( s) Use of Transformations in Active BP Filter Designs

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1 Use of Trnsformtons n Actve BP Flter Desgns even Mjt nd Držen Juršć Fculty of Electrcl Engneerng nd omutng, Unsk, H- Zgreb, rot. neven.mjt@fer.hr; drzen.jursc@fer.hr Abstrct In ths er trnsformtons, whch re very often used n desgn of ctve flters, re summrzed, nd used for revelng the reltons between dfferent clsses of ctve flter sectons. Among vrous trnsformtons we dstngush: frequency trnsformtons nd network trnsformtons. As frequency trnsformton exmles we consder the well known low-ss to bnd-ss (LP BP) nd less known lossy low-ss to bndss (LLP-BP) frequency trnsformton. As exmles of network trnsformtons, whch re erformed by network elements substtuton we consder () lossy LP BP network trnsformton: to obtn BP flter t substtutes ech resstor of the LP rototye flter by seres crcut, nd ech cctor by rllel crcut; nd () comlementry trnsformton: ssve network n negtve feedbck loo s trnsformed nto ssve network n ostve feedbck loo of oertonl mlfer.. Introducton The desgn of BP flters s usully erformed usng the well-known LP-BP frequency trnsformton led to n LP rototye flter trnsfer functon. orresondng network LP-BP trnsformton cn be led only n the desgn of ssve L flters. Snce t s rectnce trnsformton resultng wth the use of nductnces, t cn not be led for the desgn of ctve flters. One ossble soluton s lossy LP-BP (LLP-BP) trnsformton whch s resented n ths er []. Another network trnsformton we del wth, belongs to the grou of comlementry []-[] trnsformtons. It reltes BP flter crcuts bsed on ostve feedbck to those bsed on negtve feedbck. As consequence ther ssve senstvty roertes re closely relted, s well s ther gn-senstvty roducts (GSP). Ths trnsformton smlfes the flter relzton rocedures, s well s otmzton of flter senstvtes.. Lossy LP-BP Trnsformton The dvntge of ssve-l BP flter relzton les n the exstence of rectnce LP-BP trnsformton, whch defnes the BP flter structure nd enbles strghtforwrd relzton rocedure. An L BP flter s obtned by relcng ech nductor L LP of n LP rototye flter, by seres connecton of L nd nd ech cctor LP by rllel combnton of L b nd b. Such n element trnsformton, gves unque BP structure, followng drectly from LP-BP frequency trnsformton s LP = ( s +ω ) Bs. () It trnsforms the comlex frequency vrble s LP nto rtonl functon roducng the trnsfer functon of BP flter wth the center frequency ω nd the bndwdth B. The lcton of the LP-BP frequency trnsformton () to some n th -order low-ss rototye flter gves symmetrcl n th -order BP flter whose trnsfer functon cn be reresented s roduct of n nd -order trnsfer functons n K ( ω / q ) s TBP ( =. () = s + ( ω / q ) s + ω LP-BP frequency trnsformton () s redly used n ctve BP flter desgns, but lcton of the corresondng LP-BP rectnce trnsformton mkes no sense. One ossble soluton s slght modfcton clled lossy LP-BP trnsformton led to the network relzng trnslted LP rototye trnsfer functon. The bsc de s the followng. The orgnl LP rototye trnsfer functon s trnsformed by ntroducng the shfted comlex frequency vrble = s LP + δ, () BP trnsfer functon s creted by lyng the LLP-BP trnsformton to,.e. s + ω = + δ. () B s ote tht the combnton of () nd () gves the sme BP flter trnsfer functon s the conventonl LP- BP trnsformton (), led to the orgnl LP trnsfer functon. However, relzton of BP flter s enbled lyng the corresondng mednce trnsformton to the modfed LP rototye flter. The rocedure cn be brefly summrzed through the followng stes: Gven LP rototye trnsfer functon T( - hoose the shft constnt δ - Fnd new trnsfer functon T ()=T(=T (s+δ) - elze LP rototye hvng the trnsfer functon T () - Aly LLP-BP to T () nd obtn T BP ( - Perform the LLP-BP mednce trnsformton to LP rototye flter. The rocedure s strghtforwrd nd t s descrbed n more detl through the exmles tht follow.. Second-Order BP flter onsder st -order LP flter crcut n Fg..

