ELECTRONIC FILTERS Celo Joé Faria de Araújo, M.Sc.
A Ideal Electronic Filter allow ditortionle tranmiion of a certain band of frequencie and ure all the remaining frequencie of the ectrum of the inut ignal. The frequency ectrum i a rereentation of amlitude veru frequencie of thi ignal. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc.
CLASSIFICATION Electronic Filter Claification in term of Technology Ued Claification in term of function accomlihed Claification in term of Reone-Function Paive Filter Low-Pa Filter Butterworth Filter Active Filter High-Pa Filter Chebyhev Filter Digital Filter Band-Pa Filter Elitic or Cauer Filter Band-Reject Filter Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc.
FUNCTION ACCOMPLISHED K K H ( j ) Ideal Real K K H ( j ) Real Ideal c Low-Pa Filter (log cale) c High-Pa Filter (log cale) K K H ( j ) Ideal Real K K H ( j ) Real Ideal (log cale) c o c c o c Band-Pa Filter Band-Reject Filter (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 3
ACTIVES FILTERS Advantage: Sure ue of inductor; Eay Deign of Comlex Filter by cacade of imle tage; A coniderable amlification of inut ignal i oible; Deign Flexibility. Diadvantage: Power uly i neceary; The frequency reone i limited by active device (O-Am, Tranitor) frequency reone; It not often aly in medium and high ower ytem. Deite thee diadvantage it widely ued in everal alication, uch a: telecommunication and indutrial intrumentation. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4
STANDARD REQUIREMENTS FOR DESIGN OF APPROXIMATION FUNCTION LOW-PASS FILTER (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 5 Lo db Amin Tranition Paband Paband Amax Stoband
NON-FLAT STOPBAND REQUIREMENTS FOR LOW-PASS FILTER (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 6 A min A min Lo db Paband Amax
STANDARD REQUIREMENTS FOR DESIGN OF APPROXIMATION FUNCTION HIGH-PASS FILTER Amin (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 7 Lo db Stoband Tranition Paband Paband Amax
NON-FLAT STOPBAND REQUIREMENTS FOR HIGH-PASS FILTER Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 8 A min Lo db Paband A min A min Amax (log cale)
STANDARD REQUIREMENTS FOR DESIGN OF APPROXIMATION FUNCTION BAND-PASS FILTER Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. (log cale) 9 Lo db Amin o o 3 4 B Stoband Tranition Paband Tranition Paband Paband Stoband Amax 3 4 B i the aband width of BP filter o i the center (geometric mean) of the aband of BP filter
NONSYMMETRICAL REQUIREMENTS FOR BAND-PASS FILTER o o Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. (log cale) 0 A min Lo db 3 ' 4 B Paband A min Amax 3 4 B i the aband width of BP filter o i the center (geometric mean) of the aband of BP filter 4
STANDARD REQUIREMENTS FOR DESIGN OF APPROXIMATION FUNCTION BAND-REJECT FILTER o o 3 4 B Amin Tranition Paband Stoband Tranition Paband Paband Paband (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Lo db Amax 3 4 B i the aband width of BR filter o i the center (geometric mean) of the toband of BR filter
NONSYMMETRICAL REQUIREMENTS FOR BAND-REJECT FILTER o o 3 4 B (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Lo db A max A min A max 3 3 4 B i the aband width of BR filter o i the center (geometric mean) of the toband of BR filter '
A SECOND-ORDER GAIN FUNCTION FOR LOW-PASS FILTER GAIN V V O IN Q j Sloe = 40dB/decade Gain (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 3 Lo db 0dB
A SECOND-ORDER GAIN FUNCTION FOR HIGH-PASS FILTER GAIN V V O IN Q j Sloe = -40dB/decade Gain (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4 Lo db 0dB
A SECOND-ORDER GAIN FUNCTION FOR BAND-PASS FILTER GAIN V V O IN Q Q j Sloe = 40dB/decade Gain (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 5 Lo db Sloe = -40dB/decade 0dB
A SECOND-ORDER GAIN FUNCTION FOR BAND-REJECT FILTER GAIN V V O IN Q j Gain (log cale) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 6 Lo db 0dB
A SECOND-ORDER GAIN FUNCTION FOR LOW-PASS NOTCH FILTER V V GAIN O z ; z IN Q j jz Gain (log cale) -jz Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 7 Lo db z
A SECOND-ORDER GAIN FUNCTION FOR HIGH-PASS NOTCH FILTER V V GAIN O z ; z IN Q j jz Gain (log cale) -jz Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 8 Lo db z
GAIN EQUALIZERS Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 9
DELAY EQUALIZERS T o v t ) v ( t ) V ( ) V ( ) o ( in T o o in e H ( ) H V V o e in ( ( ) ) T o H ( j ) e j ( j ) H ( j ) T ( o T o ) d delay d ( ) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 0
Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. DELAY EQUALIZERS Second-order (all-a) IN O Q Q V V GAIN ) ( ) ( 0 ) ( Q arc tg Q arc tg j H db j H ) ( Q Q d d delay j Gain
EQUALIZATION OF CABLE DELAY Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc.
