From 231f59f73426c953922d1a7cbf59124d4ca9449e Mon Sep 17 00:00:00 2001 From: Carsten Wulff Date: Thu, 1 Feb 2024 13:23:39 +0100 Subject: [PATCH] update l5 --- Makefile | 4 +- ex/dt.py | 50 + lectures/l05_sc.md | 369 +- media/diff_ota.png | Bin 0 -> 104652 bytes media/diff_ota_bias.png | Bin 0 -> 60817 bytes media/l00_SAR9B_CV.png | Bin 0 -> 322441 bytes media/l05_fund1.pdf | Bin 0 -> 100019 bytes media/l05_fund1.svg | 22 + media/l05_fund2.pdf | Bin 0 -> 100019 bytes media/l05_fund2.svg | 29 + media/l05_fund3.pdf | Bin 0 -> 100017 bytes media/l05_fund3.svg | 19 + media/l5_dtfig.pdf | Bin 0 -> 148117 bytes media/l5_dtfig.svg | 581 ++ media/l5_fir.pdf | Bin 0 -> 289709 bytes media/l5_fir.svg | 42 + media/l5_iir.pdf | Bin 0 -> 104479 bytes media/l5_iir.svg | 11160 ++++++++++++++++++++++++++++++++++++++ media/l5_novl.pdf | Bin 0 -> 212420 bytes media/l5_novl.svg | 111 + media/l5_scamp.pdf | Bin 0 -> 161554 bytes media/l5_scamp.svg | 65 + media/l5_scex.pdf | Bin 0 -> 171769 bytes media/l5_scex.svg | 181 + media/l5_scfig.pdf | Bin 0 -> 212329 bytes media/l5_scfig.svg | 46 + media/l5_scifig.pdf | Bin 0 -> 182327 bytes media/l5_scifig.svg | 43 + media/l5_scint.pdf | Bin 0 -> 182258 bytes media/l5_scint.svg | 59 + media/l5_scintro1.pdf | Bin 0 -> 117269 bytes media/l5_scintro1.svg | 54 + media/l5_scintro2.pdf | Bin 0 -> 117269 bytes media/l5_scintro2.svg | 52 + media/l5_sdomain.pdf | Bin 0 -> 136951 bytes media/l5_sdomain.svg | 26 + media/l5_sh.pdf | Bin 0 -> 300198 bytes media/l5_sh.svg | 111 + media/l5_shaaf.pdf | Bin 0 -> 300395 bytes media/l5_shaaf.svg | 138 + media/l5_subsample.pdf | Bin 0 -> 134329 bytes media/l5_subsample.svg | 113 + media/l5_sw1.pdf | Bin 0 -> 117672 bytes media/l5_sw1.svg | 91 + media/l5_sw2.pdf | Bin 0 -> 117679 bytes media/l5_sw2.svg | 117 + media/l5_sw3.pdf | Bin 0 -> 117683 bytes media/l5_sw3.svg | 152 + media/l5_zdomain.pdf | Bin 0 -> 137010 bytes media/l5_zdomain.svg | 36 + media/l5_zunstable.pdf | Bin 0 -> 289855 bytes media/l5_zunstable.svg | 36 + media/lewis.png | Bin 0 -> 125768 bytes pdf/aic.tex | 4 +- 54 files changed, 13609 insertions(+), 102 deletions(-) create mode 100644 ex/dt.py create mode 100644 media/diff_ota.png create mode 100644 media/diff_ota_bias.png create mode 100644 media/l00_SAR9B_CV.png create mode 100644 media/l05_fund1.pdf create mode 100644 media/l05_fund1.svg create mode 100644 media/l05_fund2.pdf create mode 100644 media/l05_fund2.svg create mode 100644 media/l05_fund3.pdf create mode 100644 media/l05_fund3.svg create mode 100644 media/l5_dtfig.pdf create mode 100644 media/l5_dtfig.svg create mode 100644 media/l5_fir.pdf create mode 100644 media/l5_fir.svg create mode 100644 media/l5_iir.pdf create mode 100644 media/l5_iir.svg create mode 100644 media/l5_novl.pdf create mode 100644 media/l5_novl.svg create mode 100644 media/l5_scamp.pdf create mode 100644 media/l5_scamp.svg create mode 100644 media/l5_scex.pdf create mode 100644 media/l5_scex.svg create mode 100644 media/l5_scfig.pdf create mode 100644 media/l5_scfig.svg create mode 100644 media/l5_scifig.pdf create mode 100644 media/l5_scifig.svg create mode 100644 media/l5_scint.pdf create mode 100644 media/l5_scint.svg create mode 100644 media/l5_scintro1.pdf create mode 100644 media/l5_scintro1.svg create mode 100644 media/l5_scintro2.pdf create mode 100644 media/l5_scintro2.svg create mode 100644 media/l5_sdomain.pdf create mode 100644 media/l5_sdomain.svg create mode 100644 media/l5_sh.pdf create mode 100644 media/l5_sh.svg create mode 100644 media/l5_shaaf.pdf create mode 100644 media/l5_shaaf.svg create mode 100644 media/l5_subsample.pdf create mode 100644 media/l5_subsample.svg create mode 100644 media/l5_sw1.pdf create mode 100644 media/l5_sw1.svg create mode 100644 media/l5_sw2.pdf create mode 