Periodic Structures. Filter Design by the Image Parameter Method

Transcription

1 Prioic Structurs a Filtr sig y th mag Paramtr Mtho ECE53: Microwav Circuit sig Pozar Chaptr 8, Sctios 8. & 8. Josh Ottos /4/

2 Microwav Filtrs (Chaptr Eight) microwav filtr is a two-port twork us to cotrol th frqucy rspos at a crtai poit i a microwav systm y proviig trasmissio at frqucis withi th passa of th filtr a attuatio i th stopa of th filtr. Typical frqucy rsposs iclu low-pass, high-pass, apass, a a-rct charactristics. pplicatios ca fou i virtually ay typ of microwav commuicatio, raar, or tst a masurmt systm.

3 Prioic Structurs (Sctio 8.) ifiit trasmissio li or wavgui prioically loa with ractiv lmts is rfrr to as a prioic structur... Prioic structurs ca tak various forms, pig o th trasmissio li mia ig us. Oft th loaig lmts ar form as iscotiuitis i th li, ut i ay cas thy ca mol as lump ractacs across a trasmissio li... Prioic structurs hav passa a stopa charactristics similar to thos of filtrs; thy fi applicatios i travligwav tus, masrs, phas shiftrs, a atas. 3

4 Uit cll Each uit cll of this li cosists of a lgth of trasmissio li with a shut suscptac across th mipoit of th li; th suscptac is ormaliz to th charactristic impac, o. f w cosir th ifiit li as ig compos of a casca of itical two-port tworks, w ca rlat th voltags a currts o ithr si of th th uit cll ug th C matrix: C 4

5 Tal 4. (or isi covr of Pozar) * * f a twork is rciprocal, C 5 Rciprocal Ntworks: twork is sai to rciprocal if th voltag apparig at port u to a currt appli at port is th sam as th voltag apparig at port wh th sam currt is appli to port. Exchagig voltag a currt rsults i a quivalt fiitio of rciprocity. gral, a twork will rciprocal if it cosists tirly of liar passiv compots (that is, rsistors, capacitors a iuctors). gral, it will ot rciprocal if it cotais activ compots such as grators. p.3, Mahmoo Nahvi, Josph Emiistr, Schaums outli of thory a prolms of lctric circuits, McGraw-Hill Profssioal,

6 C C (ot irctio of ) (8.) / : xampl rfrshr multiplicatio matrix 6

7 7

8 C β l k β l k 8

9 C ) ( ) ( ) ( ) ( C c (8.) 9

10 ( ) ( ) ( ) ( ) z z z z For a wav propagatig i th z irctio, Sic th structur is ifiitly log, th voltag a currt at th th trmials ca iffr from th voltag a currt at th trmials oly y th propagatig factor, C (8.3) (8.4) C From (8.), For a otrivial solutio, th trmiat of th aov matrix must vaish: ) ( C Sic C, ) ( (8.5) (8.6) C

11 ) ( ( ) ) ( h β α & β α β α h h h From (8.), (8.7) (8.8)

12 Hyprolic Fuctio Rfrshr h

13 h hα β hα β α β Sic th right-ha si of (8.8) is purly ral, or α Cas#: Propagatig, No-ttuatig > PSSN β β α Cas#: ttuatig, No-Propagatig > STOPN β, π hα (8.9a) (8.9) pig o frqucy a ormaliz suscptac, th prioically loa li will xhiit ithr passas or stopas a thrfor act as a filtr. Rmmr that th quatios ar for & wavs fi at trmials of uit clls a o t cssarily scri coitios at othr poits alog th li. Ths ar similar to loch wavs. 3

14 loch givs his am to th charactristic impac of ths wavs: From (8.5), ( ) (8.) From (8.6), ( ) ( ) ± ( ) 4 ± m ( ) 4 So w ca solv for So w ca solv for th two solutios of th loch impac: Sic th uit cll is symmtrical, ± (8.) ± (8.) 4

15 From (8.) w s that is always purly imagiary. f α, β (passa), th, for symmtrical tworks: ( ) h h hα β hα β β ± ± (8.) shows that will ral. α, β f (stopa), th, for symmtrical tworks: h hα ± ± (8.) shows that will imagiary. This situatio is similar to that for th wav impac of a wavgui, which is ral for propagatig mos a imagiary for cutoff, or vasct, mos. 5

16 W arlir assum that th structur was ifiitly log, ut to implmt this filtr w will to trmiat th li. f th loa impac os t match our loch impac, thr will rflctios, which will ivaliat our arlir work. β β β β (8.4) L L Γ To avoi rflctios, L must match, which is ral for a losslss structur opratig i a passa. f cssary, a quartr-wav trasformr ca us tw th prioically loa li a th loa. 6

17 kβiagrams (Wavgui) β k k c k v p c β β v g β c k β For k< k c, thr is o solutio for β (rilloui iagram) 7

18 kβ iagram (Prioically Loa Li Exampl) β (8.9) 8

19 mag Paramtr Mtho of Filtr sig (Sctio 8.) C i C Sic C Figur 8.7 shows a aritrary, rciprocal two-port twork with imag impacs fi as follows: i iput impac at wh is trmiat with i i iput impac at wh is trmiat with i C i i C i C C i i i i W wat i i a i C i C f symmtric, i i 9

20 i i C ( C) ( C) C Ci

21 Two importat typs of two-port tworks ar th T a π circuits, which ca ma i symmtric form. Tal 8. list th imag impacs a propagatio factors, alog with othr usful paramtrs, for ths two tworks.

22 c c c 4, LC C L C L it k C L R LC c, c it R it R wh

23 L c, R LC C k Thr ar oly two paramtrs to choos (L a C), which ar trmi y th cutoff frqucy a th imag impac at zro frqucy. Ths rsults ar oly vali wh th filtr sctio is trmiat i its imag impac, which is a fuctio of frqucy a is ot likly to mach a giv sourc or loa impac. ts attuatio is t vry goo i th stopa. L c, R k LC C 3

24 To improv our sig from th costat-k filtr, w ar goig to try th m-riv filtr. Rplac with a with whr m Choos to kp it th sam: m m it ( ) m m m m m m 4

25 C L, For a low-pass filtr, ( ) L m m Cm Lm 4, So th m-riv impacs will : 4 ( ) ( ) 4 c c m m m m L Cm Lm LC c ( ) 4 c c m 5

26 4 ( ) 4 c c m f w rstrict < m <, th ths rsults show that is ral a > for > Thus th stopa gis at, as for th costat-k sctio. Howvr, wh, whr, c m c c coms ifiit. Th m-riv sctio has a vry sharp cutoff ut th th attuatio crass as To hav ifiit attuatio as w ca casca it with a costat-k sctio. 6

27 Th m-riv T-sctio was sig so that its imag impac was itical to that of th costat-k sctio (ipt of m), so w still hav th prolm of a ocostat imag impac. ut a π- sctio s imag impac os p o m. y austig m as, w ca us this to optimiz our match. iπ ( m ) c c R 7

28 To fit from th π-sctio s aility to kp a rlativly costat imag impac ut still match up with costat-k or sharp cutoff T-sctio, w will isct a π-sctio. 8

29 9

30 3

31 3

32 ackup Slis

33 C C C C

34 ) ( ) ( ) ( ) ( C C C