ee247 lecture 7
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EECS 247 Lecture 7: Filters © 2007 H.K. Page 1
EE247 Lecture 7
• Automatic on-chip filter tuning (continued from last lecture)– Continuous tuning
• Reference integrator locked to a reference frequency• DC tuning of resistive timing element
– Periodic digitally assisted filter tuning• Systems where filter is followed by ADC & DSP, existing hardware
can be used to periodically update filter freq. response• Continuous-time filters
– Highpass filters– Bandpass filters
• Lowpass to bandpass transformation• Example: 6th order bandpass filter• Gm-C BP filter using simple diff. pair
EECS 247 Lecture 7: Filters © 2007 H.K. Page 2
Summary last lecture• Continuous-time filters
– Opamp MOSFET-RC filters– Gm-C filters
• Frequency tuning for continuous-time filters– Trimming via fuses or laser – Automatic on-chip filter tuning
• Continuous tuning– Utilizing VCF built with replica integrators– Use of VCO built with replica integrators– Reference integrator locked to reference frequency
EECS 247 Lecture 7: Filters © 2007 H.K. Page 3
SummaryReference Integrator Locked to Reference Frequency
Feedback forces Gm to vary so that :
S2 S3Gm
C1 C2
Vref
A
int g
int g0
C1 N / fclkGmor
Gm fclk / NC1
τ
ω
= =
= =
Tuning error due to gm-cell offset voltage resolved
Advantage over previous schemes:
fclk can be chosen to be at much higher frequencies compared to filter bandwidth (N >1)
Feedthrough of clock attenuated by filter
EECS 247 Lecture 7: Filters © 2007 H.K. Page 4
DC Tuning of Resistive Timing Element
Vtune Tuning circuit Gm replica of Gm used in filter
Rext used to lock Gm to accurate off-chip R
Feedback forces Gm=1/Rext
Issues with DC offset
Account for capacitor variations in this gm-C implementation by trimming
Rext.
-
+ -
+
I
I
Gm
Ref: C. Laber and P.R. Gray, “A 20MHz 6th Order BiCMOS Parasitic Insensitive Continuous-time Filter and Second Order Equalizer Optimized for Disk Drive Read Channels,” IEEE Journal of Solid State Circuits, Vol. 28, pp. 462-470, April 1993
EECS 247 Lecture 7: Filters © 2007 H.K. Page 5
Digitally Assisted Frequency Tuning Example:Wireless Receiver Baseband Filters
• Systems where filter is followed by ADC & DSP– Take advantage of existing digital signal processor capabilities to
periodically update the filter critical frequency
– Filter tuned only at the outset of each data transmission session (off-line/periodic tuning) – can be fine tuned during times data is not transmitted
RF Amp
Osc.
A/D Digital Signal
Processor (DSP)
A/D
π 2IF Stage ( 0 to 2 )
EECS 247 Lecture 7: Filters © 2007 H.K. Page 6
Example: Seventh Order Tunable Low-Pass OpAmp-RC Filter
EECS 247 Lecture 7: Filters © 2007 H.K. Page 7
Digitally Assisted Filter Tuning Concept
Assumptions:– System allows a period of
time for the filter to undergo tuning (e.g. for a wireless transceiver during idle periods)
– An AC (e.g. a sinusoid) signal can be generated on-chip whose amplitude is a function of an on-chip DC source
• AC signal generator outputs a sinusoid with peak voltage equal to the DC signal source
• AC Signal Power =1/2 DC signal power @ the input of the filter
VP AC=VDC
EECS 247 Lecture 7: Filters © 2007 H.K. Page 8
Digitally Assisted Filter Tuning Concept
VP AC=VDC
AC signal @ a frequency on the roll-off of the desired filter frequency response(e.g. -3dB frequency)
Provision can be made during the tuning cycle, the input of the filter is disconnected from the previous stage (e.g. mixer) and connected to:
1. DC source2. AC source
under the control of the DSP
( )desiredAC DC 3dBV V sin 2 f tπ −= ×
EECS 247 Lecture 7: Filters © 2007 H.K. Page 9
Digitally Assisted Filter Tuning Concept
VP AC=VDC
EECS 247 Lecture 7: Filters © 2007 H.K. Page 10
2ΔΔ
Practical Implementation of Frequency TuningAC Signal Generation From DC Source
Vout
Clock
ClockB
Vout0+Δ
−Δ
Δ Vout=
Clock=high
+Δ Δ Vout= −Δ
ClockB=high
EECS 247 Lecture 7: Filters © 2007 H.K. Page 11
Δ 2ΔΔ
DC Measurement AC Measurement
A/D 4bit
10MHz
Digital Signal
ProcessorDSP1616
40MHzVref+Vref-
Filter
Register
CH
OP
TUN
E
FREQ
.C
ON
T.
