Synthesis of Multiplexers Based on Coupled Resonator Structures Using Coupling Matrix Optimization

Transcription

1 The Islamic Uiversity Of Gaza Postgraduate Studies Faculty of Egieerig Departmet of Commuicatio Systems Egieerig Master Thesis Sythesis of Multiplexers Based o Coupled Resoator Structures Usig Couplig Matrix Optimizatio Prepared by: Deeb A Tubail Supervisor Dr. Talal F. Skaik A thesis submitted for the Master degree of Sciece i Commuicatio Egieerig December ه

2 Abstract I this thesis, desig techiques of coupled resoator circuits used i sythesizig two port filters ad three port diplexers are developed to sythesize N port multiplexers. Novel geeral structures are proposed here ad they ca achieve a arbitrary umber of chaels, differet resposes ad various properties ad characteristics. The sythesis of the proposed multiplexers is based o optimizatio approach where the coupligs coefficiets betwee resoators preseted by couplig matrix are foud from optimizatio techiques by miimizig a cost fuctio. The cost fuctio which is utilized i this thesis has bee used previously i literatures. Scatterig parameters formulas are derived to suit the N port multiplexers. Differet structures with various properties ad resposes are give ad their results prove the ability of the geeral structure to achieve a massive scale of iterestig characteristics ad demads. The geeral structure is a cascade of diplexers which may reduce the complexity of the structure especially durig the optimizatio. This structure has lot of advatages, it has o limits for umber of chaels ad it has o extra resoators or exteral uctios ad power distributio etwork, also it has a small size. II

3 الملخص ف هز انشسانت تى تطى ش تق اث تص ى دوائش انش ان تشابطت ان ستخذيت ف تص ى ان ششحاث ث ائ ت ان ىافز نب اء األجهضة يتعذدة ان افز. وكزنك تى تقذ ى ه كه ت غ ش يأنىفت عايت نألجهضة يتعذدة ان افز قادسة عهى تص ى أي عذد ي انق ىاث باستجاباث يختهفت ويىاصفاث يتعذدة. تص ى هز األجهضة يتعذدة ان افز عت ذ عهى إ جاد أفضم يعايالث تشابط ب دوائش انش نتحقق أقم ق ت نذائشة انكهفت ح ث أ هز انذانت تى انحصىل عه ها ي دساساث سابقت. يعادالث ع اصش انتبذد تى اشتقاقها نت اسب األجهضة يتعذدة ان افز. تى تقذ ى أيثهت يت ىعت كبشها عهى قذسة هزا انه كم انجذ ذ انعاو عهى تح ق األ ىاع ان ختهفت ي االستجابت ب ىاصفاث وخصائص يتعذدة. انه كم ان قتشح ك اعتباس عذد يتتابع ي األجهضة ثالث ت ان افز ي ا ساعذ ف تقه م تعق ذ انه كم خصىصا ف ع ه اث تحق ق األيثه ت. هزا انه كم حقق انعذ ذ ي ان ضاث ان ت ثهت ف عذو يحذود ت عذد انق ىاث وكزنك عذو وجىد أي دائشة س إضاف ت أو وصهت إضاف ت أو شبكت نتىص ع انقذسة وأضف إنى رنك صغش حجى انه كم. III

4 Ackowledgmet I would like to thak my supervisor Dr. Talal F. Skaik for his cooperatio ad support. His ideas ad help have bee a sigificat factor for success i my thesis. Also I would like to express my respects to the electrical egieerig departmet at the Islamic uiversity i Gaza. Fially,I am highly idebted to my family ad frieds for their ecouragemets. IV

5 Table of Cotets Chapter Itroductio.... Overview of multiplexers ad their applicatios.... Overview of classical Aalog Filters Butterworth filters Chebyshev filter Elliptic filter Literature review Thesis motivatio Thesis overview... 9 REFERENCES... Chapter Coupled Resoator Circuits.... Itroductio.... Derivig Couplig Matrix of N-port Networks Circuits with magetically coupled resoators Circuits with electrically coupled resoators Geeral couplig matrix Coclusio... 5 REFERENCES... 6 Chapter 3 Sythesis of Multiplexers usig couplig Matrix Optimizatio Itroductio Optimizatio Frequecy trasformatio Derivatio of cost fuctio Multiplexer with the ovel topology Coclusio V

6 REFERENCES Chapter 4 Numerical Examples for Coupled Resoator Multiplexers Examples of multiplexers with ovel Topology Example : No-cotiguous arrow bad four chaels multiplexer with = 8, r = Example : No-cotiguous arrow bad four chaels multiplexer with =, r = Example 3: No-cotiguous bad four chaels multiplexer with =, r = Example 4: No-cotiguous bad four chaels multiplexer with Quasi-Elliptic resposes =, r = Example 5: No-cotiguous bad four chaels multiplexer cosists of two chaels with Quasi elliptic respose ad the other two chaels with Chebyshev respose ad =, r = Example 6: No-cotiguous bad four chaels multiplexer cosists of two chaels with Quasi elliptic respose ad the other two chaels with Chebyshev respose ad =, r = Example 7: No-cotiguous arrow bad six chaels multiplexer with =, r = Example 8: No-cotiguous arrow bad four chaels multiplexer with = 8, r = Example 9: No-cotiguous bad four chaels multiplexer with = 6, r =, r = Coclusio REFERENCES chapter Coclusio ad Future Work Coclusio Future Work... 8 VI

7 Chapter Itroductio. Overview of multiplexers ad their applicatios The term microwaves may be used to describe electromagetic (EM) waves with frequecies ragig from 3 MHz to 3 GHz, which correspod to wavelegths (i free space) from m to mm. The EM waves with frequecies above 3 GHz ad up to 3 GHz are also called millimeter waves because their wavelegths are i the millimeter rage ( mm). Therefore, by extesio, the RF/microwave applicatios ca be referred to as commuicatios, radar, avigatio, radio astroomy, sesig, medical istrumetatio, ad others that explore the usage of frequecy spectrums i the rage of, say, 3 khz up to 3 GHz. For coveiece, some of these frequecy spectrums are further divided ito may frequecy bads. Filters play importat roles i may RF/microwave applicatios. They are used to separate or combie differet frequecies. The electromagetic spectrum is limited ad has to be shared; filters are used to select or cofie the RF/microwave sigals withi assiged spectral limits. Emergig applicatios such as wireless commuicatios cotiue to challege RF/microwave filters with ever more striget requiremets higher performace, smaller size, lighter weight, ad lower cost. Depedig o the requiremets ad specificatios, RF/microwave filters may be desiged as lumped elemet or distributed elemet circuits; they may be realized i various trasmissio lie structures, such as waveguide, coaxial lie, ad microstrip. The recet advace of ovel materials ad fabricatio techologies, icludig moolithic microwave itegrated circuit (MMIC), microelectromechaic system (MEMS), micromachiig, high-temperature supercoductor (HTS), ad low-temperature cofired ceramics (LTCC), has stimulated the rapid developmet of ew microstrip ad other filters. I the meatime, advaces i computer-aided desig(cad) tools such as full-wave electromagetic (EM) simulators have revolutioized filter desig. May ovel microstrip filters with advaced filterig characteristics have bee demostrated []. Multiplexers (MUXs) are used i commuicatio system applicatios, where there is a eed to separate a widebad sigal ito a umber of arrowbad sigals (RF chaels). Chaelizatio of the allocated frequecy bad allows flexibility for the flow of commuicatio traffic i a multiuser eviromet. Amplificatio of idividual chaels also eases the requiremets o the high-power amplifiers (HPAs), eablig them to operate at relatively high efficiecy with a acceptable degree of oliearity. Multiplexers are also employed to provide the opposite fuctio, that is, to combie several arrowbad chaels ito a sigle widebad composite sigal for trasmissio via a commo atea. Multiplexers are, therefore, referred to as chaelizers or combiers. Due to the reciprocity of filter etworks, a MUX ca also be cofigured to separate the trasmit ad receive frequecy bads i a commo device, referred to as a duplexer or diplexer. Multiplexers have may applicatios such as i satellite payloads,

8 wireless systems, ad electroic warfare (EW) systems[]. Figure (.) shows the fuctio of multiplexer i satellite commuicatio as chaelizers or combiers[3]. Figure (.): Multiplexers i satellite commuicatio. Covetioally, multiplexer is achieved by usig a set of bad pass filters (usually kow as chael filters), ad a eergy distributio etwork. The chael filters pass frequecies withi a specified rage, ad reect frequecies outside the specified boudaries, ad the distributio etwork divides the sigal goig ito the filters, or combies the sigals comig from the filters. There are several approaches i desigig ad implemetig multiplexers. The most commo cofiguratios are maifold coupled, circulator-coupled, ad hybrid-coupled multiplexers []. The most commoly used distributio cofiguratios are E- or H- plae -furcated power dividers[4,5], circulators [6] ad maifold structures[7,8]. Figure (.) shows the cofiguratio of -chael multiplexer with a : divider multiplexig etwork, ad figure (.3) depicts a circulator cofiguratio, where each chael cosists of a bad pass filter ad a chael-droppig circulator. The power divider cofiguratios ca be desiged for multiplexers with widebad chaels or large chael separatio [4]. The circulator cofiguratios have o iteractio betwee chael filters ad they are simple to tue. They provide flexibility i addig ew chaels or replacig the chael filters by differet filters without disruptig the whole desig. However, they exhibit relatively higher losses sice sigals pass through the circulators i successio, causig extra loss per trip[]. I maifold cofiguratios, chael filters are coected by trasmissio lies: microstrip, coaxial, waveguide, etc. ad T-uctios. The cofiguratio of the maifold multiplexer is show i figure (.4). Maifold cofiguratios provide low isertio loss ad high power hadlig capability. However, they have complex desig, ad they do ot have the flexibility i addig chaels to a existig multiplexer, or