2 V V = II α= + =/(- α) =/α = F ( β-) G = Fg.. st -order LP crcut. ssve ctve The voltge trnsfer functon T( for ths crcut s ( ) σ K T ( = = = =. V s + ( ) s + σ ( s s ) () It hs negtve rel ole s =-σ=-() -. Substtutng s=-δ (6) new LP rototye functon T () s obtned σ σ σ σ T ( ) = = = = s + σ + σ δ + Γ ; (7) = Γ Γ = σ δ. (7b) The ole of T ( s shfted to the rght for n mount δ. It cn be moved even n the rght-hlf lne. In tht cse the st -order crcut wth n oertonl mlfer, shown n Fg., hs to be used for relzton of T (. Fg.. s-vrble trnslton. Pole shft, -lne The voltge trnsfer functon T () for ths crcut s ( α ) βσ ( α) βσ βσ δ T ( ) = = = =, (8) V + ( αβ ) σ + σ δ + Γ where the shft δ equls αβσ. For the smlcty resons we wll use normlzed elements of the LP rototye,.e. =, =, gvng σ= nd δ=αβ. The result s ( α) β β δ T ( ) = = =. (9) V + αβ + δ At ths ont we ntroduce the mednce trnsformton whch substtutes ech resstor of the LP rototye flter by seres crcut, nd ech cctor by rllel crcut, s shown n Fg.. / + s = +, + sb () / s s Fg.. mednce trnsformton. onsequently the normlzed vrble of the LP trnsfer functon s trnsformed nto rtonl functon,.e. s + /( bb ) b = = = + + () σ s /( b ) b omrng () nd () we obtn ω =, B =, δ = + b. () bb b b Ths rocedure results by nd -order BP flter shown n Fg., whch s tye A crcut descrbed n []. b b b =II = / α= α + = /(- α) V b b = ( -) G= F β Fg.. Second-order ctve- BP flter crcut. The voltge trnsfer functon T BP ( of ths crcut s ( α) β ( α) βbs TBP ( = = = () V s + ω s + Bs + ω + δ αβ + Bs The vlue of δ s not rbtrry. It s lmted by the resstnce nd cctnce rtos. From () we hve δ δ ω b δ δ ω = m = ±. () b B B The exresson under the squre roots s ostve f δ ω B = δ mn = q. () The constnt δ s, therefore, lmted by the rto of the centrl frequency nd the bndwdth of the BP flter. There re mny degrees of freedom n relzton of () by choosng rmeters α, β nd δ. The mn crteron s to mnmze the trnsfer functon senstvtes. As shown n [], ths s the cse when () s stsfed,.e. b = b = δ mn. (6) Desgn Exmle: As n llustrton consder nd -order BP flter, wth the center frequency f =khz, bndwdth B=Hz, nd the ss-bnd gn K=. The desgn s erformed by the followng desgn rocedure: ) For gven nd -order BP flter ole Q, choose δ. Usng () we clculte δ mn = ω B = q =. (7) Let us denote the cctnce rto s ρ,= b /, nd the resstnce rto s r= / b. In order to nlyze ther nfluence on senstvtes to comonent tolernces, nd to fnd n otml δ, three relztons corresondng to two vlues of δ re nlyzed. Frst we choose δ=δ mn =, nd from (6) we hve ρ=r=δ mn /= (crcut ). ext we choose δ=6 nd usng () obtn ρ=.6 nd r=.76 for lus sgn of the squre root (crcut ), nd ρ=.76 nd r=.6 (crcut ) for mnus sgn. Let us roceed wth clculton of crcut no. ) Fnd the new LP rototye wth shfted oles by δ: Usng (6), new functon T () wth Γ=- s obtned. To relze negtve Γ we hve to clculte gn β. hoosng α=., from (9) t follows β=δ/α=. ) elze BP flter crcut comonents. We choose b =nf, nd usng () = b /ρ=9f, =/(B b )=6.7kΩ nd b =B/( ω )=8.kΩ. The remnng elements re: = /(-α)=7kω; = /α= 7kΩ. Let G =kω, then F = G (β-)=kω. The element vlues for ths cse nd for the other two re gven n Tble. Fg. shows the trnsfer functon mgntude α(ω)=log T(jω) [db].