625k
Hz
Practical Implementation of Frequency Tuning
EECS 247 Lecture 7: Filters © 2007 H.K. Page 12
2ΔΔAC
Measurement
Practical Implementation of Frequency TuningEffect of Using a Square Waveform
( ) ( )n 1,3,5,..
4Vin sin n tnt π ω=
∞ Δ= ∑
• Input signal chosen to be a square wave due to ease of generation• Filter input signal comprises a sinusoidal waveform @ the fundamental
frequency + its odd harmonics:
Key Point: The filter itself attenuates unwanted odd harmonics Inaccuracy incurred by the harmonics negligible
( ) ( )4 1Vout sin t2
t π ωΔ= ×
EECS 247 Lecture 7: Filters © 2007 H.K. Page 13
Simplified Frequency Tuning Flowchart
EECS 247 Lecture 7: Filters © 2007 H.K. Page 14
Digitally Assisted Offset Compensation
EECS 247 Lecture 7: Filters © 2007 H.K. Page 15
Filter Tuning Prototype Diagram
EECS 247 Lecture 7: Filters © 2007 H.K. Page 16
EECS 247 Lecture 7: Filters © 2007 H.K. Page 17
Chip Photo
EECS 247 Lecture 7: Filters © 2007 H.K. Page 18
Measured Tuning Characteristics
EECS 247 Lecture 7: Filters © 2007 H.K. Page 19
Off-line Digitally Assisted Tuning• Advantages:
– No reference signal feedthrough since tuning does not take place during data transmission (off-line)
– Minimal additional hardware– Small amount of programming
• Disadvantages:– If acute temperature change during data transmission,
filter may slip out of tune!• Can add fine tuning cycles during periods of data is not
transmitted or received
Ref: H. Khorramabadi, M. Tarsia and N.Woo, “Baseband Filters for IS-95 CDMA Receiver Applications Featuring Digital Automatic Frequency Tuning,” 1996 International Solid State Circuits Conference, pp. 172-173.
EECS 247 Lecture 7: Filters © 2007 H.K. Page 20
Summary: Continuous-Time Filter Frequency Tuning
• Trimming• Expensive & does not account for temperature and supply etc… variations
• Automatic frequency tuning– Continuous tuning
• Master VCF used in tuning loop– Tuning quite accurate– Issue reference signal feedthrough to the filter output
• Master VCO used in tuning loop– Design of reliable & stable VCO challenging– Issue reference signal feedthrough
• Single integrator in negative feedback loop forces time-constant to be a function of accurate clock frequency
– More flexibility in choice of reference frequency less feedthrough issues• DC locking of a replica of the integrator to an external resistor
– DC offset issues & does not account for integrating capacitor variations– Periodic digitally assisted tuning
– Requires digital capability + minimal additional hardware– Advantage of no reference signal feedthrough since tuning performed off-line
EECS 247 Lecture 7: Filters © 2007 H.K. Page 21
Integrator Based High-Pass Filters1st Order
• Conversion of simple high-pass RC filter to integrator-based type by using signal flowgraph technique
