--- title: "SC inverting lossy, inverting and non-inverting integrators" subject: "Filter Design" chapter: 9 key: "semester_5/filter_design:sc-lossy-integrator-integrators" source: https://www.bctengineer.com/q/semester_5/filter_design/sc-lossy-integrator-integrators site: BCT Engineering --- # SC inverting lossy, inverting and non-inverting integrators Filter Design · Chapter 9: Switched-Capacitor Filters - Asked in: 2071 Chaitra, 7 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sc-lossy-integrator-integrators - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-9 ## Question What is switched capacitor filter? How inverting lossy integrator, integrator and non-inverting integrator can be realized using switched capacitor? Explain with necessary diagrams and transfer functions. ## Short answer A switched capacitor filter replaces resistors with capacitors and MOS switches, using a two-phase clock. An inverting lossy integrator has T(s) = -(f_c C1/C_F)/(s + f_c C2/C_F). An inverting integrator gives T(s) = -f_c C1/(s C_F). A non-inverting integrator is realized by reversing the switched capacitor charge, yielding T(s) = +f_c C1/(s C_F). ## Answer #### Switched capacitor filter A **switched capacitor (SC) filter** is an active filter in which every resistor of an active-RC filter is replaced by a capacitor and MOS switches driven by a two-phase non-overlapping clock. The switched capacitor behaves like a resistor $R = 1/(f_cC)$, so the filter needs only op-amps, capacitors and switches, all easily built on one MOS chip. **SC resistor.** A capacitor $C$ switched between nodes 1 and 2 by non-overlapping clocks transfers $\Delta q = C(V_1 - V_2)$ each period $T_c$, so $I = f_cC(V_1 - V_2)$ and $R_{eq} = 1/(f_cC)$. ```text phi1 phi2 V1 ----o/o----+----o/o---- V2 | === CR | GND ``` #### Inverting lossy integrator Active-RC form: input resistor $R_1$, feedback $R_2 \parallel C_F$: $$ T(s) = -\frac{1/(R_1C_F)}{s + 1/(R_2C_F)} $$ Replacing $R_1$ and $R_2$ by switched capacitors $C_1$ and $C_2$ ($1/R = f_cC$): $$ T(s) = -\frac{f_cC_1/C_F}{s + f_cC_2/C_F} $$ ```text CF +-----||-----+ | | +--[C2-SC]---+ | | | |\ | Vi -[C1-SC]--+---|-\ | | >-----+---> Vo +--|+/ | |/ GND ``` DC gain $= -C_1/C_2$ and pole $\omega_p = f_cC_2/C_F$, both set by capacitor ratios. It is the first-order low-pass section used in SC ladders and biquads. #### Integrator (inverting) The active-RC integrator has $T(s) = -\frac{1}{sRC_F}$. Replacing $R$ by a switched capacitor $C_1$: $$ T(s) = \frac{V_o}{V_i} = -\frac{f_cC_1}{sC_F} = -\frac{1}{s}\cdot\frac{C_1}{C_F}f_c $$ ```text CF +-----||-----+ phi1 | phi2 | Vi ---o/o--+-o/o-+ |\ | | +--|-\ | ===C1 | >--+---> Vo | +--|+/ GND | |/ GND ``` The integrator time constant $C_F/(f_cC_1)$ is set by a capacitor ratio. (In the z-domain, the stray-insensitive version gives $H(z) = -\frac{C_1}{C_F}\cdot\frac{1}{1 - z^{-1}}$.) #### Non-inverting integrator Here the switched capacitor is connected so that its charge is **reversed** before it reaches the op-amp: during $\phi_1$, $C_1$ is charged to $V_i$ (top plate to $V_i$, bottom plate to ground); during $\phi_2$, the top plate is grounded and the bottom plate is connected to the virtual ground. Charge $-C_1V_i$ is pushed into $C_F$, so the switched capacitor acts as a **negative resistor** $R = -1/(f_cC_1)$: $$ T(s) = \frac{V_o}{V_i} = +\frac{f_cC_1}{sC_F} $$ ```text phi1 C1 phi2 CF Vi ---o/o--+-||-+--o/o--+ +---||---+ | | | | | GND -o/o--+ +-o/o- +----+ |\ | phi2 | phi1 +--|-\ | GND | >--+--> Vo +--|+/ | |/ GND ``` (Left plate: $V_i$ on $\phi_1$, ground on $\phi_2$; right plate: ground on $\phi_1$, op-amp input on $\phi_2$.) In the z-domain: $H(z) = +\frac{C_1}{C_F}\cdot\frac{z^{-1}}{1 - z^{-1}}$. A non-inverting integrator removes the need for an extra inverter in biquads, which is a big advantage of SC over active-RC. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What makes the lossy integrator different from the ideal integrator? A. It has a feedback switched capacitor C2 in parallel with CF, giving a finite DC gain -C1/C2 B. It has no feedback capacitor C. It uses a non-overlapping clock only D. It has two op-amps Answer: A. The extra feedback resistance (switched capacitor C2) limits the DC gain to -C1/C2 and creates a pole at fc·C2/CF. (Key point: Lossy integrator) 2. What is the time constant of the SC inverting integrator? A. fc·C1/CF B. C1/(fc·CF) C. CF/(fc·C1) D. fc·CF/C1 Answer: C. T(s) = -fc·C1/(s·CF), so the time constant is CF/(fc·C1), set by a capacitor ratio. (Key point: Inverting integrator) 3. In the SC non-inverting integrator, what charge reaches CF each period? A. +C1·Vi B. -C1·Vi C. 0 D. C1·Vi/2 Answer: B. C1 is charged to Vi on φ1 and its plates are swapped on φ2, so the reversed charge -C1·Vi goes to CF. This acts as a negative resistance. (Key point: Non-inverting integrator charge) 4. What is the advantage of the SC non-inverting integrator in biquads? A. It removes the need for an extra inverter B. It removes the need for a clock C. It increases the resistor values D. It makes the response independent of fc Answer: A. A non-inverting integrator gives a positive sign directly, so no extra inverting stage is needed, an advantage over active-RC. (Key point: Advantage of non-inverting integrator) ## Close variants set in other exams - [SC summer, inverting and non-inverting integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-summer-integrators) — Filter Design, chapter 9, asked 4× ## Related in Filter Design - [SC equivalents: inverting summer, lossy and non-inverting integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-equivalent-summer-lossy-integrator) — Filter Design, chapter 9 - [SC resistor, summing integrator and inverting lossy integrator](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-resistor-summing-lossy-integrator) — Filter Design, chapter 9 - [Why switched capacitors in IC; summer and integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-ic-summer-integrators) — Filter Design, chapter 9 - [SC applications; summer and integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-applications-summer-integrators) — Filter Design, chapter 9, asked 3× - [Why SC in IC; resistor simulation; summer and integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-resistor-summer-lossy-noninverting) — Filter Design, chapter 9 - [Switched capacitor filter for (s+200)(s+800)/(s+400)²](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-filter-transfer-function-200-800) — Filter Design, chapter 9, asked 4× ## Related in other subjects - [Virtual short circuit and inverting integrator](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Avirtual-short-integrator) — Basic Electronics Engineering, chapter 4