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EE 541 Lecture Aid
#7 Fall
Semester 2010
Introduction To Active Filter NetworksDr. John Choma,
Professor Of Electrical EngineeringMing Hsieh Department of Electrical Engineering
Powell Hall Of Engineering (PHE) Room #620University of Southern CaliforniaUniversity Park; Mail Code: 0271
Los Angeles, California 90089-0271(213) 740-4692 [Office]johnc@usc.edu [E-Mail]
www.jcatsc.com [Course Notes]
EE 541 Lecture Aid #7 Active Filter Introduction 381
Overview Of LectureO Operational Transconductor Topologies For Filter Networks NMOS Floating Voltage CellBasic ConceptCircuit Realization
COMFET Floating Voltage Cell COMFET Linear Transconductor NMOS Linear Transconductor
O Sallen-Key Filter Basic Architecture ShortfallsPotential InstabilityFinite Amplifier Output Resistance
4-Pole Butterworth ExampleO Other Filter Architectures Delyiannis-Friend Bandpass Filter Miller Integrator ckerberg-Mossberg Biquadratic Filter
EE 541 Lecture Aid #7 Active Filter Introduction 382
NMOS Floating Voltage Cell
O Requirements M1 And M2 Matched Substrates Appropriately Back Biased
O Analysis
O Result Linearity Of Differential I/O Relationship
With Respect To Differential Input Voltage Transconductance Tunable By Vx
( ) ( )2 2n nd1 1 2 x hn d 2 2 1 x hnK KW WI V V V V I V V V V2 L 2 L = + = +
+
Vx
V1 M1Id1
Vgs1+ +
Vx
V2M2Id2
Vgs2+
Vss
Note:Vgs1 = V1 V2 + Vx Vgs2 = V2 V1 + Vx
( )( )
d1 d 2 me 1 2
me n x hn
I I G V V
WG 2K V V
L
=
EE 541 Lecture Aid #7 Active Filter Introduction 383
Floating Voltage Realization
O AnalysisO Comments M3 And M4 Behave As Nominally Constant Floating Voltage Sources All Transistors Matched Except For Indicated Gate Aspect Ratios Substrates Are Reverse Biased Currentsid1, id2 Are Signal CurrentsIQ Is A Quiescent Current Only nChannel Transistors Used In Signal Paths
V1M1
i Id1 Q+
Vx+
V2M2
i + Id2 Q
Vss
M3
kI iQ d2M4
kI iQ d1
Vx+
x k x k
(k 1)I+ Q (k 1)I+ Q
+VddRequirement:
kIQ >> |id1 |, |id2 |
( ) ( )2 QnQ x hn x hn n2IK kW
kI V V V V2 L K W L
+
( ) ( ) ( )( )
( )
d1 Q d 2 Q me 1 2
me n x hn
n Q
i I i I G V V
WG 2K V V
L
8K W L I
+ + = =
=
EE 541 Lecture Aid #7 Active Filter Introduction 384
COMFET Floating Voltage Cell
O Analysis
O Differential Output CurrentO Comments Linear Differential I/O Relationship Effective Transconductance, Gme, Tunable Via Vx Wide Tunability Range Owing To Vhe = Vhn + Vhp
Id1M1a
M1b
+
Vx
V1Id2
M2a
M2b
+
Vx
V2
Id1 Id2
Parametric Review:
pnnn n pp p
n p
WWK K K K
L L
he hn hpV V V= +( )2ned ge heKI V V2= ne nn pp
1 1 1K K K
+
( ) ( )2 2ne ned1 1 2 x he d 2 2 1 x heK KI V V V V I V V V V2 2= + = + ( )( )
d1 d 2 me 1 2
me ne x he
I I G V V
G 2K V V
= =
EE 541 Lecture Aid #7 Active Filter Introduction 385
