Kesim ötesi dalga klavuzu filtreleri

Abstract

Mikrodalga filtre devreleri genel olarak geçirme bandı içinde önceden belirlenen bir zayıflama karakteristiğine uyacak biçimde tasarlanmış ve mikrodalga enerjisini bu özellik uyarınca yüke aktaran iki kapılı pasif düzenler olarak tanımlanabilir. Geçirme bandındaki zayıflama karakteristiği genellikle kayıpsız ve resiprok reaktif elemanlarla gerçekleştirilir. Geçirme bandı dışında ise enerjinin büyük bölümü girişe yansıtılır. Pratikte bu amaçla dalga kılavuzları kulla nıldığında genelde bir süreksizlik tanımlanmaktadır. Bu problemlerde birçok durumda sadece dominant modun yayıldığı, üst dereceden diğer modların ise sönümlü olduğu haller göz önüne alınmaktadır. Bu çalışmada ele alınan yapı sadece dominant modun propagasyonuna izin veren iki eş dalga kılavuzu arasındaki kesimdeki dalga kılavuzundan oluşmaktadır. Çalışmamızda önce bu yapı için alan bağıntıları yazılarak, sınır koşulları yardımıyla denklemler elde edilmiş ve bu eşitlikler matematik kullanım açısından bilgisayara elverişli hale getirilecek şekilde moment metodunun kullanımı ile çözülmüştür. Daha sonra bu çözümü hızlandırmak amacıyla analitik ifadeler incelenerek önemli basitleşmeler sağlayan bir yaklaşık yöntem geliştirilmiştir. Son olarak ele alınan yapı ve 4. Bölümde verilen direkt kuple rezonatörlü filtre tasarım yöntemi kullanılarak bir ve üç elemanlı band geçiren filtreler tasarlanmıştır ve bu tip filtre için bir yaklaşık model önerilmiştir. Sayısal sonuçlar, bu çalışmada önerilen yöntemin özellikle yüksek Q'lu filtrelerin tasarımı açısından avantajlı olabileceğini göstermektedir.

