HO11_315aSP09_SC_part3

HO11_315aSP09_SC_part3 - Switched Capacitor Filters Part...

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Unformatted text preview: Switched Capacitor Filters Part III Boris Murmann Stanford University murmann@stanford.edu Copyright © 2009 by Boris Murmann EE315A ― HO #11 B. Murmann 1 Nonidealities in Switched Capacitor Filters • Finite amplifier gain • Finite amplifier bandwidth and slew rate p • Thermal noise – From SC resistor emulation – From amplifiers • Parasitic capacitance – Use parasitic insensitive configurations • Amplifier offset voltage and flicker noise – Often not an issue – If problematic use “correlated double sampling” • C Covered l t i thi course d later in this • Switch charge injection and clock feedthrough – Use “bottom plate sampling” • S EE315B See B. Murmann EE315A ― HO #11 2 SC Filter Nonidealities Sufficient to understand errors in the discrete time signal samples Must look at errors in the continuous time domain Much more complicated; signals are sums of reconstruction pulses, continuous time feedthrough signals, etc. Focus mostly on this aspect in the following discussion EE315A ― HO #11 B. Murmann 3 Finite Gain (1) n-1 n n+1 2 a0 1 t/Ts n-3/2 n-1/2 n+1/2 t/Ts Qs QI n-1/2 Cs·Vi(n-1/2) CI·Vo2(n-1)·[1+1/a0] n Cs·Vo2(n)/a0 CI·Vo2(n)·[1+1/a0] = CI·Vo(n-1)·[1+1/a0] + Cs·Vi(n-1/2) - Cs·Vo(n)/a0 B. Murmann EE315A ― HO #11 4 Finite Gain (2) 1 − ⎧⎡ 1⎤ 1 Cs ⎫ ⎪ ⎪ Vo 2 ( z )CI ⎨ ⎢1 + ⎥ ⎡1 − z −1 ⎤ + ⎬ = Vi ( z )Cs z 2 ⎦ a C ⎪ ⎪ ⎣ a0 ⎦ ⎣ 0 I ⎭ ⎩ − 1 Vo 2 ( z ) Cs C 1 z 2 = ≅ s Vi ( z ) CI ⎡ 1⎤ 1 Cs CI ⎡ 1⎤ 1 Cs −1 ⎢1 + ⎥ ⎡1 − z ⎤ + ⎢1 + ⎥ sTs + ⎣ ⎦ a C a0 CI 0 I ⎣ a0 ⎦ ⎣ a0 ⎦ ≅ Cs 1 sCITs ⎡ 1⎤ 1 Cs ⎢1 + ⎥ s + a0 CITs ⎣ a0 ⎦ Compare to active RC integrator with finite a0: A(s ) = −ω0 1 ⎛ 1⎞ ω s ⎜1+ ⎟ + 0 ⎝ a0 ⎠ a0 Bottom line: approximately same gain requirements as active RC EE315A ― HO #11 B. Murmann 5 Finite Bandwidth (1) • First order result – SC filters have much smaller amplifier bandwidth requirements than active RC counterparts K. Martin and A. Sedra, "Effects of the op amp finite gain and bandwidth on the performance of switched-capacitor filters," IEEE Trans. Circuits and Systems, vol. 28, no. 8, pp. 822-829, Aug. 1981. B. Murmann EE315A ― HO #11 6 Finite Bandwidth (2) • Unfortunately, this first order result relies on perfectly linear y, p y behavior in the amplifiers • As we will see later, the amplifiers do not settle linearly when large signals are present • As a result, it turns out that the bandwidth must be overdesigned significantly to meet typical linearity requirements • We will revisit this question once we have a better handle on the amplifier settling behavior (at the transistor level) • Everything considered it turns out that the bandwidth considered, requirements in SC filters are comparable to those in active RC realizations EE315A ― HO #11 B. Murmann 7 Noise Analysis Example Vin 1 VC1 2 Vout φ1 C1 • C2 φ2 Partition this P titi thi problem i t several steps bl into l t • First understand noise in samples acquired during φ1 • Next look at φ2 φ − Average current into infinitely large C2 − Noise spectrum and total noise of φ2 samples with finite C2 B. Murmann EE315A ― HO #11 8 Sampling Circuit • Questions – What is the