Id1 IQ IQ
M1a
M1b
V1
Id1
M4b
M3a
IQ
Vss
Va
Id2
M2a
M2b
V2
Id2
M3b
M4a
IQ
Vb
+Vdd
+V
x
+ Vx
COMFET Linear Transconductor
O Analysis
O Results
( )( ) ( )
2 QneQ x he x he 1 b 2 a
ne2 2ne ne
d1 1 a he d 2 2 b he
2IKI V V V V V V V V
2 K
K KI V V V I V V V
2 2
= = + = =
= = ( )
( )d1 d 2 me 1 2
me ne x he ne Q
I I G V V
G 2K V V 8K I
= = =
EE 541 Lecture Aid #7 Active Filter Introduction 386
Comments On COMFET Linear OTA
O All COMFET Pairs Must Be Matched M1aM1b Matched To M2aM2b M3aM3b Matched To M4aM4b Ideally, All n-Channel And p-Channel Devices Respectively Matched
O Signals Linear Differential I/O Relationship Inner COMFETs Do Not Conduct Signal Currents Inner COMFETs Conduct Current IQ, Which Controls Effective
Transconductance, GmeO Biasing All Substrates Back Biased Not Especially Amenable To Low Voltage Applications
O Applications Moderate Speed OTA For OTA-C Filter Applications Class AB Stage To Improve Slew Rate Of CMOS Op-Amps
EE 541 Lecture Aid #7 Active Filter Introduction 387
NMOS Linear Transconductor
O No Signal Currents In Transistors M3-M4 And M7-M8O Voltage VQ Biases Gate Source Terminals Of M3-M4 And M7-M8 Controls Effective Differential Transconductance
M3
M7
M4
M8
M6
Vss
V1 V2
V VQ ss
M5
M1
Ida
M2
Idb
Vx
Iss
+Vdd
+
V
Q
+ VQ
M1, M2, M5, M6Are Matched
EE 541 Lecture Aid #7 Active Filter Introduction 388
NMOS Transconductor Analysis
( ) ( )( ) ( )
( ) ( )
22n nda 1 x hn 2 Q x hn
22n ndb 2 x hn 1 Q x hn
da db n Q 1 2 me 1 2
K KW WI V V V V V V V
2 L 2 LK KW W
I V V V V V V V2 L 2 L
WI I K V V V G V V
L
= + = +
= =
M3
M7
M4
M8
M6
Vss
V1 V2
V VQ ss
M5
M1
Ida
M2
Idb
Vx
Iss
+Vdd
+
V
Q
+ VQ
EE 541 Lecture Aid #7 Active Filter Introduction 389
Sallen-Key Active RC Lowpass Filter
O Topology Lowpass Structure Bandpass And Highpass
Structures Can Be Realized Lowpass Version Common In
Baseband CommunicationSystem Applications
O Amplifier Simple Local Feedback Amplifier With Closed Loop Gain Of K Desirable To Design Amplifier For Unity GainMaximizes Bandwidth And Unity Gain FrequencyMaximizes Linearity Because Of Reduced Output SwingAvoids Network Instability Issues
Note Positive Feedback Through Capacitance C1 Network Can Oscillate For Large Open Loop Voltage Gain
Amplifier Has Parasitic Output Resistance (Ro ) And Parasitic Output Capacitance (Co )
+ K
R2R1
C2
C1
VoutVin
Vi
EE 541 Lecture Aid #7 Active Filter Introduction 390
Co
R2R1
C2
C1
VoutVinVi
Ro+
KVi
Sallen-Key Lowpass Equivalent Circuit
O ModelO Parameters Resistances:
Capacitances: Amplifier: Assume K = 1 Normalized Frequency:
p = sRC = sR1 C1
O Transfer Function, H(p) = Vout /Vin
+ K
R2R1
C2
C1
VoutVin
Vi2 1
o r
R NR NR
R k R
==
2 1
o c
C MC MC
C k C
==
( ) ( ) ( ) ( )2
r r2 3
r c r r c r c
1 pk p k MN.