in
s CV Ros CV 1 R
=+
oV
R
C
inV
EECS 247 Lecture 7: Filters © 2007 H.K. Page 22
1st Order Integrator Based High-Pass FilterSGF
oV
R
C
inV+ VC - +
VR-
IC
IRV V VR in C1V IC C sC
V Vo R1I VR R R
I IC R
= −
= ×
=
= ×
=
1
1R
1sC
RICI
CV
inV
1−1
SFG
oV1VR
EECS 247 Lecture 7: Filters © 2007 H.K. Page 23
1st Order Integrator Based High-Pass FilterSGF
1sC R
−
oVinV 1 1oV
R
C
inV
oVinV
∫ -
SGF
Note: Addition of an integrator in the feedback path High pass frequency shaping
+
+
+ VC- + VR
-
EECS 247 Lecture 7: Filters © 2007 H.K. Page 24
Addition of Integrator in Feedback Path
oVinV
∫ -
a
1/sτ
Let us assume flat gain in forward path (a)Effect of addition of an integrator in the feedback path:
+
+
in
in
int gpole o
V aoV 1 af
sV aos sV 1 a / 1 / a
azero@ DC & pole @ a
ττ τ
ω ωτ
=+
= =+ +
→ = − = − ×
Note: For large forward path gain, a, can implement high pass function with low corner frequency without the need for large value of C or RAddition of an integrator in the feedback path zero @ DC + pole @ axω0
intg
This technique used for offset cancellation in systems where the low frequency content is not important and thus disposable
EECS 247 Lecture 7: Filters © 2007 H.K. Page 25
( )H jω
( )H jω
Lowpass Highpass
ω
( )H jω
ωω
Q<5
Q>5
• Bandpass filters two cases:1- Low Q (Q < 5)
Combination of lowpass & highpass
2- High Q or narrow-band (Q > 5)Direct implementation
ω
( )H jω
+
Bandpass Filters
Bandpass
Bandpass
EECS 247 Lecture 7: Filters © 2007 H.K. Page 26
Narrow-Band Bandpass FiltersDirect Implementation
• Narrow-band BP filters Design based on lowpass prototype• Same tables used for LPFs are also used for BPFs
Lowpass Freq. Mask Bandpass Freq. Mask
cc
s s2 s1c B2 B1
ss Qs
ωω
Ω Ω − ΩΩ Ω − Ω
⎡ ⎤× +⎢ ⎥⎣ ⎦
⇒ ⇒
EECS 247 Lecture 7: Filters © 2007 H.K. Page 27
Lowpass to Bandpass TransformationLowpass pole/zero (s-plane) Bandpass pole/zero (s-plane)
From: Zverev, Handbook of filter synthesis, Wiley, 1967- p.156.
PoleZero
EECS 247 Lecture 7: Filters © 2007 H.K. Page 28
Lowpass to Bandpass Transformation Table
From: Zverev, Handbook of filter synthesis, Wiley, 1967- p.157.
'
'
'
'
1
1
1 1
r r
r
r
r
r
r r
C QCRRL
QC
RL QL
CRQC
ω
ω
ω
ω
= ×
= ×
= ×
= ×
C
L
C’
LP BP BP Values
L CL’
Lowpass filter structures & tables used to derive bandpass filters
' 'C &L are normilzed LP values
filterQ Q=
EECS 247 Lecture 7: Filters © 2007 H.K. Page 29
Lowpass to Bandpass TransformationExample: 3rd Order LPF 6th Order BPF
• Each capacitor replaced by parallel L& C• Each inductor replaced by series L&C
oVL2 C2
RsC1
C3inV RLL1 L3
RsC1’ C3’
L2’
inV RL
oV
Lowpass Bandpass
EECS 247 Lecture 7: Filters © 2007 H.K. Page 30
Lowpass to Bandpass TransformationExample: 3rd Order LPF 6th Order BPF
'1 1
0
1 '01
2 '02
'2 2
0
'3 3
