--- title: "SC summer, inverting and non-inverting integrators" subject: "Filter Design" chapter: 9 key: "semester_5/filter_design:sc-summer-integrators" source: https://www.bctengineer.com/q/semester_5/filter_design/sc-summer-integrators site: BCT Engineering --- # SC summer, inverting and non-inverting integrators Filter Design · Chapter 9: Switched-Capacitor Filters - Asked in: 2076 Chaitra, 1+5 marks; 2080 Baisakh, 2+4 marks (asked 2 times) - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sc-summer-integrators - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-9 ## Question What is switched capacitor filter? How summer, inverting integrator and non-inverting integrator can be realized using switched capacitor? Explain with necessary diagrams and transfer function. ## Short answer Switched-capacitor filters replace resistors with capacitors and MOS switches, giving Req=1/(Cfc). Using a four-switch branch, inverting integrator: H(s)≈ –(C1 fc/C2)/s; non-inverting integrator: H(s)≈ (C1 fc/C2)/s by swapping input phases. Summer: multiple branches feeding an op-amp with feedback capacitor CF; with reset across CF, output is –(C1/CF V1 + C2/CF V2). ## Answer #### Switched-capacitor filter A **switched-capacitor (SC) filter** is an active filter in which the resistors of an active-RC filter are replaced by capacitors switched by MOS transistors driven by a clock. Each switched capacitor behaves like a resistor $R = 1/(Cf_c)$, so the time constants depend only on capacitor ratios and the clock frequency. The basic SC resistor (charge $C_1(V_1 - V_2)$ moved each clock period) gives $R_{eq} = 1/(C_1f_c)$. The building blocks below use the **parasitic-insensitive** four-switch branch: capacitor $C_1$ with two switches on each plate. ```text S1 C1 S3 Vi o---/ ---+---| |---+---/ ---o to (-) of op-amp | | (virtual ground) / S2 / S4 | | GND GND ``` #### Inverting integrator Switching: $\phi_1$: S2 and S4 closed (C1 discharged). $\phi_2$: S1 and S3 closed: $C_1$ charges to $V_i$ through the virtual ground, so charge $C_1V_i$ is pushed onto the feedback capacitor $C_2$ with negative sign. $$ H(z) = -\frac{C_1}{C_2}\cdot\frac{1}{1 - z^{-1}} \qquad\approx\qquad H(s) = -\frac{C_1 f_c}{C_2}\cdot\frac{1}{s} $$ #### Non-inverting integrator Change the phases of the left switches: $\phi_1$: S1 and S4 closed ($C_1$ charges to $V_i$ with its right plate grounded). $\phi_2$: S2 and S3 closed (left plate grounded, right plate to the virtual ground). The charge is now delivered with the opposite polarity, one half-period later: $$ H(z) = +\frac{C_1}{C_2}\cdot\frac{z^{-1}}{1 - z^{-1}} \qquad\approx\qquad H(s) = +\frac{C_1 f_c}{C_2}\cdot\frac{1}{s} $$ | Switch | Inverting integrator | Non-inverting integrator | |---|---|---| | S1 (input side) | $\phi_2$ | $\phi_1$ | | S2 (left plate to ground) | $\phi_1$ | $\phi_2$ | | S3 (right plate to op-amp) | $\phi_2$ | $\phi_2$ | | S4 (right plate to ground) | $\phi_1$ | $\phi_1$ | Circuit for both integrators (the same box, with the switch phases of the table above): ```text +-----| C2 |-----+ | | Vi o--[SC branch C1]--+---(-) | | A -----+--o Vo GND--(+) ``` #### Summer (and summing integrator) Connect several SC branches $C_1, C_2, \dots$ (one per input) to the same virtual-ground node. The charges add: ```text +------------| CF |------------+ | (reset switch across CF| | for the summer) | | |\ | V1 o--[SC C1]+------------|-\ | | | >---------------+--o Vo V2 o--[SC C2]+ +----|+/ | |/ GND ``` Each box [SC Ci] is the four-switch branch shown above. - With a feedback capacitor $C_F$ only: **summing integrator** $V_o = -\dfrac{f_c}{s}\left(\dfrac{C_1}{C_F}V_1 + \dfrac{C_2}{C_F}V_2\right)$. - With $C_F$ reset each period (a switch across $C_F$), or with an SC resistor in feedback: **summer (amplifier)** $V_o = -\left(\dfrac{C_1}{C_F}V_1 + \dfrac{C_2}{C_F}V_2\right)$. A branch wired with non-inverting phasing gives a + sign for that input, so sums and differences are both possible with one op-amp. All gains are capacitor ratios, hence accurate. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the approximate s-domain transfer function of the SC inverting integrator? A. +(C1 fc/C2)/s B. -(C1 fc/C2)/s C. -C2/(C1 fc s) D. -C1/C2 Answer: B. H(z) = -(C1/C2)/(1 - z^-1), which approximates -(C1 fc/C2)(1/s). (Key point: Inverting integrator) 2. How is the non-inverting integrator obtained from the same branch? A. Swap C1 and C2 B. Remove the feedback capacitor C. Change the phasing of the left switches (S1, S4 in phi1; S2, S3 in phi2) so charge arrives with opposite polarity D. Double the clock frequency Answer: C. Changing the switch phases delivers the charge with the opposite sign, giving H(z) = +(C1/C2) z^-1/(1 - z^-1). (Key point: Non-inverting integrator) 3. How is a summer built with SC circuits? A. Put the inputs through one common switch B. Use inductors at the virtual ground C. Add the clock signals D. Connect several SC branches to the same virtual-ground node so their charges add Answer: D. Charges from each branch add at the virtual ground; with a reset feedback capacitor Vo = -(C1/CF V1 + C2/CF V2). (Key point: Summer) 4. Which circuit is used for the building blocks to reduce parasitic effects? A. A single grounded capacitor with one switch B. A four-switch branch with two switches on each plate of C1 C. A resistor-capacitor ladder D. A capacitor without switches Answer: B. The four-switch branch (S1-S4) makes the transfer depend only on C1/C2. (Key point: Parasitic-insensitive branch) ## Close variants set in other exams - [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× - [SC inverting lossy, inverting and non-inverting integrators](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-lossy-integrator-integrators) — Filter Design, chapter 9, asked 3× ## 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 - [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 - [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 - [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 - [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× - [Active filter; non-inverting (s+8)/(s+2)](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Aactive-filter-noninverting-s-plus-8) — Filter Design, chapter 6, asked 5× ## Related in other subjects - [Transfer function of op-amp integrator-type circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-r1-r2-c) — Control Systems, chapter 2 - [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