1 p M N 1 k k 1 p MN k M N 1 k k 1 M MN p k k MN
H(p)
+ ++ + + + + + + + + + +
=
EE 541 Lecture Aid #7 Active Filter Introduction 391
Transfer CharacteristicO Transfer Relationship
IdealizedRo = 0 kr = 0Co = 0 kc = 0Function
O Comparisons Ideal Response Is Second Order With No Finite Frequency Zeros Actual Response Is Third Order With Two, Likely Complex, ZerosZeros Precipitated By Finite Amplifier Output ResistanceZeros Generate Partial Notch At High FrequenciesOutput Amplifier Capacitance Does Not Affect ZerosOutput Capacitance Impacts Self-Resonant Frequency, Bandwidth, And
Quality Factor
( )I 21
H (p)1 pM N 1 p MN
=+ + +
( ) ( ) ( ) ( )2
r r2 3
r c r r c r c
1 pk p k MN
1 p M N 1 k k 1 p MN k M N 1 k k 1 M MN p k k MN
H(p)
+ ++ + + + + + + + + + +
=
EE 541 Lecture Aid #7 Active Filter Introduction 392
2
2 2Q
b
1 11 1 1
f 2Q 2Q RC
MN MN
+ + = =
Design-Oriented AnalysisO Idealized Function First Order Analysis Pre-CAD Optimization
O Alternative Form Normalized Self Resonance
Filter Quality FactorO 3-dB Bandwidth Butterworth
fQ = 1Resultant
Bandwidth
General BandwidthRelationship
( )o o
outI 2
in R ,C 0
V 1H (p)
V 1 pM N 1 p MN==
+ + +
o o
outI 2
in R ,C 0
o o
V 1H (p)
V p p1
Q y y=
= + +
Q1 N
N 1 M= +
Q 1 2=
b1
RC MN
=
2b
b Q2 2o
y 1 1 RC MN 1 1 1 f
y 2Q 2Q
= = + +
o o RCy 1 MN= =
EE 541 Lecture Aid #7 Active Filter Introduction 393
Bandwidth Function
O Bandwidth Factor Plot fQ Is Bandwidth Function Maximizes At About 1.5
For Large Quality Factor Equals Unity For MFM
Second Order Response Sensitive To Q For Low
Values Of QO Resistor Ratio Plot R/Ro Defines Minimum
Circuit -To- Amplifier Output Resistance RatioFor Partial Notch Frequency 10-Times Larger Than 3-dB Bandwidth Sensitive to Small Q Requires Large Ratio For Large Q
0
50
100
150
200
250
0.50 1.00 1.50 2.00 2.50 3.00Filter Quality Factor, Q
0.00
0.32
0.64
0.96
1.28
1.60Minimum R/R o Bandwidth Function, f Q
R/R o
f Q
EE 541 Lecture Aid #7 Active Filter Introduction 394
Frequency Response
O MFM Case ConsideredO Parasitic Effects Reduction In Bandwidth By
About 17% Response ShapeNon-Monotonic At High
Signal FrequenciesPartial Notching Is
Observed Parasitic Example Values
Are Reasonable InPractical Electronics
0.00
0.20
0.40
0.60
0.80
1.00
0.01 0.10 1.00 10.00 100.00Normalized Frequency
Ideal:kr = kc = 0
Nonideal:kr = 0.12kc = 0.10
Gain Magnitude (Volts/Volt)
EE 541 Lecture Aid #7 Active Filter Introduction 395
0
0.2
0.4
0.6
0.8
1
0.1 1 1.9 2.8 3.7 4.6 5.5Resistance Ratio, N
Capacitance Ratio, M
Q = 0.5
Q = 0.7
Q = 1.0
Q = 1.5
Optimal Element Ratio
O Quality FactorO Design Constraint Avoid Large Ratio Of
Resistances AndCapacitances Difficult To Realize
Accurately On ChipO Observations M Is Maximized At About
