Design of Band-pass Filters Using Parallel Coupled Lines and. Discrete-Time Domain Techniques

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1 Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques Ln-Chuan sa Department of Electronc Engneerng Lunghwa Unversty of Scence and echnology Abstract A new formulaton s presented to desgn parallel coupled lne (PCL) flters at mcrowave frequences. A cascade connecton of multple-secton coupled lnes forms a band-pass flter wheren each secton of PCL has the same electrcal length. As a result the transfer functons of flters are formulated n the doman. he flter structures are obtaned by usng optmaton algorthm n whch the values of even-mode and odd-mode characterstc mpedances of coupled lnes are adjusted so that the transfer functons of coupled-lne flters are close to those of deal prototype flters. wo band-pass flters are realed n the form of mcrostrp lnes and ther frequency responses are measured to valdate ths new formulaton. Keywords:flter mcrostrp lne parallel coupled lne doman. Introducton Mcrowave flters []-[] are two-port networks used n an electronc system capable of allowng transmsson of sgnals over the pass-band and rejectng unwanted harmoncs over the stop-band. Dfferent knds of approxmatons lke Butterworth [3] Chebyshev [4] and Ellptc functon [5] have been proposed and wdely used as models for mcrowave-flter synthess. he motvaton of ths study s to present the parameters of parallel coupled lnes n the doman [6]-[7]. A general procedure for desgnng mcrowave flters known as the nserton loss method starts wth lumped elements n low-pass flters. A lnk between dstrbuted elements used n the mcrowave range and ther lumped-element counterparts used at lower frequences s establshed va Rchard s transformaton and Kuroda s denttes [8]. he dstrbuted elements can be mplemented n varous formats such as wavegude coaxal lne mcrostrp lne strplne and delectrc resonator. he mcrostrp lnes of PCLs are easy to fabrcate and ther operatng bandwdths are often less than % when they are used to mplement flters such as band-pass flter band-stop flter etc.

2 龍華科技大學學報第二十三期 7.9 In ths paper we present the basc deas about the constructon of cascaded coupled-lne networks used as flters. In partcular each PCL has the same couplng length. We therefore adopt the dscrete-tme sgnal processng (DSP) technque [9]-[] and the characterstcs of each PCL can thus be expressed n the doman. In ths context we express the chan scatterng parameter matrx of a PCL n the doman and we fnd that each PCL contrbutes a ero at (dc). he cascade connecton of several PCLs forms a mult-secton PCL and the chan scatterng parameter matrx of the entre network s the sequental multplcaton of chan scatterng parameter matrx of each ndvdual PCL. As a result the transfer functon of cascaded PCLs has multple eros at. o desgn a Butterworth or Chebyshev band-pass flter we employ a dgtal flter as an deal flter whch also has eros at. After removng all eros of transfer functons of both the deal flter and multsecton PCL the remanders of both transfer functons are autoregressve (AR) processes. We then use optmaton algorthm [] to tune even-mode and odd-mode characterstc mpedances of each PCL confguraton so that the transfer functon of entre PCLs can be as close to that of the deal flter as possble. It s pertnent to pont out that very small separaton between two parallel lnes s hard to reale n practcal mplementaton. herefore we put lmts on the upper bound and the lower bound of even-mode and odd-mode characterstc mpedances of each PCL. wo types of band-pass flters are mplemented n mcrostrp formats. Both theoretcal and measurement results of flters prove the valdty of ths formulaton.. Formulaton Fg. shows a two port network where a() and b() are the ncdent wave and reflected wave at port one and a() and b() are the ncdent wave and reflected wave at port two. hese waves are nterrelated through the chan scatterng parameters mn mn of a two port network as follows: a() b() b() a() () A PCL can become a two-port network by termnatng any two ports of the four-port confguraton n ether short-crcuted or open-crcuted formats. As a result there are ten commonly elaborated combnatons []. Fg. shows a four-port PCL where l s the physcal length and I (34) s the termnal current at the respectve port. In partcular the upper and lower lnes n Fg. have the same electrcal characterstcs. Fg. A two-port devce Fg. One secton of the parallel coupled lne