0
3 '03
1
1
1 1
1
1
C QCRRL
QC
CRQLRL QL
C QCRRL
QC
ω
ω
ω
ω
ω
ω
= ×
= ×
= ×
= ×
= ×
= ×
oVL2 C2
RsC1
C3inV RLL1 L3
Where:C1
’ , L2’ , C3
’ Normalized lowpass valuesQ Bandpass filter quality factor ω0 Filter center frequency
EECS 247 Lecture 7: Filters © 2007 H.K. Page 31
Lowpass to Bandpass TransformationSignal Flowgraph
oVL2 C2
RsC1
C3inV RLL1 L3
1- Voltages & currents named for all components2- Use KCL & KVL to derive state space description 3- To have BMFs in the integrator form
Cap voltage expressed as function of its current VC=f(IC)
Ind current as a function of its voltage IL=f(VL)4-Use state space description to draw SFG5- Convert all current nodes to voltage
EECS 247 Lecture 7: Filters © 2007 H.K. Page 32
Signal Flowgraph6th Order Bandpass Filter
1
*RRs
− *1
1sC R
1−
*
1
RsL
−
1−
1
*RRL
−*3
1sC R
*
3
RsL
−*
2
1sC R
−*
2
RsL
1−
Note: each C & L in the original lowpass prototype replaced by a resonatorSubstituting the bandpass L1, C1,….. by their normalized lowpass equivalent from page 29The resulting SFG is:
1
V1’
V2
V3’
V1
V2’
VoutVinV3
EECS 247 Lecture 7: Filters © 2007 H.K. Page 33
Signal Flowgraph6th Order BPF versus 3rd Order LPF
1−
*RRs
−*
1
1sC R
1
*RRs
− *1
1sC R
1−
*
1
RsL
−
1−
1
*RRL
−*
3
1sC R
*
3
RsL
−*
2
1sC R
−*
2
RsL
1
V1’
V2
V3’
V1
V2’
VoutVinV3
inV 1 1V oV1−1
1− 1V1’ V3’V2’
*
2
RsL
*RRL
−
V2
*3
1sC R
LPF
BPF
EECS 247 Lecture 7: Filters © 2007 H.K. Page 34
Signal Flowgraph6th Order Bandpass Filter
1
*RRs
− 0
1'QCs
ω
1−
'1 0QC
sω
−
1−
1
*RRL
−'3
0
Q Csω'
3 0Q C
sω
−2 0
'QL
sω
−
0
2'QLs
ω
1−
• Note the integrators different time constants• Ratio of time constants for two integrator in each resonator ~ Q2
Typically, requires high component ratiosPoor matching
• Desirable to modify SFG so that all integrators have equal time constants for optimum matching.
• To obtain equal integrator time constant use node scaling
1
V1’
V2
V3’
V1
V2’
VoutVin V3
EECS 247 Lecture 7: Filters © 2007 H.K. Page 35
Signal Flowgraph6th Order Bandpass Filter
'1
1QC
−
'2
1QL
*
'1
R 1Rs QC
− ×
0s
ω
1−
0s
ω−
'2
1QL
−
'3
1QC
*
3
R 1RL QC
− ×
0s
ω0s
ω−
0s
ω−0
sω
• All integrator time-constants equal• To simplify implementation choose RL=Rs=R*
1
V1’/(QC1’)
V2 /(QL2’)
V3’/(QC3’)
V1 V3
V2’
VinVout
EECS 247 Lecture 7: Filters © 2007 H.K. Page 36
Signal Flowgraph6th Order Bandpass Filter
'2
1QL
'1
1QC
− 0s
ω
1−
0s
ω−
'2
1QL
−
'3
1QC
'3
1
QC−0
sω
0s
ω−
0s
ω−0
sω
'1
1QC
−
Let us try to build this bandpass filter using the simple Gm-C structure
1VinVout
EECS 247 Lecture 7: Filters © 2007 H.K. Page 37
Second Order Gm-C FilterUsing Simple Source-Couple Pair Gm-Cell
• Center frequency:
• Q function of:
Use this structure for the 1st and the 3rd resonatorUse similar structure w/o M3, M4 for the 2nd resonatorHow to couple the resonators?
