Analysis of structures formed with shunt capacitor seperated by transmission lines

Abstract

Bobin ve kondansatorlerden olusan merdiven devreler hakkinda literaturde bircok calisma bulunmaktadir. Fakat genellikle agir ve buyuk olduklari icin uretilebilecek deger araliklarinin sinirli olmasi ve mikrodalga frekanslardaki uretim gucluklerinden dolayi tasarlanan devrelerde bobin olmasi istenmez dagitilmis elemanlar (iletim hatlari) ile yaklasik olarak gerceklenmeye calisilirlar. Toplu elemanlarin dagitilmis elemanlara cevrilmesinde Richards donusumu kullanilir. Paralel bagli kondansatorlerin arasindaki seri bobinleri esit uzunlukta iletim hatlari ile degistirelim. Bu iletim hatlari Birim Eleman (BE) olarak isimlendirilir. Eger hat parcalarinin uzunlugu ceyrek dalga boyu olarak secilirse admitans inverterleri elde edilir. Bu yapilar ozellikle dar bantli (Bu tezde yukarida kisaca aciklanan yapilardaki iletim hatlarinin literaturdekinin aksine ceyrek dalga boyu uzunlugunda olma zorunlulugu yoktur. Bu sayede genis bantli devreler tasarlanabilmektedir. Ayrica kondansatorler arasindaki iletim hatlari sadece kondansatorleri ayiran elemanlar olarak yer almayip devrenin istenen cevabi vermesi icin devre elemani olarak kullanilmaktadir. Eger devrede toplu eleman (paralel kondansator) olmasi istenmezse Richards donusumu kullanilarak acik-uclu hat parcalari ile degistirilebilirler. Sonuc olarak elde edilen devre mikrostrip uretimi icin son derece elverisli bir yapi olacaktir. Bu tezde yukarida aciklanan devre yapisinin literaturdeki ilk analizi yapilmistir. Pratik acidan cok onemli bu yapinin iki-degiskenli tanimi detayli olarak verilmis gelistirilen algoritma ile bu yapi kullanilarak bir cok askeri ve ticari uygulamada yer alabilecek genisbant empedans uyumlastirma devresi tasarimlari gerceklestirilmistir. Bu iv tasarimlarin yapilabilmesi icin devreyi tanimlayan iki-degiskenli polinomlarin katsayilari tek-degiskenli sinir polinomlarinin katsayilari kullanilarak hesaplanmistir. Tum bu katsayi ifadeleri devredeki eleman sayisina yani devre derecesine baglidir. Devrede en fazla uc kondansator ve iki hat parcasi bulundugunda katsayi iliskileri elde edilebilmistir. Literature bu katsayi iliskileri kazandirilmistir. Toplu eleman iceren kisim sadece paralel bagli kondansatorlerden olustugu icin (dejenere devre) algoritma sonucunda elde edilen iki-degiskenli polinomlar kullanilarak kondansator degerlerinin hesaplanmasi icin yeni bir yaklasim gelistirilmistir. Ayrica gelistirilen algoritma ile normalize edilmis prototip uyumlastirma devresi tasarimlari yapilmaktadir. Dolayisiyla tasarimci tarafindan uygulamanin gerektirdigi frekans ve empedans degerlerine uygun normalizasyon sayilari secilerek tasarlanan devrenin istenen frekans bolgesinde ve empedans seviyesinde calismasi saglanabilmektedir.There are many works in literature about ladder networks containing inductors and capacitors. But usually it is not desired for the designed circuit to have inductors since they are heavy, bulky and available only for a limited range of values and are difficult to implement at microwave frequencies, they are approximated with distributed components. Richard's transformation is used to convert lumped elements to transmission line sections. Now consider a low-pass lumped ladder network. If the series inductors between the shunt capacitors are replaced with equal length transmission lines, a practically important mixed structure is obtained. Since the lengths of all the transmission lines are the same, these lines are called commensurate lines or unit elements (UE). It is very practical to fabricate this structure. If the transmission lines are quarter wavelength long, they are referred to as admittance inverters. These structures are useful especially for narrowband (<10%) bandpass and bandstop filters. In this thesis, as opposed to the structures existing in the literature and explained above briefly, it is not necessary to have quarter wavelength transmission lines. So it is possible to design broadband circuits. Also the transmission lines separating the parallel capacitors are not redundant elements, they are used as circuit elements effective for the desired response. Additionaly if it is preferred not to have shunt capacitors, they can be replaced with open-ended stubs via Richard's transformation. So the resultant circuit is extremely suitable for microstrip fabrication. In this thesis, the analysis of the mentioned mixed structure has been performed first time in the literature in the following manner. The description of the structure by means of two frequency variables (one for shunt capacitors and one for transmission lines) has been detailed. Then broadband matching networks for military and commercial applications have been designed by using this practically important mixed structure via the algorithm that has been developed. In the algorithm, the explicit expressions for the coefficients of the descriptive two-variable polynomials in terms of the coefficients of the single variable boundary polynomials have been derived for various numbers of elements. These coefficient relations have been obtained first time in the literature. Since the lumped section contains only shunt capacitors (a degenerate network), it is impossible to use the two-variable polynomials to calculate the capacitor values. So a synthesis algorithm for the structure has been developed to be able to get the capacitor values from the two variable polynomials. Normalized prototype circuits can be designed via the developed algorithm. So the prototype circuit can be denormalized via the frequency and impedance normalization numbers selected by the designer considering the interested frequency band and impedance level.