rms noise in the VC1 samples? p – What is the spectrum of the discrete time sequence representing these samples? EE315A ― HO #11 B. Murmann 9 Noise Samples • The sample values VC1(n) correspond to the instantaneous values of the noise process in φ1 • From Parseval's theorem, we know that the time-domain power of this process is equal to its power spectral density integrated over all f ll frequencies i 2 v C1 1 = 4kTR ⋅ Δf 1 + sRC1 ∞ 2 var [VC1( n )] = vC1,tot = ∫ 4kTR ⋅ 0 B. Murmann 2 2 1 kT df = C1 1 + j 2πf ⋅ RC1 EE315A ― HO #11 10 Spectrum of Noise Samples • Strategy – Realize that discrete time noise samples are essentially instantaneous values (mTs apart) of the continuous time noise process during φ1 – Spectrum follows from Fourier transform of the process' autocorrelation function (Wiener Khintchin) (Wiener-Khintchin) • Samples show no correlation white spectrum • Samples are correlated colored spectrum EE315A ― HO #11 B. Murmann 11 Analysis (1) • Calculate autocorrelation function Autocorrelation of resistor noise (white) Rxx ( τ ) = δ ( 0 ) ⋅ 2kTR Impulse response of RC filter 1 −t / RC h (t ) = e RC Autocorrelation of filtered noise Ryy ( τ ) = Rxx ( τ ) ∗ h ( τ ) ∗ h ( −τ ) τ τ kT − RC Ryy ( τ ) = e C1 ∴ Ryy ( n ) = B. Murmann kT − e C1 n⋅mTs RC Covariance of samples separated by n clock cycles EE315A ― HO #11 12 Analysis (2) • Apply discrete time Fourier transform ∞ X ( ω) = ∑ Ryy ( n ) e j ω⋅nTs −∞ X (f ) = 2 kT fs C1 1 − e −2M ⎛ f 1 − 2e − M cos ⎜ 2π fs ⎝ ⎞ −2 M ⎟+e ⎠ M= “number of time constants in T i mTs” mTs RC1 M=1 M=3 M=5 2 1.5 • s X(f) / (2/f *kT/C) ( 1 2.5 Spectrum of noise samples is essentially “white” for M>3 1 0.5 0 0 0.1 01 0.2 02 0.3 03 0.4 04 0.5 05 f/f s EE315A ― HO #11 B. Murmann 13 Example Waveforms M= mTs 0.4 ⋅ 1μs = ≅ 12 RC1 100k Ω ⋅ 314fF • Large M (small RC1) means that the waveform “settles” accurately to the present input; the previous state is lost • Means that noise from cycle to cycle is uncorrelated spectrum p B. Murmann EE315A ― HO #9 white 14 “Noise Folding” (1) • The noise PSD of the samples is approximately PSDS = • 2 kT fs C1 The noise PSD of the resistor that causes the noise is PSDR = 4kTR • The ratio o t e t o PSDs is e at o of the two S s s PSDS M 1 1 = = =M PSDR 2fs RC1 2m • for m = 0.5 This increase in the noise PSD is due to aliasing or “folding” of noise from higher frequencies into the band from 0…fs/2 EE315A ― HO #11 B. Murmann 15 “Noise Folding” (2) fENB = 4kTR 1 4RC1 Ken Kundert, “Simulating Switched-Capacitor Filters with SpectreRF,” http://www designersSpectreRF http://www.designersguide.org/Analysis/sc-filters.pdf 2 kT fs C1 B. Murmann EE315A ― HO #11 16 R4 0.4*Ts/C1/M Simulation Schematic C1 = 1pF Ts = 1us M = 1, 2, 6, 10 B. Murmann EE315A ― HO #11 17 PNOISE Setup “Number of sidebands” – typically ~20…200 to handle noise folding properly. Fast switches properly more sidebands needed. Be sure to set “maxacfrequency” in the PSS analysis options to a correspondingly large value. “timedomain” means simulator computes spectrum of discrete time noise samples ( (no need to correct for “sinc”) ) sampling instant B. Murmann EE315A ― HO #11 18 Simulation Result Noise PSD Noise Integral ≅64uVrms M=1 M=1 M=7 M=7 EE315A ― HO #11 B. Murmann 