---
title: "SC filter applications and resistor simulation"
subject: "Filter Design"
chapter: 9
key: "semester_5/filter_design:sc-resistor-simulation-applications"
source: https://www.bctengineer.com/q/semester_5/filter_design/sc-resistor-simulation-applications
site: BCT Engineering
---

# SC filter applications and resistor simulation

Filter Design · Chapter 9: Switched-Capacitor Filters

- Asked in: 2078 Bhadra, 1+1+5 marks; 2069 Chaitra, 3+3 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-resistor-simulation-applications
- Whole chapter: https://www.bctengineer.com/md/semester_5/filter_design/chapter-9

## Question

What is a switched capacitor filter? What are its applications? How can you simulate a resistor using switched capacitor? Explain with necessary derivations.

## Short answer

A switched-capacitor filter replaces resistors with capacitors and MOS switches clocked at fc, giving Req = 1/(C fc). Applications: PCM codecs, anti-aliasing filters, audio equalizers. Resistor simulation: capacitor C switches between V1 and V2; charge Δq = C(V1–V2) per clock period gives average current I = C fc(V1–V2), so Req = 1/(C fc) = T/C.

## 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 capacitors, MOS switches and op-amps are all made on one MOS chip.

#### Applications

- Voice-band filters in PCM telephone codecs, modems and fax.
- Anti-aliasing/reconstruction filters and sigma-delta modulators in data converters.
- Audio equalisers, tone decoders, DTMF receivers.
- Clock-tunable universal filter ICs and programmable filters.
- Filters inside mixed-signal ICs (biomedical, sensor interfaces).

#### Simulating a resistor with a switched capacitor

```text
         phi1        phi2
  V1 o---/ ----+---- / ---o V2
               |
              === C
               |
              GND
```

Two non-overlapping clock phases $\phi_1$ and $\phi_2$ at frequency $f_c = 1/T$:

- During $\phi_1$ the capacitor connects to $V_1$: charge $q_1 = CV_1$.
- During $\phi_2$ it connects to $V_2$: charge $q_2 = CV_2$.

Charge moved from $V_1$ to $V_2$ in one clock period: $\Delta q = C(V_1 - V_2)$. Average current:

$$
I = \frac{\Delta q}{T} = C f_c (V_1 - V_2) \quad\Rightarrow\quad R_{eq} = \frac{V_1 - V_2}{I} = \frac{1}{C f_c} = \frac{T}{C}
$$

This holds when $f_c$ is much higher than the signal frequencies.

**Example:** $C$ = 1 pF switched at $f_c$ = 100 kHz gives $R_{eq} = 1/(10^{-12} \times 10^5)$ = 10 MΩ, which would need a huge area as a diffused resistor.

**Use in an integrator:** replacing the input resistor of an RC integrator by $C_1$ gives

$$
\frac{V_o}{V_i} = -\frac{1}{sR_{eq}C_2} = -\frac{C_1 f_c}{C_2}\cdot\frac{1}{s}
$$

so the time constant $C_2/(C_1f_c)$ depends only on a capacitor ratio and the clock, which is very accurate.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. A capacitor C is switched between V1 (phi1) and V2 (phi2). What charge moves per clock period?
   A. C(V1 + V2)
   B. C V1 only
   C. C(V1 - V2)
   D. C/(V1 - V2)
   Answer: C. q1 = C V1 and q2 = C V2, so the transferred charge is C(V1 - V2). (Key point: Charge transfer)

2. From I = C fc (V1 - V2), the equivalent resistance is:
   A. Req = C fc
   B. Req = T C
   C. Req = fc/C
   D. Req = 1/(C fc) = T/C
   Answer: D. Req = (V1 - V2)/I = 1/(C fc). (Key point: Equivalent resistance derivation)

3. What resistance does 1 pF switched at 100 kHz simulate?
   A. 10 MOhm
   B. 100 kOhm
   C. 1 MOhm
   D. 10 kOhm
   Answer: A. 1/(10^-12 x 10^5) = 10^7 ohm = 10 MOhm. (Key point: Example value)

4. For an integrator with input SC capacitor C1 and feedback C2, the transfer function is:
   A. -C2 fc/(C1 s)
   B. -(C1 fc/C2)(1/s)
   C. +C1/(C2 fc s)
   D. -C1 C2 fc / s
   Answer: B. Replacing R by 1/(C1 fc) in -1/(sRC2) gives -(C1 fc/C2)/s, which depends on a capacitor ratio and the clock. (Key point: SC integrator)

## Related in Filter Design

- [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
- [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
- [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×
- [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
- [SC filter applications; design for 100–400 rad/s peak](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Asc-applications-bode-100-400) — 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×

## Related in other subjects

- [Resistor and capacitor with half voltage](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Aadjustable-resistor-25uf-0-8a) — Basic Electrical Engineering, chapter 6