Electronic_Filters_1565961125903_10698.ppt

Eectronic filters

1. ELECTRONIC FILTERS
Celso José Faria de Araújo, Dr.

2. 1
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
CONCEPTS
A “Ideal Electronic Filters” allow distortionless
transmission of a certain band of frequencies and suppress
all the remaining frequencies of the spectrum of the input
signal.
The frequency spectrum is a representation of amplitude
versus frequencies of this signal.

3. 2
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
CLASSIFICATION
Electronic Filters
Classification
in terms of
Technology Used
Classification
in terms of function
accomplished
Classification
in terms of
Response-Function
Passive
Filter
Active
Filter
Digital
Filter
Low-Pass
Filter
High-Pass
Filter
Band-Pass
Filter
Band-Reject
Filter
Butterworth
Filter
Chebyshev
Filter
Eliptic or
Cauer Filter

4. 3
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
FUNCTION ACCOMPLISHED

(log scale)
c
K

2
K

Ideal
Real
)(jH
Low-Pass Filter


(log scale)
c
K

2
K

Ideal
Real
)(jH
High-Pass Filter

(log scale)
o
K

2
K

Ideal
Real
)(jH
Band-Pass Filter

c2 c1

(log scale)
o
K

2
K

Ideal
Real
)(jH
Band-Reject Filter
c1 c2

5. 4
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
ACTIVES FILTERS
Advantage:
Suppressuseofinductors;
EasyDesignofComplexFiltersbycascadeofsimplestage;
Aconsiderableamplificationofinputsignalispossible;
DesignFlexibility.
Disadvantages:
Powersupplyisnecessary;
Thefrequencyresponseislimitedbyactivedevices(Op-Amps,
Transistors)frequencyresponse;
It’snotoftenapplyinmediumandhighpowersystem.
Despite these disadvantages it’s widely used in several application, such
as: telecommunication and industrial instrumentation.

6. 5
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
STANDARD REQUIREMENTS FOR
DESIGN OF APPROXIMATION
FUNCTION
LOW-PASS FILTER

(log scale)
p s
Loss dB

Amax
Amin
Passband
Stopband
Transition
Passband

7. 6
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
NON-FLAT STOPBAND
REQUIREMENTS FOR LOW -PASS
FILTER

(log scale)
p s1
Loss dB

Amax
A
min1

Passband A
min2

s2

8. 7
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Amin

(log scale)
s p
Loss dB

Amax
Passband
Stopband
Transition
Passband
STANDARD REQUIREMENTS FOR
DESIGN OF APPROXIMATION
FUNCTION
HIGH-PASS FILTER

9. 8
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A
min1


(log scale)
s1 p
Loss dB

Amax
Passband
s2
A
min1

A
min2

NON-FLAT STOPBAND
REQUIREMENTS FOR HIGH -PASS
FILTER

10. 9
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
STANDARD REQUIREMENTS FOR
DESIGN OF APPROXIMATION
FUNCTION
BAND-PASS FILTER

(log scale)
Loss dB

Passband Stopband
Transition
Passband
Transition
Passband
Stopband
3 1 2 4
Amax
Amin 21
2

o 12 B 43
2

o
•Bis the passband width of BP filter
•
ois the center (geometric mean) of the
passband of BP filter

11. 10
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
NONSYMMETRICAL REQUIREMENTS
FOR BAND-PASS FILTER

(log scale)
Loss dB

Passband
3 1 2 4
’
Amax
4

A
min2

A
min1
21
2

o 12 B '
43
2

o
•Bis the passband width of BP filter
•
ois the center (geometric mean) of the
passband of BP filter

12. 11
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
STANDARD REQUIREMENTS FOR
DESIGN OF APPROXIMATION
FUNCTION
BAND-REJECT FILTER

(log scale)
Loss dB

Transition
Passband
Transition
Passband
1 3 4 2
Amax
Amin
Stopband
Passband Passband 21
2

o 12 B 43
2

o
•Bis the passband width of BR filter
•
ois the center (geometric mean) of the
stopband of BR filter