3 o. b r b ρ β δ δ mn ) ) ) Tble omonent vlues of nd -order flters from Fg. ( n kω, n nf). the low senstve crcut to the comonent tolernces. Further senstvty reducton cn be cheved lyng the mednce terng to the LP rototye [6]. V = b = = = b =r = ρ b = /r +β = / b ρ (c) (d) Fg. Mgntude, nd -(d) results of Monte rlo nlyss of the flters n Tble. Monte rlo smulton results re resented n Fg. -(d). The reltve element chnges re ssumed s uncorrelted rndom vrbles, wth zero-men Gussn dstrbuton nd % stndrd devton. Fg. 6 Senstvtes of the BP flters n Tble. The stndrd devton (relted to the Shoeffler senstvte of the gn vrton wth resect to the elements vrtons α= T BP (ω) / T BP (ω), s clculted for the flter exmles n Tble nd shown n Fg. 6. We see tht the best results re obtned f δ=δ mn,.e. for the crcut no., whle the two remnng crcuts hve smlr but worse senstvtes.. Fourth-Order BP flter onsder the nd -order LP flter shown n Fg. 7. = Fg. 8. Fourth-order BP flter crcut. Hgher-Order BP flter Hgh-order BP flters cn be relzed usng the sme rocedure. There s, however, dfference between the flters tht use odd-order nd even-order LP rototye. V = II < α< α= + = /(- α) = = / α = / µ =r =r = (- µ ) =/ ρ = + =/ ρ µ = F ( βµ -) G = Fg. 9. Modfed rd -order LP flter rototye To construct 6 th -order BP flter, rd -order LP rototye flter shown n Fg. 9 s used. = II < α< α= = / α + = /(- α) V 6 6 µ = II = = µ- µ = + = = / µ (- µ ) +βµ Fg.. The 6 th -order BP flter crcut It s slghtly modfed stndrd sngle-mlfer rd - order LP flter confgurton. Feedbck resstor s dded to enble shft of rel ole together wth comlex oles. esultng BP flter s shown n Fg.. A th -order LP rototye for the relzton of n 8 th - order BP flter s shown n Fg.. Its trnsfer functon hs only comlex-conjugte oles nd ther δ-shft needs no ddtonl feedbck connectons. = = =r =/ ρ =r =r µ = =r V =/ ρ =/ ρ = F G( -) β G V =/ρ =( F β -) G= Fg. 7. nd -order low-ss flter. The BP flter shown n Fg. 8 s obtned by the element substtuton () to tht crcut. Snce the nd - order LP flter rototye for the desgn of th -order BP flter, hs comlex ole r we do not need to shft ny rel oles, nd modfy t s we dd n the st -order LP rototye cse. As shown n [] mnml δ rovdes Fg.. The th -order LP rototye flter. The element substtuton () led to ths crcut, gves the BP flter shown n Fg β V 8 8 Fg.. The 8 th -order BP flter crcut.

4 . omlementry trnsformton Evluton of ctve- flter qulty nvolves vrous rmeters such s: smle relzblty, reetblty, ossblty of strghtforwrd rocedure of rmeter clculton, smll number of comonents, low ower consumton, low nose, nd the most often, low flter senstvty to ssve nd ctve comonent tolernces. Actve- sngle-mlfer flters wth network n the feedbck loo stsfy most of these erformnces. However, clculton of flter elements nd otmzton of senstvtes cn be very tedous nd tme consumng. Some flter confgurtons cn be obtned from nother ones wth known erformnce usng certn network trnsformtons. Also, f the orgnl flter s otmzed, the trnsformton cn gve n otmzed crcut s well, mkng the relzton rocedure smle. We y n ttenton to trnsformton belongng to grou of comlementry trnsformtons [], []. As n exmle nd -order ctve BP flter secton s used. V V V V V V Fg.. Sngle oertonl mlfer ctve flter Second-order ctve flters re often relzed s sngle mlfer form shown n Fg., wth n network n the negtve feedbck loo. Usng the equvlent crcut shown n Fg. we cn wrte the equtons for the voltges t the nodes nd V = tv + tv + tv (8) V = tv + tv + tv where V tj = ; =,, j =,,, k j (9) V j V k = re the voltge trnsfer functons corresondng to j th nut nd th outut node of network. It s known tht [] V V = V V tj = ; =,. () V V V + V - =A(V-V ) Fg.. Actve flter secton equvlent crcut. Dfferent confgurtons nd trnsfer functons of the comlete secton cn be obtned by connectng the nut nodes,, or to the secton nut or mlfer s outut or to the ground. We shll consder two cses. se I: Frst we consder crcut resented n Fg., where the node s the nut, s ground node nd node s connected