M1,2m
oint g
M1,2mM 3,4m
g2 C
gQg
ω = ×
=
EECS 247 Lecture 7: Filters © 2007 H.K. Page 38
Coupling of the Resonators1- Additional Set of Input Devices
Coupling of resonators:Use additional input source coupled pairs for the highlighted integrators For example, the middle integrator requires 3 sets of inputs
'2
1QL
'1
1QC
− 0s
ω
1−
0s
ω−
'2
1QL
−
'3
1QC
'3
1
QC−0
sω
0s
ω−
0s
ω−0
sω
'1
1QC
−
1VinVout
EECS 247 Lecture 7: Filters © 2007 H.K. Page 39
Example: Coupling of the Resonators1- Additional Set of Input Devices
int gC
Add one source couple pair for each additional input
Coupling level ratio of device widths
Disadvantage extra power dissipation
oV
maininV
+-
+
-
M1 M2M3 M4
-
+
couplinginV
+
--+
+
-
MainInput
CouplingInput
EECS 247 Lecture 7: Filters © 2007 H.K. Page 40
Coupling of the Resonators2- Modify SFG Bidirectional Coupling Paths
' '1 2
1Q C L
'1
1QC
− 0s
ω
inV 1−
0s
ω−
' '3 2
1Q C L
−
'1
' '3 2
CQC L
3
1QC'
−0s
ω0s
ω−
0s
ω−0
sω
1' 'Q C L1 2
−
Modified signal flowgraph to have equal coupling between resonators• In most filter cases C1
’ = C3’• Example: For a butterworth lowpass filter C1’ = C3’ =1 & L2’=2• Assume desired overall bandpass filter Q=10
outV1
EECS 247 Lecture 7: Filters © 2007 H.K. Page 41
Sixth Order Bandpass Filter Signal Flowgraph
γ
1Q
− 0s
ω
inV 1−
0s
ω−
1Q
−0s
ω0s
ω−
0s
ω−0
sω
outV1γ−
γγ−
1Q 2114
γ
γ
=
≈
• Where for a Butterworth shape
• Since Q=10 then:
EECS 247 Lecture 7: Filters © 2007 H.K. Page 42
Sixth Order Bandpass Filter Signal FlowgraphSFG Modification
1Q
−0s
ω
inV 1−
0s
ω−
1Q
−0s
ω0s
ω−0
sω
−0s
ω
outV1
γ−
20s
ωγ ⎛ ⎞⎜ ⎟⎝ ⎠
×
γ−
20s
ωγ ⎛ ⎞⎜ ⎟⎝ ⎠
×
EECS 247 Lecture 7: Filters © 2007 H.K. Page 43
Sixth Order Bandpass Filter Signal FlowgraphSFG Modification
20 1
ωω
⎛ ⎞ ≈⎜ ⎟⎝ ⎠
For narrow band filters (high Q) where frequencies within the passband are close to ω0 narrow-band approximation can be used:
Within filter passband:
The resulting SFG:
2200
js
ωωω
γ γ γ⎛ ⎞⎛ ⎞ = ≈⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
× × −
EECS 247 Lecture 7: Filters © 2007 H.K. Page 44
Sixth Order Bandpass Filter Signal FlowgraphSFG Modification
1Q
−0s
ω
inV 1−
0s
ω−
1Q
−0s
ω0s
ω−0
sω
−0s
ω
outV1
γ−
γ−
γ−
Bidirectional coupling paths, can easily be implemented with coupling capacitors no extra power dissipation
γ−
EECS 247 Lecture 7: Filters © 2007 H.K. Page 45
Sixth Order Gm-C Bandpass FilterUtilizing Simple Source-Coupled Pair Gm-Cell
Parasitic cap. at integrator output, if unaccounted for, will result in inaccuracy in γ
k
int g k
int gk
k int g
C2 C C
2 CC 1 1
2C C13
1 / 14
γ
γγ
+=
×
×=
−
→ =
=
EECS 247 Lecture 7: Filters © 2007 H.K. Page 46
Sixth Order Gm-C Bandpass FilterNarrow-Band versus Exact
Frequency Response Simulation
Q=10
Regular Filter
Response
Narrow-Band Approximation
EECS 247 Lecture 7: Filters © 2007 H.K. Page 47
Simplest Form of CMOS Gm-CellNonidealities
• DC gain (integrator Q)
• Where a denotes DC gain & θ is related to channel length modulation by:
• Seems no extra poles!
