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EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 1
EE247Lecture 10
• Switched-capacitor filters– Switched-capacitor integrators
• LDI integrators• Effect of parasitic capacitance• Bottom-plate integrator topology
– Resonators– Bandpass filters– Lowpass filters
• Termination implementation• Transmission zero implementation
– Switched-capacitor filter design considerations– Effect of non-idealities– Switched-capacitor filters utilizing double sampling technique
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 2
Summary of last lecture• Switched-capacitor filters
– Tradeoffs in choosing sampling rate– Effect of sample and hold – Switched-capacitor network electronic noise – Switched-capacitor integrators
• DDI integrators• LDI integrators
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 3
LDI Switched-Capacitor Integrator
( )
( )
1/ 2s1I
j T / 2s e s 1j T j T / 2 j T / 2I I
sI
sI
C j TC 1 zin
C CC C1
CC
CC j T
Vo2 z( z ) , z eV
e e e
1j 2 sin T / 2
T / 21sin T / 2
ωω ω ω
ω
ω
ωω ω
−−
−− − +
−
− −
= − × =
= − × = ×
= − ×
= − ×
CI
Ideal Integrator Magnitude Error
No Phase Error! For signals at frequencies << sampling freq.
Magnitude error negligible
-
+
Vin
Vo2
φ1 φ2CI
Cs
φ2
LDI
LDI (Lossless Discrete Integrator) same as DDI but output is sampled ½clock cycle earlier
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 4
Switched-Capacitor Filter Built with LDI Integrators( )ωjH
Zeros Preserved
Frequency (Hz) 2fs ffsfs /2
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 5
Switched-Capacitor IntegratorParasitic Capacitor Sensitivity
Effect of parasitic capacitors:
1- Cp1 - driven by opamp o.k.
2- Cp2 - at opamp virtual gnd o.k.
3- Cp3 – Charges to Vin & discharges into CI
Problem parasitic capacitor sensitivity
-
+
VinVo
φ1 φ2CI
CsCp3
Cp2
Cp1
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 6
Parasitic InsensitiveBottom-Plate Switched-Capacitor Integrator
Sensitive parasitic cap. Cp1 rearrange circuit so that Cp1 does not charge/discharge
φ1=1 Cp1 grounded
φ2=1 Cp1 at virtual ground
Solution: Bottom plate capacitor integrator
Vi+
Cs-
+ Vo
CI
Cp1
Cp2
φ1 φ2
Vi-
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 7
Bottom Plate Switched-Capacitor Integrator
12
12
1s s1 1I I
s s1 1I I
C Cz zC C1 z 1 z
C Cz 1C C1 z 1 z
−−
− −
−
− −
− −
− −− −
Note: Different delay from Vi+ &Vi- to either output
Special attention needed for input/output connections to ensure LDI realization
Vi+
Cs-
+ Vo
CIφ1 φ2
Vi-
Vi+on φ1
Vi-on φ2
Vo1on φ1
Vo2on φ2
φ1
φ2
Vo2
Vo1
Output/Inputz-Transform
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 8
Bottom Plate Switched-Capacitor Integratorz-Transform Model
12
12
11 1
1 1
z z1 z 1 z
z 11 z 1 z
−−− −
−
− −
− −
−− −
121
z1 z
−
−−
12z−
Input/Output z-transformVi+
Cs-
+ Vo
CIφ1 φ2
Vi-
φ1
φ2
Vo2
Vo1
Vi+
Vi- 12z+ Vo2
Vo1
LDI
s IC C
s IC C−
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 9
LDI Switched-Capacitor Ladder Filter
CsCI
12
1z
1 z
−
−−
-+
+ - + -3
1sτ 4
1sτ 5
1sτ
12z
−12z
+
12z
+12z
−
CsCI
−
CsCI
−CsCI
−CsCI
CsCI
Delay around integrator loop is (z-1/2 . z+1/2 =1) LDI function
12
1z
1 z
−
−−
12
1z
1 z
−
−−12z
−
12z
+
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 10
Switched-Capacitor LDI Resonator
2s
ω
1s
ω−
C1 1f1 sR C Ceq1 2 2C1 3f2 sR C Ceq3 4 4
ω