13. 12
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
NONSYMMETRICAL REQUIREMENTS
FOR BAND-REJECT FILTER

(log scale)
Loss dB

1 3 4 2
A
min

3
’
A
max2

A
max1
21
2

o 12 B 4
'2
3

o
•Bis the passband width of BR filter
•
ois the center (geometric mean) of the
stopband of BR filter

14. 13
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR LOW-PASS FILTER22
2
p
p
p
p
IN
O
s
Q
s
V
V
GAIN






(log scale)
p
Loss dB

Slope = 40dB/decade
0dB

j
Gain

15. 14
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR HIGH-PASS FILTER22
2
p
p
pIN
O
s
Q
s
s
V
V
GAIN





(log scale)
p
Loss dB

Slope = -40dB/decade
0dB

j
Gain

16. 15
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR BAND-PASS FILTER22
p
p
p
p
p
IN
O
s
Q
s
s
Q
V
V
GAIN






(log scale)
p
Loss dB

Slope = -40dB/decade
0dB
Slope = 40dB/decade

j
Gain

17. 16
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR BAND-REJECT FILTER22
22
p
p
p
p
IN
O
s
Q
s
s
V
V
GAIN







(log scale)
p
Loss dB

0dB

j
Gain
p

18. 17
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR LOW-PASS NOTCH FILTERpz
p
p
p
z
IN
O
s
Q
s
s
V
V
GAIN 






 ;
22
22

j
Gain
jz
-jz
p

(log scale)
p
Loss dB

z

19. 18
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
A SECOND-ORDER GAIN FUNCTION
FOR HIGH-PASS NOTCH FILTERpz
p
p
p
z
IN
O
s
Q
s
s
V
V
GAIN 






 ;
22
22

j
Gain
jz
-jz
p

(log scale)
Loss dB

p z

20. 19
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
GAIN EQUALIZERS

21. 20
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
DELAY EQUALIZERS)()(
oino Ttvtv  osT
ino esVsV

 )()( o
sT
in
o
e
sV
sV
sH


)(
)(
)( oTj
ejH



)( )()(1)(  
o
TjHjH  )(


d
d
delay

22. 21
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
DELAY EQUALIZERS
Second-order (all-pass)22
22
p
p
p
p
p
p
IN
O
s
Q
s
s
Q
s
V
V
GAIN






 )()(0)(
2222







 

































p
p
p
p
p
p
Q
tgarc
Q
tgarcjHdBjH  
 
2
2
2
22
22
2)(

















p
p
p
p
p
p
Q
Qd
d
delay

j
Gain
p

23. 22
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
EQUALIZATION OF
CABLE DELAY

24. 23
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Filter pole-zero patterns

25. 24
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Bode Plot Approximation Technique
 100 400
Loss dB

3dB
35dB 3
2/400/100
35353



order
oitavasradsrad
dBdBdB  










22
100
100
100
)(
s
Q
ss
K
sH
p 6
20
101
100100
)( 

K
K
sH
s  
13
100
100
100
10
log20
100
22
6











p
js
p
Q
s
Q
ss )100100)(100(
10
)(
22
6


sss
sH
Frequency (rad/s)
Loss (dB)

26. 25
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
BUTTERWORTH APPOXIMATION
Requirements: 
pA
max
sA
min2
1,0
1,0
log
110
110
log
max
min



















p
s
A
A
n

 110
1,0

má xA
 nkondeeS
n
nk
j
k 2....,,2,1
12
2












 
Order: Correction Factor:
Roots of the normalized filters:
Roots of a third-order normalized filters

j Gain
-1 1
/3
The poles are just the roots at left half plane
n 1/H(S)
1 S+1
2 S
2
+1,414S+1
3 (S
2
+S+1)(S+1)
4 (S
2
+0,76537S+1)(S
2
+1,84776S+1)
5 (S
2
+0,61803S+1)(S
2
+1,61803S+1)(S+1)








p
n
sS


1
Denormalization:

27. 26
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Magnitude response for Butterworth filters of various order with = 1. Note that as the order increases, the response approaches the
ideal brickwall type transmission.