3 Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques Fg. 3 Fundamental crcut of PCL he coupled lnes n Fg. can be decomposed nto an even-mode exctaton and an odd-mode exctaton [3]. he overall crcut performance s the summaton of the crcut responses due to both even-mode and odd-mode exctatons. he mpedance parameters concerned are as follows []: e + o j tanθ e o jtanθ e o jsnθ e + o jsnθ (. a) (. b) (. c) (. d) where e s the characterstc mpedance of even-mode exctaton o s the characterstc mpedance of odd-mode exctaton. Notce that θ s the electrc length of coupled-lne wthθ β l ; β s the propagaton constant and l s the physcal length. We can get mpedance parameters of coupled lnes n the doman by settng jθ e. We obtan c( + ( ) ) c( + ) ( ) where c 4 ( / / c ( ) c. e + o c e o ) (3. a) (3. b) (3. c) (3. d) Our object s to desgn a flter by usng the parameters n the doman. For such a crcumstance we set I I 4. he crcut arrangement n Fg. s thus changed to that shown n Fg. 3. he four-port mpedance matrx equatons reduce to: V V 3 3 I I I I 3 3. (4. a) (4. b) he four-port parallel coupled-lnes now can be treated as a two port network. Wth a proper notaton modfcaton the two-port mpedance matrx can be expressed as follows c ( + ) ( ) / c ( ) / c ( ) c ( + ) ( ) where 3 3 and 33. Furthermore we can convert the mpedance matrx nto the chan scatterng matrx (or matrx) of a two-port network [][4] wth proper algebrac manpulaton. he chan scatterng matrx s employed here because of ts sutablty for the cascade connecton of two or more (5)

4 龍華科技大學學報第二十三期 7.9 two-port networks. he correspondng matrx of PCL s: 8c / ( l ) n n (6) p where s the reference characterstc mpedance and l (4c + c + 4 ) + (c 4c 8 ) + ( 4c + c + 4 ) n ( c 4 ) + (c 4c + 8 ) + ( c 4 ) p (4c c 4 ) + (c + 4c + 8 ) + ( 4c c 4 Notce that the denomnator of each matrx element has the term ( ) and the numerator of each matrx element s cast n the form of α + α + α where α α and α are real numbers and they are functons of even-mode odd-mode and reference characterstc mpedances. he overall chan-scatterng parameter matrx of a cascade connecton of parallel coupled lnes s the sequental multplcaton of the chan scatterng matrx of each PCL.e. Network M ( ) ( ) ( ) ( ) ). (7) where M s the number of PCLs ( ) ( ) ( ) ( ) and are the matrx elements representng the -th element. If a crcut conssts of M sectons of PCLs the matrx element crcut s as follows network M a network ( ) M / M ( ) () of the overall (8) where a s a real number and a s determned by reference characterstc mpedance even-mode characterstc mpedance e and odd-mode characterstc mpedance o of every PCL. When the output port of the cascade network s properly termnated we have a(). he transfer functon () of such a network becomes b() ( ) a() a() network ( ) M / ( M a ) M M / where represents tme delay of all PCLs. Equaton () reveals that () has M eros at (dc) whch are contrbuted by M sectons of PCLs. If all eros n equaton () are removed the remanng part of () can be regarded as an autoregressve (AR) process M / multpled by. If we express ths correspondng AR process of ( ) as AR () we then have where A a. AR ( ) M A (9) () An AR process AR () s solely charactered by the coeffcents A and these coeffcents are determned by the modal characterstc mpedances of coupled lnes. If the ero locatons of () of the network are the same as that of an deal flter F () we could make () AR approxmate a correspondng AR process F AR () of the deal flter F() by adjustng the values of coupled-lne mpedances. hs assures that the characterstc of ()

5 Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques s close to that of F() 3. Expermental Results Frst of all we select a dscrete-tme flter F() that satsfes the specfcaton of requred frequency response. F AR () s obtaned by dvdng F() by the terms producng the eros of (). If both ( ) and F() have the same ero locatons () can be expressed as follows F AR F AR ( ) N A () where N s the order of the denomnator of F() A s the denomnator coeffcents of F( ). If () and F() do not have the same eros locatons we fnd the equvalent AR process F AR () by parametrc modelng technques. Snce the magntude of the delay term M / n equaton (9) s equal to we can neglect ts nfluence on the magntude consderaton. We let AR () approach F AR () by adjustng the values of even-mode characterstc mpedances and odd-mode characterstc mpedances of PCLs n a sense that M( o mnmed. If the coeffcent dfference between AR () and F AR () s small enough we argue that the transfer functon () of the PCL network and the system functon F() of the deal flter have smlar magntude responses n the frequency doman. A ) A A. Butterworth Band-pass Flter A Butterworth band-pass flters s s presented n ths subsecton. he central frequency of the flter s 3 GH and the bandwdth s 5%. A dscrete-tme Butterworth flter prototype s gven as follows [9] b j j F ( ) 4 a 4 j () where { bj j 4}e-4 { } and { a 4}{ }. Note that the flter prototype F() shown n () s usually classfed as a hgh-pass flter when t s used for dscrete-tme sgnal processng. However when we unfold the magntude functon F () on the frequency axs F () s a perodc functon wth a perod of π. he flter prototype F() s a band-pass flter wth ts central frequency equal to π. he flter prototype F( ) has four eros at whch are to be realed by four sectons of PCLs. We mnme the value of ( A A ) by adjustng the even-mode and 8 o odd-mode characterstc mpedances of PCLs. As stated prevously a very small separaton between two parallel lnes s hard to reale n practcal mplementaton. As a result we set the lmt so that the gap se s larger than 4 ml (.mm). Note that the gap se of each PCL s determned by even-mode and odd-mode characterstc mpedances of PCL [5]. We therefore confne the upper bound and lower bound of even-mode and odd-mode characterstcs