( )
M 1,2m
M 1,20 load
M 1,2
gag g
2LaV Vgs th
L
θ
θλ
=+
=−
=
Small Signal Differential Mode Half-Circuit
EECS 247 Lecture 7: Filters © 2007 H.K. Page 48
CMOS Gm-Cell High-Frequency Poles
• Distributed nature of gate capacitance & channel resistance results in infinite no. of high-frequency poles
Cross section view of a MOS transistor operating in saturation
Distributed channel resistance & gate capacitance
EECS 247 Lecture 7: Filters © 2007 H.K. Page 49
CMOS Gm-Cell High-Frequency Poles
• Distributed nature of gate capacitance & channel resistance results in an effective pole at 2.5 times input device cut-off frequency
High frequency behavior of an MOS transistor operating in saturation region
( )
M 1,2
M 1,2
effective2
i 2 i
effectivet2
M 1,2M 1,2m
t 2
1P1
P
P 2.5
V Vgs thg 3C2 / 3 WL 2 Lox
μ
ω
ω
∞
=
≈
≈
−= =
∑
EECS 247 Lecture 7: Filters © 2007 H.K. Page 50
Simple Gm-Cell Quality Factor
( )M 1,2effective2 2
V Vgs th15P4 L
μ −=( )M 1,2
2LaV Vgs thθ
=−
• Note that phase lead associated with DC gain is inversely prop. to L• Phase lag due to high-freq. poles directly prop. to L
For a given ωο there exists an optimum L which cancel the lead/lag phase error resulting in high integrator Q
( )( )
i1 1o pi 2
2M1,2 o
M1,2
int g. 1real
V Vgs th L1 4int g. 2L 15 V Vgs th
Q
a
Q
ω
θ ωμ
∞
=
≈−
−≈ −
−
∑
EECS 247 Lecture 7: Filters © 2007 H.K. Page 51
Simple Gm-Cell Channel Length for Optimum Integrator Quality Factor
( )1/ 32
M1,2
o
V Vgs th. 15opt. 4Lθμ
ω
⎡ ⎤−⎢ ⎥≈ ⎢ ⎥⎢ ⎥⎣ ⎦
• Optimum channel length computed based on process parameters (could vary from process to process)
EECS 247 Lecture 7: Filters © 2007 H.K. Page 52
Source-Coupled Pair CMOS Gm-Cell Transconductance
( ) ( )
( )
1/ 22i i
d ssM1,2 M1,2
i M 1,M 2dm
iM 1,2
di
i
v v1I I 1V V V V4gs th gs th
v INote : For small gV V vgs thINote : As v increases or the v
ef fect ive transconductance decreases
⎧ ⎫Δ⎡ ⎤ Δ⎡ ⎤⎪ ⎪Δ = ⎢ ⎥ − ⎢ ⎥⎨ ⎬− −⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭
Δ⎡ ⎤ Δ→ =⎢ ⎥− Δ⎢ ⎥⎣ ⎦ΔΔ Δ
For a source-coupled pair the differential output current (ΔId)as a function of the input voltage(Δvi):
i i1 i2
d d1 d2
v V V
I I I
Δ = −
Δ = −
EECS 247 Lecture 7: Filters © 2007 H.K. Page 53
Source-Coupled Pair CMOS Gm-Cell Linearity
Ideal Gm=gm
• Large signal Gm drops as input voltage increasesGives rise to nonlinearity
EECS 247 Lecture 7: Filters © 2007 H.K. Page 54
Measure of Linearity
ω1 ω1 3ω1 ωω
2ω1−ω2 2ω2−ω1
Vin Voutω1 ωω2 ω1 ωω2
Vin Vout2 31 2 3
23
1
3
2 43 5
1 1
.............
3 . .3
1 ......4
3 .
