5 Filter Part 2 (Lecture 19, 20, 21, 22).pdf

5 Filter Part 2 (Lecture 19, 20, 21, 22).pdf5 Filter Part 2 (Lecture 19, 20, 21, 22).pdf5 Filter Part 2 (Lecture 19, 20, 21, 22).pdf

1. Filters
Basic Realization of Second Order Filter

2. Second Order LCR Resonator
Use of this resonator to derive circuit realizations for the
various second-order filter functions
The Resonator Natural Modes
Thenaturalmodesoftheparallelresonancecircuitcanbe
determinedbyapplyinganexcitationthatdoesnotchange
thenaturalstructureofthecircuit.
Resonatorisexcitedwithacurrentsource‘I’connectedin
parallel
Circuitisconcernedwithanindependentidealcurrent
sourceisequivalenttoanopencircuit
Findthepolesofanyresponsefunction

3. Second Order LCR Resonator
The Resonator Natural Modes
V
oacrosstheresonatorastheresponseandthusobtainthe
responsefunctionV
o/I=Z
Z is the impedance of the parallel resonance circuit
In terms of the admittance Y

4. Second Order LCR Resonator
Alternative way of exciting LCR resonator
NodexofinductorLhasbeendisconnectedfrom
groundandconnectedtoanidealvoltagesourceV
i
Idealindependentvoltagesourceisequivalenttoa
shortcircuit
Notalterthenaturalstructureoftheresonator
SelectVoastheresponsevariableandfindthetransfer
functionV
0/V
i

5. Second Order LCR Resonator
Realization of Transmission Zeros
TransferfunctionV
0/V
iwouldbeaccordingtothefilter
specification
Nodeslabeledx,y,orzcanbedisconnectedfromground
andconnectedtoV
iwithoutalteringthecircuit’snatural
modes
Circuittakestheformofavoltagedivider

6. Realization of the Low-Pass Function
Node x is disconnected from ground and connected to Vi
sLbecomes infinite at s = ∞
shunt impedance becomes zero at s = ∞

7. Realization of the High-Pass Function
Node y is disconnected from ground and connected to V
i
Series capacitor introduces a transmission zero at s = 0 (dc)
Shunt inductor introduces another transmission zero at s = 0
(dc)
ω
0and Q are the natural mode parameters
a
2is the high-frequency transmission
Value of a
2can be determined from the circuit by observing
that as s approaches ∞
S
2

8. Realization of the Band-Pass Function
Onezeroats=0isrealizedbytheshuntinductor,andone
zeroats=∞isrealizedbytheshuntcapacitor
Centerfrequencyω
0theparallelLC-tunedcircuitexhibits
aninfiniteimpedanceandthusnocurrentflowsinthe
circuit
At,ω=ω
0,V
o=V
i;Center-frequencygainofthebandpass
filterisunity

9. Realization of the Notch Functions
Disconnecting both nodes x and y from ground and connecting
them together to V
i
The impedance of the LC circuit becomes infinite
at ω= ω
0= 1/√(LC)

10. Realization of the Notch Functions
To obtain a notch-filter realization in which the notch
frequency ω
nis arbitrarily placed relative to ω
0
Simple notch-filter when ω
n= ω
0
(ω
n)
2
= 1/L
1C
1
(ω
0)
2
= 1/(L
2││L
1) (C
2+C
1)

11. Realization of Low Pass Notch Functions
(ω
n)
2
= 1/L
1C
1
(ω
0)
2
= 1/(L
2││L
1) (C
2+C
1)
(ω
0)
2
= 1/L
1(C
2+C
1)

12. Realization of High Pass Notch Functions
(ω
n)
2
= 1/L
1C
1
(ω
0)
2
= 1/(L
2││L
1) (C
2+C
1)
(ω
0)
2
= 1/(L
2││L
1)C

13. Realization of the All-Pass Function
All-passrealizationwithaflat
gainof0.5
Disadvantageoflackinga
common groundterminal
betweentheinputandthe
output

14. Inductance Simulation
Inductor Replacement

15. Antoniuoinductance simulation
circuit

16. Second-Order Active Filters Based
on Inductor Replacement
The Op Amp–RC Resonator
Thenodefromwhereoutput
istakenisnotveryconvenient
totakeanoutput.Iftheoutput
loadishighthenthefilter
characteristicswillbechanged
accordingtotheload.
So, it is
convenient
to connect
a buffer
amplifier
to the
output of
the filter