ω
= = ×
= = ×
Resonator Signal Flowgraph
φ1 φ2
φ2 φ1
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 11
Fully Differential Switched-Capacitor Resonator
φ1 φ2
φ1 φ2
• Note: Two sets of S.C. bottom plate networks for each differential integrator
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 12
Switched-Capacitor LDI Bandpass FilterUtilizing Continuous-Time Termination
0s
ω
0s
ω−
C C3 1f0 s sC C4 2C2Q CQ
ω = × = ×
=
Bandpass FilterSignal Flowgraph
CQ
Vi
Vo-1/QVo2Vi
Vo2
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 13
s-Plane versus z-PlaneExample: 2nd Order LDI Bandpass Filter
σ
jωs-plane z-plane
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 14
Switched-Capacitor LDI Bandpass FilterContinuous-Time Termination
0.1 1 10f0
0
-3dBΔf
Frequency
Mag
nitu
de (d
B)
C1 1f f0 s2 C2f0f Q
C C1 Q1 fs2 C C2 4
π
π
= ×
Δ =
= ×
Both accurately determined by cap ratios & clock frequency
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 15
Fifth Order All-Pole LDI Low-Pass Ladder FilterComplex Conjugate Terminations
•Complex conjugate terminations (alternate phase switching)
Termination Resistor
Termination Resistor
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 16
Fifth-Order All-Pole Low-Pass Ladder FilterTermination Implementation
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 17
Sixth-Order Elliptic LDI Bandpass Filter
TransmissionZero
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 18
Use of T-Network
High Q filter large cap. ratio for Q & transmission zero implementationTo reduce large ratios required T-networks utilized
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
2 1 4
1 2 1 3 4
V C CV C C C C
= − ×+ +
2 1
1 2
V CV C
= −
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 19
Sixth Order Elliptic Bandpass FilterUtilizing T-Network
•T-networks utilized for:• Q implemention• Transmission zero implementation
Q implementation
Zero
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 20
Effect of Opamp Nonidealities on Switched Capacitor Filter Behaviour
• Opamp finite gain
• Opamp finite bandwidth
• Finite slew rate of the opamp
• Non-linearity associated with opamp output/input characteristics
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 21
Effect of Opamp Non-IdealitiesFinite DC Gain
Input/Output z-transformVi+
Cs-
+ Vo
CIφ1 φ2
Vi-DC Gain = a
sI s 1
I
1
Cs C C
s C a
o
o a
int g
1H( s ) f
s f
H( s )s
Q a
ωω
≈ −+ ×
−≈
+ ×
⇒ ≈
• Finite DC gain same effect in S.C. filters as for C.T. filters• If DC gain not high enough lowing of overall Q & droop in passband
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 22
Effect of Opamp Non-IdealitiesFinite Opamp Bandwidth
Input/Output z-transformVi+
Cs-
+ Vo
CIφ1 φ2
Vi-Unity-gain-freq.
= ft
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
Assumption-Opamp does not slew (will be revisited)Opamp has only one pole only exponential settling
Vo
φ2
T=1/fs
settlingerror
time
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 23
Effect of Opamp Non-IdealitiesFinite Opamp Bandwidth
actual idealIk k 1
I s
I t
I s s
t s
C1 e e ZH ( Z ) H ( Z )
C CC f
where kC C f
f Opamp unity gain frequency , f Clock frequency
π
− − −⎡ ⎤− + ×≈ ⎢ ⎥+⎣ ⎦
= × ×+
→ − − →
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
Input/Output z-transformVi+
Cs-
+ Vo
CIφ1 φ2
Vi-Unity-gain-freq.