9 chagig a chael sice this requires a ew desig. Also, tuig the whole multiplexer ca be time cosumig []. Other multiplexer cofiguratios based o coupled resoators without exteral eergy distributio etworks have also bee proposed i literature. Star-uctio multiplexers are cosidered a geeral approach to the sythesis of microwave multiplexers presetig a star-uctio topology (with a resoatig uctio) [9]. Figure (.5) shows geeral architecture of the resoat star-uctio multiplexer[9]. Figure (.6) shows a geeral four-chael star uctio multiplexer topology []. The grey circle i Figure (.6) represets a resoat uctio, a extra resoator i additio to the resoators formig the filters. This multiplexer does ot iclude exteral uctios like the covetioal multiplexers, which makes miiaturizatio possible. Moreover, it has fewer coectios to the resoatig uctio tha the star -uctio multiplexers [,]. Figure (.): Cofiguratio of multiplexer with a : divider multiplexig etwork. Figure (.3): Cofiguratio of circulator-coupled multiplexer. 3

10 Figure (.4): Cofiguratio of maifold-coupled multiplexer. Figure (.5): Geeral architecture of the resoat star-uctio multiplexer. Figure (.6): Geeral four-chael star-uctio multiplexer topology. 4

11 . Overview of classical Aalog Filters... Butterworth filters The first family of aalog filters are the butter-worth filters which are called maximally flat filters. The butter-worth filter is desiged to have as flat a frequecy respose as possible i the pass-bad. A butter-worth filter of order is a low pass aalog filter with the followig squared magitude respose [6]. Figure (.7) preset the squared magitude respose of a low pass butter-worth filter. Figure (.7): Squared Magitude Respose of a Low-pass butterworth Filter.... Chebyshev filter The magitude resposes of Butterworth filters are smooth ad flat because of the maximally flat property. However, a drawback of the maximally flat property is that the trasitio bad of a Butterworth filter is ot as arrow as it could be. A effective way to decrease the width of the trasitio bad is to allow ripples or oscillatios i the pass-bad or the stop-bad. They are two types of Chebyshev, whe the ripple is i the pass-bad, it's called Chebyshev (type I) ad whe the ripple is i the stop-bad, it's called Chebyshev (type II) [6]. Figure (.8) (a) show Squared Magitude Resposes of a Chebyshev type I ad figure (.8) (b) Squared Magitude Resposes of a Chebyshev type II. 5

12 Figure (.8): (a) Squared Magitude Resposes of a Chebyshev type I. (b) Squared Magitude Resposes of a Chebyshev type II. 6

13 ..3. Elliptic filter The last classical low-pass aalog filter is the elliptic or Cauer filter. Elliptic filters are filters that are equiripple i both the pass-bad ad the stop-bad. The elliptic filter has the arrowest trasitio edge amog all types [6]. As the ripple i the stopbad approaches zero, the filter becomes a type I Chebyshev filter. As the ripple i the pass-bad approaches zero, the filter becomes a type II Chebyshev filter ad fially, as both ripple values approach zero, the filter becomes a Butterworth filter. Figure (.9) show Squared Magitude Resposes of a elliptic filter. Figure (.9): Squared Magitude Respose of a elliptic Filter..3 Literature review There have bee several techiques proposed to sythesize multiplexers : classical methods as maifold coupled, circulator-coupled, ad hybrid-coupled multiplexers ad coupled resoator circuit as moder techique also is used i sythesis of multiplexers. Here is a list of some previous researches iterested i desigig multiplexers by coupled resoat circuit as follows:. I [9], a ovel method for the polyomial sythesis of microwave star uctio multiplexers with a resoatig uctio has bee preseted. The chael filters ca be arbitrarily specified, icludig the assigmet of trasmissio zeros. A iterative procedure has bee developed for the evaluatio of the characteristic polyomials of the multiplexer, which are subsequetly used for 7

14 computig the polyomials associated with the chael filters; these polyomials are the employed for sythesizig the filters like they were detached from the multiplexer. I this way, the results of the sythesis process are ot costraied to a specific cofiguratio, which must be oly compatible with the assiged trasmissio zeros. However star uctio multiplexer has extra uctio i additio to the resoators which costruct multiplexer's chael.. I [], the author has preseted a ovel desig of coupled resoator star uctio multiplexer which has bee desiged at the X-bad with four o cotiguous chaels. The multiplexer topology is based o coupled resoator structure, ad it cosists of thirtee waveguide cavities, oe of which serves as a resoatig uctio. The multiplexer has reduced umber of coectios to the resoatig uctio ad also has smaller size tha other covetioal multiplexers, as it does ot cotai maifolds, circulators...etc. This desig cotais extra resoator which is't used i costructig multiplexer's chael. 3. I [], the author has proposed ovel topologies of star uctio multiplexers with resoatig uctios. These proposed topologies have a advatage that the umber of coectios to the resoatig uctio is reduced ad thus allowig multiplexers with more chaels to be implemeted. A optimizatio techique is used to sythesize the couplig matrix of the proposed multiplexers i this paper. However the resoatig uctios have a fewer coectio ad it is a extra resoator which icrease the size of multiplexer. 4. I [3], the authors have preseted a three chael multiplexer formed exclusively by coupled microwave resoators, ust like filters but with lager umber of ports. This multiplexer oly has three chaels ad this is usuitable for applicatios which eed more chaels. 5. I [4], a ovel procedure for sythesizig arrow bad triplexers for base statio combiers has bee preseted. The desig results have bee represeted by the couplig coefficiets ad exteral Qs of the filters costitutig the combier. The sythesis algorithm is very fast ad it allows to obtai a quasi equiripple respose i the three pass bads. This multiplexer talks about special case three chaels while lots of applicatios demad larger umber of chaels. 6. I [5], desig techiques used for two-port coupled resoator circuits has bee exteded to desig three-port microwave compoets such as power dividers with arbitrary power divisio ad diplexers with ovel topologies. The sythesis of these devices employs similar couplig matrix optimizatio techiques to those of coupled resoator filters. The three port devices oly has two chael but this is't suitable for applicatios that eed more chaels. 8

15 .4 Thesis motivatio May techiques have bee used i desigig multiplexer. Each techique differet from others ad has some advatages ad disadvatages as metioed i sectio.. The thesis addresses the developmet of a geeral ovel topology for N- chaels multiplexer by coupled resoators circuits without usig exteral distributios etworks ad hece compact multiplexers cab be desiged. Desigig multiplexers i covetioal techiques is achieved by usig a set of bad pass filters (usually kow as chael filters), ad a eergy distributio etwork (uctio) which is used to divide the icomig sigals ito the N chaels. Extesive work has bee reported i literature o miiaturizatio of multiplexers usig specific types of compact resoators or usig folded structures. However, the use of exteral uctios i the structures of these diplexers might ivolve desig complexity. Desig techiques for multiplexers based o coupled resoator structures without exteral uctios have also bee preseted i literature. These structures are miiaturized sice there are o exteral uctios or extra resoators i additio to the resoators formig filters. Coupled resoator circuits with multiple chaels are addressed i this thesis to sythesize compact ovel topologies for multiplexers with reduced desig complexity ad with o practical costraits i realizatio. Figure (.) illustrates a proposed geeral structure for N chael multiplexer without ay extra resoator or ay extra uctio. The isolatio betwee chaels chages by chagig umber of resoators per chael ad chagig the positio of chaels..5 Thesis overview The obective of this research work is sythesis of coupled-resoator circuits with multiple outputs (N chaels) by extedig the desig techiques used for threeport coupled resoator diplexers (two chael) proposed i[,5]. Figure (.) shows a topology for a two-chael coupled resoator diplexer, where the circles represet resoators ad the lies likig the resoators represet coupligs. Sythesis methods of coupled resoator diplexers have bee preseted i literature. The work i this thesis exteds the theory of two-chael coupled resoator diplexer to N- chaels coupled resoator circuits, such as the geeral etwork show i figure (.). This eables sythesis of other passive microwave compoets made of coupled resoators such as multi chael multiplexers. I this thesis a geeral ovel topology of multiplexer will be preseted ad multiplexers based o the ovel topologies with differet umber of chaels ad differet umber of resoators will be preseted. I chapter two circuits with both electrical ad magetic couplig are preseted. A detailed derivatio of the couplig matrix of multiple coupled resoators with multiple outputs is also preseted. The relatios betwee the scatterig parameters for N port etwork i additio to the geeral couplig matrix are also preseted i this chapter. These equatios i chapter are used as a basis to the sythesis of N chaels multiplexers i the ext chapters. I chapter three, frequecy trasformatio, derivatio of cost fuctio ad optimizatio are preseted. After that i chapter four, various examples for differet coupled resoators will be preseted whereby the couplig matrix obtaied from 9

16 optimizatio will be give as well as the ideal multiplexer respose of the scatterig parameters. The fial chapter provides summary ad coclusios draw from this work. Figure (.): A Geeral ovel N chael multiplexer topology. Figure (.): The structure of coupled resoators diplexer

17 REFERENCES. J.S. Hog ad M.J. Lacaster, "Microstrip filters for RF/microwave applicatios", Wiley,. R. Camero, C. Kudsia ad R. Masour, "Microwave filters for commuicatio systems", Wiley, D. Roddy, " Satellite Commuicatios", McGraw-Hill, J. A. Ruiz-Cruz, J. R. Moteo-Garai, J. M. Rebollar, ad S. Sobrio, "Compact full ku-bad triplexer with improved E-plae power divider," Progress I Electromagetics Research, Vol. 86, pp. 39-5, J. Dittloff, J. Borema, ad F. Ardt, "Computer aided desig of optimum E- or H-plae N-furcated waveguide power dividers," i Proc. Europea Microwave Coferece, Sept. 987, pp R.R. Masour, et al., "Desig cosideratios of supercoductive iput multiplexers for satellite applicatios," IEEE Trasactios o Microwave Theory ad Techiques, vol. 44, o. 7, pt., pp. 3-9, J. D. Rhodes ad R. Levy, "Desig of geeral maifold multiplexers," IEEE Trasactios o Microwave Theory ad Techiques, vol. 7, o., pp. -3, R.J. Camero ad M. Yu, "Desig of maifold coupled multiplexers," IEEE Microwave Mag., vol. 8, o. 5, pp , Oct G. Macchiarella ad S. Tamiazzo, "Sythesis of star-uctio multiplexers", IEEE Trasactios o Microwave Theory ad Techiques, Vol. 58, No., ,.. T. Skaik, "Sythesis of Coupled Resoator Circuits with Multiple Outputs usig Couplig Matrix Optimizatio", PhD thesis, Uiversity of Birmigham, March.. T. Skaik, "Star-Juctio X-bad Coupled Resoator Multiplexer," Proceedigs of the th Mediterraea Microwave Symposium, Istabul, Turkey, Sept.. T. Skaik, " Novel Star Juctio Coupled Resoator Multiplexer Structures," Progress i Electromagetics Research Letters, Vol. 3, pp. 3-, April. 3. A. Garcfa-Lamperez, M. Salazar-Palma, ad T. K. Sarkar, "Compact Multiplexer Formed by Coupled Resoators with Distributed Couplig", IEEE Ateas ad Propagatio Society Iteratioal Symposium, USA, 5, pp G. Macchiarella, S. Tamiazzo, "Desig of Triplexer Combiers for Base Statios of Mobile Commuicatios", The IEEE MTT-S Iteratioal Microwave Symposium, USA, May. 5. T. Skaik, M. Lacaster, ad F. Huag, Sythesis of multiple output coupled resoator microwave circuits usig couplig matrix optimizatio, IET Joural of Microwaves, Atea & Propagatio, vol.5, o.9, pp. 8-88, Jue. 6. R. J. Schillig ad S. L. Harris," Fudametals of Digital Sigal Processig Usig MATLAB", Cegage Learig,5.