17. Second Order Filter by Op-amp RC
Resonator

18. Second Order Filter by Op-amp RC
Resonator

19. Second Order Filter by Op-amp RC
Resonator

20. Second Order Filter by Op-amp RC
Resonator
X
X

21. Second Order Filter by Op-amp RC
Resonator

22. Problem 4
Identifythetypeoffilter.
Alsofindω
0andQ

23. Second Order Filer by Op-amp RC
Resonator
Sallen-Keytopology
MultipleFeedbacktopology
Twointegratorlooptopology

24. Sallen-Key Filter
TheSallen–Keytopologyisanelectronicfiltertopology
usedtoimplementsecond-orderactivefiltersthatis
particularlyvaluedforitssimplicity
Voltage-controlledvoltage-source(VCVS)filtertopology
Practicallyinfiniteinputimpedanceandzerooutput
impedance
HighQfactor
BestgainaccuracybecauseitsgainisnotdependonRC
componentvalue
IntroducedbyR.P.SallenandE.L.Key

25. Sallen-Key Filter
GenericSallen–KeyFilterstructure

26. Sallen-Key Low Pass Filter
R
4
R
3
V
x V
y

27. Sallen-Key High Pass Filter
R
4
R
3

28. Sallen-Key Band Pass Filter
R
5
R
4
R
3
R
2

29. Sallen-Key Band Pass Filter
Gain (K)=1+(R
B/R
A)
Centre frequency (f
0)=1/(2πRC)
Q-factor: 1/(3-K)
Gain at f
0=K/(3-K)

30. Sallen-Key Band Reject Filter
Gain (K)=1+(R
B/R
A)
Centre frequency (f
0)=1/(2πRC)
Pass-band gain=K
Q-factor: 1/[2(2-K)]

31. Problem 5
Design a 2
nd
order unity gain Butterworth Sallen-Key
Low pass filter at ω
0= 1 kHz

32. Cascading of two filter stages
(Butterworth)
Ref: Operational Amplifiers … by David A Bell
For 1.5 dbf
c1 = f
c/0.65 For 1.5 dB f
c2 = f
c/0.8
1.5=20log1+
??????
??????
??????
??????1
2
1
2
1.5=20log1+
??????
??????
??????
??????2
4
1
2
fc : Cut-off frequency of 3
rd
order filter
f
c1and f
c2are the individual cut-off frequency of 1
st
and 2
nd
order stages.

33. Second-Order Active Filters Based on the
Two-Integrator-Loop Topology
Anotherfamilyofopamp–RCcircuitsthat
realizesecond-orderfilterfunctions
Twointegratorsconnectedincascadeinan
overallfeedbackloop
Two-integrator-loopbiquadraticcircuitor
biquadcommonlydesignedwithsecond-
orderhigh-passtransferfunction

34. Derivation of the Two-Integrator-Loop Biquad

35. Derivation of the Two-Integrator-Loop Biquad

36. Derivation of the Two-Integrator-Loop Biquad
Two-integrator-loopbiquadrealizesthethreebasicsecond-
orderfilteringfunctions,LP,BP,andHP,simultaneously.
Thisversatilityhasmadethecircuitverypopularandhasgiven
itthenameuniversalactivefilter.

37. Circuit Implementation of Two-Integrator-
Loop Biquad
ReplaceeachintegratorwithaMillerintegratorcircuit
having
Replacethesummerblockwithanop-ampsumming
circuitthatiscapableofassigningbothpositiveand
negativeweightstoitsinputs.
Theresultingcircuit,knownastheKerwin–Huelsman–
NewcomborKHNbiquad
Selectsuitablypracticalvaluesforthecomponentsof
ω0,Q,andK

38. Two Integrator loop topology
(Practical Circuit)

39. Two Integrator loop topology
(Practical Circuit)

40. Two Integrator loop topology
(Practical Circuit)

41. Two Integrator loop topology
(Practical Circuit)

42. Two Integrator loop topology (Practical Circuit)

43. Realization of All Filter (Practical
Circuit)
All Pass Filter

44. Realization of Notch Filter (Practical
Circuit)
All Pass Filter
A notch is obtained by selecting R
B= ∞ and

45. Alternate Two Integrator loop
Circuit (Tow-Thomas method)
Ratherthanusingtheinputsummertoaddsignalswith
positiveandnegativecoefficients,introduceanadditional
inverterinthecircuit
Nowallthecoefficientsofthesummerhavethesamesign.
Performthesummationatthevirtual-groundinputofthefirst
integrator
Summingweightsof1,1/Q,andKarerealizedbyusing
resistancesofR,QR,andR/K,respectively.
High-passfunctionisnolongeravailable

