ELECTRICAL CIRCUITS 10. PART II BAND PASS BUTTERWORTH AND CHEBYSHEV

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1 45 ELECTRICAL CIRCUITS 0. PART II BAND PASS BUTTERWRTH AND CHEBYSHEV Introduction Bnd p ctive filter re different enough from the low p nd high p ctive filter tht the ubject will be treted eprte prt. Thi ection will exmine the bic bnd p function, focu on one of the mny ctive implementtion nd then explin how to implement it Butterworth or Chebyhev ny order. Bic Bnd P Form A imple pive circuit tht yield the bic bnd p chrcteritic i given by Figure 0. Figure 0 Pive Bnd P implementtion The ytem function for the pive bnd p circuit of Figure 0 i given by Eqution 45. H ( ) L R L LC R (45) The generl form of the imple bnd p function i given by Eqution 46: H ( ) (46) The frequency chrcteritic of Eqution 46 re illutrted in Figure. The numertor nd denomintor re illutrted eprtely the dhed line nd then dded (mgnitude db dd) to obtin the totl reult. berve tht the reult i nrrow pek reching up to 0 db on top of houlder tht re in eence only t order. Firt order becue the run up to the pek i +0db/dec nd the rundown from the pek i

2 46 0db/dec. The electivity of the pek i et by the contribution from the denomintor prt of the plot t the criticl frequency. berve tht even though tht me contnt i in the numertor term, it impct i more ubtle in tht it cue, in the totl repone, the pek to jut rech up to 0db. Eqution 46 i given in n lternte form Eqution 47. It i the form of Eqution 47 we will work with for thi development. Figure Bode Sketch of Eqution 46 H ( ), (47) i the preferred wy to decribe the bnd p function. Additionlly, offer n importnt performnce meure of the function een in Eqution 48. Where: i the center frequency in rd. i the difference in frequency cro the function t 3db. (48) i the difference frequency cro the houlder of the function t the hlf power point or 3db. Figure illutrte thi on ketch of bnd p function.

3 47 Figure nd for ketch of bnd p function The circuit tht relize ctively Eqution 47 i given by Figure 3. The ytem function for thi circuit i given by Eqution 49. Figure 3 ctive circuit to implement Eqution 47. H ( ) R C RC K RC (4 K) (49) R f Where: K Rin (50) 4 K (5)

4 48 K 4 ( i pecified, we olve for K ) (5) RC (5) R C (Here we pick C nd compute R ) (5) K Gin 4 K (Thi i the expreion for exce gin) (53) Eqution 50 give the gin K for the op mp, Eqution 5 give the for Eqution 49, Eqution 5 give the (the filter center frequency) for Eqution 49 nd Eqution 53 give the gin of Eqution 49 t the center frequency. A een by Eqution 5 the op mp gin K et the circuit which i independent of R or C but, by product of tht pect i exce center frequency gin. In the originl bnd p function, Eqution 49, the center frequency gin i 0db. Here it i not. Thi exceive gin mut be dipoed of it cn become quite lrge nd the op mp will not hve enough dynmic rnge to hndle it. The clenet wy to dipoe of thi gin i to turn the firt input reitor into voltge divider w done in the n even Chebychev low nd high p implementtion. It will be hown tht the n even Chebyhev ttenution of the db requirement will lo pply the bnd-p implementtion. For ingle tge or Butterworth implementtion, the voltge divider circuit of Figure 4 h n output impednce of R nd neutrlize the exce gin of Eqution 53. Ripple Figure 4 reitor ttenution circuit for circuit 3 The reltionhip for R, Rb re given by Eqution 54,b: R K 4 K R (54) R b K R 4 K K 4 K (54b)

5 49 Figure 5 illutrte thi voltge divider incorported into the circuit of Figure 3. Figure 5 Gin elimintion incorported into ctive bnd-p circuit In the deign of ctive bnd-p filter, the order n, the type (Butterworth, Chebyhev) the center frequency, nd the, would ll hve to be pecified or elected. To develop thi deign proce we conider 4 ce. The firt ce re Butterworth nd Chebyhev both with n = nd then Butterworth nd Chebyhev of n higher order. Ce Butterworth Bnd-P of rder n = We ume tht we re given the center frequency, nd the. The Butterworth polynomil for n = from Tble i given by Eqution 55: D (55) Thi polynomil i converted to bnd-p vi the following proce: o o (56)

