5 Filter Part 1 (16, 17, 18).pdf5 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

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).pdf5 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 & Circuit Design

2. Introduction of Filter
Filter is a broad area of electronics also an
independent subject
Oldest Technology: Filter with inductor &
capacitor called passive LC filter
LC filter works well in high frequency
Low frequency application : DC to 10 KHz
required inductors are bulky and provide
non ideal characteristics
So, filter design without inductor is an
interesting issue

3. Introduction of Filter
Passive LC Filter
Active RC Filter
Op-amp based RC filter
Switch Capacitor Filter

4. Filter Transmission
Filter transfer function
Passband:Passingsignalwhosefrequencyspectrum
lieswithinmagnitudeoftransmission
Stopband:Frequencybandoverwhichtransmissionis
zero
<Gain function>
<Attenuation function>
<Transfer function with phase>
<Transfer function >

5. Filter Types

6. Specifications for physical filter
circuit
Physicalcircuitcannotrealizetheidealized
characteristics
PhysicalCircuitcannotprovideconstanttransmission
atallthepassbandfrequencies
Alsophysicalcircuitcannotprovidezerotransmission
atallthestopbandfrequenciesi.e.sometransmission
overstopband
Transmissionofphysicalcircuitcannotchangeabruptly
attheedgeofthepassband.
Transmissionbandextendsfromthepassbandedgeω
P
tostopbandedgeω
S.
ω
S/ω
Pismeasuredtounderstandthesharpnessof
thelowpassfilterresponsecalledSelectivityFactor

7. Filter Specifications (Low pass)
Realisticspecificationforthetransmissioncharacteristicsof
lowpassfilter
Maximum
deviation in
passband
transmission
Stop band signal
must be attenuated
by at least A
min
Pass-band edge ω
P
Maximum allowed variation in pass-band transmission A
max
Stop-band edge ω
S
Maximum required stop-band attenuation A
min

8. Towards Ideal Filter (Low Pass )
Selectivityratioω
S/ω
Ptowardsunity
LowerA
max
HigherA
min
Toachievetheabovespecificationfiltercircuitshouldbe
higherorderandcomplexandexpensive
Filterdesignmustbecomplicatedifboththemagnitude
andphasespecified
Ripplepeakatpassbandaswellasstopbandmustbe
equalcalledequi-ripplecharacteristics

9. Filter Transfer function
The degree of the denominator, N, is the filter order
For the filter circuit to be stable, the degree of the numerator must
be less than or equal to that of the denominator M ≤ N
Numerator and denominator coefficients, a
0, a
1, . . . , a
Mand b
0, b
1, . . . ,
b
N−1, are real numbers.
Thenumeratorroots,z
1,z
2,...,z
M,arethetransferfunction
zeros,ortransmissionzeros
Denominatorroots,p
1,p
2,...,p
N,arethetransferfunctionpoles,
orthenaturalmodes
Eachtransmissionzeroorpolecanbeeitherarealoracomplex
number
Complexzerosandpoles,however,mustoccurinconjugatepairs.
(1)
(2)

10. Filter transfer function (LP)-‘Zeros’
Zerosareusuallyplacedonthejωaxisatstopbandfrequencies
Infiniteattenuation(zerotransmission)attwostopbandfrequencies:
ω
l1andω
l2.
Thefilterthenmusthavetransmissionzerosats=+jω
l1ands=
+jω
l2
Sincecomplexzerosoccurinconjugatepairs,theremustalsobe
transmissionzerosats=−jω
l1ands=−jω
l2.
Thus the numerator polynomial of this filter will have the factors
(s + j ω
l1)(s −j ω
l1)(s + j ω
l2)(s −j ω
l2)
Can be written as (s
2
+ ω
l1
2
)(s
2
+ ω
l2
2
) If, S=jω, then ω=ω
l1&
ω=ω
l2

11. Filter transfer function (LP)-‘Zeros’
Transmissiondecreasestoward-∞asωapproaches∞.
Thusthefiltermusthaveoneormoretransmissionzerosats
=∞.
Numberoftransmissionzerosats=∞isthedifference
betweenthedegreeofthenumeratorpolynomial,M,andthe
degreeofthedenominatorpolynomial,N,ofthetransfer
function
N−Mzerosats=∞

12. Filter transfer function (LP)-‘Poles’
Forafiltercircuittobestable,
allitspolesmustlieintheleft
halfofthesplane,andthusp
1,
p
2,...,p
Nmustallhavenegative
realparts.
Assumedthatfilterisoffifth
order(N=5).
Twopairsofcomplex-conjugate
polesandonereal-axispole,for
atotaloffivepoles.
Allthepoleslieinthepassband
thatgivesthefilteritshigh
transmissionatpassband
frequencies.
Thefivetransmissionzerosare
ats=∞,±jω
l1,±jω
l2,