18 Chapter Coupled Resoator Circuits. Itroductio Coupled resoator circuits are of importace for desig of RF/microwave filters (ad multiplexers), i particular the arrow-bad pass filters that play a sigificat role i may applicatios. There is a geeral techique for desigig coupled resoator filters i the sese that it ca be applied to ay type of resoator despite its physical structure. It has bee applied to the desig of waveguide filters, dielectric resoator filter, ceramic comblie filters, microstrip filters, supercoductig filters, ad micromachied filters. This desig method is based o couplig coefficiets of iter coupled resoators ad the exteral quality factors of the iput ad output resoators []. The geeral couplig matrix is of importace for represetig a wide rage of coupled-resoator filter topologies. It ca be formulated either from a set of loop equatios or from a set of ode equatios. This leads to a very useful formula for aalysis ad sythesis of coupled-resoator filter circuits i terms of couplig coefficiets ad exteral quality factors [].. Derivig Couplig Matrix of N-port Networks I a coupled resoator circuit, eergy may be coupled betwee adacet resoators by a magetic field or a electric field or both. The couplig matrix ca be derived from the equivalet circuit by formulatio of impedace matrix for magetically coupled resoators or admittace matrix for electrically coupled resoators []. This approach has bee used to derive the couplig matrix of coupled resoator filters, ad it is adopted i [] to derive geeral couplig matrix of a N-port -coupled resoators circuit. Magetic couplig ad electric couplig will be cosidered separately ad later a solutio will be geeralized for both types of coupligs [].... Circuits with magetically coupled resoators Cosiderig oly magetic couplig betwee adacet resoators, the equivalet circuit of magetically coupled -resoators with multiple ports is show i figure (.) [], where i represets loop curret, L, C deote the iductace ad capacitace, ad R deotes the resistace (represets a port). It is assumed that all the resoators are coected to ports, ad the sigal source is coected to resoator. It is also assumed that the couplig exists betwee all the resoators.

19 3 Figure (.): Equivalet circuit of magetically -coupled resoators i N-port etwork Usig Kirchoff's voltage law, the loop equatios are derived as follows, s e i L i L i C L R i L i C L R i L i C L R i L i L (.) where L L M i i ad L ab=l ba deotes the mutual iductace betwee resoators a ad b. The matrix form represetatio of these equatios is as follows, ) ( ) ( ) ( ) ( ) ( ) ( s e i i i i C L R L L L L C L R L L L L C L R L L L L C L R

20 (.) or equivaletly [Z].[i]=[e], where [Z] is the impedace matrix. Assumig all resoators are sychroized at the same resoat frequecy LC, where L = L = L = = L - = L ad C = C = C = = C - = C the impedace matrix [Z] ca be expressed by Z LFBW. Z where FBW is the fractioal badwidth, ad Z is the ormalized impedace matrix, give by, Z R P L( FBW ) L L FBW L( ) L FBW L L FBW L L FBW R P L( FBW ) L( ) L FBW L L FBW L( ) L FBW L( ) L FBW R P L( FBW ) L( ) L FBW L L FBW L L FBW L( ) L FBW R P L( FBW ) (.3) where P FBW is the complex low pass frequecy variable. Defiig the exteral quality factor for resoator i as Q L R couplig coefficiet as approximatio, Z is simplified to M i Li L, ad assumig ei i ad the for arrow bad Z qe m m( m P ) m P qe m m ( ) m m P q e( ) m ( ) ( ) ( ) m m m q e ( ) P (.4) 4

21 where q ei is the scaled exteral quality factor couplig coefficiet m i M i FBW q Qei FBW ad mi is the ormalized ei. The etwork represetatio for the circuit i figure (.), cosiderig N-ports, is show i figure (.), where a, b, a, b, a 3, b 3, a ad b are the wave variables, V, I, V, I, V 3, I 3, V ad I are the voltage ad curret variables ad i is the loop curret. It is assumed that port is coected to resoator, port is coected to resoator, port 3 is coected to resoator 3, ad port N is coected to resoator N. Figure (.): Network represetatio of N-port circuit The relatioships betwee the voltage ad curret variables ad the wave variables are defied as follows [3], V N ( ) R an bn ad I N (a N b N ) (.5) R Solvig the equatios (.5) for a N ad b N, the wave parameters are defied as follows, 5

22 6 ) ( N N N I R R V a ad ) ( N N N I R R V b (.6) where N is the port umber, ad R correspods to R for port, R for port, R 3 for port 3, ad R N for port N. It is oticed i the circuit i figure (.) that I =i, I = - i, I 3 = - i 3, I N = - i N, ad V =e s -i R. Accordigly, the wave variables may be rewritte as follows R e a s R R i e b s a R i b 3 a R i b N a N N N R i b (.7) The S-parameters are foud from the wave variables as follows, s a a a e R i a b S N... 3 s a a a e i R R a b S N... 3 s a a a e i R R a b S N s N N a a a N N e i R R a b S N... 3 (.8) Solvig (.) for currets loops,. Z L FBW e i s. Z L FBW e i s 3 3. Z L FBW e i s

23 i N e L FBW s. Z N (.9) ad by substitutio of equatios (.9) ito equatios (.8), we have, S R L. FBW Z S S 3 R R Z L. FBW RR3 Z 3 L. FBW S N RR N L. FBW Z N (.) I terms of exteral quality factors S q e Z q ei L. FBW R i, the S-parameters become, S S 3 q q e e q e e3 Z q Z 3 e Z N S N (.) q q en where q e, q e, q e3 ad q en are the ormalized exteral quality factors at resoators,, 3 ad N respectively. I case of asychroously tued coupled-resoator circuit, resoators may have differet resoat frequecies, ad extra etries m ii are added to the diagoal etries i Z to accout for asychroous tuig as follows, 7

24 Z qe P m m m m ( ) q e m P m m m ( ) q e( ) m ( ) m( ) P m( m ( ) )( ) q m m m e ( ) P m (.)... Circuits with electrically coupled resoators The couplig coefficiets itroduced i the previous sectio are based o magetic couplig. This sectio presets the derivatio of couplig coefficiets for electrically coupled resoators i a N-port circuit, where the electric couplig is represeted by capacitors []. The ormalized admittace matrix Y will be derived here i a aalogous way to the derivatio of the Z matrix i the previous sectio []. Show i figure (.3) is the equivalet circuit of electrically coupled - resoators i a N-port etwork, where i s represets the source curret, v i deotes the ode voltage, ad G represets port coductace. Figure (.3): Equivalet circuit of electrically -coupled resoators i N-port etwork. It is assumed here that all resoators are coected to ports, ad the curret source is coected to resoator. Also, it is assumed that all resoators are coupled to each other. The solutio of this etwork is foud by usig Kirchhoff s curret law, which states that the algebraic sum of the currets leavig a ode is zero. Usig this law, the ode voltage equatios are formulated as follows, 8

25 9 s i v C v C v C v L C G ) ( ) ( v C v C v L C G v C ) ( v L C G v C v C v C (.3) where C C M i i ad C ab=c ba deotes the mutual capacitace betwee resoators a ad b. The previous equatios are represeted i matrix form as follows, ) ( ) ( ) ( ) ( ) ( ) ( s i v v v v L C G C C C C L C G C C C C L C G C C C C L C G (.4) or equivaletly [Y].[v]=[i], where [Y] is the admittace matrix. Assumig all resoators are sychroized at the same resoat frequecy LC, where L=L =L = =L - =L ad C=C =C = =C - =C, the admittace matrix [Y] ca be expressed by Y Y CFBW., (.5) where FBW is the fractioal badwidth, ad Y is the ormalized impedace matrix, give by,

26 Y G P C( FBW ) C C FBW C( ) C FBW C C FBW C C FBW G P C( FBW ) C( ) C FBW C C FBW C( ) C FBW C( ) C FBW G P C( FBW ) C( ) C FBW C C FBW C C FBW C( ) C FBW G P C( FBW ) (.6) where P is the complex low pass frequecy variable. Defiig the exteral quality factor for resoator i as Q C G couplig coefficiet as approximatio, Y is simplified to M i Ci C, ad assumig ei i ad the for arrow bad Y qe m m( m P ) m P qe m m ( ) m m P q e( ) m ( ) ( ) ( ) m m m q e ( ) P where q ei is the scaled exteral quality factor couplig coefficiet m i M i FBW q (.7) Qei FBW ad mi is the ormalized ei. The etwork represetatio for the circuit i figure (.3), cosiderig N-ports, is show i figure (.4), where a, b, a, b, a 3, b 3, a ad b are the wave variables, V, I, V, I, V 3, I 3, V ad I are the voltage ad curret variables ad i is the loop curret. It is assumed that port is coected to resoator, port is coected to resoator, port 3 is coected to resoator 3, ad port N is coected to resoator N.