100644 media/l5_sw2.svg create mode 100644 media/l5_sw3.pdf create mode 100644 media/l5_sw3.svg create mode 100644 media/l5_zdomain.pdf create mode 100644 media/l5_zdomain.svg create mode 100644 media/l5_zunstable.pdf create mode 100644 media/l5_zunstable.svg create mode 100644 media/lewis.png diff --git a/Makefile b/Makefile index 0fbf018..2fbb362 100644 --- a/Makefile +++ b/Makefile @@ -11,8 +11,8 @@ FILES = l00_diode \ l02_esd \ l03_refbias \ l04_afe \ -# l05_sc \ - l06_adc \ + l05_sc \ +# l06_adc \ l07_vreg \ l08_pll \ l09_osc \ diff --git a/ex/dt.py b/ex/dt.py new file mode 100644 index 0000000..d414b7e --- /dev/null +++ b/ex/dt.py @@ -0,0 +1,50 @@ +#!/usr/bin/env python3 +# +import numpy as np +import matplotlib.pyplot as plt + +Hann = True + +#- Create a time vector +N = 2**13 +t = np.linspace(0,N,N) + +#- Create the "continuous time" signal with multiple sinusoidal signals and some noise +f1 = 233/N +fd = 1/N*119 +x_s = np.sin(2*np.pi*f1*t) + 1/1024*np.random.randn(N) + 0.5*np.sin(2*np.pi*(f1-fd)*t) + 0.5*np.sin(2*np.pi*(f1+fd)*t) + +#- Create the sampling vector, and the sampled signal +t_s_unit = [1,1,0,0,0,0,0,0] +t_s = np.tile(t_s_unit,int(N/len(t_s_unit))) +x_sn = x_s*t_s + +#- Convert to frequency domain with a hanning window to avoid FFT bin +#- energy spread +if(Hann): + w = np.hanning(N+1) +else: + w = np.ones(N+1) +X_s = np.fft.fftshift(np.fft.fft(np.multiply(w[0:N],x_s))) +X_sn = np.fft.fftshift(np.fft.fft(np.multiply(w[0:N],x_sn))) + + + +plt.subplot(2,2,1) +plt.ylabel("Time Domain") +plt.plot(x_s) +plt.xlabel("Continuous time, continuous value") +plt.subplot(2,2,2) +plt.plot(x_sn) +plt.ylabel("Frequency Domain") +plt.xlabel("Discrete time, continuous value") +plt.subplot(2,2,3) +plt.plot(20*np.log10(np.abs(X_s))) +plt.subplot(2,2,4) +plt.plot(20*np.log10(np.abs(X_sn))) + +fig = plt.gcf() +fig.set_size_inches(12, 7) +plt.tight_layout() +plt.savefig(f"l5_dt.pdf") +plt.show() diff --git a/lectures/l05_sc.md b/lectures/l05_sc.md index e045b54..84bd145 100644 --- a/lectures/l05_sc.md +++ b/lectures/l05_sc.md @@ -1,16 +1,17 @@ -footer: Carsten Wulff 2023 +footer: Carsten Wulff 2024 slidenumbers:true autoscale:true theme: Plain Jane, 1 text: Helvetica header: Helvetica +date: 2024-02-09 -## TFE4188 - Introduction to Lecture 5 +## TFE4188 - Lecture 5 # Switched-Capacitor Circuits --- @@ -100,12 +101,13 @@ resistors. But how large is the on-resistance of the transistor switch? Would that also affect our precision? -But is the calibration step linear when we add the transistors. If we have a non-linear +But is the calibration step linear with addition of the transistors? If we have a non-linear calibration step, then we cannot use gradient decent calibration algorithms, nor can we use binary search. -Analog designers need to deal with an almost infinite series of "But", and -the experienced designer will know when to stop, when is the "But what if" not +Analog designers need to deal with an almost infinite series of "But". + +The experienced designer will know when to stop, when is the "But what if" not a problem anymore. The most common error in analog integrated circuit design is a "I did not imagine that @@ -113,7 +115,7 @@ my circuit could fail in this manner" type of problem. Or, not following the lin But if we follow all the "But"'s we will never tapeout! -Although active-RC filters are great for linearity and easy to drive, but if we need accurate time constant, there are better alternatives. +Active-RC filters are great for linearity, but if we need accurate time constant, there are better alternatives. --> @@ -148,12 +150,159 @@ Same as Active-RC, Gm-C need calibration to get accurate pole or zero frequency. --- -# Switched capacitor circuits +# Switched capacitor + +--- + + +![left