Finite-Difference-Frequency-Domain Simulation of Electrically Large Microwave Structures using PML and Internal Ports

Transcription

1 Fte-ffeece-Fequecy-oa Sulato of Electcally Lage Mcowave Stuctues usg PML ad teal Pots vogelegt vo MSc Eg Podyut Kua Talukde aus Netakoa Bagladesh vo de Fakultät V - Elektotechk ud foatk - de Techsche Uvestät Bel zu Elagug des akadesche Gades okto de geeuwsseschafte - -g - geehgte ssetato Pootosausschuss: Vostzede: Pof -g Heo Heke Gutachte: Pof -g Wolfgag Hech Gutachte: Pof -g Geog Böck Tag de wsseschaftlche Ausspache: 3 Jaua 9 Bel 9-83 Fedad-Bau-sttut fü Höchstfequeztechk, Bel-Adleshof

2

3 To y paets Gta Ra Roy ad Khageda Nath Talukde, y sste Mouta Talukde ad y wfe Adt Gupta

4 v

5 Ackowledgeets sceely ackowledge the Fedad-Bau-sttut fü Höchstfequeztechk, Bel FBH fo povdg e the oppotuty to pusue y Ph a debted to all the dvduals of the Mcowave Egeeg epatet fo the coopeato ad a healthy tea wok cessatly expess y gattude to Pof -g Wolfgag Hech fo hs geeous teachg Hs sghtful scetfc appoach dected e to pofessoals ad helped e to udestad the coplex aspects of the thess wok a gateful to -g Faz-Josef Schückle fo hs gudace ad ubouded suppot thoughout y Ph wok Thaks to hs pesece ad hs valuable advces o the thess wok ad pesoal attes My specal thaks go to pl Math Rae Schludt fo hs valuable suggestos ad easyto-udestad explaatos of the F3 solve gatefully eebe the fedly ad valuable assstace of Host Zschele, -g Maco Kuze, -g Thoste Tschle ad -g Mchael Buchta at the begg of y wok May thaks to -g Ndh Chatuved fo he fedly suggestos dug wtg of the thess Fally, heatly ackowledge y spatos, y paets, y sste ad y wfe fo the love ad affectos Specal thaks to y wfe fo he pesoal otvato ad suppot the ost stugglg oets of y lfe Podyut Kua Talukde 639 v

6 v

7 Cotets Abstact Kuzfassug 3 toducto 5 Motvato 5 Fte ffeece Fequecy oa FF Method 9 3 Asotopc PML Absobg Bouday Codto 4 teal Pots PML Paaetes ad Covegece 3 Motvato ad Geeal Cosdeatos 3 Ovelappg PML Walls 8 3 PML Cell Sze 3 3 Sgle PML Wall: Mco-stp Stuctue 4 3 Multple PML Walls: Patch Atea 9 33 Slot Atea 34 4 PML Layeed Meda 35 5 poved Nuecal Solve 4 6 Cocluso 44 3 Wavegude Pot Sulato wth PML 45 3 Physcal ad Atfcal Modes 45 3 PML Modes ad the Couplg to othe Modes 5 33 Cocluso 55 4 teal Pot 57 4 Relaxato Cuet teal Pot 58 4 Coducto Cuet teal pot Le Cuet Paastc Effects of Coducto Cuet teal Pots Aalytcal Foulato fo the Paastc ductace Sulatos wth teal Pot T-jucto HEMT Stuctue Cocluso 79 v

8 5 4-Quadat Slot Atea 8 5 toducto 8 5 Atea esg 8 53 Sulato Results wth FF Realzato ad Measueet Results 9 55 Cocluso 95 6 Suay 97 7 Appedx 99 7 FF scetzato 99 7 Exctato by Relaxato Cuet teal Pot Souce 73 Exctato by Coducto Cuet teal Pot Souce 5 74 Coducto Cuet teal Pots as Luped Eleets 7 75 eebeddg Coducto Cuet teal Pot Paastcs 9 76 Exctato wth Wavegude Pots 77 poveets the F3 Solve 4 78 Acoys ad sybols 6 8 Lteatue 9 9 Cuculu Vtae 3 v