27 Figure (.4): Network represetatio of N-port circuit. It is oticed i the circuit i figure (.4) that V =v, V = v, V 3 =v 3, V N = V N, ad I =i s -v G. Accordigly, the wave variables may be rewritte as follows, a is G vg is b G a b v G a 3 b3 v3 G3 a N bn vn GN (.8)

28 The S-parameters are foud from the wave variables as follows, S b a a a... a 3 N vg i s S b a a a3... an G G i s v S 3 b a 3 a a3... an G G i s 3 v 3 S N b a N a a3... an G G i s N v N (.9) Solvig (.4) for voltage odes v is C. FBW Y v v 3 is CFBW is C. FBW Y Y 3 v N i C FBW s. Y N (.) ad by substitutio of equatios (.) ito equatios (.9), we have, S S S G C. FBW 3 Y GG Y CFBW GG3 Y 3 C. FBW S N GG N C. FBW Y N (.)

29 I terms of exteral quality factors q ei CFBW G i, the S-parameters become, S qe Y S S 3 q q e e q e e3 Y q Y 3 e Y N S N (.) q q en To accout for asychroous tuig, the ormalized admittace matrix will have extra terms m ii i the pricipal diagoal, ad it will be idetical to the ormalized impedace matrix i equatio (.)...3. Geeral couplig matrix The derivatios i the previous sectios show that the ormalized admittace matrix of electrically coupled resoators is idetical to the ormalized impedace matrix of magetically coupled resoators. Accordigly, a uified solutio may be formulated regardless of the type of couplig. I cosequece, the S parameters of a N-port coupled resoator circuit may be geeralized as, S q e A S S 3 q q e e q e e3 A q A 3 e A N S N (.3) q q en 3

30 The matrix [A] is give below A qe q e( ) P qe m m( m ) m m ( ) ( )( ) m ( ) m m ( ) m (.4) The formulae i (.3) ad (.4) will be used as a basis to sythesize N-port coupled resoator multiplexer i the ext chapters. For completeess, the geeral formulae of the scatterig parameters ca be derived aalogously to the previous derivatios, ad they are give by S xx A xx, q ex A S xy xy. (.5) q q ex ey The iverse of the matrix [A] ca be described i terms of the adugate ad determiat by employig Cramer s rule for the iverse of a matrix, A ad A, A A A (.6) where ad is the adugate of the matrix [A], ad A is its determiat. Notig that the adugate is the traspose of the matrix cofactors, the (x,) elemet of the iverse of matrix [ A] is: A xy where cof xy A A A 4 (.7) cof xy is the (x,y) elemet of the cofactor matrix of [A]. By substitutio of (.7) ito (.5), the followig equatios are obtaied, S S xx xy q q ex ex q cof xx ey cof A A xy A A (.8)

31 The coupled resoator compoets may be sythesized usig differet ways: aalytic solutio to calculate the couplig coefficiets, or full sythesis usig EM simulatio tools, whereby the dimesios of the physical structure are optimized, or optimizatio techiques to sythesize the couplig matrix [m].the use of full-wave EM simulatio is very time cosumig whe compared to couplig matrix optimizatio that requires sigificatly less computatioal time. Couplig matrix optimizatio techiques similar to those used to sythesize coupled-resoator filters will be utilized i the curret work to produce the couplig matrix etries of the proposed coupled resoator multiplexers. The etries of the couplig matrix [m] are modified at each iteratio i the optimizatio process util a optimal solutio is foud such that a scalar cost fuctio is miimized. Optimizatio techiques ad cost fuctio formulatio will be discussed i Chapter 3..3 Coclusio The derivatio of the couplig matrix of multiple coupled resoators with multiple outputs has bee preseted. A uified solutio has bee preseted for both electrically ad magetically coupled resoators. The relatioships betwee the scatterig parameters ad the couplig matrix of a N-port coupled resoator circuit have bee formulated. The equatios i this chapter will be used as a basis i the sythesis of coupled resoator multiplexers i the ext chapters. 5

32 REFERENCES. J.S. Hog ad M.J. Lacaster, "Microstrip filters for RF/microwave applicatios". New York: Wiley,.. T Skaik, "Sythesis of Coupled Resoator Circuits with Multiple Outputs usig Couplig Matrix Optimizatio", PhD thesis, Uiversity of Birmigham,. 3. M. Radmaesh, RF & Microwave Desig Essetials, Authorhouse, 7. 6

33 Chapter 3 Sythesis of Multiplexers usig couplig Matrix Optimizatio 3. Itroductio This chapter presets the sythesis of multiplexers usig couplig Matrix Optimizatio. I chapter, a geeral overview has bee preseted about covetioal multiplexers ad their advatages ad disadvatages. All of these multiplexers have differet eergy distributio etwork. I chapter, the theory of coupled resoator circuits for multiple ports has bee preseted ad couplig matrix ad scatterig parameters for multiple ports etwork have bee derived. I this chapter, frequecy trasformatio is preseted, after that based o theory i chapter two, the cost fuctio is formulated to be used i the optimizatio algorithm. Novel coupled resoator topologies are proposed i this chapter ad their sythesis based o couplig matrix optimizatio will be show. Numerical examples will be show i the ext chapter. 3. Optimizatio Optimizatio theory is a body of mathematical results ad umerical methods for fidig ad idetifyig the best cadidate from a collectio of alteratives without havig to explicitly eumerate ad evaluate all possible alteratives. The process of optimizatio lies at the root of egieerig, sice the classical fuctio of the egieer is to desig ew, better, more efficiet, ad less expesive systems as well as to devise plas ad procedures for the improved operatio of existig systems. The power of optimizatio methods to determie the best case without actually testig all possible cases comes through the use of a modest level of mathematics ad at the cost of performig iterative umerical calculatios usig clearly defied logical procedures or algorithms implemeted o computig machies. The developmet of optimizatio methodology will therefore require some facility with basic vector matrix maipulatios, a bit of liear algebra ad calculus, ad some elemets of real aalysis. We use mathematical cocepts ad costructios ot simply to add rigor to the proceedigs but because they are the laguage i terms of which calculatio procedures are best developed, defied, ad uderstood. Because of the scope of most egieerig applicatios ad the tedium of the umerical calculatios ivolved i optimizatio algorithms, the techiques of optimizatio are iteded primarily for computer implemetatio. However, although the methodology is developed with computers i mid, we do ot delve ito the details of program desig ad codig. Istead, our emphasis is o the ideas ad logic uderlyig the methods, o the factors ivolved i selectig the appropriate techiques, ad o the cosideratios importat to successful egieerig applicatio []. 7

34 The problem of optimizatio may be formulated as miimizatio of a scalar obective fuctio U, where U is also kow as a error fuctio or cost fuctio because it represets the differece betwee the performace achieved at ay stage ad the desired specificatios. For example, i the case of a microwave filter, the formulatio of U may ivolve the specified ad achieved values of the isertio loss ad the retur loss i the pass bad, ad the reectio i the stop bad. I this thesis, the optimizatio process aims to miimize the cost fuctio which is specified by the reflectio loss, trasmissio loss ad the locatios of reflectio zeros. The outputs of optimizatios are couplig coefficiets ad the locatios of reflectio zeros. U. This does ot Optimizatio problems are usually formulated as miimizatio of cause ay loss of geerality, sice the miima of a fuctio U correspod to the maxima of the fuctio U. Thus, by a proper choice of U, ay maximizatio problem may be reformulated as a miimizatio problem. is the set of desigable parameters whose values may be modified durig the optimizatio process. I microwave filters, firstly elemets of could be the values of capacitors ad iductors for a lumped-elemet or low pass prototype filter, or they could be couplig coefficiets for a coupled resoator circuit. But at last, elemets of could directly iclude the physical dimesios of a filter, which are realized usig microstrip or other microwave trasmissio lie structures. Usually, there are various costraits o the desigable parameters for a feasible solutio obtaied by optimizatio. For istace, available or achievable values of lumped elemets, the miimum values of microstrip lie width, ad coupled microstrip lie spacig that ca be etched. The elemets of defie a space. A portio of this space where all the costraits are satisfied is called the desig space D. I the optimizatio process, we look for optimum value of iside D []. I microwave coupled resoator optimizatio problems, as well as real world optimizatio problems, the cost fuctio of may variables will have several local miima, oe of them is the global miimum. Local optimizatio methods are used to fid a arbitrary local miimum, which is relatively straightforward. However, fidig the global miimum is more challegig ad global optimizatio methods ca be used. Local optimizatio algorithms strogly deped o the iitial values of the cotrol parameters. The iitial guess should be give as a iput to the algorithm that will seek a local miimum withi the local eighborhood of the iitial guess. However, this local miimum is ot guarateed to be the global miimum. Global optimizatio algorithms geerally do ot require iitial guess for the cotrol variables, as they geerate their ow iitial values, ad they seek the global miimum withi the etire search space. I compariso to local methods, global optimizatio methods are much slower ad may take hours or eve days to fid the optimal solutio for problems with tes of variables. Global algorithms ted to be utilized whe the local algorithms are ot adequate, or whe it is of great importace to fid the global solutio [3]. Lots of optimizatio methods have bee developed for solvig costraied ad ucostraied optimizatio problems. Direct Search Optimizatio is a approach used i solvig optimizatio problems. It makes repeated use of evaluatio of the obective fuctio ad does ot require the derivatives of the obective fuctio. Two typical types of the direct search method are described as follows. Powell s method is a powerful direct search method for multidimesioal optimizatio. A geetic algorithm is the other type of the direct search method which starts with a iitial set of radom 8