Filter ole-zero attern Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 3
Bode Plot Aroximation Technique 35dB 6 0 H ( ) ( 00)( 00 00 ) 00 400 Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4 Lo db 3dB 3 db 35 db 35 db 00 rad order H ( ) / 400 rad / oitava 3 K 00 00 00 Q K ( ) 00 00 H K 0 0 6 Frequency (rad/) 0 log 00 0 6 00 Q 00 00 3 Q j Lo (db)
BUTTERWORTH APPOXIMATION Requirement: A max A min 0, A min 0 log 0, max 0 A Order: n Correction Factor: 0 0, A máx log Root of a third-order normalized filter Root of the normalized filter: Gain j S j k n n k e onde k,,..., n /H(S) S+ S +,44S + 3 (S +S+)(S+) 4 (S + 0,76537S + ) (S +,84776S + ) 5 (S + 0,6803S + ) (S +,6803S + ) (S + ) n /3 - Denormalization: Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. S n The ole are jut the root at left half lane 5
Magnitude reone for Butterworth filter of variou order with =. Note that a the order increae, the reone aroache the ideal brickwall tye tranmiion. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 6
Grahical contruction for determining the ole of a Butterworth filter of order N. All the ole lie in the left half of the -lane on a circle of radiu 0 = (/) /N, where i the aband deviation arameter : A max /0 0 (a) the general cae, (b) N =, (c) N = 3, (d) N = 4. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 7
BUTTERWORTH APPOXIMATION Examle Requirement: = 00 rad/ A max = 0.5 db = 400 rad/ A min = db order n log 0 0 : log 0, 0, x x 0.5 400 00.73 Correction factor : 0 0, x 0.5 0.35 Normalized H ( S ) S.44 S Denormalization Denormalized H : S ( ) n 0.0059 877.4 39.6 877.4 Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 8
CHEBYSHEV APPOXIMATION Requirement: A max A min Order : n 0 coh 0 coh 0, 0, A A min máx where Root of the normalized filter: Root of a third-order normalized filter Gain j coh inh n The ole are jut the root at left half lane Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. S k k k k en co j k where n n k k k enh coh 0,,,..., n n enh enh n inh inh n 0 A máx 0 Denormalization : S 9,
Sketche of the tranmiion characteritic of a rereentative even- and odd-order Chebyhev filter. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 30
CHEBYSHEV APPOXIMATION (Order from Plot) Requirement: A max A min Lo of LP Chebyhev aroximation for A max = 0.5dB Lo of LP Chebyhev aroximation for A max = 0.50dB Lo of LP Chebyhev aroximation for A max = db Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 3
CHEBYSHEV APPOXIMATION (Polynominal from Table) Requirement: A max A min n Numerator of /H(S) A max = 0.5dB Denominator (k) S + 4.08 4.08 S +.79668S +.403.05405 3 (S + 0.767S +.33863) (S+0.767).070 4 (S + 0.4504S +.695) (S +.063S + 0.45485) 0.535 5 (S + 0.7005S +.09543) (S + 0.70700S + 0.5364) (S + 0.43695) 0.5676 n Numerator of /HS) A max = 0.50dB Denominator (k) S +.8678.8678 S +.456S +.560.4338 3 (S + 0.6646S +.445) (S+0.6646) 0.7570 4 (S + 0.3507S +.0635) (S + 0.84668S + 0.3564) 0.35785 5 (S + 0.393S +.03578) (S + 0.5865S + 0.47677) (S + 0.3633) 0.789 n Numerator of /H(S) A max = db Denominator (k) S +.9653.9653 S +.09773S +.05 0.986 3 (S + 0.4947S + 0.9940) (S+0.4947) 0.4930 4 (S + 0.7907S + 0.98650) (S + 0.67374S + 0.7940) 0.4565 5 (S + 0.789S + 0.9883) (S + 0.4684S + 0.4930) (S + 0.8949) 0.83 Denormalization : S Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 3
