circuit). ## Short answer Sensitivity S_x^y=(x/y)(∂y/∂x) quantifies drift from component tolerances, crucial for reliable design. In the Tow-Thomas LP biquad, ω_o sensitivities are −1/2 to R1,R4,C1,C2 and 0 to others. Q sensitivity is 1 to R2, 1/2 to C1, −1/2 to R1,R4,C2. DC gain H=R1/R3 has S_{R1}^H=1, S_{R3}^H=−1. All sensitivities are ≤1 in magnitude. ## Answer #### 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 - 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 ($V_3 = -V_2$). ```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 $\omega_o$:** $\omega_o = R_1^{-1/2}R_4^{-1/2}C_1^{-1/2}C_2^{-1/2}$, 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. | Element | $S^{\omega_o}$ | $S^{Q}$ | |---|---|---| | $R_1$ | −1/2 | −1/2 | | $R_2$ | 0 | 1 | | $R_3$ | 0 | 0 | | $R_4$ | −1/2 | −1/2 | | $C_1$ | −1/2 | 1/2 | | $C_2$ | −1/2 | −1/2 | | Inverter $r$ | 0 | 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. ## 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 are the wo sensitivities in the Tow-Thomas biquad? A. -1/2 for R1, R4, C1, C2 and 0 for R2, R3 B. -1/2 for R2 and R3 only C. 1 for R2 and 0 for others D. +1/2 for all R and C Answer: A. wo = (R1R4C1C2)^(-1/2), so those four give -1/2; R2 and R3 do not appear. (Key point: wo sensitivities) 4. DC gain H = R1/R3. What is the overall conclusion? A. S to R1 = -1, S to R3 = 1, and sensitivities grow with Q B. S to R1 = 1, S to R3 = -1; all sensitivities are at most 1 and independent of Q C. S to R1 = 1/2, S to R3 = -1/2; sensitivity grows with Q D. Gain does not depend on R1 or R3 Answer: B. H = R1/R3 gives +1 and -1. All sensitivities are small and independent of Q: low-sensitivity circuit. (Key point: Gain and conclusion) ## Close variants set in other exams - [Importance of sensitivity; ωo of Sallen-Key LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-wo-sallen-key-importance) — Filter Design, chapter 7, asked 13× ## Related in Filter Design - [Sensitivity of 0.1; ωo of Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-value-0-1-tow-thomas) — Filter Design, chapter 7, asked 4× - [Sensitivity of ωo and Q in Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-wo-q-tow-thomas) — Filter Design, chapter 7, asked 3× - [Sensitivity of -3; Q of Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-minus-3-q-tow-thomas) — Filter Design, chapter 7, asked 2× - [Tow-Thomas band pass biquad design](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atow-thomas-bandpass-design) — Filter Design, chapter 6 - [Tow-Thomas high pass derivation; LPF design](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Atow-thomas-highpass-derive-lpf-design) — Filter Design, chapter 6, asked 3× - [Sensitivity of Q in Tow-Thomas LPF](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asensitivity-q-tow-thomas) — Filter Design, chapter 7, asked 6× ## Related in other subjects - [RC low pass filter operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Arc-low-pass-filter) — Basic Electronics Engineering, chapter 1, asked 5× - [Low-pass FIR window design, αs = 40 dB](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Afir-lowpass-window-0-3-0-5-40) — Digital Signal Analysis and Processing, chapter 5, asked 2× - [Impulse response of ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-response) — Signals and Systems, chapter 5, asked 5×