to the oertonl mlfer outut,.e. V = Vn ; V = ; V = () V nd V re the voltges t the vrtul short crcut,.e. V V. () Usng (8), () nd () the overll trnsfer functon s V t t T ( = = = () Vn V t t se II: If we nterchnge the outut nd ground termnl of the crcut from Fg., we obtn crcut n Fg., wth the node of the network connected to the oertonl mlfer outut nd grounded node. V V V V V V V - =A(V-V ) V + Fg.. Trnsformed ctve flter crcut. ote tht the voltge source must lso be nverted n the rocess of exchngng termnls. Ths cnnot be done n rctce, but the equvlent result,.e. reversng the outut voltge olrty, cn be cheved by nvertng the nut termnls of the oertonl mlfer. Therefore, the rctcl crcut hs the form shown n Fg. 6. For ths secton we cn wrte V = Vn V = V = () ombnng (8), () nd () gves V t t T ( = = = () V V t t V V V V n V V Fg. 6. omlementry ctve flter secton. omrng () nd () nd lyng () we get + =, (6) T T.e. trnsfer functons recrocls re comlementry. They hve dfferent denomntors nd equl numertors. Therefore, f the frst crcut s BP flter, nother one s BP flter s well. The reresenttve exmles re two well-known BP nd -order crcuts. The crcut n Fg. 7 s the Sngle- Amlfer-Bqud (SAB), wth n network n the negtve feedbck loo of the oertonl mlfer nd corresonds to the toology resented n se I bove. V Fg. 7. Actve flter SAB secton omlementry to the SAB secton s Sllen nd Key (SAK) secton shown n Fg. 8.

5 V Fg. 8. omlementry SAK ctve flter secton SAK secton uses ostve feedbck of the oertonl mlfer nd corresonds to the se II bove. Generlly both crcuts hve the trnsfer functon T( ω ( s q = = = K = K ω V D( s + s + s + q s s + ω (7) Assumng tht ssve networks n the mlfer s ostve (SAB) nd negtve (SAK) feedbck loos re equl, we wll use the followng trnsfer functon of the ssve network where Tˆ( ω n( ˆ qˆ = = k ω d( s + qˆ s s + ω, (8) ˆ k = ; qˆ ( + ) + = ( + ) + ; (8b) ω = ; = =. + The trnsfer functon for the SAB crcut hs the form ( αβtˆ( T ( = = (9) D( βtˆ( nd for the SAK crcut ( αβtˆ( T ( = = () D( β( α) Tˆ( They must be dentcl,.e. T (=T (=T(, nd comrng ther numertors nd denomntors we obtn α = α ( α ); β = β ( α ). () Fnlly, the coeffcents of the trnsfer functon (7) re gven by K = ( ) α β kˆ q qˆ; =,. ω = ω ; =,. q ˆ ( ˆ = q b k) ; =,. (c) where b =β, nd b =β (-α ). Desgn Exmle: As n llustrton consder nd -order BP flter wth f =khz, B=Hz nd K=. For the SAB desgn we choose ρ= / =, r= / =, =nf. Other comonents followng from (8b) re gven n lne of Tble. o. α r ρ β ) ) Tble. SAB flter comonent vlues ( n kω, n nf). From (c) β =., nd wth =kω we clculte =(β -) =.kω. From α =.. Fnlly we clculte = /α =9.kΩ, nd = /(- α )=9.79kΩ. Flter no. n Tble s the del mednce terng relzton,.e. wth ρ=r=. The desgn of the SAK secton s now very smle. Usng the elements of the SAB secton from Tble, we only need to clculte the new vlues of constnts α nd β. Thus for crcut no. usng () we obtn α = α α + =.667; β = β( α + ) =. () The resstors,, nd re: = /α =9kΩ, = /(-α )=8.kΩ, =(β -) =kω. For SAK no. we hve α =., β =., =.kω, =.9kΩ, nd =.kω. Actve senstvtes cn be reduced mnmzng the GSP n the flter desgn. Pssve senstvtes re reduced usng mednce terng [7]. It cn be shown tht otmzton of SAB crcut gves the otmum for SAK crcut s well [8]. The results of usng Monte rlo smultons for the SAB secton, re shown n Fg. 9. Better results re obtned for the mednce-tered crcut,.e. for the crcut no.. Fg. 9 Monte rlo runs of SAB BP flters M runs for the SAK secton re resented n Fg.. Better senstvty s obtned for the crcut no., gn. Fg. Monte rlo runs of SAK BP flters Shoeffler senstvtes of both SAB nd SAK crcuts re resented n Fg.. We cn conclude tht two comlementry crcuts n Fg. 7 nd Fg. 8 hve very smlr chrcterstcs concernng senstvty to ssve flter comonents. SAB crcut shows somewht lower senstvty. Fg. Shoeffler senstvtes of SAB nd SAK flters. ombnng L-LP-BP nd omlementry trnsformtons onsder the st -order crcut wth n oertonl mlfer shown n Fg.. If we ly comlementry trnsformton to ths crcut we obtn the crcut n Fg.