CHEBYSHEV APPOXIMATION Examle Requirement: = 00 rad/ A max = 0.5 db = 600 rad/ A min = 0 db 600 00 3 from grah n 3 From table H(S) normalized i obtained H ( S ) 0.7570 ( S 0.6646 S.445)( S 0.6646) Denormaliz ation : S 00 Denormalized H ( ) 575600 ( 5.3 45698)( 5.3) Banda Detail Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 33
ELLIPTIC (CAUER) APPOXIMATION Requirement: A max A min H() = Lo Function Denormalization : S Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 34
ELLIPTIC APPOXIMATION Examle Requirement: = 00 rad/ A max = 0.5 db = 600 rad/ A min = 0 db 600 3 from table n 00 From table H(S) normalized i obtained ( S 7.4858) H ( S ) 0.083974 ( S.3575 S.5553) Denormaliz ation : S 00 Denormalized H ( ) 0.083974 ( ( 6994) 7.4 6.8) Banda Detail Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 35
Phae and delay characteritic Characteritic of a fourth-order Chebyhev (A max = 0.5dB); (a) Lo, (b) Delay, (c) Ste inut, (d) Ste reone. Characteritic of a fourth-order Butterworth (A max = 3dB); (a) Lo, (b) Delay, (c) Ste reone. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 36
H(S) normalized i H ( S ) B B BESSEL APPROXIMATION n n (0) ( S ) n 3 4 5 Where B n (S) i the nth order olynomial which i defined by the following recurive equation B B 0 ( ( S S ) ) S B n ( S ) ( n ) B n ( S ) S B n ( S ) Beel Aroximation Function in Normalized and Factored Form Denominator Numerator S+ S +3S+3 3 (S +3.6778S+6.45944)(S+.39) 5 (S +5.794S+9.403) (S +4.0758S+.4878) 05 (S +6.7039S+4.75) (S +4.64934S+8.563) (S+3.64674) 945 and Lo of LP Beel Aroximation T o Denormalization S T o T o Delay (%) T T o 00% 50 n n e n ( n ) Delay of LP Beel Aroximation Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 37
BESSEL APPROXIMATION Examle Requirement: a) The delay mut be flat within ercent of the DC value u to KHz. b) The attenuation at 6KHz mut exceed 5dB. Solution. Try a fourth-order Filter. % 0 =.9 (from delay Beel aroximation lot) = (6/).9 = 5.7 Attenuation i only db = 5.7 (from lo Beel aroximation lot). Try a fifth-order Filter. % 0 =.5 (from delay Beel aroximation lot) = (6/).5 = 7.5 Attenuation i 9.5 db = 5.7 (from lo Beel aroximation lot) From table : H S ) (S 6.7039S Denormalization ( 4.75) (S 945 4.64934S 8.563) (S 3.64674).5 4 T o.989 0 ec S T K o H ( ) 3.370 3.608 0 Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4 8 0 3.608 0 8.338 4.5894 4 0 8 0 4.5894 0 8.8335.8335 0 4 0 4 Delay (%) T 00% khz 0,5% f T o 38
DELAY EQUALIZERS Function Aroximation GAIN V V O IN a a b b b GAIN V V O IN N a i a b b i i i i The number of delay ection N and their defining arameter (ai, bi) for aroximating a given delay hae are uually obtained by comuter otimization. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 39
DELAY EQUALIZERS Examle Delay Equalization of a fouth-order Chebyhev (A max =0.5dB, aband edge = rad/ec) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 40
FREQUENCY TRANSFORMATIONS HP, BP or BR requirement LP requirement HLP() H () HP HBP() HBR() Block diagram of the frequency tranformation rocedure Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4