28. 27
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Graphical construction for determining the poles of a Butterworth filter of order N. All the poles lie in the left half of the s-plane on a
circle of radius 
0
= 
p
(1/)
1/N
, where is the passband deviation parameter :
(a)the general case, (b)N= 2, (c)N= 3, (d)N= 4. 110
10/
max

A


29. 28
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
BUTTERWORTH APPOXIMATION
Example
Requirements: 
p= 100 rad/s A
max= 0.5 dB 
s= 400 rad/s A
min= 12 dB273.1
100
400
log
110
110
log
:
2
5.01,0
121,0


















x
x
norder 35.0110:
5.01,0

x
factorCorrection  ssSationDenormaliz
p
n
0059.0:
1











 1414.1
1
)(
2


SS
SHNormalized 4.287276.239
4.28727
)(
2


ss
sHedDenormaliz

30. 29
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.

j
Gain
The poles are just the roots at left half plane





 

11
cosh
1
sinh
n






 

11
1
sinh
n
sinh
CHEBYSHEV APPOXIMATION
Requirements: 
pA
max
sA
min110
1,0

má xA

Roots of the normalized filters:
Roots of a third-order normalized filters
p
s
s
s
A
A
wherenOrder
má x

















1
1,0
1,0
1
cosh
110
110
cosh
:
min 















































1
senh
1
cosh
21
2
cos
1
senh
1
senh
21
2
sen
12,...,2,1,0
1
1
nn
k
nn
k
nkwherejS
k
k
kkk p
s
SationDenormaliz

:

31. 30
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Sketches of the transmission characteristics of a representative even-and odd-order Chebyshev filters.

32. 31
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
CHEBYSHEV APPOXIMATION
(Order from Plot)
Requirements: 
pA
max
sA
min
Loss of LP Chebyshev approximation for A
max= 0.25dB1
p
p
s
s
p 


 Loss of LP Chebyshev approximation for A
max= 0.50dB Loss of LP Chebyshev approximation for A
max= 1dB

33. 32
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
CHEBYSHEV APPOXIMATION
(Polynominal from Table)
Requirements: 
pA
max
sA
min
A
max= 0.25dB
A
max= 0.50dB
A
max= 1dB
n
Numerator of 1/HS) Denominator (k)
1 S+2.86278 2.86278
2 S
2
+1.42562S+1.51620 1.43138
3 (S
2
+0.62646S+1.14245)(S+0.62646) 0.71570
4 (S
2
+0.35071S+1.06352)(S
2
+0.84668S+0.356412) 0.35785
5 (S
2
+0.22393S+1.03578)(S
2
+0.58625S+0.47677)(S+0.362332) 0.17892
n
Numerator of 1/H(S) Denominator (k)
1 S+1.96523 1.96523
2 S
2
+1.09773S+1.10251 0.98261
3 (S
2
+0.49417S+0.99420)(S+0.49417) 0.49130
4 (S
2
+0.27907S+0.98650)(S
2
+0.67374S+0.27940) 0.24565
5 (S
2
+0.17892S+0.98831)(S
2
+0.46841S+0.42930)(S+0.28949) 0.12283
n Numerator of 1/H(S) Denominator (k)
1 S+4.10811 4.10811
2 S
2
+1.79668S+2.11403 2.05405
3 (S
2
+0.76722S+1.33863)(S+0.76722) 1.02702
4 (S
2
+0.42504S+1.16195)(S
2
+1.02613S+0.45485) 0.51352
5 (S
2
+0.27005S+1.09543)(S
2
+0.70700S+0.53642)(S+0.43695) 0.25676p
s
SationDenormaliz

:

34. 33
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
CHEBYSHEV APPOXIMATION
Example
Requirements: 
p= 200 rad/s A
max= 0.5 dB 
s= 600 rad/s A
min= 20 dB33
200
600
 ngraphfrom
s 200
:
s
SationDenormaliz  )62646.0)(14245.162646.0(
71570.0
)(
2