6 , wth n network n the negtve feedbck loo. The voltge trnsfer functon T () for ths crcut s ( α) βσ ( α) δ ( α) Γ T ( ) = = = =. () V + β σ + σ δ + ( ) Γ = II =/(- α) V =/α ( β-) Fg.. omlementry st -order shfted ole crcut We choose gn σ= for smlcty. The trnsfer functon T () s defned by () hs the sme form s T () n (8). To relze T ()=T () ther numertors nd denomntors must be dentcl,.e. β = αβ ; α = / α. () To rovde the sme ole shft for vlue of δ, s n the ostve feedbck cse, we need to clculte gn β. Alcton of L-LP-BP trnsformton to tht crcut gves new BP flter crcut shown n Fg.. It s comlementry to the one n Fg. nd we exect tht t hs senstvty roertes very smlr to the orgnl crcut. onsequently, otmzng GSP nd mnmzng senstvty to ssve comonents by usng mnmum δ n flter desgn cn be nherted from the orgnl crcut. V = II = /(- α) = / α b b ( β-) Fg.. omlementry nd -order ctve- BP crcut. For the crcuts n Tble we clculted new vlues for gn β. Thus for no. usng () we hve β =, nd for crcuts no., we hve β =6. We used element vlues from Tble nd new gn β to bult crcuts s n Fg.. Monte rlo runs nd Shoeffler senstvtes re resented n Fg. nd Fg., resectvely. We cn conclude tht mn. senstvty s obtned for the crcut no. (mnmum δ), gn. Two comlementry crcuts n Fg. nd Fg. hve very smlr senstvty roertes to ssve comonents. (c) Fg. nd -order BP flter s n Fg. : Mgntude. -(d) Monte rlo runs. (d) Fg. Senstvtes of nd -order BP flters from Fg.. omrng ll relzed nd -order BP flters t cn be concluded tht both SAB nd SAK toologes hve better senstvtes thn toologes obtned usng L- LP- BP trnsformton. In ll cses flters wth negtve feedbck lwys hve lower senstvtes.. onclusons Lossy LP-BP trnsformton enbles BP flter relztons usng drect comonent trnsformton nd strghtforwrd desgn rocedure. omlementry trnsformton reltes the desgns, whch use ssve flter sub-network n ostve nd negtve feedbck loo of OA. omonent vlues clculted for one toology cn be used for clculton of the comlementry toology crcut. Ther Gn-Senstvty- Products (GSP) nd ssve senstvty erformnces re lso closely relted. eferences [] D. Juršć,. Mjt G. S. Moschytz, Desgn of th - Order Bnd-Pss Actve- Flters Usng Lossy LP-BP Trnsformton, Proc. ETD, Esoo, Fnlnd, Vol., Aug. 8 -,,. 7-. [] G. S. Moschytz, P. Horn, Otmzng Two ommonly Used Actve-Flter Buldng Blocks Usng the omlementry Trnsformton, Electronc rc. nd Syst., vol., no.,. -, July 977. []. Flege, omlementry Trnsformton of Feedbck Systems, IEEE Trns. on rcut Theory, T-,. 7-9, 97. [] D. Hlbermn, Inut nd Ground s omlements n Actve Flters, IEEE Trns. on rcut Theory, T-,. -7, 97. [] G. S. Moschytz, Lner Integrted etworks: Desgn. ew York: Vn ostrd enhold o., 97. [6] D. Juršć, G.S. Moschytz,. Mjt, Low-Senstvty, Low-Power th -Order Bnd-Pss Actve- Allole Flter Usng Imednce Terng, Proc. IES, Msd, Mlt, Vol., Set. -,, [7] D. Juršć, G. S. Moschytz,. Mjt Low-Senstvty SAB Bnd-Pss Actve- Flter Usng Imednce Terng, Proc. of the IEEE Int. Sym. on rc. nd Syst., ISAS, Sydney, Austrl, Vol.,. 6-6, My 6-9,. [8]. Mjt, V. Čosć, omlementry Trnsformtons nd Senstvty of Actve Flter Sectons, Inf- Telecom-Automton ITA o. -, vol. 8,. 9-, July-Dec., 989.