19 Back to Lowpass Example • Circuit in non-overlap phase between φ1 and φ2: Noise from previous cycle kT C1 • During φ2, the following will happen − Noise charge on C1 will redistribute − Noise from previous cycle stored on C2 will redistribute − Noise generated by R will move charge back and forth between C1 and C2 B. Murmann EE315A ― HO #11 20 Simplified Analysis (1) • Goal: Calculate rms noise current into C2, assuming C2 – Vout is essentially a “virtual ground” virtual ground i2 φ1 Vin R VC1 kT C1 • infinity Vout C1 C2 At the end of φ2, C1 will be completely discharged, and therefore 2 iC 1 = 2 QC1 kTC1 = Ts2 Ts2 EE315A ― HO #11 B. Murmann 21 Simplified Analysis (2) Vin φ1 i2 R VC1 C1 • Vout C2 The noise from R induces a noise charge of kTC1 on C1 − This noise is separate and independent of the noise that was already stored on C1 (from φ1) 2 iR = 2 QC1 kTC1 = Ts2 Ts2 2 2 i 2 = iC1 + i R = 2kTC1fs2 i2 2 1 = 2kTC1fs2 = 4kTC1fs = 4kT Δf fs Ravg B. Murmann EE315A ― HO #11 Noise PSD in fs/2 is the same as that of a physical resistor! 22 Elaborate Analysis (1) • The previous result indicates that (at least for C2 infinity) a switched capacitor behaves roughly like a resistor in terms of the average noise current • In order to compute the spectrum of the noise samples taken at φ2 more work is needed • First, take a closer look at the φ1 noise and realize that we can directly refer its PSD to the input – Allows us to re-use the transfer function that we already know 2 v in kT 2 ≅ Δf C1 fs Input PSD 2 v out Δf Output PSD noiseless EE315A ― HO #11 B. Murmann 23 Elaborate Analysis (2) T − jω s 2 v e H( j ω) = out = C vin 1 + 2 1 − e − j ωTs C1 ( ) 2 T − jω s 2 2 2 v out v in e = ⋅ Δf Δf 1 + C2 1 − e − j ωTs C1 ( v Ravg = Ts C1 v2 1 v2 1 ≅ ∫ in ⋅ df = in Δf 1 + j ωR avg C2 Δf 4RavgC2 0 ≅ • 2 2 ∞ 2 out ) v2 1 ≅ in Δf 1 + j ωR avg C2 kT 2 fsC1 1 kT ⋅ ⋅ = C1 fs 4C2 2 C2 At the output, the noise from φ1 is lowpass filtered and contributes a total output noise of approximately kT/C2 B. Murmann EE315A ― HO #11 24 Elaborate Analysis (3) • Next look at noise introduced during φ2 kT v = ⎛ C1C2 ⎞ ⎜ ⎟ ⎝ C1 + C2 ⎠ 2 x v 2 out ⎛ C1 ⎞ kT = ⎜ ⎟ ⎛ C1C2 ⎞ ⎝ C1 + C2 ⎠ ⎜ ⎟ ⎝ C1 + C2 ⎠ ⎛ CC ⎞ 2 q x = kT ⎜ 1 2 ⎟ ⎝ C1 + C2 ⎠ EE315A ― HO #11 B. Murmann 25 Elaborate Analysis (4) • The noise charge qx can be referred to an equivalent noise voltage on C1, and subsequently referred to the input 2 vC 1 = • 2 q x kT ⎛ C1C2 ⎞ kT ⎛ C2 ⎞ kT = 2⎜ ⎟= ⎜ ⎟≅ 2 C1 C1 ⎝ C1 + C2 ⎠ C1 ⎝ C1 + C2 ⎠ C1 Complete model 2 2 v out v in 1 ≅ Δf Δf 1 + j ωRavgC2 2 v in kT 2 ≅2 Δf C1 fs White i Whit input PSD t Colored output PSD noiseless B. Murmann EE315A ― HO #11 26 2 R2 300K R1 300K Lowpass Simulation Circuit C1 = 68.83 fF C2 = 1 pF F fs = 1 MHz f-3dB = 10 kHz EE315A ― HO #11 B. Murmann 27 Simulation Result Noise PSD Noise Integral R1 noiseless R2 noiseless B. Murmann EE315A ― HO #11 28 sres p1! ideal R2 300K R1 300K Experiment: Reset Output During φ1 noiseless • Expecting to see – White noise spectrum – Total integrated noise power equal to v 2 out 2 2 ⎛ C1 ⎞ ⎛ C1 ⎞ kT ⎜ ⎟ + ⎜ ⎟ = 21.7μVrms ⎛ C1C2 ⎞ ⎝ C1 + C2 ⎠ ⎝ C1 + C2 ⎠ ⎜ ⎟ φ1 noise referred to output ⎝ C1 + C2 ⎠ kT = C1 φ 2 noise EE315A ― HO #11 B. Murmann 29 Simulation Result Noise PSD