6 50 H ( ) o o (57) Notice tht the order of Eqution 57 i double tht of Eqution 55. In generl the order of the bnd-p function will be double tht of the generting polynomil. Eqution 57 i relized by Eqution 49 nd the circuit of Figure 5. Which lo get rid of the exce gin. Eqution 5 obtin the required gin K nd Eqution 50 i ued to deign the op mp R nd R to et tht gin. Eqution 5 ue the pecified to deign R nd C. The in f R nd Rb tht dipoe of the exce gin re deigned with Eqution 54 & 54b. We re redy for n exmple. Exmple 5 Butterworth t order bnd-p, =5, =000rd Uing Eqution 5 we clculte K = Uing Eqution 50 R f =9.434k nd R in =0k. Uing Eqution 5 nd picking C =0.uf, R =4.4k. Uing Eqution 54, Gin Exce = Now uing Eqution 54 & 54b R =985.86k, R b=4.348k. Figure 6 how the finl chemtic nd Figure 7 how the PSPICE model. In the PSPICE model the gin K w implemented imple voltge controlled ource with gin K. The repone of the PSPICE model i given in Figure 8.

7 5 Figure 6 Butterworth t order bnd-p Exmple 5 Figure 7 PSPICE model of Exmple 5 If one etimte the 3db point from the houlder of the repone in Figure 8 we get 56Hz nd 6.5Hz, where the center frequency in Hz i bout 59Hz. Uing the reltion of Eqution 48to etimte : It hould be 5, but not bd for eyeblling plot.

8 5 Figure 8 Frequency repone for PSPICE model of Exmple 5 Ce Chebyhev Bnd-P rder n = We ume tht we re given the center frequency, nd the. The Chebyhev polynomil for n = from Tble i given by Eqution 55: D.004 (58) We will ue.004 for the derivtion: Thi polynomil i converted to bnd-p vi the following proce: o o (59) Since it Chebyhev nd the order i odd, the gin in the p-bnd i et to one. Thu Eqution 60 give the finl reult.

9 53 H ( ) o o (60) In exmining Eqution 57 oberve tht the Chebyhev implementtion h reulted in the circuit being given by : Cheb Spec, n (6) The circuit i the me the pecified. We re redy to ue the deign Eqution to obtin circuit prmeter for Figure 3. Plugging in the modified from Eqution 6 into Eqution 5: K 4 ( i pecified, we olve for K ) (6) The other Eqution re repeted here unchnged: R f K (pick R f nd Rin Rin to obtin K ) (50) R (Here we pick C nd compute R ) (5) C R K 4 K R (54) R b K R 4 K K 4 K (54b) We re now redy for deign exmple. Exmple 6

10 54 Chebyhev t order bnd-p, =5, =000rd, D. 004 With. 004 nd uing Eqution 6 we clculte K = Uing Eqution 50 R =9.433k nd R =0k. Uing Eqution 5 nd picking C =0.uf, R =4.4k. Uing f in Eqution 54, Gin Exce = Now uing Eqution 54 & 54b R = k, R b =4.3484k. Figure 9 how the finl chemtic nd Figure 30 how the PSPICE model. In the PSPICE model the gin K w implemented imple voltge controlled ource with gin K. The repone of the PSPICE model i given in Figure 3. Figure 30 Chebyhev t order bnd-p Exmple 6 Figure 3 PSPICE model for Exmple 6

11 55 Figure 3 Frequency repone for Exmple 6 With. 004 for the Chebyhev nd for Butterworth, there i not lot of difference between thee for t order. We now exmine higher order nd the reult re more intereting. Higher rder Butterworth Bnd-P For n greter thn t order, the polynomil if n i even, will be multiple nd order nd if n i odd will be t order long with multiple nd order. Any t order fctor will reult in reliztion exctly like Ce. The nd order fctor will yield eprte t order implementtion t hifted frequencie nd ltered. The following derivtion will develop the pecific. Conider the generl form of the Butterworth polynomil: b (63) We ue the me pproch the t order to obtin the bnd-p, given nd :