35 cofiguratios ad uses a process similar to biological evolutio to improve upo them. The set of cofiguratios is called the populatio. Durig each iteratio, called a geeratio, the cofiguratios i the curret populatio are evaluated usig some measure of fitess []. The Nelder-Mead simplex algorithm is a eormously popular direct search method for multidimesioal ucostraied miimizatio. It's especially popular i the fields of chemistry, chemical egieerig, ad medicie. This method attempts to miimize a scalar-valued oliear fuctio of real variables usig oly fuctio values, without ay derivative iformatio (explicit or implicit) [4]. This algorithm is applied by usig "fmisearch" fuctio available i MATLAB to solve optimizatio problems i this thesis. I a gradiet-based optimizatio method, the derivatives of a obective fuctio with respect to the desigable parameters are used. The primary reaso for the use of derivatives is that at ay poit i the desig space, the egative gradiet directio would imply the directio of the greatest rate of decrease of the obective fuctio at that poit []. Lots of papers talk about sythesizig filters or diplexers usig optimizatio approach. I [6], The authors have used optimizatio approach i the their paper to sythesize a three chael multiplexer exclusively by coupled microwave resoators. Also i [7] the authors have sythesized diplexers by applyig couplig matrix optimizatio techiques to those of coupled resoator circuits. After that, i [8] the author proposed ovel topologies of star-uctio multiplexers with resoatig uctios. A optimizatio techique has bee used to sythesize the couplig matrix of the proposed multiplexers i this paper. Also i [9], the author preseted a ovel desig of coupled resoator star-uctio multiplexer which was desiged at the X- bad with four o cotiguous chaels by optimizatio method. Fially i [], the authors have sythesized filters by optimizatio a cost fuctio based o the Hausdorff distace betwee the template sets (the sets of zeros ad poles of template filter reflectio ad trasmissio characteristics). 3.3 Frequecy trasformatio The specificatios of a multiplexer are usually give i the bad-pass frequecy domai, i which the real multiplexer operates. As metioed earlier, the desig of the proposed multiplexers takes place i the ormalized frequecy domai as a low pass prototype. Therefore, a frequecy trasformatio from bad pass frequecy domai to ormalized frequecy domai is eeded. This sectio presets frequecy trasformatio formulas of bad pass multiplexer with give specificatio to a low pass prototype. Equatio (3.) is used to calculate the ormalized value for each edge i the multiplexer's chaels. A illustratio of the frequecy mappig is show i figure (3.). The frequecy edges of the bads of the multiplexer are ω, ω, ω 3,... ad ω. These frequecies are mapped ito low pass prototype frequecies (x) usig the followig trasformatio formula [3,5]. 9

36 x m m m Figure (3.): Low pass to bad pass trasformatio (3.) To map the bad edges x x to, ad x x to x (3.) x (3.3) Solvig equatios (3.) ad (3.3) yields, (3.4) x where FBW (3.5) FBW The value of the low pass cutoff frequecy x is ormally take as π radia/sec, ad the values of x m ca ow be foud from equatio (3.) [3]. 3

37 3.4 Derivatio of cost fuctio A cost fuctio that is used i the optimizatio of the couplig matrix of coupled resoator multiplexer is formulated here. For two ports filter i [] S S s s F E s s s s P (3.6) E Substitutig (3.6) i the eergy coservatio formula S S (3.7) ad maipulatig, the followig expressio is achieved E F P (3.8) where E, F ad P are all ormalized such that their highest degree coefficiets are equal to oe. "This shows that apart from the, scalig factor, a combiatio of ay two polyomials ca completely describe the respose. The problem is resolved by aalytically calculatig the exteral quality factors. Settig these factors to their correct values at the outset of the optimizatio cacels the eed to ivoke the ripple at ay stage of the sythesis procedure. Cosequetly, a combiatio of ay two polyomials ca completely determie the respose as log as the quality factors are predetermied. As far as speed is cocered, polyomials F ad P prove most efficiet to use i the cost fuctio" []. The cost fuctio formulated as Ω R i P R N (Sti ) F(Sr ) where R is the umber of reflectio zeros., (3.9) This is derived i chapter (4) i [3] for diplexers ad utilized here for multiplexers. For a coupled resoator multiplexer, the reflectio ad trasmissio fuctios may be defied i terms of polyomials as follows, S s F E s s, 3

38 s s PN S N s (3.) E where the roots of F(s) correspod to the reflectio zeros, the roots of P N (s) correspod to the trasmissio zeros of the filter frequecy respose at ports N, ε is a ripple costat, ad E(s) roots correspod to the pole positios of the filterig fuctio. The iitial cost fuctio may be writte i terms of the characteristic polyomials as follows, Ω M T N N i P N (S ti ) R F(S r ) R( M ) v F E S S pv pv LR (3.) where M is the umber of ports, Sti are the frequecy locatios of trasmissio zeros of S N, T N is the umber of the trasmissio zeros of S N, R is the total umber of resoators i the multiplexer, L R is the desired retur loss i db (L R <), ad S ad S are the frequecy locatios of the reflectio zeros ad the peaks' frequecy values of S i the pass bad. The last term i the cost fuctio is used to set the peaks of s s F Ss to the required retur loss level. It is assumed here that all chaels of the E multiplexer have the same retur loss level. Recall from sectio (..3), that for a N-port etwork of multiple coupled resoators, the scatterig parameters are expressed the i terms of the geeral matrix [A] (equatio (.8)) as follows, S S xx xy q ex q cof xx ex q ey A A cof xy A A r pv (3.) where x is the port coected to resoator x, y is the port coected to resoator y, Δ A is the determiat of the matrix [A] ad cof xy is the cofactor of matrix [A] evaluated by removig the x th row ad the y th colum of [A]. By equatig (3.) ad (3.), the polyomials P (s), P (s), F(s) ad E(s) are expressed i terms of the geeral matrix [A] as follows, PN ( s). cof q F s A s N e q en A s. cof q e A s 3

39 E s A s (3.3) By substitutio of the polyomials i equatio (3.3) ito equatio (3.), the cost fuctio is ow expressed i terms of determiats ad cofactors of the matrix [A] ad the exteral quality factors as follows, M TN N i.cof N q e q A s en ti.cof R A sr qe A s R( M ) r.cof As pv v q. s e A pv LR, (3.4) where q e ad q en are the exteral quality factors at ports ad N respectively, Δ A (S=x( is the determiat of the matrix [A] evaluated at the frequecy variable x, ad cof ab ([A(s=v)]) is the cofactor of matrix [A] evaluated by removig the a th row ad the b th colum of [A] ad fidig the determiat of the resultig matrix at the frequecy variable s=v. The first term i the cost fuctio is used if the multiplexer characteristics cotai trasmissio zeros. However, for a Chebyshev respose, this term may be used to miimize the trasmissio of each chael at the pass bad of the other chael, thus icreasig the isolatio betwee chael ports. Cosequetly, the frequecy locatios s ti are chose to be the bad edges of the chael at port m. I aother word, this term is used i Chebyshev whe much steeper edges of the chaels are eeded. The low pass frequecy positios of the reflectio zeros of the multiplexer are iitially set to be equally spaced i the optimizatio algorithm, ad later these positios are moved util equiripple level at specified isertio loss is achieved. The iitial guess of the locatios of reflectio zeros withi a multiplexer chael is preseted here. For a multiplexer chael with edges of (x,x ) Hz, the leftmost reflectio zero is located at (x +.)i Hz, ad the rightmost reflectio zero is located at (x -.)i Hz. The other reflectio zeros are equally spaced betwee (x +.)i ad (x -.)i with frequecy spacig as follows: ( x x m.4), (3.5) where m is the total umber of reflectio zeros withi a multiplexer chael. The values of the exteral quality factors are umerically calculated, ad their values are set at the begiig of the algorithm This reduces the optimizatio parameters set ad improves the covergece time. 33

40 q The ormalized exteral quality factors of these filters are related by: q exx e x x q e N i q ei qex x qe (3.6) m where q is the ormalized exteral quality factor of the filter with ex x edges of x ad x, N is total umber of chaels ad q e± is the ormalized exteral quality factor of the filter with edges of ±, that ca be calculated from the g- values. For a symmetrical multiplexer with chael edges (x,x ), the ormalized exteral quality factor at port (m) are calculated from first equatio i equatio (3.6), ad the ormalized exteral quality factor at the commo port is calculated from secod equatio i equatio (3.6). The variables that eed to be optimized i the optimizatio algorithm are the couplig coefficiets ad also the frequecy locatios of the reflectio zeros. 3.5 Multiplexer with the ovel topology Two geeral coupled resoator multiplexer topologies are proposed here. They are show i figures (3.) ad (3.3) ad they ca achieve Chebyshev ad quasi-elliptic resposes respectively. Firstly, the desig started with simple structure for four Chebyshev chaels ad it was cosisted of eight resoators ad oe resoator per chael, the the total umber of resoators was icreased by icreasig the umber of resoators per chaels. After that, quasi-elliptic resposes was achieved. Fially, the umber of chaels was icreased ad six chaels multiplexer was give as a example. As show i figure (3.) ad figure (3.3) where figure (3.) represets a geeral structure proposed for multiplexer havig Chebyshev respose ad figure (3.3) represets a geeral structure proposed for multiplexer havig quasi elliptic respose, the multiplexer cosists of resoators where is the total umber of resoators, ad c is the umber of the upper arms or lower arms. These resoators are distributed o c chael. The lower arms are mirror of the upper arms (symmetrical chaels) ad each arm represets a chael cosisted of r resoators. I figure (3.4), the chaels i the positive side are from upper arms ad the chaels i the egative side are from lower arms. Figure (3.4) shows that i this topology the umber of chaels ca be a arbitrary umber which is depedet o the umber of arms meaig that it ca be 34

41 icreased by icreasig the umber of arms. Also the umber of resoators per arm r ca chage the isolatio betwee chaels ad improve the selectivity. I other words icreasig the umber of resoators per arm icreases the isolatio. Resoators i each arm should have differet self-resoat frequecies (M ii ) to separate the multiplexer chaels from each other. The resoators i the vertical brach should have differet self-resoat frequecies to achieve disoit frequecy bads at the differet chaels. Cosequetly, for the high frequecy chaels to be at upper arms (ports), the resoators above the uctio resoator should have positive frequecy offsets (M ii > ), ad for the low frequecy chael to be at lower arms (ports), the resoators below the uctio resoator should have egative frequecy offsets ( M ii < ). This is supposed to reduce the complexity by achievig symmetry but i geeral chaels' self resoat frequecies may be iterchaged betwee some chaels to achieve some advatages such as iterchagig the positios of chaels to improve the isolatio. It should be oted that the work o ovel multiplexer topologies started from a simple eight resoator structure, with = 8 ad r =, followed by other experimets o addig arms to icrease chaels, also followed by other experimets o addig resoators to the vertical arms util arrivig to the geeralized topology give i figure (3.). The uctio resoator takes importat part i power distributio ad also it cotributes to the filter trasfer fuctio. Figure (3.): geeral structure proposed for multiplexer havig Chebyshev respose 35

