--- title: "SC applications; summer and integrators" subject: "Filter Design" chapter: 9 key: "semester_5/filter_design:sc-applications-summer-integrators" source: https://www.bctengineer.com/q/semester_5/filter_design/sc-applications-summer-integrators site: BCT Engineering --- # SC applications; summer and integrators Filter Design · Chapter 9: Switched-Capacitor Filters - Asked in: 2076 Asoj, 1+1+5 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sc-applications-summer-integrators - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-9 ## Question What is switched capacitor filter? What are the applications of switched capacitor? How summer, inverting integrator and non-inverting integrator can be realized using switched capacitor? Explain with necessary diagrams and transfer function. ## Short answer A switched capacitor filter uses capacitors and switches to simulate resistors, enabling monolithic integration. Applications: telephone codecs, modems, tunable filters, sigma‑delta converters. Inverting summer: Vo = -Σ(Ci/CF)Vi. Inverting integrator: H(s) = -(fc C1)/(s CF). Non-inverting integrator: by reversing charge, H(s) = +(fc C1)/(s CF). ## 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. #### Applications - Voice-band filters in telephone systems (PCM codec anti-aliasing and reconstruction filters, DTMF tone decoders). - Modems, audio equalizers and speech processing ICs. - Clock-tunable (programmable) filters: changing $f_c$ shifts all frequencies, e.g. MF10, MAX7400 type ICs. - Mixed-signal ICs: sigma-delta ADCs, sample-and-hold, data acquisition and biomedical front ends. #### Realization of summer and integrators Basis: a capacitor $C$ switched at $f_c$ acts as $R = 1/(f_cC)$ (charge $C(V_1 - V_2)$ moved per period). So each resistor of an active-RC circuit is replaced by a switched capacitor. **Inverting summer.** The active-RC summer gives $V_o = -\left(\frac{R_F}{R_1}V_1 + \frac{R_F}{R_2}V_2\right)$. Replacing each resistor by a switched capacitor ($R = 1/(f_cC)$) gives $R_F/R_i = C_i/C_F$, so $$ V_o = -\left(\frac{C_1}{C_F}V_1 + \frac{C_2}{C_F}V_2\right) $$ ```text +--[CF-SC]---+ | | V1 -[C1-SC]--+ |\ | +---|-\ | V2 -[C2-SC]--+ | >-----+---> Vo +--|+/ | |/ GND ``` The gains depend only on capacitor ratios. **Inverting integrator.** 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. Replacing each resistor of the inverting summer by a switched capacitor gives which output? A. Vo = -(C1/CF)V1 - (C2/CF)V2 B. Vo = +(C1/CF)V1 + (C2/CF)V2 C. Vo = -(CF/C1)V1 - (CF/C2)V2 D. Vo = -fc(C1V1 + C2V2) Answer: A. RF/Ri becomes Ci/CF, so the gains are capacitor ratios and the output is inverted. (Key point: SC summer gain) 2. What is the transfer function of the SC inverting integrator? A. T(s) = -sCF/(fc·C1) B. T(s) = +fc·C1/(s·CF) C. T(s) = -C1/(fc·CF) D. T(s) = -fc·C1/(s·CF) Answer: D. Replacing R by 1/(fc·C1) in -1/(sR·CF) gives -fc·C1/(s·CF), with time constant CF/(fc·C1). (Key point: SC inverting integrator) 3. How does the SC non-inverting integrator get a positive sign? A. It adds an extra op-amp inverter B. The charge on C1 is reversed before reaching the op-amp, giving a negative resistance C. It uses a larger feedback capacitor D. It swaps the clock frequency Answer: B. C1 is connected so charge -C1·Vi enters CF. This acts as R = -1/(fc·C1), so T(s) = +fc·C1/(s·CF). (Key point: Non-inverting integrator principle) 4. Which is a typical application of SC filters? A. High-power RF transmitters B. Voice-band filters in telephone codecs C. Mains power transformers D. Optical amplifiers Answer: B. SC filters are used in voice-band telephone codecs, modems, audio ICs and mixed-signal circuits such as sigma-delta ADCs. (Key point: Applications) ## 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 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× - [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×