SSS
SH )3.125)(456983.125(
5725600
)(
2


sss
sH
edDenormaliz
From table H(S) normalized isobtained
Bandpass Details

35. 34
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
ELLIPTIC (CAUER) APPOXIMATION
Requirements: 
pA
max
sA
minp
s
s


 p
s
SationDenormaliz

:
H(s) = Loss Function

36. 35
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
ELLIPTIC APPOXIMATION
Example
Requirements: 
p= 200 rad/s A
max= 0.5 dB 
s= 600 rad/s A
min= 20 dB23
200
600
 ntablefrom
s 200
:
s
SationDenormaliz  )55532.135715.1(
)48528.17(
083974.0)(
2
2



SS
S
SH )8.622124.271(
)699411(
083974.0)(
2
2



ss
s
sH
edDenormaliz
From table H(S) normalized isobtained
Bandpass Details

37. 36
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Phase and delay characteristics
Characteristics of a fourth-order Chebyshev (A
max= 0.5dB);
(a) Loss, (b) Delay, (c) Step input, (d) Step response.
Characteristics of a fourth-order Butterworth (A
max= 3dB);
(a) Loss, (b) Delay, (c) Step response.

38. 37
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
BESSEL APPROXIMATION
H(S) normalized is)(
)0(
)(
SB
B
SH
n
n

Where B
n(S) is the nth order polynomial which
is defined by the following recursive equation1)(
1)(
1
0


SSB
SB )()()12()(
2
2
1 SBSSBnSB
nnn 
and
n Denominator Numerator
1S+1 1
2S
2
+3S+3 3
3(S
2
+3.67782S+6.45944)(S+2.32219) 15
4(S
2
+5.79242S+9.14013) (S
2
+4.20758S+11.4878) 105
5(S
2
+6.70391S+14.2725) (S
2
+4.64934S+18.15631) (S+3.64674)945
Bessel Approximation Function in Normalized and Factored Form
Loss of LP Bessel Approximations
Delay of LP Bessel ApproximationsoT p
p
s
s



Denormalizationp
p
oo
TsTS


 





























)12(
1
2
2
2
2
50%100(%)
n
n
n
p
p
o
p
p
e
nT
T
Delay





39. 38
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
1.Try a fourth-order Filter. 1% 0 … 
p= 1.9 (from delay
Bessel approximations plots)

s= (6/2)1.9 = 5.7
Attenuation is only 22 dB 
p= 5.7 (from loss Bessel
approximations plots)
BESSEL APPROXIMATION
Example
Denormalizationo
p
p
o
sTS
K
T 



sec10989.1
22
5.2
4

Requirements:
a)The delay must be flat within 1 percent of the DC value up to 2KHz.
b)The attenuation at 6KHz must exceed 25dB.
Solution
2.Try a fifth-order Filter. 1% 0 … 
p= 2.5 (from delay
Bessel approximations plots)

s= (6/2)2.5 = 7.5
Attenuation is 29.5 dB 
p= 5.7 (from loss Bessel
approximations plots)3.64674)(S 18.15631)4.64934S(S 14.2725)6.70391S(S
945
)(
22

SH
From table :4
4
842
8
842
8
108335.1
108335.1
105894.4s10338.2s
105894.4
10608.3s10370.3s
10608.3
)(







s
sH %52,0%100(%)
2



kHzfoT
T
Delay

40. 39
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
DELAY EQUALIZERS
Function Approximationbass
bass
V
V
GAIN
IN
O



2
2 b

 
 


N
i ii
ii
IN
O
bsas
bsas
V
V
GAIN
1
2
2
The number of delay sections N and their defining
parameters (ai , bi) for approximating a given
delay shape are usually obtained by computer
optimization.