A microwave filter is a passive two-port which provides the transmission of microwave energy to a matched load in accordance with a prescribed attenuation versus frequency response. Design of the microwave filter begins with the definition of the pass-band, that is, the frequency range in which the insertion loss does not exceed some specified value. Outside the pass band, the insertion loss must be as high as practicable. To minimize insertion loss or attenuation within the pass-band microwave filters are usually made with lossless and reciprocal elements. When waveguides are used in practice, it is always necessary to introduce discontinuities. The presence of a discontinuity gives rise to a reflected wave and a storage of reactive energy in the vicinity of the discontinuity because of excitation of higher-order waveguide modes which are evanescent, i.e., decay exponentially with distance from the discontinuity. Under most conditions, in practice, only the dominant mode can propagate in the guide. A microwave filter is a cascade connection of several sections. Here we will consider lengths of evanescent waveguides as generic sections. The structure consists of two identical TEjo mode waveguides connected via a nonpropagating waveguide section of length das shown in Fig.l. TE 10 L Matched Termination oi-»z Figure 1: Evanescent mode waveguide section. We assume TE-.Q mode incident from left and that the other pert i's match terminated [1].. Due to symmetry of the discontinuity we conclude that the excited higher order modes are TE only. a mo J (v) For TE " modes field components read; mo Ez-0 (la) V-Hm Cos(iHl)e tJ3z (lb) ±J3z y m nr ' (1c) E*-0 xm (Id) * H* - E* Ym f(x) e xm m m nr ' 1^-0 ym ±j$z (le) (If) where ± indicates propagation in ± z directions and m sin (- x) n= 1,2,... a wy am m Jwy Y-="J 62 - ^-(7r/a)2 m Na ' o kp **= a)2ey By using equations (1) we can write field components for the three regions as (see Fig.l), for z<0 (vi) -JBz J3z °° _ anz -JBz J8z °° anz for 0<zd + -jB(z-d) » -a(z-d) Ey=Elfle \İ2EnVne (2-e) -j3(z-d) co -a (z-d) Hx~Wle *£ EnVne (2'f) EaV Hal ancl Ea2'Ha2 ^P^5^* electric and magnetic fields at z=0 and z=d respectively. Since the fields are continuous at z=0 and z=d and E&i, E ~= 0 xGtap.a, ) we obtain two coupled integral equations. We can use the method of moments to solve them, as explained in appendix C. After choosing f. as expansion functions we write, Eal * £ Ctft (3. a) Ea2 * ^ Dtft (3,b) al (vii) F â 2 /T (-l)m(- ) - Sin(£7tt) l>m * (£t)2- m2 Using above definitions we write, N P N D.S. ZVi,r * L E Y Fn Fn. ? Y.Coth(ai.ci)6.]C.- Z Y -LİÜE 11lV t=l L n=l n n'v "'* t * v'tJ * t=l ^inhC^d) (4a) ° = A [ 1 Vn.vFn.t*VottKd)«v.*]V * Yt ^^ W t_l n_l t=l Sinh(atd) In matrix notation, with obvious definitions one has, [U] - [Q] [C] -[R] [D] (5a) [0] - [ . denote the module and phase of S,, for the i.th symmetric lossless 2-port S-, respectively. Introducing a frequency variable, V"anbnbn-1 *= ] '2 N * ^\ (10> one can obtain an explicit expression for the input reflection oefficier in" bl K coefficient p. of the filter network PN Pin- h /- (Ilia) where P^ and QN are A polynomials which are given by, (ix) P = I r V a N_1 N N+1 N « i n 4i Vi+ z z z r r-r n-l i-1 n * n=1 m=n+1 p=m+1 n m p n-1 p_m n a"_4 n a. i=l n~1 ill "p-j N-3 N-2 N-l 'N N+i + z z 2 i zrrrrr n=l m=n+l p=m+l r=p+l s=r+l n m P r s' n-l p-m s-r ", N N+l m.n qn" 1+ t- î r"r n a n-l m-n+1- n m i^i m-i N-2 N-l n N+l + z 2 r irrrr n-l m=n+l p=m+i r-=p+l n m P r " nt-n r-p (11. c) n Am. n a + i~l m~i j==1 Ar-j + --- less ratT?o6 1^*11°" ^ ' °f the C1>cu1t 1s exP^ssed via the power 1= 10£og.PLR(dB); P^ (H Pin|2)~ W^A.b) and b can be expressed as, P^s-band msertlon loss, respectively A= e (11. d) OJ-'-O) b- si"-[_^,.K)J (i] (x) For given design objectives A(w0) is fixed and one can derive via equations (11) a set of nonlinear equations. An iterative algorithm can be used to solve these equations for the unknowns T- in (9b) [2]. Noting the interconnection of r. with S-,-, of the cutoff wavequide section as defined by (7b) and the relation between the electrical length 6^ of the propagating unimodal waveguide sections and the argument of S,, with A as given by equations (9a), (9b) and (10) one can complete the design of the cut-off waveguide filter. Two approximations are implicit in the above derivations: 1. An (w) - A (w) n = 1,2,..,N n 2. rn (q)) -rn(oj0) n ~1,2,..,N (12) To test the validity of these approximations we have worked out a design example with the objectives of 0.5 dB equal-ripple within ±100 MHz band centered at 10 GHz, using standard WG90 as the propagating section and a cut-off waveguide which is obtained from WG90 by shortening the grater dimension by 10 mm.s. The dimensions of the designed 3. element filter and its calculated response are depicted in Figures 2a. and 2b. Clearly, the required specifications are met with remarkable accuracy. On the other hand, being a high Q device one would expect rather strong correlation between the response and the physical dimensions of the filter. To obtain a rough estimate on the effect of dimensional tolerances.we have chosen to work out numerical tests by lengthening and for shorting all dimensions simultaneously. Our calculations indicate that the required machining tolerances should be in the order of ± 10 ym's for the presented example. It should also be noted that conductor losses are not accounted for in the present formulation of the analysis. Finally we have worked out a simple network model for the cut off waveguide section consisting of two inductive discontinuities separated by a cut-off model transmission line section. As can be seen from calculated response given in Figure 2b, the model is not sufficiently accurate although it..greatly simplifies the calculations.</z