= ft
Vo
φ2
T=1/fs
settlingerror
time
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 24
Effect of Opamp Finite Bandwidth on Filter Magnitude Response
Magnitude deviation due to finite opamp unity-gain-frequency
Example: 2nd
order bandpass with Q=25
fc /ft
|Τ|non-ideal /|Τ|ideal (dB)
fc /fs=1/32fc /fs=1/12
Active RC
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 25
Effect of Opamp Finite Bandwidth on Filter Magnitude Response
Example:For 1dB magnitude response deviation:1- fc/fs=1/12
fc/ft~0.04ft>25fc
2- fc/fs=1/32fc/ft~0.022
ft>45fc
3- Cont.-Timefc/ft~1/700
ft >700fc fc /ft
|Τ|non-ideal /|Τ|ideal (dB)
fc /fs=1/32fc /fs=1/12
Active RC
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 26
Effect of Opamp Finite Bandwidth Maximum Achievable Q
fc /ft
Max. allowable biquad Q for peak gain change <10%
C.T. filters
Oversampling Ratio
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 27
Effect of Opamp Finite Bandwidth Maximum Achievable Q
fc /ft
C.T. filters
Example:For Q of 40 required Max. allowable biquad Q for peak gain change <10%
1- fc/fs=1/32fc/ft~0.02
ft>50fc
2- fc/fs=1/12fc/ft~0.035
ft>28fc
3- fc/fs=1/6fc/ft~0.05
ft >20fcRef: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-
Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 28
Effect of Opamp Finite Bandwidth on Filter Critical Frequency
Critical frequency deviation due to finite opamp unity-gain-frequency
Example: 2nd
order filterfc /ft
Δωc /ωc
fc /fs=1/32
fc /fs=1/12
Active RC
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 29
Effect of Opamp Finite Bandwidth on Filter Critical Frequency
fc /ft
Δωc /ωc
fc /fs=1/32
fc /fs=1/12
Active RC
Example:For maximum critical frequency shift of <1%
1- fc/fs=1/32fc/ft~0.028
ft>36fc
2- fc/fs=1/12fc/ft~0.046
ft>22fc
3- Active RCfc/ft~0.008
ft >125fc
C.T. filters
Ref: K.Martin, A. Sedra, “Effect of the Opamp Finite Gain & Bandwidth on the Performance of Switched-Capacitor Filters," IEEE Trans. Circuits Syst., vol. CAS-28, no. 8, pp. 822-829, Aug 1981.
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 30
Opamp Bandwidth Requirements for Switched-Capacitor Filters Compared to Continuous-Time Filters
• Finite opamp bandwidth causes phase lag at the unity-gain frequency of the integrator for both type filters
Results in negative intg. Q & thus increases overall Q and gain @ results in peaking in the passband of interest
• For given filter requirements, opamp bandwidth requirements muchless stringent for S.C. filters compared to cont. time filters
Lower power dissipation for S.C. filters (at low freq.s only due to other effects)
• Finite opamp bandwidth causes down shifting of critical frequencies in both type filters– Since cont. time filters are usually tuned tuning accounts for frequency
deviation– S.C. filters are untuned and thus frequency shift could cause problems
specially for narrow-band filters
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 31
Sources of Distortion in Switched-Capacitor Filters
• Distortion induced by finite slew rate of the opamp
• Opamp output/input transfer function non-linearity- similar to cont. time filters
• Distortion incurred by finite setting time of the opamp
• Capacitor non-linearity- similar to cont. time filters
• Distortion due to switch clock feed-through and charge injection
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 32
What is Slewing?
oV
Vi+Vi-
φ2
Cs
-
+
Vin
Vo
φ2CI
Cs
CI
CL
Assumption:Integrator opamp is a simple class Atransconductance type differential pairwith fixed tail current, Iss=const.
Iss
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 33
oV
Vi+Vi-
φ2
Cs
CI
Iss
What is Slewing?