46. Alternate Two Integrator loop
Circuit (Two-Thomas method)

47. Alternate Two Integrator loop
Circuit (Two-Thomas method)

48. Modified Tow-Thomas method
Ratherthanusingafourthopamptorealizethefinite
transmissionzerosrequiredforthenotchandall-pass
functions,aswasdonewiththeKHNbiquad,aneconomical
schemecanbeemployedwiththeTow–Thomascircuit.

49. Modified Two-Thomas method
Two-integrator-loopbiquadsareextremely
versatileandeasytodesign.However,their
performanceisadverselyaffectedbythe
finitebandwidthoftheopamps.

50. Switched Capacitor
Switched-capacitorfilterusedinanintegral
circuitenvironmentwhereresistorsareexpensive
andnoteasilycontrolled
Switched-capacitormaysimulatebyusing
capacitorandMOSFETs.
Acceptabledegreeofaccuracyandlessexpensive
Switched-capacitordealwithtransferofcharges
ratherthanvoltages

51. Switched Capacitor Basics

52. Switched Capacitor Basics

53. Switched Capacitor in Miller’s Integrator
Theswitched-capacitorfiltertechniqueisbasedonthe
realizationthatacapacitorswitchedbetweentwocircuit
nodesatasufficientlyhighrateisequivalenttoaresistor
connectingthesetwonodes.
Considertheactive-RCintegratori.e.Millerintegrator
ReplacedtheinputresistorR
1byagroundedcapacitorC
1
togetherwithtwoMOStransistorsactingasswitches
ThetwoMOSswitchesaredrivenbyanonoverlappingtwo-
phaseclock.
Clockfrequency ismuchhigherthanthe
frequencyoftheinputsignalv
i

54. Switch Capacitor filter in Miller’s
Integrator

55. Design of a Practical Circuit by
Switch Capacitor topology

56. Two-Thomas Circuit
C
1
C
2
R
3
R
4
R
5
R
6
r
r
V
i
-V
lp
V
bp

57. Two-Thomas Circuit by Switch Capacitor topology
C
1 C
2
C
3
C
4C
5
C
6
V
bp
-V
lp
-

58. Design of a Practical Circuit by
Switch Capacitor topology
Onlyanintroductiontoswitched-capacitorfilters
Manysimplifyingassumptionstaken
Themostimportantbeingtheswitched-capacitor–resistor
equivalence
Thisequivalenceiscorrectonlyatf
c=∞
Approximatelycorrectforf
c>>f
Switched-capacitorfiltersdesigncanbecarriedoutexactly
usingz-transformtechniques.

59. Problem 7
DesignaswitchcapacitorbasedMiller’sIntegratorwithclock
frequency100kHz,Vi=1V,C1=1pFandC2=10pF.
(a)Whatchargeistransferredforeachcycle.
(b)Whatistheaveragecurrentdrawnfromtheinputsource
(c)ForC2capacitor,whatchangeofvoltagewillbeexpectedat
theoutputateachclockpulse.
(d)Foranop-ampthatsaturatesat±10Vandfeedback
capacitorinitiallydischarged,howmanyclockwouldit
taketosaturatetheop-amp.
(e)Whatistheaverageslopeofthestaircaseoutput
voltageproduced.

60. Problem 8
DesignaswitchcapacitorbasedTwo-Thomascircuitfor
mainlyflatresponselowpassfilterwithw
0=10
4
rad/sand
unityDCgain.Useaclockfrequencyof100kHz,C
1=C
2=10
pF.Find,C
3,C
4,C
5andC
6.

61. Problem 8
DesignaswitchcapacitorbasedTwo-Thomascircuitformainly
flatresponselowpassfilterwithw
0=10
4
rad/sandunityDCgain.
Useaclockfrequencyof100kHz,C
1=C
2=10pF.Find,C
3,C
4,C
5
andC
6.

62. ENDofFILTER