One For N = 1 M Is Very Sensitive To N
For N < 1 Select N Slightly Larger
Than MSmall N Implies SensitivityLarge N Implies Small M
Q1 N
N 1 M= + ( )22
NM
Q N 1=
+
EE 541 Lecture Aid #7 Active Filter Introduction 396
Multi-Pole Sallen-Key Lowpass FilterO Normalized Transfer Function Of
Sallen-Key KernelO De-Normalized
Transfer Function
O Four-PoleFunction
O Realization
2
o o
1H(p)
p p1
Q y y
= + +
+
R2aR1a
C2a
C1a
Vout
ViaK=1
+K=1
R2b
R1b
C2b
C1b
Vib
Vin
2 2Qa Qa Qb Qb
a ba ba b bb bb
1H(s)
f s f s f s f s1 11 1
Q Q
= + + + +
2Q Q
b b
1H(s)
f s f s11
Q
= + +
EE 541 Lecture Aid #7 Active Filter Introduction 397
2 2
b b b b
1H(s)
s s s s1 1.848 1 0.765
= + + + +
Butterworth 4-Pole Filter
O Butterworth Transfer FunctionBandwidth = b
O Four-Pole Sallen-Key
O Design Requirements First Stage Quality Factor: Qa = 1/1.848 = 0.541 Second Stage Quality Factor: Qb = 1/0.765 = 1.307 First And Second Stage Bandwidths Why Are First And Second Stage Bandwidths
Identical And Equal To The Overall Filter Bandwidth?
2 2Qa Qa Qb Qb
a ba ba b bb bb
1H(s)
f s f s f s f s1 11 1
Q Q
= + + + +
ba bbb
Qa Qb
f f= =
EE 541 Lecture Aid #7 Active Filter Introduction 398
2
Q 2 21 1
f 1 1 12Q 2Q
= + +
Butterworth 4-Pole Design Example
O Specifications 3-dB Bandwidth: b = 2(300 MHz) Amplifier Output Resistance: Ro = 50 Filter Output Port Capacitance: Co = 30 fF Design For kr = Ro /R = 0.04
O STEP #1: Calculate fQa And fQb fQa = 0.7195 For Qa = 0.541 fQb = 1.390 For Qb = 1.307O Step #2: Calculate Resistance Values R = R1a = R1b = Ro /kr = 1.25 K Choose N1a = N1b = 1.15 (Slightly Larger Than Sensitivity Peak) R2a = NaR1a = 1.438 K R2b = NbR1b = 1.438 K
EE 541 Lecture Aid #7 Active Filter Introduction 399
Design Example, ContdO Step #3: Calculate Capacitance Ratios Ma = 0.8494 For Qa = 0.541 And Na = 1.15 Mb = 0.1457 For Qb = 1.307 And Nb = 1.15
O Step #4: Bandwidth Correction For Output Capacitance Design For a Bandwidth That Is 15% Larger Than Specification Design For b = 2(340 MHz) Determine Frequency Parameters, ba And bbba = fQab = 2(244.6 MHz)bb = fQbb = 2(472.6 MHz) Observation: One Of The Two Stages Must Operate At A Frequency
That Is Considerably Larger Than The Overall Filter BandwidthO Step #5: Calculate Filter Capacitances C1a = 378.9 fF For Ma = 0.8494, Na = 1.15, R = 1.25 K, fQa = 0.7195 C1b = 914.7 fF For Mb = 0.1457, Nb = 1.15, R = 1.25 K, fQb = 1.390 C2a = MaC1a = 321.8 fF C2b = MbC1b = 133.3 fF
( )22N
MQ N 1
=+
b Q RC MN f=
EE 541 Lecture Aid #7 Active Filter Introduction 400
Finalized DesignO Schematic Diagram
O Element Values Resistances Are In Ohms Capacitances Are In Femtofarads
O Simulation Accounts For Amplifier Output Port Resistance Accounts For Output Port Capacitance
+