6 龍華科技大學學報第二十三期 7.9 mpedances of coupled lnes. he even-mode and odd-mode characterstc mpedances of ths four-secton PCL are ( )-( )-( )-( ) ohms. In each parenthess the frst number s the even-mode characterstc mpedance and the second number s the odd-mode characterstc mpedance. When even-mode and odd-mode characterstc mpedances are gven we get the lne wdth and gap se of each PCL by usng HP LneCalc tool [6]. he lne wdth and gap se can also be obtaned by usng analytcal formulatons [5]. Ideally each secton of a PCL has the physcal length of a quarter wavelength at 3 GH. o consder frngng capactance effect the physcal length of each secton of PCL has been modfed accordngly [7]. Our program development envronment s under Matlab sgnal processng toolbox and Optmaton toolbox []. he optmaton method nvolves the fmncon functon n Matlab envronment wheren the optmaton program conssts of one man functon and a sub-functon executng the optmaton. A typcal program s composed about command lnes. Fg. 4 shows the physcal layout of the mcrostrp flter. he total length of the flter excludng the 5 ohms reference lne on both sdes s 68.3 mm. he flter was fabrcated on a Durod substrate havng thckness 3 ml (.79mm) and relatve delectrc constant.5. On both sdes of the flter we place 5 ohms reference lnes. Fg. 5 shows the measured scatterng parameters S and S of the network shown n Fg. 4. he nserton loss s.9 (db) n the pass-band whch s due to both conductor and substrate losses [8]. Fg. 4 Fabrcated four-secton PCL Butterworth band-pass flter wth a center frequency of 3 GH. Magntude Response (db) Ideal S Measured S Measured S Frequency (GH) Fg. 5 Measured reflected and transferred scatterng coeffcents at nput and output ports of the band-pass flter shown n Fg. 4. B. Chebyshev Band-pass Flter A dscrete-tme Chebyshev band-pass flter s elaborated n ths subsecton. he rpple n the pass-band s.5 (db) the central frequency s 4 GH and the bandwdth s %. A dscrete-tme Chebyshev flter prototype s gven as follows [9] where { b j 6 b j j F ( ) 6 a j (3) j 6}e-6 { } and { a 6}{

7 Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques.839}. he flter prototype F() has sx eros at whch can be realed by sx sectons of PCLs. We mnme the value of ( A ) o A by adjustng even-mode and odd-mode characterstc mpedances of all PCLs. he even-mode and odd-mode characterstc mpedances of ths sx-secton PCL are ( )-( )-( )-( )-( )-( ) ohms. In each parenthess the frst number s the even-mode characterstc mpedance and the second number s the odd-mode characterstc mpedance. Ideally each secton of PCL has physcal length of a quarter wavelength at 4 GH. However the physcal length of each secton of PCL should be modfed to count for the frngng capactance effect. Fg. 6 shows the physcal layout of the mcrostrp flter and the total length of the flter excludng the 5 ohms reference lne on both sdes s 77.9 mm. he flter s fabrcated on a Durod substrate havng thckness 3 ml (.79mm) and relatve delectrc constant.5. Fg. 7 shows the measured scatterng parameters S and S of the network shown n Fg. 6. he nserton loss s. (db) n the pass-band whch s manly due to both conductor and substrate losses [8]. Obvously the measured results are n good agreement wth the numercal values va HP ADS smulaton tool [9]. 4. Concluson he parameters of parallel coupled lnes were expressed n the doman. In addton a new formulaton was developed to desgn flters by usng doman transfer functon and optmaton method. he close agreement between theoretcal values and expermental results had llustrated the valdty of ths new formulaton. Acknowledgements hs work was supported by the Natonal Scence Councl R.O.C. under Grant NSC95--E-6-3. Fg. 6 Fabrcated sx-secton PCL Chebyshev type I band-pass flter wth a center frequency of 4 GH. Magntude Response (db) Ideal S Measured S Measured S Frequency (GH) Fg. 7 Measured reflected and transferred scatterng coeffcents at nput and output ports of the band-pass flter shown n Fg Reference. Gaobao Xao; Yashro K.; Nng Guan; Ohkawa S. An effectve method for desgnng nonunformly coupled transmsson-lne flters IEEE rans. Mcrowave heory ech. vol. 49 pp.7-3 June.. M. L. Roy et. Al. he contnuously varyng transmsson-lne tecnque-applcaton to flter desgn IEEE rans. Mcrowave heory ech. vol. 47 pp Sept. 999.

