Unformatted text preview: Switched Capacitor Filters
Part III Boris Murmann
Stanford University
[email protected]
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)
n1 n n+1 2 a0 1
t/Ts n3/2 n1/2 n+1/2 t/Ts Qs QI n1/2 Cs·Vi(n1/2) CI·Vo2(n1)·[1+1/a0] n Cs·Vo2(n)/a0 CI·Vo2(n)·[1+1/a0]
= CI·Vo(n1)·[1+1/a0] + Cs·Vi(n1/2)  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
switchedcapacitor filters," IEEE Trans. Circuits and
Systems, vol. 28, no. 8, pp. 822829, 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 timedomain 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)
(WienerKhintchin)
• 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 SwitchedCapacitor
Filters with SpectreRF,” http://www designersSpectreRF http://www.designersguide.org/Analysis/scfilters.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 nonoverlap 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 reuse 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
f3dB = 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 antialiasing 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, “SwitchedCapacitor Circuit
Design,” Proceedings of the IEEE, vol. 71, no. 8, pp. 941966, Aug. 1983 • D.L. Fried, "Analog sampledata filters," IEEE J. SolidState Circuits, vol. 7, no.
4, pp. 302304, 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. SolidState Circuits, pp.10141023, Dec. • T.C. Choi, "HighFrequency CMOS SwitchedCapacitor 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 HighQ Bandpass Filters for IF
PR
SwitchedCapacitor High Q
Applications," IEEE J. SolidState Circuits, pp. 924933, Dec. 1986 • K. Martin and A. Sedra, "Effects of the op amp finite gain and bandwidth on the
performance of switchedcapacitor filters," IEEE Trans. Circuits and Systems,
vol. 28, no. 8, pp. 822829, Aug. 1981 • K.L. Lee, “Low Distortion SwitchedCapacitor 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, “Strayinsensitive switchedcapacitor filters based on
the bilinear z transform,” Electronics Letters, vol. 19, pp. 365366, June 1979 • R. Castello,
R Castello and P R Gray "A high performance micropower switchedcapacitor
P.R. Gray,
highperformance
switched capacitor
filter," IEEE J. SolidState Circuits, vol. 20, no. 6, pp. 11221132, Dec. 1985 • J. H. Fischer, "Noise sources and calculation techniques for switched capacitor
pp
g
filters," IEEE J. SolidState Circuits, vol. 17, no. 4, pp. 742752, Aug. 1982 • C.A. Gobet and A. Knob, "Noise analysis of switched capacitor networks," IEEE
Trans. Circuits and Systems, vol. 30, no. 1, pp. 3743, 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.508521, Apr. 1989. • R. Schreier, et al., "Designoriented 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 SwitchedCapacitor Filters with SpectreRF,”
p
g
g
g
y
p
http://www.designersguide.org/Analysis/scfilters.pdf B. Murmann EE315A ― HO #11 33 ...
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This note was uploaded on 08/13/2009 for the course EE 315 taught by Professor Borismurmann during the Spring '09 term at Stanford.
 Spring '09
 BorisMurmann

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