14381250
321.8
378.9
Vout
ViaK=1
+K=1
1438
1250
133.3
914.7
Vib
Vin Vab
EE 541 Lecture Aid #7 Active Filter Introduction 401
Frequency Response Simulation
-80
-70
-60
-50
-40
-30
-20
-10
0
10
0.01 0.1 1 10
I
/
O
G
a
i
n
(
d
e
c
i
b
e
l
s
)
Signal Frequency (GHz)
Two-StageFilter
First-StageFilter
Second-StageFilter
EE 541 Lecture Aid #7 Active Filter Introduction 402
Frequency Response Comments
O Simulated Bandwidth Is 301.9 MHzO Simulated Low Frequency Gain Is 1 (0 dB)O Observations Partial Notching In Each Stage And In Overall Filter Overall Filter Response Is Flat In PassbandFirst Stage Has Anticipated Inferior 3-dB BandwidthSecond Stage Has Anticipated Large BandwidthSecond Stage Peaking Compensates For First Stage Roll Off
First Stage Displays No Peaking Because Of Low Q (0.541) Second Stage Projects Peaking Because Of High Q (1.307)
EE 541 Lecture Aid #7 Active Filter Introduction 403
Pulse Response SimulationO HSPICE Simulation Dashed Is Input Red Is Output
O Input Pulse Train 1 Volt Amplitude 10 pSEC Rise Time 10 pSEC Fall Time 1 nSEC Initial Delay 20 nSEC Pulse Width 40 nSEC Period
O Observation Overshoot/Undershoot
Is About 10%--Not Surprising For Butterworth Filter Settling Time To 0.5% Of Steady State Is 6.4 nSEC (Not Bad For a
300 MHz Lowpass Filter)
-0.2
0
0.2
0.4
0.6
0.8
1.0
1.2
0 20 40 60 80 100 120
Time (nSEC)
I
/
O
R
e
s
p
o
n
s
e
s
(
v
o
l
t
s
)
EE 541 Lecture Aid #7 Active Filter Introduction 404
+ KR2
R1
R3
C2
C1Vout
VinV /Kout Vx
Sallen-Key Bandpass FilterO Schematic Diagram Amplifier Presumed IdealZero Output ResistanceZero Output CapacitanceZero Input CapacitanceInfinitely Large Input ResistanceFrequency-Invariant Gain (K)
I/O Transfer Function Is H(s) = Vout /VinO Equilibrium Equations Eliminate Node Voltage
Variable Vx Solve For TransferFunction H(s) = Vout /Vin Cast Transfer Function IntoCanonic Bandpass FormCenter Frequency Is oBandwidth B Is o /QGain At Center Frequency
Is H(j o )
out out2 x
2
x in out x out1 x 2 x
1 2
V VsC V 0
KR K
V V V V VsC V sC V 0
R K R
+ = + + + =
( ) ( )oo oo
22 2oo
o o
H jH j
QQH(s)s ss s 1Q Q
= = + + + +
EE 541 Lecture Aid #7 Active Filter Introduction 405
Bandpass Filter Transfer FunctionO Transfer Relationship
O Network Stability Resistance R3 Establishes Positive
Feedback Denominator s-Term Coefficient
Can Be Negative For Large Gain KNetwork Instability Is Precluded If
Denominator s-Term Coefficient Is Greater Than ZeroStability Requirement
Easier To Satisfy For Large Resistance R3 Larger R3 Implements Smaller Amount Of
Closed Loop FeedbackK =1 Desirable
Assures Network Stability Supportive Of Broadband Amplifier Response Capability
+ KR2
R1
R3
C2
C1Vout
VinV /Kout Vx
( )( ) ( ) ( )
2 3 2 1 3out
in 21 3 1 2 3 1 2 21 3 2 1 2