41. 40
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
DELAY EQUALIZERS
Example
Delay Equalization of a fouth-order Chebyshev
(A
max=0.25dB, passband edge = 1 rad/sec)

42. 41
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
FREQUENCY TRANSFORMATIONS
Block diagram of the frequency transformation procedure
HP, BP or BR
requirements
HHP(s)
HBP(s)
HBR(s)
LP
requirements
HLP(s)

43. 42
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
HIGH-PASS FILTERS
Requirementss
p
ss
pp
AA
AA








1
maxmax
minmin
NormalizedLPHP
H(S)
LPNormalized is obtained from LP
Requirements Normalized
H(S)
LPnormalized can be transformed
to a high-pass function by the
frequency transformations
S
p

 s
S
LPHP
pSHsH 

 )()(
Example:2/500
1/1000
3
max
3
max
15
min
15
min
NormalizedLPHP




s
p
s
srad
s
p
srad
p
dBAdBA
dBAdBA



 )1)(1(
1
)(
2


SSS
SH
LP
Butterworth)10)(1010(
)(
3632
3


sss
s
sH
HP s
S
LPHP SHsH 1000)()(


1K rad/s -> -3dB
500 rad/s -> -18dB

44. 43
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
BAND-PASS FILTERS
Requirements




34
12
4321
1
maxmax
minmin
NormalizedLPBP






s
po
AA
AA
H(S)
BPNormalized is obtained from LP
Requirements Normalized
H(S)
LPnormalized can be transformed
to a band-pass function by the
frequency transformations
s
S
o


22

 s
s
S
LPBP oSHsH


22)()( 


Example:08.3
5001000
2751818
/18182
'
/2752
1
/10002
/5002
5.0
max
5.0
max
20
min
20
min
NormalizedLPBP
4
3
2
1











s
srad
srad
p
srad
srad
dBAdBA
dBAdBA



 55532.135715.1
48528.17
083947.0)(
2
2



SS
S
SH
LP
Elliptic141027334
14284
1089.31042.81048.51026.4
1089.31012.2
084.0)(



ssss
ss
sH
HP s
s
S
LPBP SHsH
5002
5000004
22)()(




A
max= 0.5dB A
min= 20 dB
Passband = 500 Hz to 1000 Hz
Stopbands = DC to 275 Hz and 2000 Hz to 

45. 44
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
BAND-REJECT FILTERS
Requirements34
12
4321
1
maxmax
minmin
NormalizedLPBR











s
po
AA
AA
H(S)
BRNormalized is obtained from LP
Requirements Normalized
H(S)
LPnormalized can be transformed
to a band-reject function by the
frequency transformation22
o
s
s
S



 22
)()(
os
s
SLPBP SHsH





Example:5
00023000
00010006
/30002
/20002
'
1
/60002
/10002
1
max
1
max
20
min
20
min
NormalizedLPBP
4
3
2
1











s
srad
srad
p
srad
srad
dBAdBA
dBAdBA



 10251.109773.1
98261.0
)(
2


SS
SH
LP
Chebyshev1612234
1624
105.610107.409136893510531279.720
105.6103473741011.
0.89)(



ssss
ss
sH
HP 60000004
50002
22
)()(





s
s
S
LPBP SHsH
A
max= 1dB A
min= 20 dB
Passbands = below 1000 Hz and above 6000 Hz
Stopband = 2500 Hz to 3000 Hz

46. 45
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
First-order filters.

47. 46
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
First-order all-pass filter.

48. 47
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Second-order filtering functions.

49. 48
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Second-order filtering functions. (continued 1)

50. 49
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Second-order filtering functions. (continued 2)

51. 50
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
Realization of various second-order filter functions using the LCR resonator: (a)general structure, (b)LP, (c)HP, (d)BP, (e)notch
at 
0
, (f)general notch, (g)LPN (
n

0
), (h)LPN as s , (i) HPN (
n
< 
0
).

52. 51
Electronic Filters
Electrical Circuits -Celso José Faria de Araújo, M.Sc.
REFERENCES
SEDRA,AdelS.andSMITH,KennethC.Microelectronic
Circuits.OxfordUniversityPress.
DARYANANI, Gobind.PrinciplesofActiveNetwork
SynthesisandDesign.JohnWiley&Sons.
LATHI,B.P.SignalProcessing&LinearSystems.Berkeley-
CambridgePress.
RUSTON,HenryandBORDOGNA, Joseph.Electric
Networks:functions,filters,analysis.MacGraw-Hill.
NOCETIFILHO,Sidnei.FiltosSeletoresdeSinais.Editora
daUFSC.