12 56 b b b b b b ) ( b b T (64) Eqution 64 i 4 th order nd mut be relized by nd order implementtion. Thu Eqution 64 mut be fctored vi MATLAB (or imilr) rooter function. The yntx for MATLAB with TBD for the ctul number i follow: % Fctoring the econd order bnd-p wo=tbd ; q=tbd ; b=tbd ; A=*b*wo/q; B=(+q^-)*wo^; C=*b/q*wo^3; D=wo^4; r=root([,a,b,c,d]) The root will be complex conjugte they will hve to be reembled econd order product nd ech implemented per Figure 5. The generl form of the root obtined from the denomintor of Eqution 64 i given by Eqution 65: ) )( )( )( ( jb jb jb jb (65) The reembled pir re given by Eqution 66 : b nd b (66)

13 57 We tke one of thee nd cll it typicl, but thi proce mut be done for both prt of Eqution 66: b (67) We cn obtin n nd for ech of thee. THEY WILL NT BE THE RGINAL AS SPECIFIED, but will be unique to the individul qudrtic piece of Eqution 66. Tht nd re follow: b The reultnt bnd-p for one of thee qudrtic piece i the me Eqution 47, repeted here: H ( ) (68) (47) Agin, the nd for ech of thee i given by Eqution 68. The deign Eqution of 49 through 5 will be ued to initilly deign Figure 3. There will till be exce gin to dipoe of but it will NT be pecified by Eqution 53. The overll gin tht mut be normlized i obtined by tking the mgnitude of the product of Eqution 49 for ech nd every qudrtic t the pecified overll filter center frequency, NT THE INDIVIDUAL FR EACH F THESE CIRCUITS! Eqution 69 illutrte thi point: Gin SPEC n i R i C i RiCi K i RiCi (4 K i ) j SPEC (69) Uing the gin obtined in Eqution 69, tht gin will be plit uniformly mongt ll the tge nd dipoed of in ditributed mnner. Eqution 70 give the gin A, to be dipoed of in ech tge. A Gin n SPEC the gin neutrliztion of Eqution 54 nd 54b become: (70) R AR (7)

14 58 R b AR A (7b) Ech tge will be relized Figure 5 with ll the tge ccded together to provide the over ll repone of the bnd-p. Conider the following deign exmple. Exmple 7 Butterworth bnd-p, =000, =5, n =3 The n =3 polynomil i given by Eqution 63: D3 ( )( (.5) ) (7) The firt order piece trnform directly into Eqution 47, repeted below. H ( ) (47) Thi i exctly like Exmple 5 except tht the R nd Eqution 69 through 7 Thu from Exmple 5: K = R =9.434k f R in =0k C =0.uf R =4.4k Rb will be determined lter vi We ue the ubcript to denote the firt tge. Additionlly, we pick the cpcitnce to be the me for ll tge, C =0.uf. Thi firt tge, non-normlized, reult in the firt of 3 ccded bnd-p function like Eqution 49, repeted here Eqution 73 with the ubcript for the prmeter: H ( ) R C RC K RC (4 K ) (73)

15 59 The qudrtic piece of Eqution 7: ( (.5) ), i proceed uing the methodology of Eqution 63 through 65. The MATLAB code (with the TBD vlue plugged in) needed to obtin the root to emble Eqution 65 i follow: % Fctoring the econd order bnd-p wo=000 ; q=5 ; b=.5 ; A=*b*wo/q; B=(+q^-)*wo^; C=*b/q*wo^3; D=wo^4; r=root([,a,b,c,d]); The MATLAB reult from thi code re: r =.0e+003 * i i i i Thee root plugged into Eqution 65 yield qudrtic form of Eqution 66. Ech of thee will reult in non-normlized gin bnd-p ctive ection of Figure 3 nd Eqution 49. There i gin to be neutrlized but it will be tken cre of t the lt tep. Eqution 68 give the nd tht i pecific to ech pir of root. The MATLAB code to do thi i follow: % wo nd q for root pir =b(rel(r())); = b(rel(r(3))); wo=b(r()); wo=b(r(3)); q=wo/(*); q=wo/(*); The reult re follow: wo =.075e+003 wo = q = q =