1 3
sKR R C R RVH(s)
V R R C C R K 1 R R C1 s s R R R C C
R R
+= = + + + + +
3 1 2 2
2 2 1
R C C RK 1
R C R
+ < + +
EE 541 Lecture Aid #7 Active Filter Introduction 406
Sallen-Key Bandpass Filter MetricsO Tuned Center Frequency
O Filter Quality Factor
O Center Frequency GainO Comments Quality Factor And Center Frequency Gain Adjustable Through K
Without Altering Bandpass Center Frequency High Q Can Be Implemented Through K Within Stability Constraint K=1 Ensures Network Stability But No Metric Adjustability Is
Conveniently Possible And Center Frequency Gain Is Smaller Than One Circuit Can Be Operated With Phase Inversion (K < 1)Smaller Network Quality FactorSmall Center Frequency Gain Magnitude
( )o 1 3 2 1 21
R R R C C=
( )( ) ( )1 3 1 2 3 1 21 3 1 2 3 1 2 2
R R R R R C CQ
R R C C R K 1 R R C
+= + +
( )( )
o1 1 2 1
2 2 3
KH j
R C C R1 K 1
R C R
= + +
EE 541 Lecture Aid #7 Active Filter Introduction 407
Q-Enhancement In Sallen-Key BandpassO Stability
RequirementO Quality Factor
Q1 Infinite Q (Instability) For K = 1+Kc Q-Enhancement Is Certainly Possible (Q = 10Q1 For K = 1+0.9Kc )
O Realization OfQ-Enhancement Op-Amp Realization Ideal Op-Amp PresumedInfinite Input ResistanceZero Output ResistanceInfinite Open Loop Voltage Gain
3 31 2 2 1 2 2c c
2 2 1 2 2 1
R RC C R C C RK 1 1 K ; K
R C R R C R
+ + < + + = + +
( )( ) ( )
( )1 3 1 2 3 1 2 1 3 3 1 1c 1 2 21 3 1 2 3 1 2 2
c
R R R R R C C R R R C Q1Q 1 K1 K K R R CR R C C R K 1 R R C 1
K
+ + = = = + + + +
+ K
(K 1)R
V /Kout Vout
+
Op-AmpV /Kout
Vout
R
( )1 3 3 1K 11
c 1 2 2
R R R C1Q Q
K R R C=+=
EE 541 Lecture Aid #7 Active Filter Introduction 408
Negative Feedback Sallen-Key BandpassO Schematic DiagramO Equilibrium Equations
O Filter Transfer Function
O Filter Performance Metrics Center Frequency Center Frequency Gain
Filter Quality Factor
+
IdealVx Vout
VinR1
R2
C1
C2
0
Op-Amp( )
out1 x
2
x in1 x 2 x out
1
VsC V 0
R
V VsC V sC V V 0
R
+ =
+ + =
( )out 2 1 2in 1 1 2 1 2 1 2V sR C
H(s)V 1 sR C C s R R C C
= = + + +
o1 2 1 2
1R R C C
=
( ) 2 1o1 1 2
R CH j
R C C
= +
1 22
1 1 2
C CRQ
R C C
= +
EE 541 Lecture Aid #7 Active Filter Introduction 409
Delyiannis-Friend Bandpass FilterO Schematic Diagram Positive Feedback Around Op-AmpKR (1K)R Resistive PathRequires Constraint On KK=0 Implies No Op-Amp FeedbackIndicated Node Voltages Presume
Network Stability Equilibrium Equations
O Transfer Function
+
IdealVx Vout
KVout
KVout Vin
R1R2
(1 K)R
C
Op-Amp
KR
C
( ) ( )( ) ( )
outout x
2
x inx out x out
1
K 1 VsC KV V 0
R
V VsC V KV sC V V 0
R
+ =
+ + =
out 2
2 2in 21 1 2
1
V sR C1H(s)
V 1 K R K1 2sR C 1 s R R C2R 1 K
= = + +
PositiveFeedback
EE 541 Lecture Aid #7 Active Filter Introduction 410
Delyiannis-Friend Bandpass MetricsO Transfer
