Switched Capacitor Circuits

Posted on Edited on In

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Switched-Capacitor Filter

switched-Capacitor Resistor

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[https://youtu.be/G0lzrMll-Ho], [https://youtu.be/eMOFMjuKiJQ]

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Due to not taking loading \(C_2\) into account, actual switched-capacitor filter deviate from equivalent \(R_{SC}\) + \(C_2\) low pass filter as \(f_p\) approaching to \(f_s\)

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image-20260320225532492 \[ \color{red}H(z) =\frac{V_{OUT}(z)}{V_{IN}(z)}=\frac{C_1z^{-1/2}}{C_1+C_2}\frac{1}{1-\frac{C_2}{C_1+C_2}z^{-1}} \]

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[https://youtu.be/eMOFMjuKiJQ]

Track-and-Hold (TH)

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image-20260301153439749 \[ H_\mathrm{TH}(f) \approx \frac{1}{2} \left( 1 + \mathrm{sinc}\left( \frac{f}{2f_s} \right)e^{-j\pi f T_s/2} \right) \]

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PSS+PAC SImulaiton [https://youtu.be/VLdcY76V9Ss]

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Sample-and-Hold (SH)

Zero-Order Hold

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Given \(\color{red}T_p = T_s\) \[ H_\mathrm{SH}(f) \approx \mathrm{sinc}\left( \frac{f}{f_s} \right)e^{-j\pi f T_s} \]

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image-20240928101832121 \[ h_{ZOH}(t) = \text{rect}(\frac{t}{T} - \frac{1}{2}) = \left\{ \begin{array}{cl} 1 & : \ 0 \leq t \lt T \\ 0 & : \ \text{otherwise} \end{array} \right. \] The effective frequency response is the continuous Fourier transform of the impulse response \[ H_{ZOH}(f) = \mathcal{F}\{h_{ZOH}(t)\} = T\frac{1-e^{j2\pi fT}}{j2\pi fT}=Te^{-j\pi fT}\text{sinc}(fT) \] where \(\text{sinc}(x)\) is the normalized sinc function \(\frac{\sin(\pi x)}{\pi x}\)

The Laplace transform transfer function of the ZOH is found by substituting \(s=j2\pi f\) \[ H_{ZOH}(s) = \mathcal{L}\{h_{ZOH}(t)\}=\frac{1-e^{-sT}}{s} \]

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Clock Feedthrough

aka. LO leakage

TODO 📅

Integrator

TODO 📅

[https://www.eecg.utoronto.ca/~johns/ece1371/slides/10_switched_capacitor.pdf]

[https://www.seas.ucla.edu/brweb/papers/Journals/BRWinter17SwCap.pdf]

[https://class.ece.iastate.edu/ee508/lectures/EE%20508%20Lect%2029%20Fall%202016.pdf]

reference

Boris Murmann. EE315A VLSI Signal Conditioning Circuits [pdf]

Kwantae Kim. ELEC-E3530 Integrated Analog Systems D (5 ECTS) [video] [github]

R. S. Ashwin Kumar, Analog circuits for signal processing [https://home.iitk.ac.in/~ashwinrs/2022_EE698W.html]

R. Gregorian and G. C. Temes. Analog MOS Integrated Circuits for Signal Processing. Wiley-Interscience, 1986 [pdf]

Christian-Charles Enz. "High precision CMOS micropower amplifiers" [pdf]

Negar Reiskarimian. CICC 2025 Insight: Switched Capacitor Circuits [https://youtu.be/SL3-9ZMwdJQ]