9 Abstact Fo EM sulatos of coplex MMC stuctues, the usage of PML Pefectly Matched Laye absobg bouday codtos s evtable PML povdes a eflectoless ope bouday codto ad thus helps to keep the esh sze wth a easoable lt O the othe had, stuctues copletely eclosed by PML walls do ot allow exctato by wavegude pots but eque so-called teal pots Such pots, whch ae toduced betwee two o oe esh pots sde the stuctue, facltate also to clude luped eleets o ete etwoks thus eablg oe to study actve ccut, fo stace Ths wok teats both the PML bouday codto ad the teal pots, the faewok of the fequecy-doa Fte-ffeece ethod FF t s show that the pesece of PML walls elages the agtude age of the syste atx coeffcets, ecouteed whe calculatg the felds Ths akes the syste atx ll-codtoed ad ceases the ube of teatos whe solvg the syste ad ay eve ede ths PML totally uusable fo ost pactcal applcatos t s foud that by avodg ay ovelappg PML walls ad by akg the PML cell szes the lagest oes the whole esh the hgh cout of the ube of teatos ca be loweed dastcally The ode spectu at wavegude pots wth lateal PML walls cossts of atfcal PML ad physcal odes We fd that the couplg betwee PML odes ad physcal odes ust be aalyzed thooughly ode to flte the atfcal PML odes out a elable way Regadg the teal pot, a le-cuet foulato poves to be the oly pactcal choce The paastc ductaces assocated wth t ae foud to be of sgfcat fluece A closedfo expesso fo these ductaces s developed ode to detee the values so that t ca be deebedded fo the esults The optzed PML ad teal pot foulatos ae vefed fo the exaple of a 4 GHz slot atea wth tegated fot-ed MMC, sutable fo shot age coucatos ad seso etwoks - -

10 - -

11 Kuzfassug Be Beechuge des elektoagetsche Vehaltes vo koplexe MMC-Stuktue st de Vewedug vo absobeede Radbedguge Fo des sogeate Pefectly Matched Laye PML uveedlch Allsetg duch PML-Wäde eflexosfe abgeschlossee Raubeeche helfe de Azahl de sketseugszelle zu eduzee Se lasse abe auch kee Welleletetoe t Öffuge PML-Regoe zu dese Fälle sd tee Toe efodelch eatge Toe elaube daübe haus das Ebette dskete Eleete ud gaze Netzweke ud dat zb de Aalyse aktve Stuktue ese Abet behadelt sowohl de PML-Radbedgug als auch de tee Toe ud beschebt h Vehalte Rahe des Fte-ffeeze-Vefahes Fequezbeech FF abe wede sbesodee de uewüsche paastäe Effekte ausfühlch utesucht Zu Lösug des be de FF-Methode auftetede Glechugssystes, das de Zusaehag de elektoagetsche Felde de zu beechede Stuktu beschebt, wd e teatves SSOR-Vefahe beutzt e Kovegez deses Vefahes wd seh stak vo de Kodto de Matx bestt Ee schlecht kodtoete Matx vegößet de Zahl de teatoe ud dat de Rechezet betächtlch Lede ttt dese Fall be de Awedug vo PML-Wäde deutlch Escheug Totzde ka de Zahl de teatoe auf e veüftges Maß beschäkt wede, we bestte Rchtle egehalte wede dese Abet wude veschedee Möglchkete zu Reduzeug asoste hohe teatoszahle utesucht ud bewetet, was vo goße paktsche Nutze st Stuktue, be dee de PML-Wäde als Beadug ees Welleletes egesetzt wede, besteht das Modespektu zb des etspechede Welleletetoes aus de physkalsch svolle sowe aus uphyskalsche, küstlche PML-Mode e küstlche Mode lasse sch häufg duch geegete Ktee heausflte Abe Bespele zege, daß vele Fälle das Sepaee physkalsche ud küstlche Mode u ach ee ufassede Betachtug öglch st e Beschebug tee Toe ka t veschedee Asätze duchgefüht wede Tatsächlch zegt sch abe, daß u de Asatz t de Lestofouleug paktkabel st Nebe de ee pleeteug wede de volegede Abet auch de paastäe Effekte we de t de tee To veküpfte duktvtäte dskutet ud Abschätzugsfoel etwckelt Abschleßed wd de Esatz de optete PML ud de tee Toe a Bespel ee 4 GHz-Schltzatee t tegete Foted-MMC, vewedba fü Kuzsteckekoukato ud Seso-Netze, deostet - 3 -