fit](../media/l05_fund1.pdf) + + + + + + $$Q_{\phi2\$} = C_1 V_{GND} = 0$$ + + + + $$Q_{\phi1\$} = C_1 V_{I}$$ + + + + $$ Z_{I} = (V_{I} - V_{GND})/I_{I}$$ + + + + $$ I_{I} = \frac{Q}{dt} = Q f_{\phi}$$ + + + + + $$ Z_{I} = \frac{V_{I} - V_{GND}}{\left(Q_{\phi1$} - Q_{\phi2$}\right) f_{\phi}}$$ + + + + $$ Z_{I} = \frac{V_{I}}{\left(V_{I} C - 0 \right) f_{\phi}} = \frac{1}{C_1 f_\phi}$$ + + + + + +--- + + + +![left fit](../media/l05_fund2.pdf) + + + + +$$ Z_{I} = \frac{V_{I} - V_{O}}{\left(Q_{\phi1$} - Q_{\phi2$}\right) f_{\phi}}$$ + +$$ Q_{\phi1\$} = C_1 (V_I - V_O)$$ + +$$ Q_{\phi2\$} = 0 $$ + +$$ Z_{I} = \frac{V_{I} - V_{O}}{\left(C_1 (V_I - V_O)\right) f_{\phi}} = \frac{1}{C_1 f_\phi}$$ + + + + + +--- + + + +![left fit](../media/l05_fund3.pdf) + +$$ Z_{I} = \frac{V_{I} - V_{O}}{\left(Q_{\phi1$} - Q_{\phi2$}\right) f_{\phi}}$$ + +$$ Q_{\phi1\$} = C_1 V_I )$$ + +$$ Q_{\phi2\$} = C_1 V_O $$ + + + +$$ Z_{I} = \frac{V_{I} - V_{O}}{\left(C_1 V_I - C_1 V_O)\right) f_{\phi}} = \frac{1}{C_1 f_\phi}$$ + + + + + + +--- + + + + @@ -167,10 +316,7 @@ to create a gain. Imagine we create a circuit without the switches, and with a resistor of $R$ from input to virtual ground, and $4R$ in the feedback. Our Active-R would have a gain of $A = 4$. -You might not believe it, but the circuit below is almost the same, but we've -used switched capacitor. The complete amplifier still has a $A = 4$, but not all the time. - -The switches disconect the OTA and capacitors for half the time, but for the other half, +The switches disconnect the OTA and capacitors for half the time, but for the other half, at least for the latter parts of $\phi_2$ the gain is four. --> @@ -184,18 +330,18 @@ time interval. The circuit is discrete time. As long as all circuits afterwards also have a discrete-time input, then it's fine. An ADC can sample the output from the amplifier at the right time, and never notice that the output is shorted to a DC voltage in $\phi_1$ -Switched capacitor circuits rely on charge transfer. We charge the capacitor $4C$ to the +We charge the capacitor $4C$ to the differential input voltage in $\phi_1$ $$ Q_1 = 4 C V_{in} $$ Then we turn off $\phi_1$, which opens all switches. The charge on $4C$ will still be $Q_1$ -(except for higher order effects like charge injection from switches). +(except for higher order effects like charge injection from switches). -After a short time (overlap), we turn on $\phi_2$, closing some of the switches. -The OTA will start to force it's two inputs to be the same, and we short the left side of +After a short time (non-overlap), we turn on $\phi_2$, closing some of the switches. +The OTA will start to force its two inputs to be the same voltage, and we short the left side of $4C$. After some time we would have the same voltage on the left side of $4C$ for the -two capacitors, and another voltage on the left side of the $4C$ capacitors. The two +two capacitors, and another voltage on the right side of the $4C$ capacitors. The two capacitors must now have the same charge, so the difference in charge, or differential charge must be zero. @@ -228,8 +374,9 @@ fantastic feature. Assume we make two identical capacitors in our layout. We won integrated circuit, whether the $C_1$ is 100 fF or 80 fF, but we can be certain that if $C_1 = 80$ fF, then $C_2 = 80$ fF to a precision of around 0.1 %. -With switched capacitor amplfiers we can set an accurate gain, and we can set an accurate pole and zero frequency (as long as we have an accurate cloc and a high +With switched capacitor amplifiers we can set an accurate gain, and we can set an accurate pole and zero frequency (as long as we have an accurate clock and a high DC gain OTA). + The switched capacitor circuits do have a drawback. They are