Io
VinVmax
Imax= -Iss/2
Slope ~ gm
|VCs| > Vmax Output current constant Io=Iss/2 or –Iss/2Constant current charging/discharging CI: Vo ramps down/up Slewing
After VCs is discharged enough to have: |VCs|<Vmax Io=gm VCs Exponential or over-shoot settling
Io
Opamp Io v.s. Vin
Imax= +Iss/2
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 34
Distortion Induced by Opamp Finite Slew Rate
Multiple pole settling
One pole settling
Output Voltage
TimeSlewing SettlingSettling (multi-pole)
Vo
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 35
Ideal Switched-Capacitor Output Waveform
φ1
φ2
Vin
Vo
Vcs
Clock-
+
Vin
Vo
φ1CI
Cs
-
+
Vin
Vo
φ2CI
Cs
φ2 High Charge transferred from Cs to CI
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 36
Slew Limited Switched-Capacitor IntegratorOutput Slewing & Settling
φ1
φ2
Vo-real
Vo-ideal
Clock
Slewing LinearSettling
Slewing LinearSettling
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 37
Distortion Induced by Finite Slew Rate of the Opamp
Ref: K.L. Lee, “Low Distortion Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, Feb. 1986 (ERL Memorandum No. UCB/ERL M86/12).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 38
Distortion Induced by Opamp Finite Slew Rate
• Error due to exponential settling changes linearly with signal amplitude
• Error due to slew-limited settling changes non-linearly with signal amplitude (doubling signal amplitude X4 error)
For high-linearity need to have either high slew rate or non-slewing opamp
( )( )
( )
o s
o s
2T2ok 2r s
2T 22 oo o3 o s 3
r s r s
8 sinVHD S T k k 4
8 sin fV 8 VHD for f f HDS T 15 15S f
ω
ω
π
ππ
=−
→ = << → ≈
Ref: K.L. Lee, “Low Distortion Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, Feb. 1986 (ERL Memorandum No. UCB/ERL M86/12).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 39
Example: Slew Related Harmonic Distortion
( )o s2T
2o3r s
2oo3
r s
8 sinVHD S T 15
f8 VHD 15S f
ω
π
π
=
≈
Ref: K.L. Lee, “Low Distortion Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, Feb. 1986 (ERL Memorandum No. UCB/ERL M86/12).
Switched-capacitor filter with 4kHz bandwidth, fs=128kHz, Sr=1V/μsec, Vo=3V
12dB
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 40
Distortion Induced by Opamp Finite Slew Rate Example
-120
-100
-80
-60
-40
-20
1 10 100 1000
HD
3 [d
B]
(Slew-rate / fs ) [V]
f / fs =1/32
f / fs =1/12
Vo =1V V
o =2V
Vo =1V V
o =2V
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 41
Distortion Induced by Finite Slew Rate of the Opamp
• Note that for a high order switched capacitor filter only the last stage slewing will affect the output linearity (as long as the previous stages settle to the required accuracy)
Can reduce slew limited linearity by using an amplifier with a higher slew rate only for the last stageCan reduce slew limited linearity by using class A/B amplifiers
• Even though the output/input characteristics is non-linear as long as the DC open-loop gain is high, the significantly higher slew rate compared to class A amplifiers helps improve slew rate induced distortion
• In cases where the output is sampled by another sampled data circuit (e.g. an ADC or a S/H) no issue with the slewing of the output as long as the output settles to the required accuracy & is sampled at the right time
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 42
More Realistic Switched-Capacitor Circuit Slew Scenario
-
+
Vin
Vo
φ2CI
Cs
At the instant Cs connects to input of opamp (t=0+)Opamp not yet active at t=0+ due to finite opamp bandwidth delayFeedforward path from input to output generates a voltage spike at the output with polarity opposite to final Vo step- spike magnitude function of CI, CL , Cs