8 龍華科技大學學報第二十三期 Drod J.M.; Jones W.. Maxmally flat quarter-wavelength-coupled transmsson-lne flters usng Q dstrbuton IEEE rans. Mcrowave heory ech. volume: 45 Issue: Part: pp. 3 Dec Ch-Yang Chang; Cheng-Chung Chen; Hong-Je Huang Folded quarter-wave resonator flters wth Chebyshev flat group delay or quas-ellptcal functon response Mcrowave Symposum Dgest IEEE M-S Internatonal pp vol R. Levy and J. D. Rhodes A comb-lne ellptc flter IEEE rans. Mcrowave heory ech. vol. 9 pp. 6-9 Jan Da-Chang Chang Chng-Wen Hsue Desgn and mplementaton of flters usng transfer functons n the doman IEEE rans. Mcrowave heory ech. vol. 49 no. 5 pp May. 7. Ln-Chuan sa and Chng-Wen Hsue Dual-band band-pass flters usng equal-length coupled-seral-shunted lnes and -transform technque IEEE rans. Mcrowave heory ech. vol. 5 no. 4 pp. -7 Aprl K. Kuroda General propertes and synthess of transmsson-lne nerworks n Mcrowave Flters and Crcuts A. Mstsumoto Ed. New York: Academc 97 vol.. 9. A. V. Oppenhem R. W. Schafer Dscrete-me Sgnal Processng. Englewood Clffs NJ:Prentce-Hall S. Haykn Adaptve Flter heory. Englewood Clffs NJ: Pretce-Hall Coleman M. A. Branch A. Grace Optmaton oolbox. he Math Works Inc.User s Gude Verson. Natck MA D. M. Poar Mcrowave Engneerng nd ed. New York: Wley Robert S. Ellott An Introducton to Guded Waves and Mcrowave Crcuts. Prentce-Hall R.E Colln Foundaton for Mcrowave Engneer. New York: McGraw-Hall Inc S. Akhtarad. R. Rowbotham and P. B. Johns he desgn of coupled mcrostrp lnes IEEE rans. Mcrowave heory ech. vol. 3 no. 6 pp June D. urner R. Wlhelm W. Lemberg Lne Calc. Aglent echnologes. Palo Alto CA. 7. S. B. Cohn Parallel-coupled transmsson-lne-resonator flters IRE rans. M vol. M-6 pp. 3-3 Aprl Sheng-Yuan Lee Chh-Mng sa New cross-coupled flter desgn usng mproved harpn resonators IEEE rans. Mcrowave heory ech. vol. 48 pp Dec. 9. D. urner R. Wlhelm W. Lemberg Advanced Desgn System. Aglent echnologes. Palo Alto CA.

9 Desgn of Band-pass Flters Usng Parallel Coupled Lnes and Dscrete-me Doman echnques 使用平行耦合線及離散時間技術設計帶通瀘波器 蔡林憲 龍華科大學電子工程系 摘要新的方法使用平行耦合線設計微波瀘波器 串接多段平行耦合線形成帶通瀘波器其中每一段平行耦合線有相同的電長度 瀘波器轉移函數公式為 領域 瀘波器結構使用最佳化演算法調整平行耦合線特性阻抗的偶模及奇模阻抗值使得其轉移函數可以接理想數位瀘波器 兩個帶通瀘波器驗証此一設計方法 關鍵詞 : 瀘波器 微帶線 行平耦合線 領域