discrete time circuits. As such, we must treat them with caution, and they will always need some analog filter before to avoid a phenomena we call aliasing. @@ -242,15 +389,14 @@ some analog filter before to avoid a phenomena we call aliasing. @@ -432,13 +577,13 @@ our wanted signal (green) is undisturbed. Assume that we we're interested in the red signal. We could still use a sample rate of $f_s$. If we bandpass-filtered all but the red signal the red signal would fold on sampling, as shown in the figure below. -Remember that the [Nyquist-Shannon](https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem) states that a suffient +Remember that the [Nyquist-Shannon](https://en.wikipedia.org/wiki/Nyquist–Shannon_sampling_theorem) states that a sufficient no-loss condition is to sample signals with a sample rate of twice the bandwidth of the signal. Nyquist-Shannon has been extended for sparse signals, compressed sensing, and non-uniform sampling to demonstrate that it's sufficient for the average sample rate to be twice the bandwidth. One 2009 paper [Blind Multiband Signal Reconstruction: Compressed Sensing for Analog Signal](https://ieeexplore.ieee.org/document/4749297) is -a good place to start to devlve into the latest on signal reconstruction. +a good place to start to delve into the latest on signal reconstruction. --> @@ -511,7 +656,7 @@ The "x" shows the complex frequency where the fourier transform goes to infinity Any real circuit will have complex conjugate, or real, poles/zeros. A combination of two real circuits where one path is shifted 90 degrees in phase can have non-conjugate complex poles/zeros. -If the "x" is $a<0$, then any pertubation will eventually die out. +If the "x" is $a<0$, then any perturbation will eventually die out. If the "x" is on the $a=0$ line, then we have a oscillator that will ring forever. If the "x" is $a>0$ then the oscillation amplitude will grow without bounds, although, only in Matlab. In any physical circuit an oscillation cannot grow without bounds forever. @@ -556,7 +701,7 @@ Assume a first order filter given by the discrete time equation. $$ y[n+1] = bx[n] + ay[n] \Rightarrow Y z = b X + a Y$$ $$ y[n] = b x[n-1] + ay[n-1] \Rightarrow Y = b X z^{-1} + a Y z^{-1} $$ @@ -596,7 +741,16 @@ python example. There are smarter, and faster ways to do IIR filters (and FIR) in python, see [scipy.signal.iirfilter](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.iirfilter.html) -[iir.py](https://github.com/wulffern/aic2023/blob/main/ex/iir.py) +--> + +[iir.py](https://github.com/wulffern/aic2024/blob/main/ex/iir.py) + +--- + +![ fit ](../media/l5_iir.pdf) + + --- @@ -624,7 +776,7 @@ The IIR filter we implemented above is a low-pass filter, and the filter partial $$ H(z) = \frac{1}{3}\sum_{i=0}^2 z^{-1}$$ @@ -645,68 +796,95 @@ $$ H(z) = \frac{1}{3}\sum_{i=0}^2 z^{-1}$$ #[fit] Switched-Capacitor - --- - -## **Q:** How do switched capacitor circuits work? + + +![left fit](../media/l5_scintro1.pdf) -The $C_2$ capacitor is shorted, as such, $V_2 = 0$, which must mean that the charge on $C_2$ given by $Q_2 = 0$. +$$Q_{1\phi_1\$} = C_1 V_1$$ + + + +$$Q_{2\phi_1\$} = 0$$ + + +--- + +![left fit](../media/l5_scintro2.pdf) + + + +$$Q_{1\phi_2\$} = 0$$ + + -thus +$$ Q_{2\phi_2\$} = Q_{1\phi_1\$} = C_1 V_1 = C_2 V_2$$ + + +$$ \frac{V_2}{V_1} = \frac{C_1}{C_2}$$ -![original fit](../media/l5_scintro.pdf) ---- - -#[fit] Principles --- + ## Switched capacitor gain circuit --- - -## **Q:** How many different OTAs do you know? - ---- - + +[wulffern/sun\_sar9b\_sky130nm](https://github.com/wulffern/sun_sar9b_sky130nm) + +![original fit](../media/l00_SAR9B_CV.png) + +--- + + @@ -995,37 +1177,28 @@ The non-overlap generator is standard. 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