Butterworth LC Filter Designer

Transcription

1 Butterworth LC Filter Desiger R S = g g 4 g - V S g g 3 g R L = Fig. : LC filter used for odd-order aalysis g R S = g 4 g V S g g 3 g - R L = useful fuctios ad idetities Uits Costats Table of Cotets I. Itroductio IV. Iputs IX. Copyright ad Trademark Notice Itroductio This routie calculates the required iductaces ad capacitaces for a lowpass LC butterworth filter. It is special i that it also sizes the compoets to deliver maximum power from a give source resistace to a differet load resistace. Reactive portios of the source ad load must be remove subtracted the filter etwork, so it is usually desirable to choose a odd order etwork for maximum power trasfer. Give the lowpass LC coefficiets, the etwork ca easily be trasformed ito a badpass, highpass, otch filter based o other topologies: gm/c, active RC, etc. A butterworth filter has the followig power trasfer fuctio, where the order of the filter is give by : f A log + = Iputs Atte := 8dB Ripple := db := khz := khz R S := kω R L := kω Fig. : Filter used for eve order aalysis Stop-Bad Atteuatio Maximum Passbad Ripple Passbad corer frequecy Stopbad corer frequecy Source Impedace Load Impedace

2 Order ad -3dB Corer Frequecy Estimatio Atte l = ceil Required Order of Butterworth Filter for 3dB l droop Fuctio : Holds Atte Costat ad chooses to meet Ripple := Atte Aatfp log + Aatfs log + + while ( Aatfs Atte) ( Aatfp > Ripple) := := Hz Aatfp := log + Aatfs := log + Atte Aatfp log + Aatfs log + Hz = 5 =.585 khz Aatfp =.43 db Aatfs = 8 db Number of Poles 3 db Corer Frequecy Passbad Ripple Stopbad Atteuatio

3 Fuctio : Holds Ripple Costat ad chooses to meet Atte := Ripple Aatfp log + Aatfs log + + Ripple Aatfp log + Aatfs log + while ( Aatfs Atte) ( Aatfp Ripple) Hz := = 5 Number of Poles := Hz.45 khz Aatfp := log + Aatfp db Aatfs := log + Aatfs 94.3 db = 3 db Corer Frequecy = Passbad Ripple = Stopbad Atteuatio 3

4 Poles ad Zeros k :=.. Idex Vector for poles ω c := π s := ω k c e j ( k+ ) π Poles s T = ( i i i i ) khz.5 Im( s) ω c.5.5 Re( s) ω c 4

5 s = re re im + + re im s + + m :=.. floor ω m := + Im( s m ) Re s m c Quadratic Sectios s ω = s + + Q ω s ω ω T π s + ζ + ω = ( ) khz Quadratic ceter frequec butterworth) Q := m ω m Re s m Q T = ( ) Quadratic Q ζ m := Re( s m ) ω m ω := ieil ceil >, s, rad ceil sec ζ T = ( ) ω T π = ( ) khz Quadratic Dampig Facto First Order Sectio 5

6 Trasfer Fuctio k α k := m = cos si ( m ) π m π α T = ( ) Trasfer Fuctio Coefficiets (α always equals ) M( s) gd( f) := + d f arg := d + α k phase( p) := p arg( p) k = retur α k p wrap for i s ω c k = if k last( p) <.. last( p) j f > 3 ( + wrap) < 3 wrap wrap π i p + wrap i i wrap wrap + π i p i i p p + wrap i i k Trasfer Fuctio Group Delay Fuctio Phase Fuctio M ( f) ( S ( s) ) K = M 3 ( s) := := + p f + ε f + ( ) s ω c α max = log + ε α max ε = Magitude Trasfer Fuctio for 3dB droop Power Trasfer Fuctio for αmax db droop Trasfer Fuctio (Eq..b) 6

7 Plottig um:= i :=.. um Number of Poits for plottig Frequecy Idex Vector tart := 5 top := f := i top tart ω i := π f i s := j ω i i ag := phase M( s).6 i um tart Group Delay Startig Frequecy for Plottig Edig Frequecy for Plottig Frequecy Vector Phase Respose.4 gd( f i ).. f i khz Phase Respose ag i deg 4 6. f i khz Magitude Respose ( ) ( ) log M 3 s i log M f i ( ) log M s i 5 5. f i khz 7

8 Images Elemet Values for Differet Load ad Source Impedaces[] This method assumes the iductor is the first elemet Create Circuit Z S = kω Z L = kω Elemet Values for Differet Load ad Source Impedaces[] This method assumes the capacitor is the first elemet Create Circuit Z S = kω Z L = kω Elemet Values for Equal Source ad Load Impedaces Create Circuit Z S = kω Z L = kω Butterworth Coefficiets := ieil >, ( ss + ) k = ss k π + cos ss +, k = ss ( k ) π + cos ss + ss + ss + 8