Noise Integral Good match! B. Murmann EE315A ― HO #11 30 SC Filter Summary • Pole and zero frequencies are proportional to sampling frequency and capacitor ratios – High accuracy and stability in response – Large time constants realizable without large R, C • Compatible with operational transconductance amplifiers; no need to drive resistive loads • Amplifier gain and BW requirements comparable to active RC • Noise – SC resistor emulation has same noise as an actual resistor – Arguing about amplifier noise requires detailed analysis • Special issue in SC circuits: noise aliasing • SC filters typically require continuous time anti-aliasing and reconstruction filters – Sometimes first order RC will suffice, particularly for large fs ,p y g B. Murmann EE315A ― HO #11 31 References (1) • R. Gregorian, K.W. Martin, and G.C. Temes, “Switched-Capacitor Circuit Design,” Proceedings of the IEEE, vol. 71, no. 8, pp. 941-966, Aug. 1983 • D.L. Fried, "Analog sample-data filters," IEEE J. Solid-State Circuits, vol. 7, no. 4, pp. 302-304, Aug. 1972 • D. Senderowicz et al., “A Family of Differential NMOS Analog Circuits for PCM Codec Filter Chip ” IEEE J Solid State Circuits pp 1014 1023 Dec 1982 Chip, J. Solid-State Circuits, pp.1014-1023, Dec. • T.C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," UC Berkeley, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31) • B.-S. B S Song and P.R. Gray "Switched Capacitor High-Q Bandpass Filters for IF PR Switched-Capacitor High Q Applications," IEEE J. Solid-State Circuits, pp. 924-933, Dec. 1986 • K. Martin and A. Sedra, "Effects of the op amp finite gain and bandwidth on the performance of switched-capacitor filters," IEEE Trans. Circuits and Systems, vol. 28, no. 8, pp. 822-829, Aug. 1981 • K.L. Lee, “Low Distortion Switched-Capacitor Filters," UC Berkeley, Ph.D. Thesis, Feb. 1986 (ERL Memorandum No. UCB/ERL M86/12) B. Murmann EE315A ― HO #11 32 References (2) • K. Martin and A.S. Sedra, “Stray-insensitive switched-capacitor filters based on the bilinear z transform,” Electronics Letters, vol. 19, pp. 365-366, June 1979 • R. Castello, R Castello and P R Gray "A high performance micropower switched-capacitor P.R. Gray, high-performance switched capacitor filter," IEEE J. Solid-State Circuits, vol. 20, no. 6, pp. 1122-1132, Dec. 1985 • J. H. Fischer, "Noise sources and calculation techniques for switched capacitor pp g filters," IEEE J. Solid-State Circuits, vol. 17, no. 4, pp. 742-752, Aug. 1982 • C.-A. Gobet and A. Knob, "Noise analysis of switched capacitor networks," IEEE Trans. Circuits and Systems, vol. 30, no. 1, pp. 37-43, Jan 1983 • y J. Goette and C.-A. Gobet, "Exact noise analysis of SC circuits and an approximate computer implementation," IEEE Trans. Circuits and Systems, vol. 36, no. 4, pp.508-521, Apr. 1989. • R. Schreier, et al., "Design-oriented estimation of thermal noise in switchedcapacitor circuits," IEEE Trans Circuits and Systems I vol 52 no 11 pp 2358 circuits Trans. I, vol. 52, no. 11, pp. 23582368, Nov. 2005 • K. Kundert, “Simulating Switched-Capacitor Filters with SpectreRF,” p g g g y p http://www.designers-guide.org/Analysis/sc-filters.pdf B. Murmann EE315A ― HO #11 33 ...
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