Spike increases slewing periodEventually, opamp becomes active - starts slewing followed by subsequent settling
CL
Vo
φ2CI
CsCL
t=0+
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 43
Switched-Capacitor Circuit Opamp not Active @ t=0+
-
+
Vin
Vo
φ2CI
CsCL
Vo
φ2CI
CsCL
t=0+
( )
( )
0 0
0 0 0
0 0 0 0
0 0 0
t t I Ls Cs Cs s eq eq
I L
t t tI s Iout Cs Cs
I L s eq I L
t t t tL s I eq L s Cs Cs s L Cs Cs
t t ts Iout Cs Cs
s L I L
C CCharg e sharing : C V V C C where CC C
C C CV V VC C C C C C
Assu min g C C C C C C V V C C V V
C CV V VC C C C
Note
− +
+ + −
− + − +
+ − −
= + =+
Δ = = ×+ + +
<< << → ≈ → ≈ + → ≈
→ Δ ≈ × ≈+ +
0 0final t ts sout Cs Cs
I I
C Cthat V V VC C
+ −Δ ≈ − ≈ −
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 44
More Realistic Switched-Capacitor Circuit Slew Scenario
-
+
Vin
Vo
φ2CI
CsCL
Vo
φ2CI
CsCL
t=0+
( )0 0
0 0 0
t t I Ls Cs Cs s eq eq
I L
t t ts sCs Cs Cs
I Ls eqs
I L
C CCharg e sharing : C V V C C where CC C
C CV V V C CC C CC C
− +
+ − −
= + =+
= =+ +
+
Notice that if CL is large some of the charge stored on Cs is lost prior to opamp becoming effective operation looses accuracy
Partly responsible for S.C. filters only good for low-frequency applications
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 45
More Realistic S.C. Slew Scenario
Vo_realIncluding t=0+spike
Slewing LinearSettling
Slewing
Spike generated at t=0+
Vo_real
Vo_ideal
Slewing LinearSettling
Slewing LinearSettling
Ref: R. Castello, “Low Voltage, Low Power Switched-Capacitor Signal Processing Techniques," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, Aug. ‘84 (ERL Memorandum No. UCB/ERL M84/67).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 46
Sources of Noise in Switched-Capacitor Filters
• Opamp Noise– Thermal noise– 1/f (flicker) noise
• Thermal noise associated with the switching process (kT/C)– Same as continuous-time filters
• Precaution regarding aliasing of noise required
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 47
Extending the Maximum Achievable Critical Frequency
of Switched-Capacitor FiltersConsider a switched-capacitor resonator:
Regular sampling:Each opamp is busy settling only during one of the two clock phases
Idle during the other clock phase
φ1 φ2
φ2 φ1
Note: During φ1 both opamps are idle
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 48
Switched-Capacitor Resonator Using Double-Sampling
Double-sampling:
• 2nd set of switches & sampling caps added to all integrators
• While one set of switches/caps sampling the other set transfers charge into the intg. cap
• Opamps busy during both clock phases
• Effective sampling freq. twice THE clock freq. while opamp bandwidth requirement remains the same
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 49
Double-Sampling Issues
fclock fs= 2fclock
Issues to be aware of:- Jitter in the clock - Unequal clock phases -Mismatch in sampling caps.
parasitic passbands Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of
Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 50
Double-Sampled Fully Differential S.C. 6th Order All-Pole Bandpass Filter
Ref: Tat C. Choi, "High-Frequency CMOS Switched-Capacitor Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, May 1983 (ERL Memorandum No. UCB/ERL M83/31).
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 51
Sixth Order Bandpass Filter Signal Flowgraph
γ
1Q
− 0s
ω
inV 1−
0s
ω−
1Q
−0s
ω0s
ω−
0s
ω−0
sω
outV1γ−
γγ−
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 52
Double-Sampled Fully Differential 6th Order S.C. All-Pole Bandpass Filter
Ref: B.S. Song, P.R. Gray "Switched-Capacitor High-Q Bandpass Filters for IF Applications," IEEE Journal of Solid State Circuits, Vol. 21, No. 6, pp. 924-933, Dec. 1986.