16 60 Thee vlue re plugged into Eqution 5 nd 5 to obtin the prmeter for n Eqution 49 tht implement ech of thee qudrtic. The MATLAB yntx for the prmeter of Eqution 49 i follow: % eqn 49 prmeter % picked cp=.uf k=4-^.5/q; kb=4-^.5/q; c=.e-6; R=^.5/(wo*c); Rb=^.5/(wo*c); The reult re follow: k = kb = R =.3899e+004 Rb =.4389e+004 Thee will become more non-normlized form of Eqution 49: H ( ) R C RC K RC (4 K ) (74) H b ( ) R b C RbC K b RbC (4 K b ) (74b) The lt tep i to normlize the overll gin t the center frequency. Uing Eqution 69, the exce gin i given by Eqution 75: Gin Exce H ) ( ) H ( ) H b ( (75) jspec The MATLAB code (include ll the previou code) to do thi i follow: % Totl Exce gin compute everything then compute gin W=000; % Spec center freq

17 6 =5; % Spec filter % Firt tge from + prmeter k=4-^.5/; c=.e-6; % picked cp ll me R=^.5/(W*c); % firt tge reitnce % Now the qudrtic tge b=.5; % Butterworth qudrtic prmeter % Fctor the denomintor of Eqn 64 A=*b*W/; B=(+^-)*W^; C=*b/*W^3; D=W^4; r=root([,a,b,c,d]); % wo nd q for root pir =b(rel(r())); = b(rel(r(3))); wo=b(r()); wo=b(r(3)); q=wo/(*); q=wo/(*); % eqn 49 prmeter % picked cp=.uf k=4-^.5/q; kb=4-^.5/q; c=.e-6; R=^.5/(wo*c); Rb=^.5/(wo*c); % Now the gin R=[R,R,Rb]; K=[k,k,kb]; =j*w; n=3; f=; for i=:n y=*k(i)*r(i)*c*.5/(^*r(i)^*c^*.5+*r(i)*c*.5*(4-k(i))+); f=f*y; end g=b(f); gg=g^(/n);

18 6 The reult re follow: % Ditributed exce gin gg = % Filter ech tge reitnce R =.0e+004 *[ ] % Filter ech tge gin K = [ ] The voltge divider necery for ech tge uing Eqution 70 nd 7 i computed follow: % Ech tge ttenution reitor for i=:n r(i)=gg*r(i); rb(i)=gg*r(i)/(gg-); end Reult: % r =.0e+006 *[ ] % rb =.0e+004 *[ ] The overll circuit i 3 ccded tge of Figure 5. The vriou tge op mp gin reitor, R & R, re determined with Eqution 50. The PSPICE chemtic for the f in nlyi performed i given by Figure 33. Figure 33 PSPICE chemtic of Exmple 7

63 Figure 34 give plot of the PSPICE frequency repone for Exmple 7. Upon exmining the plot cro the 3db point the difference in frequency i bout 6.4Hz.

Figure 34 PSPICE frequency repone for Exmple 7 Higher rder Chebyhev Bnd-P A with the Butterworth, for the Chebyhev with n greter thn t order, the polynomil, if n i

Any t order fctor will reult in reliztion exctly like the form of Ce except tht the polynomil t order contnt, will be different for different odd n.