9 Fuctios := butter Atte, Ripple,,, R S Atte Aatfp log + Aatfs log + + Atte Aatfp log + Aatfs log + while ( Aatfs Atte) ( Aatfp > Ripple) for k.. g si k k π uml ieil umc ieil for il R S L g il π f il c for ic.. uml >, >, C g ic π R ic S L H C F.. umc +,, 9

10 Example x:= butter Atte, Ripple,,, R S L := x H L T = (.8.8 ) H C := x F C T = ( ) F Butterworth LC Filter Coefficiets := order of filter k = ss ( k ) π + cos ss + ss + ss + is eve

11 S Parameter Frequecy Respose S-parameter is a abbreviatio for scatterig parameters. S-parameters are a measure of the power gai of a etwork. The term scatterig comes from the cocept of a cue ball scatterig other balls as it trasfers power to them. S ij is the measure oower gai from port j to port i. Specifically, the square root oower gai. For example, S, the most useful S-parameter, is the measure of the ratio of output power to the available iput power. This is a importat setece. This setece is used to perform had calculatios. The output power is easy to explai, it is V Orms /R L. The available iput power is trickier to explai. Available iput power is the maximum power that ca be delivered from a source. For a source impedace of R S, ad a source voltage of V Srms ; the available iput power is (V Srms /) /R S. Why, because maximum power is delivered is to a resistace of R S. Thus a voltage divisio of two for the voltage, whe deliverig maximum available power. Thus the power gai of a etwork, S, is foud by dividig the two powers *V o /V S *sqrt(r S /R L ). Here S-parameters for the etwork are calculated by fidig the ABCD matrix for each elemet of the etwork, the multiplyig all the matrices to get a ABCD matrix for the system. Z := 5Ω ABCD ω ORIGIN = := ABCD prev for m.. ( legth( LC) ) Zx if LCtype = "C" j ω LC F Ω m m H Zx j ω LC if LCtype = "L" m Ω m Zx LC if LCtype = "R" m m Zx ABCDx ip = "s" m ABCDx ip = "p" m Zx ABCD ABCDx ABCD prev ABCD prev ABCD ABCD

12 ABCD to S parameter coversio with a commo impedace o all ports usig this expressio from "Microwave Egieerig" by Pozar. := A ABCD, ABCDS ABCD, Z B ABCD, C ABCD, D ABCD, A + B Ω C Z + + D Z Ω A + B Ω C Z D Z Ω ( A D B C) B Ω A + C Z + D Z Ω 8 A + B Ω C Z if + + D = Z Ω This scatterig parameter coversio routie (to covert to arbitrary source ad load impedaces) is give "Applied RF Techiques I" lecture otes, but is icorrect. A correct versio of the routie is foud o page 3 of "Microwave Amplifiers ad Oscillators," by Christia Getili. Z Sed Z Sbegi := Γ S S cov S, Z Sbegi, Z Sed, Z Lbegi, Z Led S 5 ω Z Sed + Z Sbegi Z Led Z Lbegi Γ L Z Led + Z Lbegi ( Γ L S, ) D Γ S S, Γ S A Γ S Γ L A Γ L ( Γ S ) ( Γ L ) Γ S Γ L S S,, A ( Γ L S, ) S Γ, S + Γ L S S,, A D A S A, ( Γ L ) D if A A Γ S S, A := ABCDS ABCD( ω), Z S-parameters of ideal matchig etwork with actual load ad source impedaces. Z L := R S Z S := R S S ω + B Ω + Z A S, A D S := S cov S 5 ( ω), Z, Z S, Z, Z L 5 Ohm S-parameters of ideal matchig etwork., D

13 Plots of lossy ad ideal S-paramaters of the matchig etwork verses frequecy. Be careful whe usig these plots, as they do ot reflect the chage i source ad load impedace vs. frequecy (for example if driver output impedace is capacitive). S vs. Frequecy Ripple S vs. Frequecy S vs. Frequecy Ripple Amplitude (db) 5 khz khz Atte khz 4. Frequecy (khz). frequecy (khz) 4 frequecy (khz) (, ) log S ω i S vs. Frequecy f i MHz (, ) log S ω i Refereces S vs. Frequecy f i []Microwave Electroic Circuit Techology, by Yoshihiro Koishi, Marcel Dekker Publishig, New York, 998, Filter Desig Equatios: pp 99- ]Passive ad Active Filters, Theory ad Implemetatios, by Wai-Kai Che, Joh Wiley & Sos, New York, 986, pp Copyright ad Trademark Notice All software ad other materials icluded i this documet are protected by copyright, ad are owed or cotrolled by Circuit Sage. The routies are protected by copyright as a collective work ad/or compilatio, pursuat to federal copyright laws, iteratioal covetios, ad other copyright laws. Ay reproductio, modificatio, publicatio, trasmissio, trasfer, sale, distributio, performace, display or exploitatio of ay of the routies, whether i whole or i part, without the express writte permissio of Circuit Sage is prohibited. MHz 3