Relationship
O CenterFrequency
O Quality Factor With No Feedback (K=0) With Feedback (K>0)Q-Enhancement PossibleStability Constraint Requires
Q > 0
O Center Frequency Gain
out 2
2 2in 21 1 2
1
V sR C1H(s)
V 1 K R K1 2sR C 1 s R R C2R 1 K
= = + + o
1 2
1C R R
=
o2o
K1 2Q1 K
= 2o
1K 1
1 2Q< > 1Approximate TransferFunction
+A(s) V2
V1 V /A(s)2 R
C
o
o
o
o o oB
AA(s) s1
B
A BA(s)
s s>
=+
=
o o oA B=
22 1
V1sCV 1 V
A(s) A(s)
+ = +
( )21 oV 1 sRCV 1 s
+
[ ]o2
1 o
sV A(s)V 1 sRC 1 A(s) 1 RC sRC
= + + + +
EE 541 Lecture Aid #7 Active Filter Introduction 412
Non-Ideal Nature Of Miller IntegratorO Basic Schematic DiagramO Ideal I/O Integration Ideal Transfer Function Achievable With Ideal Op-AmpAo o
O ActualIntegrator Extra LHP Pole Established At s = o,
Assuming o RC >> 1 Known As Lossy IntegratorLoss Is Negligible For Progressively Larger
Unity Gain FrequencyIntegration Is More Impaired At Progressively Higher Signal Frequencies
High Frequency Compensation Is Recommended For High Performance Active Filter ApplicationsPassive Compensation Is Straight-Forward And Nominally EffectiveActive Compensation Is Effective But Requires Additional Amplifier
+A(s) V2
V1 V /A(s)2 R
C
2
1
V 1V sRC
o
o
o
o o oB
AA(s) s1
B
A BA(s)
s s>
=+
=
o o oA B=( )o21 o osV 1 sRC
V 1 RC sRC 1 s + + +
EE 541 Lecture Aid #7 Active Filter Introduction 413
Passive Compensation Of Miller IntegratorO Basic Schematic DiagramO Transfer Relationship Equilibrium
Equation Transfer
Function
O Compensation Criterion Transfer Relationship Pole-Zero Cancellation
+A(s) V2
V1 V /A(s)2 R
Rc C
2 22 1
c
V VV V
A(s) A(s)1 RR
sC
+ +=
+
cc c
o o o
R1 R 1R C 1 R
R RC 1 C
= + =
( )( )co2
c1 oc c oc
1 1 sR CsV A(s) sRCsRC sA(s)RC RsV RC s1 1 111 sR C 1 sR C R1 sR C
+= = ++ + + + ++ + +
( )c ?2c1
o
1 1 sR CV 1sRCRsV sRC1 1R
+ + +
oRC 1>>
EE 541 Lecture Aid #7 Active Filter Introduction 414
+A(s)
A (s)c
V2V1 V /A(s)2
V /A (s)x cR
C
Vx
++
Active Compensation Of Miller IntegratorO Schematic Diagram Buffer In Feedback LoopAllows For Unidirectional
(Left -To- Right) Current FlowThrough Capacitance CIndicated Feedback Makes
Amplifier Behave As Two-Terminal Linear ResistanceTopology Therefore Mimics Passive Compensation Topology
AmplifiersDominant Pole StructuresHigh Frequency Gain Approximations:
O Equilibrium Equations Vx /V2 Is Classical Buffer
Transfer Relationship Eliminate Variable Vx And
Solve For Transfer FunctionV2 /V1
o c cA(s) s A (s) s x x c
x 2c 2 c
1 2 2x
V V A (s)V V
A (s) V A (s) 1
V V A(s) VsC V
R A(s)
= + = ++ = +
EE 541 Lecture Aid #7 Active Filter Introduction 415
Active Compensation AnalysisO Transfer Relationship
O Design Guidelines Very Large c and o Produce Ideal Integration I/O Characteristics Large cApproximationMatched Amplifiers (c = o ) Produce Ideal Integrator