13. Filter Transfer function (BP)

14. Filter Transfer function (BP)

15. Filter Transfer function (BP)
Bandpassfilter
Transmissionzerosareats=±jω
l1
andS=±jω
l2
oneormorezerosats=0andone
ormorezerosats=∞becausethe
transmissiondecreasestoward0as
ωapproaches0and∞
Assumingthatonlyonezeroexists
ateachofs=0ands=∞,thefilter
mustbeofsixthorder

16. Transfer function (All Pole Filter)
Low-passfilter
Nofinitevaluesofωatwhichthe
attenuationisinfinite(zero
transmission).
Thusitispossiblethatallthe
transmissionzerosofthisfilterare
ats=∞.
All-polefilter

17. Problem 1
Asecondorderfilterhasitspolesats=[-1/2±j(√3/2)].
Thetransmissioniszeroatw=2rad/sandisunityatDC
(w=0).Findthetransferfunction

18. Problem 2
Aforthorderfilterhaszerotransmissionatw=0,w=2rad/s
andw=∞.Thenaturalmodesare-0.1±j0.8and-0.1±j1.2find
T(s).

19. Filter Approximations
ButterworthApproximation:Maximallyflat
responseinpassband.
ChebyshevApproximation:Passbandripple
andsharpcut-off.
EllipticalApproximation:Passbandandstop
bandrippleandverysharpcut-off
BesselApproximation:Nosignaldistortionin
passband.

20. Filter Approximations

21. Butterworth Filter
Flat pass band.
This filter exhibits a monotonically decreasing transmission with
all the transmission zeros at ω= ∞
All-pole filter
Design specifications:
•A
max
•passband edge ω
p
•A
min
•stop band edge ω
S
ℇ: To determine maximum
deviation in pass band

22. Butterworth Filter
Fix the value of ℇ,
for A
max=3 dB, ℇ=1
A
max
Fix the order N for A(ω
s)≥A
min
ω
S
A
min

23. Butterworth Filter
ThedegreeofpassbandflatnessincreasesastheorderNis
increased
Nisincreasedthefilterresponseapproachestheidealbrick-
walltypeofresponse
DC Gain normalized at 1

24. Butterworth Filter: Graphical
Construction
The natural modes of an N
th
-order Butterworth filter can
be determined from the graphical construction
Natural modes lie on a circle of radius ω
0= ω
p(1/ε)
1/N
Spaced by equal angles of (П/N)
First mode at an angle (П/2N) from the +jω axis
P
1, P
2…P
Nare poles, K is setting any DC gain

25. Graphical construction for determining
the poles of Butterworth Filter

26. Graphical construction for determining
the poles of Butterworth Filter

27. Problem 3
FindtheButterworthtransferfunctionthatmeetsthe
followinglow-passfilterspecifications:f
p=10kHz,A
max=1
dB,f
s=15kHz,A
min=25dB,dcgain=1.

28. Problem 3
Solution:
A
max=1dB;ε=0.5088
If, N = 8, A( ω
s) = 22.3 dB
If, N = 9 , A( ω
s) = 25.8 dB.Select N = 9
The poles all have the same radius: ω
0
=ω
p(1/ε)
1/N
ω
0= 6.773 ×10
4
rad/s
p1 = ω
0(−cos80°+ j sin80°)
= ω
0(−0.1736 + j0.9848)

29. First-Order and Second-Order
Filter Functions
Simplestfiltertransferfunctions:firstandsecond
order.
Thesefunctionsareusefulinthedesignofsimple
filters.First-andsecond-orderfilterscanalsobe
cascadedtorealizeahigh-orderfilter.
Cascadedesignisoneofthemostpopular
methodsforthedesignofactivefilters(utilizing
opampsandRCcircuits).

30. First Order Filter
Thegeneralfirst-ordertransferfunctionisgivenby
First-orderfilterwithanaturalmodeats=−ω
0
Transmissionzeroats=-a
0/a
1
High-frequencygainthatapproachesa
1
Thenumeratorcoefficients,a
0anda
1,determinethetypeof
filter
Activerealizationsprovideconsiderablymoreversatilitythan
theirpassivecounterparts;inmanycasesthegaincanbesetto
adesiredvalue
Theoutputimpedanceoftheactivecircuitverylow,making
cascadingeasilypossible.
Theopamplimitsthehigh-frequencyoperationoftheactive
circuits.

31. Low pass (LP)
Bode Plot

32. Low pass (LP) with load

33. Low pass (LP) with load

34. High pass (HP)
Bode Plot

35. High pass (HP)

36. High pass (HP)

37. General

38. General

39. All Pass Filter
Animportantspecialcaseofthefirst-orderfilter
function
Transmissionzeroandthenaturalmodeare
symmetricallylocatedrelativetothejωaxis
Transmissionoftheall-passfilteris(ideally)constantat
allfrequencies
Phaseshowsfrequencyselectivity
All-passfiltersareusedasphaseshiftersandin
systemsthatrequirephaseshaping

40. All Pass Filter

41. All Pass Filter

42. All Pass Filter