introduction to active filter networks.pdf

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][email protected] [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


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

=


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

+ + = =

=


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

= =


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

= = =


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


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


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


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


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)

+ ++ + + + + + + + + + +

=


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)

+ ++ + + + + + + + + + +

=


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= =


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


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)


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=

+


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

= + +


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= =


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


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=


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


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


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)


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

)


+ 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

= = + + + +


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

+ < + +


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

= + +


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=+=


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

= +


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


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

QQ

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

= + + + +


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 + + +


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>>


+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)

= + = ++ = +


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

= + + + +


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


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

introduction to active filter networks.pdf

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