--- title: "Why switched capacitors in IC; summer and integrators" subject: "Filter Design" chapter: 9 key: "semester_5/filter_design:sc-ic-summer-integrators" source: https://www.bctengineer.com/q/semester_5/filter_design/sc-ic-summer-integrators site: BCT Engineering --- # Why switched capacitors in IC; summer and integrators Filter Design · Chapter 9: Switched-Capacitor Filters - Asked in: 2080 Bhadra, 2+4 marks - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/filter_design/sc-ic-summer-integrators - Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-9 ## Question Why resistors are replaced by switched capacitors in IC technology? How summer, inverting integrator and non-inverting integrator can be realized using switched capacitor? Explain with necessary diagrams and expressions. ## Short answer Resistors in ICs occupy large area and have poor absolute accuracy; switched capacitors yield compact, accurate time constants set by capacitor ratios and clock frequency, with Req=1/(fc C). Inverting summer: Vo = -Σ(Ci/CF)Vi. Inverting integrator: H(s) = -(fc C1)/(s CF). Non-inverting integrator: charge reversal yields H(s) = +(fc C1)/(s CF). ## Answer #### Why resistors are replaced by switched capacitors in IC technology - **Area:** MΩ resistors take huge silicon area; an equivalent switched capacitor is about 1 pF. - **Accuracy:** absolute R and C on chip vary about ±20 % independently, so RC products are poor. SC time constants $C_F/(f_cC_R)$ depend on **capacitor ratios** (about 0.1 % accurate) and a crystal clock. - **Temperature/voltage tracking:** ratios of matched capacitors stay constant; resistors drift. - **Tunable** by the clock frequency, and fully compatible with MOS digital circuits on the same 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 ``` #### Realization of summer and integrators **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. Why are resistors replaced by switched capacitors in ICs? A. Resistors cannot carry current in MOS B. MΩ resistors take huge area, while an equivalent switched capacitor is about 1 pF C. Capacitors are always more linear than op-amps D. Switches have zero capacitance Answer: B. Large resistors need great silicon area, but a switched capacitor of about 1 pF gives about 10 MΩ. (Key point: Area advantage) 2. What sets the time constant CF/(fc·C1) of the SC integrator? A. Absolute capacitor value only B. A capacitor ratio and the clock frequency C. Resistor tolerance D. The op-amp gain Answer: B. The time constant is a capacitor ratio divided by fc, so it is accurate and tunable by the clock. (Key point: Integrator time constant) 3. What is T(s) of the SC non-inverting integrator? A. -fc·C1/(s·CF) B. +fc·C1/(s·CF) C. +s·CF/(fc·C1) D. -C1/CF Answer: B. The reversed charge acts as a negative resistor, so the sign becomes positive: +fc·C1/(s·CF). (Key point: Non-inverting integrator result) ## Related in Filter Design - [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× - [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 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× - [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 simulate resistors with switched capacitors in MOS](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-resistor-simulation-mos) — Filter Design, chapter 9 ## 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