42 Figure (3.3): geeral structure proposed for multiplexer havig quasi elliptic respose The topology of the multiplexer has bee eforced i the optimizatio algorithm, ad the followig coditios for couplig coefficiets have bee applied to simplify the optimizatio problem:. The self-resoat frequecies (M ii ) i the lower arm are the egative of the selfresoat frequecies (M ii ) i the upper arms as below 36

43 M M c, c M M, c, c 3, 3 4, c r c r c r c r4,..., M M,,. The self-resoat frequecies (M ii ) for resoators i the horizotal lie equal zero. M, M, M......, M c c r, r 3. The couplig coefficiets betwee the resoators i upper arms are equal to those i lower arms as below M, M 3,,..., M M c r c r c r c r4, 3, Figure (3.4): Chaels i low pass prototype The badwidth of each chael ca be determied by choosig the values of edges of chaels x, x, x 3, x 4,...,x c- ad x c. Chaels may have differet characteristics, some of them have Chebyshev resposes ad others have elliptic respose. 3.6 Coclusio I this chapter, the sythesis procure of coupled resoator multiplexers is preseted. Frequecy trasformatio formulas as well as cost fuctio have bee show. A overview of umber of optimizatio methods ad algorithms i geeral have bee preseted. The frequecy trasformatio from bad pass to low pass for multiplexers has bee derived, also the derivatio of cost fuctio used i multiplexer sythesis has bee declared. Fially the proposed topologies for multiplexer with both Chebyshev ad elliptic respose were itroduced to be used i the ext chapter. Numerical examples will be preseted i the ext chapter. 37

44 REFERENCES. A. Ravidra, K. M. Ragsdell ad G. V. Reklaitis, "Egieerig Optimizatio Methods ad Applicatios",Wiley,6.. J.S. Hog ad M.J. Lacaster, "Microstrip filters for RF/microwave applicatios". New York: Wiley,. 3. T. F. Skaik, "Sythesis of Coupled Resoator Circuits with Multiple Outputs usig Couplig Matrix Optimizatio", PhD thesis, the Uiversity of Birmigham,. 4. J.C. Lagarias, J. A. Reeds, M. H. Wright, ad P. E. Wright, "Covergece Properties of the Nelder-Mead Simplex Method i Low Dimesios," SIAM Joural of Optimizatio, Vol. 9 Number, pp. -47, I. Huter, "Theory ad Desig of Microwave Filters", the IEEE, Lodo, UK, 6. A. Garcfa-Lamperez, M. Salazar-Palma, ad T. K. Sarkar, "Compact Multiplexer Formed by Coupled Resoators with Distributed Couplig", IEEE Ateas ad Propagatio Society Iteratioal Symposium, USA, 5, pp T. Skaik, M. Lacaster, ad F. Huag, Sythesis of multiple output coupled resoator microwave circuits usig couplig matrix optimizatio, IET Joural of Microwaves, Atea & Propagatio, vol.5, o.9, pp. 8-88, Jue. 8. T. Skaik, "Novel Star Juctio Coupled Resoator Multiplexer Structures", Progress i Electromagetics Research Letters, Vol. 3, pp. 3-, April. 9. T. Skaik, "Star-Juctio X-bad Coupled Resoator Multiplexer", Proceedigs of the th Mediterraea Microwave Symposium, Istabul, Turkey, Sept.. T. Kacmaor, J. J. Michalski, J. Gulgowski, " Filter Tuig ad Couplig Matrix Sythesis by Optimizatio with Cost Fuctio Based o Zeros, Poles ad Hausdorff Distace ", The IEEE MTT-S Iteratioal Microwave Symposium, Caada, Jue.. A. B. Jayyousi ad M. J. Lacaster, "A gradiet-based optimizatio techique employig determiats for the sythesis of microwave coupled filters," IEEE MTT-S Iteratioal Microwave Symposium, USA, 4, pp

45 Chapter 4 Numerical Examples for Coupled Resoator Multiplexers 4. Examples of multiplexers with ovel Topology I this chapter, ie examples of multiplexers with the proposed ovel topologies are give where each example has differet characteristics from others ad each example with differet characteristics gives a proof for validity of the ew topology. The first example is the simplest example which cosists of miimum umber of resoators i the simplest image to costruct four chaels multiplexer. I the ext example the umber of resoators per arm is icreased ad thus the total umber of resoators is icreased which declares that the isolatio betwee each chael ca be improved by this way. The third example is as the previous oe, but it has wider badwidth tha the secod example. All the first three examples have Chebyshev respose, however i the fourth example Quasi-Elliptic resposes are achieved. The isolatio i example umber four is better tha isolatio i example three because the umber of resoators i example four is larger tha that i example three, ad the chaels are sharper tha chaels i third example due to quasi elliptic respose i example four. I the fifth example both the Chebyshev ad Quasi elliptic respose are achieved i the same structure, where two chaels have Chebyshev respose ad the other two chaels have Quasi elliptic respose. I all previous examples the frequecy locatios of reflectio zeros were ot optimized, but oly were the couplig coefficiets. But i sixth example, both the couplig coefficiet ad frequecy locatio of reflectio zeros are optimized to get equal S peaks with - db retur loss. I the seveth example the umber of chaels is icreased to six chaels by icreasig umber of arms. I the eighth example the isolatio is improved by iterchagig the positio of chaels as aother way to improve the isolatio. Fially, i ith example the two ier chaels have a differet order ad umber of resoators from the two outer chaels ad the validity of the results have bee checked. All previous examples show that the umber of chaels ca be cotrolled by umber of arms, ad umber of resoators ca be chaged to get differet characteristics, also ay type of resposes ca be achieved, all that based o coupled resoators circuits. The sythesis procedure starts with frequecy trasformatio from bad pass to low pass prototype usig ormalizatio equatios or trasformatio equatios (3.-3.5). After that usig equatios (3.5, 3.6) the iitial frequecy locatio reflectio zeros ad exteral quality factors are calculated. Fially the iitial frequecy locatio reflectio zeros, iitial couplig coefficiet, retur loss, exteral couplig factors ad trasmissio zeros are etered as parameters of cost fuctio equatio (3.4) to be optimized usig optimizatio algorithms. The results obtaied from optimizatio processes are couplig coefficiets ad frequecy locatios of reflectio zeros. 39

46 4... Example : No-cotiguous arrow bad four chaels multiplexer with = 8, r =, x =.8, x =.9, x 3 =.5, x 4 =.6. The multiplexer cosists of four o cotiguous chaels as show i figure (4.), the simplest structure, ad it has eight resoators i total with oe resoator per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by uequal guard bads. The edges of chaels,, 3 ad 4 are {-.6,-.5},{-.9,-.8},{.8,.9},{.5,.6}, respectively. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e3 =q e4 =q e7 =q e8 =3.96 ad q e = The locatios of retur zeros are calculated usig equatio (3.5) as.8i,.88i,.5i ad.58i. These locatios are the iitial locatios ad they may eed optimizatio as i some ext examples. Figure (4.): Structure of multiplexer oe. Table (4.): The geeral couplig matrix of the above structure. Resoators m m m 3 m 3 m 5 3 m 3 m 33 4 m 3 -m 33 5 m 5 m 56 6 m 56 m 67 m 67 7 m 67 m 77 8 m 67 -m 77 The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.47, m 3 =.98, m 5 =.635, m 33 =.84, m 56 =.964, m 67 =.65, m 77 =.55. For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m4, m67 m68, m33 m44 ad m77 m88. The optimizatio started with te iitial values {.,.,...,} to get the best result to be optimized to get fial values. This meas that the iitial values were put i loop to get the correct begiig. I this example oly the secod term i the cost 4

47 boudaries fuctio i equatio (3.4), the term related to retur zeros, was used i optimizatio ad the others were eglected. Table (4.) is the geeral optimizatio matrix of the structure i figure (4.). The umerical optimized couplig matrix is give i table (4.) ad the multiplexer prototype respose is depicted i figure (4.). Table (4.3) displays the realized values that are achieved by optimizatio versus the targets. Table (4.): The optimized couplig matrix of multiplexer oe. Resoators Table (4.3): The Realized values versus the targets. Item Retur loss(l R ) i db x x x 3 x 4 Target Realized values Percetage of error 9.%.3% 3.8% 4.7%.9% Figure (4.): The theoretical respose of multiplexer oe. 4

48 4... Example : No-cotiguous arrow bad four chaels multiplexer with =, r =, x =.4, x =.55, x 3 =.35, x 4 =.5. The multiplexer cosists of four o cotiguous chaels as show i figure (4.3), ad it has twelve resoators i total with two resoators per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by equal guard bads. The edges of chaels,,3 ad 4 are {-.5,-.35}, {-.55,-.4},{.4,.55},{.35,.5} respectively. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e5 = q e6 = q e = q e =.3547 ad q e = The locatios of retur zeros are calculated usig equatio (3.5) as.4i,.475i,.53i,.37i,.45i ad.48i. These locatios are the iitial locatios ad they may eed optimizatio as i some ext examples. Figure (4.3): structure of multiplexer two. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.773, m 3 =.774, m 7 =.778, m 33 =.463, m 35 =.764, m 55 =.464, m 78 =.758, m 89 =.383, m 99 =.3383, m 9 =.768, m =.4. For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m4, m35 m46, m89 m8, m9 m, m33 m 44, m55 m66, m99 m ad m m. The optimizatio started with te iitial values {.,.,...,} to get the best result to be optimized to get fial values. This meas that the iitial values were put i loop to get the correct begiig. I this example oly the secod term i the cost fuctio i equatio (3.4), the term related to retur zeros, was used i optimizatio ad the others were eglected. The optimized couplig matrix is give i table (4.4) ad the multiplexer prototype respose is depicted i figure (4.4), where figure (4.4)(a) represets S, S, S 3, S 4, S 5 while figure (4.4)(b) represets isolatio betwee every two adacet chaels S 3, S 4, S 35. Table (4.5) displays the realized values that are achieved by optimizatio versus the targets. 4

49 Figure (4.4): The theoretical respose of multiplexer two: (a) Reflectio loss ad isertio loss, (b) The isolatio betwee adacet chaels. 43

50 boudaries Table (4.4): The optimized couplig matrix of multiplexer two. Resoators Table (4.5): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db x x x 3 x %.3% 3.% 3.%.9% 44