3 25 ......4 8
Vout Vin Vin Vin
amplitude rd harmonicdist compHDamplitude fundamental
Vin
amplitude rd order IM compIMamplitude fundamental
Vin Vin
α α α
αα
α αα α
= + + +
=
= +
=
= + +
EECS 247 Lecture 7: Filters © 2007 H.K. Page 55
Source-Coupled Pair Gm-Cell Linearity
( ) ( )
( )
( )
( )
1/ 22i i
d ssM 1,2 M1,2
2 3d 1 i 2 i 3 i
ss1 2
M1,2
ss3 43
M 1,2
ss5
v v1I I (1)1V V V V4gs th gs th
I a v a v a v . . . . . . . . . . . . .
Series expansion used in (1)Ia & a 0
V Vgs thIa & a 0
8 V Vgs thIa
128 V Vgs th
⎧ ⎫Δ⎡ ⎤ Δ⎡ ⎤⎪ ⎪Δ = ⎢ ⎥ − ⎢ ⎥⎨ ⎬− −⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭
Δ = ×Δ + ×Δ + ×Δ +
= =−
= − =−
= −−
65
M 1,2
& a 0=
EECS 247 Lecture 7: Filters © 2007 H.K. Page 56
Linearity of the Source-Coupled Pair CMOS Gm-Cell
• Key point: Max. signal handling capability function of gate-overdrive voltage
( ) ( )
( )
( )
2 43 5i i
1 1
1 32 4
i i
GS th GS th
i max GS th
rms3 GS th in
3a 25aˆ ˆIM 3 v v . . . . . . . . . . . .4a 8aSubst i tu t ing for a ,a ,. . . .
ˆ ˆv v3 25IM 3 . . . . . . . . . . . .32 1024V V V V
2v̂ 4 V V IM 33
ˆIM 1% & V V 1V V 230mV
≈ +
⎛ ⎞ ⎛ ⎞≈ +⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
≈ − × ×
= − = ⇒ ≈
EECS 247 Lecture 7: Filters © 2007 H.K. Page 57
Simplest Form of CMOS Gm CellDisadvantages
( )
( )( )
since
then
23 GS th
M 1,2m
oint g
o
IM V V
g2 C
W V VCg gs thm ox LV Vgs th
ω
μ
ω
−∝ −
=×
−=
−∝
•Max. signal handling capability function of gate-overdrive
•Critical freq. is also a function of gate-overdrive
Filter tuning affects max. signal handling capability!
EECS 247 Lecture 7: Filters © 2007 H.K. Page 58
Simplest Form of CMOS Gm CellRemoving Dependence of Maximum Signal Handling
Capability on Tuning
Dynamic range dependence on tuning removed (to 1st order)Ref: R.Castello ,I.Bietti, F. Svelto , “High-Frequency Analog Filters in Deep Submicron Technology ,
“International Solid State Circuits Conference, pp 74-75, 1999.
• Can overcome problem of max. signal handling capability being a function of tuning by providing tuning through :
– Coarse tuning via switching in/out binary-weighted cross-coupled pairs Try to keep gate overdrive voltage constant
– Fine tuning through varying current sources
EECS 247 Lecture 7: Filters © 2007 H.K. Page 59
Dynamic Range for Source-Coupled Pair Based Filter
( )3 1% & 1 230rmsGS th inIM V V V V mV= − = ⇒ ≈
• Minimum detectable signal determined by total noise voltage• It can be shown for the 6th order Butterworth bandpass filter
fundamental noise contribution is given by:
2o
int g
int grmsnoise
rmsmax
36
k Tv Q C
Assumin g Q 10 C 5pF
v 160 Vsince v 230mV
230x10Dynamic Range 20log 63dB160x10
3
μ
−−
≈
= =
≈=
= ≈
EECS 247 Lecture 7: Filters © 2007 H.K. Page 60
Improving the Max. Signal Handling Capability of the Source-Coupled Pair Gm-Cell
( )( )
( )( )
M 1,2ss1ss3
M 3,4
M 1,22
M 3,4
V Vgs thI b & a and thusI V Vgs th
WL b
W aL
−= =
−
=
• 2nd source-coupled pair added to subtract current proportional to nonlinear component associated with the main SCP
EECS 247 Lecture 7: Filters © 2007 H.K. Page 61
Improving the Max. Signal Handling Capability of the Source-Coupled Pair Gm
Ref: H. Khorramabadi, "High-Frequency CMOS Continuous-Time Filters," U. C. Berkeley, Department of Electrical Engineering, Ph.D. Thesis, February 1985 (ERL Memorandom No. UCB/ERL M85/19).