- Cont. time termination (Q) implementation- Folded-Cascode opamp with fu = 100MHz used- Center freq. 3.1MHz (Measured error >1%), filter Q=55- Clock freq. 12.83MHz effective oversampling ratio 8.27- Measured dynamic range 46dB (IM3=1%)
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 53
Switched-Capacitor Filter ApplicationExample: Voice-Band Codec (Coder-Decoder) Chip
Ref: D. Senderowicz et. al, “A Family of Differential NMOS Analog Circuits for PCM Codec Filter Chip,” IEEE Journal of Solid-State Circuits, Vol.-SC-17, No. 6, pp.1014-1023, Dec. 1982.
fs= 1024kHz fs= 128kHz fs= 8kHz fs= 8kHz
fs= 8kHz fs= 128kHz fs= 128kHz
fs= 128kHz
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 54
CODEC Transmit Path Lowpass Filter Frequency Response
0 2000 4000 6000 8000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
Note: fs=128kHz
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 55
CODEC Transmit Path Highpass Filter
1000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
1000010010
Note: fs=8kHz
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 56
CODEC Transmit Path Filter Overall Frequency Response
1000-50
-40
-30
-20
-10
0
Frequency (Hz)
Mag
nitu
de (d
B)
1000010010
Low Q bandpass (Q<1) filter shape Implemented with lowpass followed by highpass
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 57
CODEC Transmit Path Clocking Scheme
First filter (1st order RC type) performs anti-aliasing for the next S.C. biquad
The first 2 stage filters form 3rd order elliptic with corner frequency @ 32kHz Anti-aliasing for the next lowpass filter
The stages prior to the high-pass perform anti-aliasing for high-pass
Notice gradual lowering of clock frequency Ease of anti-aliasing
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 58
SC Filter SummaryPole and zero frequencies proportional to – Sampling frequency fs– Capacitor ratios
High accuracy and stability in responseLong time constants realizable without large R, C
Compatible with transconductance amplifiers– Reduced circuit complexity, power dissipation
Amplifier bandwidth requirements less stringent compared to CT filters (low frequencies only)Issue: Sampled-data filters require anti-aliasing prefiltering
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 59
Switched-Capacitor Filters versus Continuous-Time Filter Limitations
Considering overall effects:
• Opamp finite unity-gain-bandwidth
• Opamp settling issues
• Opamp finite slew rate
• Clock feedthru• Switch+ sampling
cap. finite time-constant
Filter bandwidthAssuming constant opamp fu
MagnitudeError
5-10MHz
S.C. Filter
Cont. Time Filter
Limited switched-capacitor filter performance frequency range
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 60
SummaryFilter Performance versus Filter Topology
_
1-5%
1-5%
1-5%
1-5%
Freq. Tolerance+ Tuning
<1%40-90dB~ 10MHzSwitched Capacitor
+-40-60%40-70dB~ 100MHzGm-C
+-30-50%50-90dB~ 5MHzOpamp-MOSFET-RC
+-30-50%40-60dB~ 5MHzOpamp-MOSFET-C
+-30-50%60-90dB~10MHzOpamp-RC
Freq. Tolerance
w/o Tuning
SNDRMax. Usable
Bandwidth
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 61
Frequency Warping• Frequency response
– Continuous time (s-plane): imaginary axis– Sampled time (z-plane): unit circle
• Continuous to sampled time transformation– Should map imaginary axis onto unit circle– How do S.C. integrators map frequencies?
12s
S.C. 11
1s2
C zH ( z )C zint
C
C j sin f Tint π
−= −−
= −
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 62
CT – SC Integrator ComparisonCT Integrator SC Integrator
int
s s
CRCf C
τ = =Identical time constants:
Set: HRC(fRC) = HSC(fSC)
⎟⎟⎠
⎞⎜⎜⎝
⎛=
s
SCsRC f
fff ππ
sin
12s
SC 11
1s2 s
C zH ( z )C zint
C
C j sin f Tint SCπ
−= −−
= −
RC1
12 RC
H ( s )s
jf
τ
π τ
= −
= −
EECS 247 Lecture 10 Switched-Capacitor Filters © 2007 H. K. Page 63
LDI Integration
⎟⎟⎠
⎞⎜⎜⎝
⎛=
s
SCsRC f
fff ππ
sin
• “RC” frequencies up to fs /πmap to physical (real) “SC”frequencies
• Frequencies above fs /π do not map to physical frequencies
• Mapping is symmetric about fs /2 (aliasing)
• “Accurate” only for fRC << fs0 0.1 0.2 0.30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
fRC /fs
f SC/f s
1/π
Slope=1