EE 508 Lecture 6 Filter Cocepts/Termiology Approximatio Problem
Root characterizatio i s-plae (for complex-cojugate roots) s ω Q 2 0 2 + s + ω0 o - relatioship betwee agle θ ad Q of root For low Q, θ is large For high Q, θ is small
Root characterizatio i s-plae (for complex-cojugate roots) Im s-plae θ s ω Q 2 0 2 + s + ω0 ω o Re for θ=90 o, Q=/ 2 roots located at s 2 2 ω0 ω0 2 = ± 4ω0 ω 0 4 2Q 2 Q = ± 2Q 2 Q ( Q ) 2 θ = ta 4
Filter Cocepts ad Termiology Frequecy scalig Frequecy Normalizatio Impedace scalig Trasformatios LP to BP LP to HP LP to BR
Frequecy Scalig ad ormalizatio Frequecy ormalizatio: s s = ω 0 Frequecy scalig: s = ω s 0 Purpose: ω 0 idepedet approximatios ω 0 idepedet sythesis Simplifies aalytical expressios for T(s) Simplifies compoet values i sythesis Use sigle table of ormalized filter circuits for give ormalized approximatig fuctio Note: The ormalizatio subscript is ofte dropped
Frequecy ormalizatio/scalig example ( ) T s 6000 = s + 6000 Defie ω 0 =6000 s s = ω 0 T( jω) ( ) 0 T s = ω s + ω 0 0 Normalized trasfer fuctio: T ( s ) = s + T ( jω)
Frequecy ormalizatio/scalig example T ( s ) = s + T ( jω) Sythesis of ormalized fuctio Vo V IN T( s) = s +
Frequecy ormalizatio/scalig example T ( s ) = s + Frequecy scalig by ω 0 (of trasfer fuctio) s = ω s 0 T ( jω) T ( s ) = s + ω0 ω T 0 ( s ) = s + ω 0 Frequecy scalig by ω 0 (of eergy storage elemets i circuit) C = C /ω0 ω T 0 ( s ) = s + ω 0
Frequecy ormalizatio/scalig example T ( s ) = s + T ( jω) ( ) 0 T s T( jω) ω = s + ω 0 ω 0 ω Frequecy scalig / ormalizatio does ot chage the shape of the trasfer fuctio, it oly scales the frequecy axis liearly The frequecy scaled circuit ca be obtaied from the ormalized circuit simply by scalig the frequecy depedet impedaces (up or dow) by the scalig factor This makes the use of filter desig tables for the desig of lowpass filters practical whereby the circuits i the table all have a ormalized bad edge of rad/sec.
Frequecy ormalizatio/scalig Example: Table for passive ladder Butterworth filter with 3dB bad edge of rad/sec T ( s ) = s + T ( jω)
Frequecy ormalizatio/scalig The frequecy scaled circuit ca be obtaied from the ormalized circuit simply by scalig the frequecy depedet impedaces (up or dow) by the scalig factor Compoet deormalizatio by factor of ω 0 Normalized Compoet Deormalized Compoet o o Other Compoets Uchaged Compoet values of eergy storage elemets are scaled dow by a factor of ω 0
Desgi Strategy Theorem: A circuit with trasfer fuctio T(s) ca be obtaied from a circuit with ormalized trasfer fuctio T (s ) by deormalizig all frequecy depedet compoets.
Example: Desig a V-V passive 3 rd -order Lowpass Butterworth filter with a bad-edge of K Rad/Sec ad equal source ad load termiatios. R S L L 3 V OUT V IN C 2 R L Filter architecture L =H L 3 =H V OUT V IN C 2 =2F Normalized filter T( s ) = s+ L =59uH L 3 =59uH V OUT V IN C 2 =38uF Deormalized filter 000 T( s ) = s+000
Example: Desig a V-V passive 3 rd -order Lowpass Butterworth filter with a bad-edge of K Rad/Sec ad equal source ad load termiatios. L =59uH L 3 =59uH V OUT V IN C 2 =38uF ( ) 000 T s = s+000 Is this solutio practical? Some compoet values are too big ad some are too small!
Filter Cocepts ad Termiology Frequecy scalig Frequecy Normalizatio Impedace scalig Trasformatios LP to BP LP to HP LP to BR
Impedace Scalig Impedace scalig of a circuit is achieved by multiplyig ALL impedaces i the circuit by a costat for trasresistace gai for dimesioless gai for trascoductace gai
Impedace Scalig Theorem: If all impedaces i a circuit are scaled by a costat θ, the a) All dimesioless trasfer fuctios are uchaged b) All trasresistace trasfer fuctios are scaled by θ c) All trascoductace trasfer fuctios are scaled by θ -
Impedace Scalig Example: ( ) T s = s+ T(s) is dimesioless Impedaces scaled by θ=0 5 ( ) T s = s+ Note secod circuit much more practical tha the first
Example: Desig a V-V passive 3 rd -order Lowpass Butterworth filter with a bad-edge of K Rad/Sec ad equal source ad load termiatios. L =59uH L 3 =59uH V OUT V IN C 2 =38uF 000 T( s ) = s+000 Is this solutio practical? Some compoet values are too big ad some are too small! Impedace scale by θ=000 K L =59mH L 3 =59mH V IN C 2 =38F K V OUT 000 T( s ) = s+000 Compoet values more practical
Typical approach to lowpass filter desig. Obtai ormalized approximatig fuctio 2. Sythesize circuit to realize ormalized approximatig fuctio 3. Deormalize circuit obtaied i step 2 4. Impedace scale to obtai acceptable compoet values
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz Note: We have ot discussed the Butterworth approximatio yet so some details here will be based upo cocepts that will be developed later T BW = 5 2 s+ 2s+
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz ω = 0 RR CC 2 2 RRCC ( ) 2 0 2 T s = RC RRCC 2 s+s + Q= Q 2 2 R RR Q 2 C C 2 8 desig variables ad oly two costraits (igorig the gai right ow)
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz If C =C 2 =C ad R =R 2 =R 0 =R, this reduces to ( ) T s = ( RC) 2 R s+s + RQ RC RC 2 ( ) 2 QR R V IN R C R C R 3 R 3 V OLP INT INT 2 A Popular Secod-Order Lowpass Filter
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz QR R V IN R C R C R 3 R 3 V OLP INT INT 2 A Popular Secod-Order Lowpass Filter ( ) T s = ( RC) 2 R s+s + RQ RC RC 2 ( ) 2 ω 0= RC RQ Q= R Normalizig by the factor ω 0, we obtai ( ) T s = Q 2 s+s + Settig R=C= obtai the followig ormalized circuit
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz Q V IN V OLP INT INT 2 A Popular Secod-Order Lowpass Filter ( ) T s = Must ow set Q = Q 2 s+s + 2 ω 0= Now we ca do frequecy scalig
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz Deormalized circuit with badedge of 4 KHz This has the right trasfer fuctio (but uity gai) Ca ow do impedace scalig to get more practical compoet values A good impedace scalig factor may be θ=000
Example: Desig a 2 d order lowpass Butterworth filter with 3dB passbad atteuatio, a dc gai of 5, ad a 3dB badedge of 4KHz Deormalized circuit with badedge of 4 KHz This has the right trasfer fuctio (but uity gai) To fiish the desig, preceed or follow this circuit with a amplifier with a gai of 5 to meet the dc gai requiremets