+A(s)
A (s)c
V2V1 V /A(s)2
V /A (s)x cR
C
Vx
++
o c cA(s) s A (s) s oRC 1>>
c2
1 c c
c
o oc
A (s)V
V A (s)A (s)1 sRC sRC
A (s) 1
1 sRC1 s 1
s RC1
= + + + =
+ ++
cc
1 s1s1
+
2
1o o o c
c
V 1 sRC 1 sRC 11 s 1V sRC1 11 ss RC1
= + + + +
EE 541 Lecture Aid #7 Active Filter Introduction 416
ckerberg-Mossberg Biquadratic FilterO Schematic
DiagramO Discussion Amplifier 3
SubcircuitActs AsPhase-Inverted,Unity GainAmplifier Amplifier 2-3
Subcircuit Acts As Integrator Without Phase Inversion All Amplifiers Presumed Matched With Very High Open Loop Gains
And/Or Very Large Unity Gain Frequencies Signal Flow Path From Vin To Vo2 Delivers Lowpass Frequency
Response Signal Flow Path From Vin To Vo1 Delivers Bandpass Frequency
Response
Vx
Vo2
VinR/k
+
Ideal
Op-Amp
+
Ideal
Op-Amp
C
QR
C Ideal
Op-Amp
+
R
R
Rx
Rx
Vo1
00
0
12
3
EE 541 Lecture Aid #7 Active Filter Introduction 417
ckerberg-Mossberg Circuit AnalysisO Equilibrium Equations
O Lowpass Transfer Characteristic
O Bandpass Transfer Characteristic Center Frequency Determined
By Inverse RC Product Center Frequency Gain Determined By Relative Resistance Ratios Quality Factor Determined By Ratio of Local To Global Feedback
Resistance Of Amplifier 1 Subcircuit
( )
x o1 x
x x o1
o2x o2 o1
in o1o2
0 V 0 V V0 1
R R V
VsCV 0 V sRCV
RV V 1
sC V 0R k R QR
+ = =
+ = = + + + =
( )o1
2in
V ksRCV 1 sRCQ
= + +
( )o2
2in
V skRCsRCV 1 sRCQ
= + +
Vx
Vo2
VinR/k
+
Ideal
Op-Amp
+
Ideal
Op-Amp
C
QR
C Ideal
Op-Amp
+
R
R
Rx
Rx
Vo1
00
0
12
3
Introduction To Active Filter NetworksOverview Of LectureNMOS Floating Voltage CellFloating Voltage RealizationCOMFET Floating Voltage CellCOMFET Linear TransconductorComments On COMFET Linear OTANMOS Linear TransconductorNMOS Transconductor AnalysisSallen-Key Active RC Lowpass FilterSallen-Key Lowpass Equivalent CircuitTransfer CharacteristicDesign-Oriented AnalysisBandwidth FunctionFrequency ResponseOptimal Element RatioMulti-Pole Sallen-Key Lowpass FilterButterworth 4-Pole FilterButterworth 4-Pole Design ExampleDesign Example, ContdFinalized DesignFrequency Response SimulationFrequency Response CommentsPulse Response SimulationSallen-Key Bandpass FilterBandpass Filter Transfer FunctionSallen-Key Bandpass Filter MetricsQ-Enhancement In Sallen-Key BandpassNegative Feedback Sallen-Key BandpassDelyiannis-Friend Bandpass FilterDelyiannis-Friend Bandpass MetricsMiller IntegratorNon-Ideal Nature Of Miller IntegratorPassive Compensation Of Miller IntegratorActive Compensation Of Miller IntegratorActive Compensation Analysisckerberg-Mossberg Biquadratic Filterckerberg-Mossberg Circuit Analysis
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