51 4..3. Example 3: No-cotiguous bad four chaels multiplexer with =, r =, x =.4, x =.75, x 3 =.55, x 4 =.9. The multiplexer cosists of four o cotiguous chaels as show i figure (4.5), ad it has twelve resoators i total with two resoators per arm. The specified retur loss is db. The chaels have a equal badwidth with wider bad width tha previous examples ad they are separated by equal guard bads. The edges of chaels,,3 ad 4 are {-.9,-.55},{-.75,-.4},{.4,.75},{.55,.9} respectively. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e5 = q e6 = q e = q e = ad q e =.66. The locatios of retur zeros are calculated usig equatio (3.5) as.88i,.75i,.57i,.73i,.575i ad.4i. These locatios are the iitial locatios ad they may eed optimizatio as i some ext examples. Figure (4.5): structure of multiplexer three. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.348, m 3 =.545, m 7 =.83, m 33 =.563, m 35 =.96, m 55 =.596, m 78 =.836, m 89 =.5746, m 99 =.459, m 9 =.3, m = For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m 4, m35 m46, m89 m8, m9 m, m33 m44, m55 m66, m99 m m. ad m The optimizatio started with example two couplig coefficiets as iitial to get fial values. I this example oly the secod term i the cost fuctio i equatio (3.4), the term related to retur zeros, was used i optimizatio ad the others were eglected. The optimized couplig matrix is give i table (4.6) ad the multiplexer prototype respose is depicted i figure (4.6), where figure (4.6)(a) represets S, S, S 3, S 4, S 5 while figure (4.6)(b) represets isolatio betwee every two adacet chaels S 3, S 4, S 35. Table (4.7) displays the realized values that are achieved by optimizatio versus the targets. 45

52 boudaries Table (4.6): The optimized couplig matrix of multiplexer three. Resoators Table (4.7): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db % x.4.4 4% x % x % x % 46

53 Figure (4.6): The theoretical respose of multiplexer three: (a) Reflectio loss ad isertio loss, (b) The isolatio betwee adacet chaels. 47

54 4..4. Example 4: No-cotiguous bad four chaels multiplexer with Quasi-Elliptic resposes =, r = 4, x =.4, x =.75, x 3 =.55, x 4 =.9. The multiplexer cosists of four o cotiguous chaels as show i figure (4.7), ad it has twety resoators i total with four resoators per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by equal guard bads. The edges of chaels,, 3 ad 4 are {-.9,-.55},{-.75,-.4},{.4,.75},{.55,.9}, respectively. The eight trasmissio zeros locate at.3i,.85i,.45i ad i. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e9 = q e = q e9 = q e = ad q e = The locatios of retur zeros are calculated usig equatio (3.5) as.88i,.85i,.75i,.6475i,.57i,.73i,.655i,.575i,.4975i ad.4i. These locatios are the iitial locatios ad they may eed optimizatio as i some ext examples. I figure (4.7), solid lies i the multiplexer represet direct couplig, ad dashed lies represet cross couplig, ad Quasi-Elliptic resposes ca be achieved. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.3489, m 3 =.439, m 33 =.549, m 35 =.3, m 39 = -.37, m 55 =.56, m 57 =.9, m 77 =.567, m 79 =.39, m 99 =.5588, m =.8746, m =.9734, m 3 =.578, m 33 =.598, m 35 =.97, m 39 = -.7, m 55 =.757, m 57 =.77, m 77 =.73, m 79 =.37, m 99 =.789. For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m 4, m35 m46, m57 m68, m79 m8, m39 m4, m3 m4, m35 m46, m57 m 68, m79 m8, m39 m4, m33 m44, m55 m66, m77 m88, m99 m, m33 m44, m55 m66, m77 m88 ad m99 m. I this example the whole cost fuctio i equatio (3.4) has bee used except the third term that optimizes the retur zeros' locatios to eforce retur loss level of db. The optimized couplig matrix is give i table (4.9) ad the multiplexer prototype respose is depicted i figure (4.8), where figure (4.8) (a) represets S, S, S 3, S 4, S 5 while figure (4.8) (b) represets isolatio betwee every two adacet chaels S 3, S 4, S 35. Table (4.8) displays the realized values that are achieved by optimizatio versus the targets. 48

55 Table (4.8): The Realized values versus the targets. Item Target Realized values Percetage of error x % x % x % boudaries x % t % t % t % Trasmissio zeros t4. % Figure (4.7): structure of multiplexer four. 49

56 Figure (4.8): The theoretical respose of multiplexer four: (a) Reflectio loss ad isertio loss, (b) The isolatio betwee adacet chaels. 5

57 Table (4.9): The optimized couplig matrix of multiplexer four. 5

58 4..5. Example 5: No-cotiguous bad four chaels multiplexer cosists of two chaels with Quasi elliptic respose ad the other two chaels with Chebyshev respose ad =, r = 4, x =.4, x =.75, x 3 =.55, x 4 =.9. The multiplexer cosists of four o cotiguous chaels as show i figure (4.9), ad it has twety resoators i total with four resoators per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by equal guard bads. The edges of chaels,,3 ad 4 are {-.9,-.55},{-.75,-.4},{.4,.75},{.55,.9} respectively. The four trasmissio zeros locate at.45i ad i. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e9 = q e = q e9 = q e =5.559 ad q e = The locatios of retur zeros are calculated usig equatio (3.5) as.88i,.85i,.75i,.6475i,.57i,.73i,.655i,.575i,.4975i ad.4i. These locatios are the iitial locatios ad they eed optimizatio as i example six. I figure (4.9), solid lies represet direct couplig, ad dashed lies represet cross couplig, ad both Quasi-Elliptic ad Chebyshev resposes ca be achieved. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.347, m 3 =.4458, m =.8663, m 33 =.5448, m 35 =.9, m 55 =.5589, m 57 =.43, m 77 =.568, m 79 =.49, m 99 =.564, m =.983, m 3 =.53, m 33 =.55, m 35 =., m 39 = -.7, m 55 =.749, m 57 =.8, m 77 =.7, m 79 =.375, m 99 =.774. For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m4, m35 m 46, m57 m68, m79 m8, m3 m4, m35 m46, m57 m68, m79 m8, m39 m 4, m33 m44, m55 m66, m77 m88, m99 m, m33 m44, m55 m 66, m77 m88 ad m99 m. I this example the whole cost fuctio i equatio (3.4) has bee used except the third term that optimizes the retur zeros' locatios to eforce retur loss level of db. Table (4.) displays the realized values that are achieved by optimizatio versus the targets. The optimized couplig matrix is give i table (4.) ad the multiplexer prototype respose is depicted i figure (4.). Table (4.): The Realized values versus the targets. Item Target Realized values Percetage of error boudaries x x x 3 x 4 t % % % % % Trasmissio zeros % t 4 5

59 Figure (4.9): structure of multiplexer five. 53

60 Figure (4.): The theoretical respose of multiplexer five. 54

61 Table (4.): The optimized couplig matrix of multiplexer five. 55

62 4..6. Example 6: No-cotiguous bad four chaels multiplexer cosists of two chaels with Quasi elliptic respose ad the other two chaels with Chebyshev respose ad =, r = 4, x =.4, x =.75, x 3 =.55, x 4 =.9. The multiplexer cosists of four o cotiguous chaels as show i figure (4.), ad it has twety resoators i total with four resoators per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by equal guard bads. The edges of chaels,,3 ad 4 are {-.9,-.55},{-.75,-.4},{.4,.75},{.55,.9} respectively. The four trasmissio zeros locate at.3i ad.85i. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e9 = q e = q e9 = q e =5.559 ad q e = The locatios of retur zeros are calculated usig equatio (3.5) as.88i,.85i,.75i,.6475i,.57i,.73i,.655i,.575i,.4975i ad.4i. These locatios are the iitial locatios ad they are optimized to get retur zeros with retur loss less tha db. The fial retur zeros' locatios are.88i,.84i,.755i,.63i,.55i,.775i,.686i,.5934i,.48i,.4i. I figure (4.), solid lies represet direct couplig, ad dashed lies represet cross couplig, ad both Quasi-Elliptic ad Chebyshev resposes ca be achieved. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.3393, m 3 =.437, m =.88, m 33 =.54, m 35 =.79, m 39 = -.86, m 55 =.5569, m 57 =.3, m 77 =.5659, m 79 =.44, m 99 =.5669, m =.66, m 3 =.569, m 33 =.54, m 35 =.38, m 55 =.76, m 57 =.5, m 77 =.76, m 79 =.54, m 99 =.73. For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m4, m35 m46, m39 m 4, m57 m68, m79 m8, m3 m4, m35 m46, m57 m68, m79 m8, m33 m44, m55 m66, m77 m88, m99 m, m33 m44, m55 m66, m77 m 88 ad m99 m. I this example all terms i the cost fuctio i equatio (3.4) are etered ito optimizatio process. Table (4.) displays the realized values that are achieved by optimizatio versus the targets. The optimized couplig matrix is give i table (4.3) ad the multiplexer prototype respose is depicted i figure (4.). 56

63 Table (4.): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db - -..% x % x % x % boudaries x 4 t % % Trasmissio zeros % t Figure (4.): structure of multiplexer six. 57

64 Figure (4.): The theoretical respose of multiplexer six. 58

65 Table (4.3): The optimized couplig matrix of multiplexer six. 59

66 4..7. Example 7: No-cotiguous arrow bad six chaels multiplexer with =, r =, x =.3, x =.4, x 3 =, x 4 =., x 5 =.7, x 6 =.8. The multiplexer cosists of six o cotiguous chaels as show i figure (4.3), ad it has twelve resoators i total with oe resoator per arm. The specified retur loss is db. The chaels have a equal badwidth ad they are separated by equal guard bads. The edges of chaels,, 3, 4, 5 ad 6 are {-.8,-.7}, {-.,-}, {-.4,-.3}, {.3,.4}, {,.}, {.8,.7} respectively. The ormalized exteral quality factors are umerically calculated usig equatio (3.6) as q e3 = q e4 = q e7 = q e8 = q e = q e = 3.96 ad q e =.6. The locatios of retur zeros are calculated usig equatio (3.5) as.3i,.38i,.i,.8i,.7i ad.78i. Figure (4.3): structure of multiplexer seve. The couplig coefficiets betwee ay adacet resoators m i ad frequecy offsets m ii are optimized usig cost fuctio i equatio (3.4) to have fial values m =.53, m 3 =.8, m 5 =.9936, m 33 =.437, m 56 =.938, m 67 =.3, m 77 =.79, m 69 =.83, m 9 =.5839, m =., m = For symmetry, some coditios were take i accout to simplify the optimizatio, these coditios are m3 m4, m67 m68, m m, m33 m44, m77 m88 ad m. m I this example oly the secod term i the cost fuctio i equatio (3.4), the term related to retur zeros, was used i optimizatio ad the others were eglected. The optimized couplig matrix is give i table (4.4) ad the multiplexer prototype respose is depicted i figure (4.4). Table (4.5) displays the realized values that are achieved by optimizatio versus the targets. 6