19 63 Figure 34 give plot of the PSPICE frequency repone for Exmple 7. Upon exmining the plot cro the 3db point the difference in frequency i bout 6.4Hz. The ocited with thi i 4.8. The idel w 5. Thi i impreive for n eyebll etimte. We now conider Chebyhev. Figure 34 PSPICE frequency repone for Exmple 7 Higher rder Chebyhev Bnd-P A with the Butterworth, for the Chebyhev with n greter thn t order, the polynomil, if n i even, will be multiple nd order nd if n i odd will be t order long with multiple nd order. Any t order fctor will reult in reliztion exctly like the form of Ce except tht the polynomil t order contnt, will be different for different odd n. The nd order fctor will yield eprte t order implementtion t hifted frequencie nd ltered. The following derivtion i lmot identicl to the Butterworth but with few ubtle difference nd it will preented in totl. Conider the generl form of the Chebyhev polynomil: ( b) c b b c (76) We ue the me pproch the t order to obtin the bnd-p, given nd.

20 64 To implify the lgebr let: c b p (76) p b p b c b b ) ( b p b T (77) Eqution 77 i 4 th order nd mut be relized by two nd order implementtion. Thu Eqution 77 mut be fctored vi MATLAB (or imilr) rooter function. The yntx for MATLAB with TBD for the ctul number i follow: % Fctoring the econd order bnd-p wo=tbd ; % pec center freq q=tbd ; % pec circuit b=tbd ; % qudrtic coef b cc=tbd ; % qudrtic coef c p=b^+cc^; % eqution 76 ubtitution A=*b*wo/q; B=(+p*q^-)*wo^; C=*b/q*wo^3; D=wo^4; r=root([,a,b,c,d]) The root will be complex conjugte they will hve to be reembled econd order product nd ech implemented per Figure 5. The generl form of the root obtined from the denomintor of Eqution 77 i given by Eqution 78: ) )( )( )( ( jb jb jb jb (78) The reembled pir re given by Eqution 79 : b nd b (79) We tke one of thee nd cll it typicl, but thi proce mut be done for both prt of Eqution 78. Note tht nd b here re the mgnitude of the rel nd imginry prt of the root nd not the originl Chebyhev polynomil contnt.

21 65 b (80) We cn obtin n nd for ech of thee. THEY WILL NT BE THE RGINAL AS SPECIFIED, but will be unique to the individul qudrtic piece of Eqution 79. Tht nd re follow: b The reultnt bnd-p for one of thee qudrtic piece i the me Eqution 47, repeted here: H ( ) (8) (47) Agin, the nd for ech of thee i given by Eqution 8. The deign Eqution of 49 through 5 will be ued to initilly deign Figure 3. There will till be exce gin to dipoe of but it will NT be pecified by Eqution 53. The overll gin tht mut be normlized i obtined by tking the mgnitude of the product of Eqution 49 for ech nd every qudrtic t the pecified overll filter center frequency, NT THE INDIVIDUAL FR EACH F THESE CIRCUITS! Eqution 69 illutrte thi point: Gin SPEC n i R i C i RiCi K i RiCi (4 K i ) j SPEC (8) Uing the gin obtined in Eqution 69, tht gin will be plit uniformly mongt ll the tge nd dipoed of in ditributed mnner. Additionlly, if n, the order of the originl Chebyhev polynomil were even then n dditionl gin fctor of the db RIPPLE mut be included. Eqution 83 give the gin A, to be dipoed of in ech tge. A n dd Gin SPEC n, A n Even Gin SPEC 0 dbripple 0 Uing the pproprite A from Eqution 83 the gin neutrliztion of Eqution 54 nd 54b become: n (83) R AR (84)

22 66 R b AR A (84b) Ech tge will be relized Figure 5 with ll the tge ccded together to provide the over ll repone of the bnd-p. Conider the following deign exmple. Exmple 8 Chebyhev bnd-p, =000, =5, n =3 The n =3 polynomil i given by Eqution 63: D3 (.986)[(.493).9038 ) (85) The firt order piece trnform into Eqution 47, uing the proce of Ce Chebyhev Bnd-P rder n = but with =.986, repeted here. K 4 ( i pecified for the filter, we olve for K ) (86) R ( i pecified for the filter, pick C nd compute R ) (87) C The MATLAB code to for Eqution 86 nd 87 i follow: % Code for Eqution 86 nd 87 wo=000; % pec center freq q=5; % pec circuit =.986; % qudrtic coef c=.e-6; % picked cp ll the me % Computing the firt tge k nd r k=4-*^.5/q; % firt tge opmp gin r=^.5/(wo*c); % firt tge filter reitnce The MATLAB reult re: k = r =.44e+004