Requirement A A HP min max LP Normalized A min A max H(S) LP Normalized i obtained from LP Requirement Normalized H(S) LP normalized can be tranformed to a high-a function by the frequency tranformation S H ( ) H ( S ) Examle: A A HP LP S HP min max 000 500 5 db 3 db rad rad / / LP A A HIGH-PASS FILTERS Normalized min 5 db max 3 db H HP ( ) H LP ( S ) S 000 H HP 500 rad/ -> -8dB 3 ( ) 3 6 ( 0 0 )( 0 3 K rad/ -> -3dB ) Butterworth H LP ( S ) ( S S )( S ) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 4
Requirement BP o A A min max 3 4 BAND-PASS FILTERS LP Normalized A min A max 4 3 H(S) BP Normalized i obtained from LP Requirement Normalized H(S) LP normalized can be tranformed to a band-a function by the frequency tranformation H Examle: o S BP ( ) H LP ( S ) o S A max = 0.5dB A min = 0 db Paband = 500 Hz to 000 Hz Stoband = DC to 75 Hz and 000 Hz to 3 4 A A min max BP 500 000 75 88 0 db 0.5 db rad rad rad / / rad / / 88 000 LP Normalized A min 0 db A max 0.5 db 75 500 3.08 H HP ( ) Ellitic H 0.084 4 H LP ( S ) 0.083947 S BP ( ) H LP ( S 4.6 0 3 4 3 ) S. 5.48 0 0 S 7.4858.3575 S.5553 4 500000 500 8 7 3.89 8.4 0 0 4 0 3.89 Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 0 4 43 '
Requirement BR o A A min max BAND-REJECT FILTERS LP Normalized A min A max 3 4 H(S) BR Normalized i obtained from LP Requirement Normalized H(S) LP normalized can be tranformed to a band-reject function by the frequency tranformation H S Examle: BP ( ) H LP ( S o ) S A max = db A min = 0 db Paband = below 000 Hz and above 6000 Hz Stoband = 500 Hz to 3000 Hz 3 4 A A BP min 0 max 000 6000 000 3000 db db rad rad rad rad / / / / LP A A o 4 6000 3000 3 Normalized min 0 db max db 000 000 5 H HP ( Chebyhev ) 0.89 H H LP ( S ) S BP ( ) H LP ( S 4 379.70 4 3 ) S 0.986.09773 S 4 473740.3 36893505 5000 6000000.05 5.60 7.409 Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 0 0 6 5.60 0 6 44 '
Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Firt-order filter. 45
Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Firt-order all-a filter. 46
Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Second-order filtering function. 47
Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. Second-order filtering function. (continued ) 48
Second-order filtering function. (continued ) Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 49
Realization of variou econd-order filter function uing the LCR reonator: (a) general tructure, (b) LP, (c) HP, (d) BP, (e) notch at 0, (f) general notch, (g) LPN ( n 0 ), (h) LPN a, (i) HPN ( n < 0 ). Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 50
REFERENCES SEDRA, Adel S. and SMITH, Kenneth C. Microelectronic Circuit. Oxford Univerity Pre. DARYANANI, Gobind. Princile of Active Network Synthei and Deign. John Wiley & Son. LATHI, B. P. Signal Proceing & Linear Sytem. Berkeley- Cambridge Pre. RUSTON, Henry and BORDOGNA, Joeh. Electric Network: function, filter, analyi. MacGraw-Hill. NOCETI FILHO, Sidnei. Filto Seletore de Sinai. Editora da UFSC. Electronic Filter Electrical Circuit - Celo Joé Faria de Araújo, M.Sc. 5