67 Figure (4.4): The theoretical respose of multiplexer seve. 6

68 Table (4.4): The optimized couplig matrix of multiplexer seve. Table (4.5): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db % x % x % x % boudaries x 4 x 5 x % 4.%.7% 6

69 Boudaries Example 8: No-cotiguous arrow bad four chaels multiplexer with = 8, r =, x =.5, x =.6, x 3 =.6, x 4 =.7. Isolatio betwee chaels ca be improved by icreasig the umber of resoator per arm. This example presets a ew way to improve the isolatio betwee chaels. The example is based o the structures i figure (4.5). Figure (4.6) shows the respose of the structure with the ormal distributio of chaels ad table (4.7) shows the optimized couplig matrix of this structure. Figure (4.7) shows the respose of the structure with iterchagig the positios of chaels i the same structure show i figure (4.5) ad table (4.8) shows the optimized couplig matrix of this structure. The iterchagig of the positios of chael 3 ad chael 4 improves the isolatio betwee the adacet chaels as appears i figure (4.7). Before iterchagig the chaels the isolatio peaks is aroud - db as show i figure (4.6), but after iterchagig the chaels the isolatio peaks is aroud - db. This improvemet i isolatio occurs due to separatig the chaels by more frequecy bad. Table (4.6) displays the realized values that are achieved by optimizatio versus the targets. Table (4.6): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db % x % x % x % x % 63

70 Figure (4.5): structure of multiplexer eight: (a) before iterchagig of chaels. (b) after iterchagig of chaels. 64

71 Figure (4.6): The theoretical respose of multiplexer eight before iterchagig of chaels: (a) Reflectio loss ad isertio loss, (b) The isolatio betwee adacet chaels. 65

72 Figure (4.7): The theoretical respose of multiplexer eight after iterchagig of chaels: (a) Reflectio loss ad isertio loss, (b) The isolatio betwee adacet chaels. 66

73 Table (4.7): The optimized couplig matrix of multiplexer eight before iterchagig of chaels. Resoators Table (4.8): The optimized couplig matrix of multiplexer eight after iterchagig of chaels. Resoators

74 4..9. Example 9: No-cotiguous bad four chaels multiplexer with = 6, r =, r = 4, x =.4, x =.65, x 3 =.45, x 4 =.95, t =.35, t =.5. This example is differet from all previous examples because it has four chaels ad each two chaels have differet badwidth, differet respose ad differet umber of resoators per arm. This meas that the geeral structure is able to sythesize multiplexers with massive scale of properties ad characteristics. As show i figure (4.8) the total umber of resoators i the whole structure is sixtee resoators ad the umber of resoators i both chaels oe ad two is two resoators but the umber of resoators i both chaels three ad four is four resoators. Chaels oe ad two have Chebyshev respose while the chaels three ad four have quasi elliptic respose due to the existece of cross couplig m 95. Figure (4.8): structure of multiplexer ie. 68

75 The optimizatio process i this example is differet from the optimizatio processes i all previous examples because i the previous examples the optimizatio processes are doe i oe step where all couplig coefficiets ad reflectio zeros' locatios were etered ito optimizatio process but i this example each diplexer (two arms) is optimized as a idividual block the most of the results are take as iitial values i the optimizatio process for the whole multiplexer. The first diplexer is optimized to get some of the couplig coefficiets that are take as iitial values m 33 =.4963, m 35 =.93, m 55 =.58, ad reflectio zeros' locatios as.4i,.55i,.63i. The secod diplexer is optimized to get some of the couplig coefficiets that are take as iitial values m 99 =.675, m 9 =.438, m 95 =.736, m =.6986, m 3 =.39, m 33 =.75, m 35 =.3, m 55 =.77, ad the optimized reflectio zeros' locatios.4533i,.538i,.76i,.8697i,.9496i will be etered ito whole optimizatio process. Table (4.9) displays the realized values that are achieved by optimizatio versus the targets. The fial values of couplig coefficiets, as show i table (4.), are m =.4543, m 3 =.4798, m 7 =.755, m 33 =.495, m 35 =.49, m 55 =.494, m 78 =.5, m 89 =.759, m 99 =.479, m 9 =.44, m =.6564, m 3 =.9, m 33 =.756, m 35 =.48, m 55 =.6787, m 95 =.95. The ormalized exteral quality factors q e5 = q e6 = 6.88, q e5 = q e6 = ad q e =.37. Table (4.9): The Realized values versus the targets. Item Target Realized values Percetage of error Retur loss(l R ) i db % x % x % x % boudaries x % t % Trasmissio zeros.5.49 % t 69

76 Figure (4.9): The theoretical respose of multiplexer ie. 7

77 Table (4.): The optimized couplig matrix of multiplexer ie. 7

78 The validity of the last example is goig to be checked. The bad pass multiplexer starts from 7 MHz to 33 MHz. The smallest ad the largest values i both rage of values of ormalized exteral quality factors ad couplig coefficiets have bee chose for implemetatio, so other values betwee them ca be guarateed to be realized. To achieve these values, the ope loop resoators have bee used ad HFSS software has bee used i simulatio. The implemetatio is goig to be doe o RT/duroid 66 substrate which has dielectric costat r of 6.5 ad thickess of.7mm. The equatios eeded for desiged are stated i chapter eight i []. The ormalized exteral couplig is calculated from equatio (4.) ke (4.) q e where q e is the ormalized exteral quality factor. The exteral couplig K e ad couplig coefficiets M i are calculated from equatio (4.) K M e i k FBW e, x mi FBW (4.) x where FBW is the fractioal badwidth which is calculated from equatio (3.5), ad x is the maximum ormalized frequecy i low-pass. S respose for a resoator with a port with weak couplig should be foud to extract the exteral couplig from physical structure, ad S respose for two coupled resoators with two weak couplig ports should be foud to extract the couplig coefficiet for two coupled resoators. Equatio (4.3) is used for calculatios of exteral couplig where 3 db ad ca be extracted from S as show i figure (4.) ad the couplig coefficiet ca be extracted for sychroous resoators by equatio (4.4) where ad are the frequecy at peaks as show i figure (4.) []. K e M i 3dB (4.3) (4.4) Table (4.) shows the specificatio of the bad-pass frequecies ad their trasformatios ito low-pass usig equatios (3.) ad (3.5) ad table (4.) shows the calculatios of FBW ad ceter frequecy usig equatio (3.5). 7

79 The miimum ad maximum exteral couplig values have bee calculated for example ie as show i table (4.3) usig equatios (4.) ad (4.). Figure (4.) shows the physical structure used i achievig exteral couplig. Equatio (4.3) is used to calculate the exteral couplig from physical structure as show i table (4.4). Figures (4.3) ad (4.4) represet respose of S for miimum ad maximum ormalized quality factor respectively. The miimum ad maximum couplig coefficiet values have bee calculated for example ie as show i table (4.5) usig (4.). Figures (4.5) ad (4.6) shows the physical structure for two coupled microstrip used i achievig miimum ad maximum couplig coefficiet respectively. Equatio (4.4) is used to calculate the couplig coefficiet for sychroous resoators from physical structure as show i table (4.6). Figure (4.7) represet respose of S for maximum couplig coefficiet. It is oticed the FBW is relatively large which meas that these values ca be achievable for arrower multiplexers. Figure (4.): Respose of S for loaded resoator. Figure (4.): S of two coupled resoators showig two frequecy peaks. 73

80 Table (4.): Normalizatio values of multiplexer's bads. Bad-pass (MHz) W 7 W 77 W W 4 94 W W W W 8 Normalized x x x x x.6569 x x x 4 Table (4.): Calculatio of FBW ad ceter frequecy. upper frequecy lower frequecy ceter frequecy FBW Table (4.3): Calculatios of exteral couplig. ormalized quality factor max value mi value q e5 q e ormalized exteral couplig exteral couplig Table (4.4): The physical dimesios ad calculatios of exteral couplig. x (mm) y (mm) w (mm) t (mm) g (mm) (GHz) 3dB (GHz) K e

81 Figure (4.): Exterally coupled microstrip resoator. Figure (4.3): Respose of S for miimum quality factor. Figure (4.4): Respose of S for maximum quality factor. 75

82 Table (4.5): Calculatios of couplig coefficiets. ormalized quality factor max value m.4543 mi value m46.49 couplig coefficiet Table (4.6): The physical dimesios ad calculatios of couplig coefficiets. x(mm) y(mm) w(mm) g(mm) s(mm) f (GHz) f (GHz) M i Figure (4.5): The physical structure of the miimum couplig coefficiet i example ie. Figure (4.6): The physical structure of the maximum couplig coefficiet i example ie. 76

83 Figure (4.7): Respose of S for the maximum couplig coefficiet. 4. Coclusio I chapter three, the sythesis procedure of coupled resoator multiplexers is preseted. The procedure has bee applied to the proposed ovel structure i ie examples i chapter four ad the reality of results have bee checked i last example. Each example is metioed to advace a advatage of the structure ad to prove the ability of structure to meet the iterestig characteristics i multiplexers. A remarkable advatage of this ovel structure is the ability of dividig the multiplexer ito smaller blocks (diplexers) ad optimizig each diplexer idividually. This decreases the complexity of optimizatio process ad save the time cosumed i optimizatio. The mai disadvatage i the ovel structure is the degradatio of isolatio betwee chael compared with covetioal multiplexer. Icreasig the umber of resoators per chael ad iterchagig the chaels positios improves the isolatio as show i examples. 77