23 67 The qudrtic piece of Eqution 85 i proceed nd fctored uing the methodology of Eqution 76 through 8. The MATLAB code tht doe tht i follow: % Fctoring the econd order bnd-p wo=000; % pec center freq q=5; % pec circuit =.986; % qudrtic coef b=.493; % qudrtic coef b cc=.9038; % qudrtic coef c p=b^+cc^; % eqution ubtitution A=*b*wo/q; B=(+p*q^-)*wo^; C=*b/q*wo^3; D=wo^4; r=root([,a,b,c,d]); The reult of the MATLAB code re: r =.0e+003 * [ i, i, i, i] The root pir re now reembled qudrtic form Eqution 79 where ech will yield filter tge per Figure 5. The MATLAB code to clculte the op mp gin nd the filter reitnce for the tge i follow: % wo nd q for root pir =b(rel(r())); = b(rel(r(3))); wo=b(r()); wo=b(r(3)); q=wo/(*); q=wo/(*); % eqn 49 prmeter % picked cp=.uf k=4-^.5/q; kb=4-^.5/q; c=.e-6; R=^.5/(wo*c); Rb=^.5/(wo*c); The reult of MATLAB: k = kb = R =.3889e+004 Rb =.4400e+004

24 68 A with the Butterworth Exmple 7 we hve 3 filter tge with non-normlized gin. Eqution 8 i ued to compute the exce gin. Noting tht the order i odd (no db ripple ttenution i needed), the pproprite form of Eqution 83 will be ued nd the ttenution reitor deigned per Eqution 84 for ech tge. The MATLAB code tht doe thi (lo include ll previou code for thi exmple) i follow: % Exmple 8 wo=000; % pec center freq q=5; % pec circuit =.986; % qudrtic coef b=.493; % qudrtic coef b cc=.9038; % qudrtic coef c c=.e-6; % picked cp ll the me % Computing the firt tge k nd r k=4-*^.5/q; % firt tge opmp gin R=^.5/(wo*c); % firt tge filter reitnce % Fctoring the econd order bnd-p p=b^+cc^; % eqution ubtution A=*b*wo/q; B=(+p*q^-)*wo^; C=*b/q*wo^3; D=wo^4; r=root([,a,b,c,d]); % wo nd q for root pir =b(rel(r())); = b(rel(r(3))); wo=b(r()); wo=b(r(3)); q=wo/(*); q=wo/(*); % eqn 49 prmeter k=4-^.5/q; kb=4-^.5/q; c=.e-6; % picked cp=.uf R=^.5/(wo*c); Rb=^.5/(wo*c); % Now the gin R=[R,R,Rb]; K=[k,k,kb]; =j*wo; n=3;

25 69 f=; for i=:n y=*k(i)*r(i)*c*.5/(^*r(i)^*c^*.5+*r(i)*c*.5*(4-k(i))+); f=f*y; end g=b(f); gg=g^(/n); % Compute the ttenution reitor for i=:n r(i)=gg*r(i); rb(i)=gg*r(i)/(gg-); end The MATLAB reult re: Gin nd filter reitor: K = [ ], R =.0e+004 *[ ] Attenution reitor: R =.0e+006 *[ ], Rb =.0e+004 *[ ] The overll circuit i 3 ccded tge of Figure 5. The vriou tge op mp gin reitor, R & R, re determined with Eqution 50. The PSPICE chemtic for the f in nlyi performed i given by Figure 35 nd the plot of the PSPICE reult re given by Figure 36. Figure 35 PSPICE circuit for Chebyhev Exmple 8

26 Figure 36 PSPICE frequency repone for Figure 35 70

is the cut off frequency in rads.

is the cut off frequency in rads. 0 ELETRIAL IRUITS 9. HIGH RDER ATIVE FILTERS (With Tle) Introdution Thi development explin how to deign Butterworth (Mximlly Flt) or heyhev (Equl Ripple) Low P, High P or Bnd P tive filter. Thi tretment