EECS 247 Lecture 7: Filters © 2007 H.K. Page 62
Improving the Max. Signal Handling Capability of the Source-Coupled Pair Gm
• Improves maximum signal handling capability by about 12dBDynamic range theoretically improved to 63+12=75dB
EECS 247 Lecture 7: Filters © 2007 H.K. Page 63
Simplest Form of CMOS Gm-Cell
• Pros– Capable of very high frequency
performance (highest?)– Simple design
• Cons– Tuning affects power dissipation– Tuning affects max. signal handling
capability (can overcome)– Limited linearity (possible to
improve)
Ref: H. Khorramabadi and P.R. Gray, “High Frequency CMOS continuous-time filters,” IEEE Journal of Solid-State Circuits, Vol.-SC-19, No. 6, pp.939-948, Dec. 1984.
EECS 247 Lecture 7: Filters © 2007 H.K. Page 64
Gm-CellSource-Coupled Pair with Degeneration
( )
( )dsV small
eff
M 3 M 1,2mds
M 1,2 M 3m ds
M 3eff ds
C Wox 2V VI 2 V Vgs thd ds ds2 L
I Wd V VCg gs thds oxV Lds1g 1 2
g g
for g g
g g
μ
μ
⎡ ⎤−= −⎣ ⎦
∂ −= ≈∂
=+
>>
≈
M3 operating in triode mode source degeneration determines overall gmProvides tuning through varing Vc
EECS 247 Lecture 7: Filters © 2007 H.K. Page 65
Gm-CellSource-Coupled Pair with Degeneration
•Pros– Moderate linearity– Continuous tuning provided
by Vc– Tuning does not affect power
dissipation
•Cons– Extra poles associated
with the source of M1,2 Low frequency
applications only
Ref: Y. Tsividis, Z. Czarnul and S.C. Fang, “MOS transconductors and integrators with high linearity,”Electronics Letters, vol. 22, pp. 245-246, Feb. 27, 1986
EECS 247 Lecture 7: Filters © 2007 H.K. Page 66
BiCMOS Gm-CellExample
• MOSFET in triode mode:
• Note that if Vds is kept constant:
• Linearity performance keep gm constantfunction of how constant Vds can be held
– Need to minimize Gain @ Node X
• Since for a given current, gm of BJT is larger compared to MOS- preferable to use BJT
• Extra pole at node X
( )
M 1m
C Wox 2V VI 2 V Vgs thd ds ds2 L
I Wd Cg Vox dsV Lgs
M 1 B1A g gx m m
μ
μ
⎡ ⎤−= −⎣ ⎦
∂= =
∂
=
B1
M1X
Iout
Is
Vcm+Vin
Vb
gm can be varied by changing Vb and thus Vds
EECS 247 Lecture 7: Filters © 2007 H.K. Page 67
Alternative Fully CMOS Gm-CellExample
• BJT replaced by a MOS transistor with boosted gm
• Lower frequency of operation compared to the BiCMOS version due to more parasitic capacitance at nodes A & B A B
+- +
-
EECS 247 Lecture 7: Filters © 2007 H.K. Page 68
• Differential- needs common-mode feedback ckt
• Freq.tuned by varying Vb
• Design tradeoffs:– Extra poles at the input device drain
junctions– Input devices have to be small to
minimize parasitic poles– Results in high input-referred offset
voltage could drive ckt into non-linear region
– Small devices high 1/f noise
BiCMOS Gm-C Integrator
-Vout+
Cintg/2
Cintg/2
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