12 - 4 -

13 toducto Motvato Moble coucatos, satellte, ada, autooble ad seso techologes have fuelled the developet of cowave tegated ccuts MMCs Fo the gowg dead of MMCs to wok at hghe fequeces, the atusato of the ccut eleets, e the actve ad passve devces, s dspesable Wthout pope ad systeatc desg ad optsato of the ccut eleets such MMC s ae ot achevable Sce post-poducto chages ae expesve ad desg cycles ae shot, accuate desg tools ae a ust Fo the desg ad sulato essetally two potat developets have eeged O the oe had, thee ae etwok-oeted desg sulatos whee actve devces, luped eleets ad sple passve dstbuted ccuts ae coected o tefaced These sulatos take to accout oly those couplg, echas kow a po ad deal ostly wth a sgle ode fo tassso les t s vey uch potat howeve to desg the actve ad passve stuctues such a way that ay cosstalk ad the depedeces betwee all odes ae cluded Pesetly sulato tools ae offeed whch allow fo deepe ad thoough vestgato of the electoagetc felds ad waves passve stuctues Such electoagetc sulato tools EM sulatos accout fo all the effects of the electoagetc feld behavou, fo exaple coss talk, suface waves, couplg effects ad a sees of othe potat ssues The capabltes of the EM sulatos have bee exteded sgfcatly dug the last decade, patly due to the advaces copute hadwae, patly due to poveets uecal atheatcs wth a vaous ube of dffeet appoaches used The ost posg developets ae Fte ffeeces F ad Fte Eleets FE, whle the begg of these appoaches sple assuptos wee ade to descbe a stuctue, the latest yeas oe poveets aose whch allow to cove the eal atue of vestgated stuctues oe accuately Oe of the ost useful advaces s the descpto of the ope bouday Usg the past electc ad agetc walls wth all the dsadvatages, eg eo due to eflectos fo these walls, seveal ew ethods have bee toduced appoxatg a ope bouday, such that waves popagatg to the decto of the bouday ae absobed by specally defed layes A pefeable ethod ths sese s the Pefectly Matched Laye PML Fg techque t was toduced by Beege [6] ad, tally, s based o the splttg of Catesa electc ad agetc feld copoets to two subcopoets leadg to a odfed set of Maxwell s equatos Soe of the ltatos of ths appoach ae dcated [7, 9] ad oe of the s that the geeatg equatos ad thus felds the PML edu ae o- Maxwella Aothe appoach of Chew ad Weedo [] s based o the copoato of the coplex coodate stetchg vaables ths case, Maxwell s equatos ae also to be odfed ad the coplex co-odates povde o physcal eag Alteatvely, toduced by Sachs et al, a PML ca be ealzed by toducg asotopc ateal popetes, wth electc ad agetc coductvtes, leadg to peeablty ad pettvty tesos [3] - 5 -

14 Chapte toducto Fg Replacg the ope boudaes of a patch atea stuctue wth the PML absobg bouday codtos PML walls ode to absob the adatos Today, PML fos a salet featue of all coo 3 EM sulato ethods, both the te doa FT [,3], TLM [44] ad the fequecy doa fte-eleet ethod [45], FF the late case, ostly the asotopc ateal appoach [8] s appled because t ca be easly pleeted wthout ay eed to odfy the basc equatos Howeve, the beefts of PML do ot coe fo fee the fequecy-doa case, the ateal tesos wose the uecal popetes of the syste of equatos to be solved, whch esults ceased CPU te the te doa, echass ae ot such clealy to be detfed but sla effects ae obseved Geeally, the deteoatos deped stogly o the ube ad paaetes of the PML layes ad occu patculaly f the PML layes ovelap, eg, at the edges ad coes of the oute bouday of the coputatoal doa Beyod ths, PML gves se to aothe poble, whch s ofte ovelooked the lteatue: dug wavegude pot sulatos, a PML bouday causes appeaace of paastc PML odes wth coplex popagato costats whch ust be dealt wth atteto ode to sepaate the fo the desed physcal odes [, 5] a PPP Powe Pat PML cteo s toduced fo ths sepaato Howeve, ths does ot gve elable esults all cases ad futhe vestgatos ae ecessay to exploe these effects Besde the PML deftos, aothe potat featue of EM sulato ethods s the pleetato of so-called teal pots, whch allow to feed the test stuctues teally Lke voltage o cuet souces teal pots ae used to excte stuctues by placg the betwee two coductg les eg sgal ad goud les sde the stuctue Ulke wavegude pots whee a coss-secto s defed o a bouday ad a ube of odes ae excted, teal pots ae based o a cuet-voltage descpto ad thus ca be used to teface the actve / luped ccut Moeove, whe a stuctue s suouded by PML walls all dectos eg Fg, t s ot possble to excte the stuctue by wavegude pots, the oly opto s the use of teal pots to excte the stuctue teally The ecessty ad use of teal pots ae to be descbed hee wth the help of a tape shaped hgh electo oblty tassto HEMT stuctue [5] as gve Fg t s potat to kow the cuet dstbuto at each the gate fge whe a couple of tasstos ae coected paallel to a coo feedg co-stp le Fg a The sae s tue - 6 -

15 Chapte toducto also fo the da fges Because f the cuet dstbutos at the gate fges ae ot sae, e the put cuets of the tasstos ae ot detcal, the pefoaces ae also ot sae eve though detcal tasstos ae used ode to check these cuet dstbutos though 3 EM sulato teal pots have to be placed betwee the gate ad souce fges as well as betwee the da ad souce fges Fg b ad c whle the co-stp les ca be excted by wavegude o teal pots A 3 sulato wth all these teal pots povdes a Z-atx Whe the output pedaces of the gate ad da fges ae kow, cuet dstbuto though the gate ad da fges elato to the put cuets at the MS les ca be calculated out of the esultg Z-atx whch s usually doe etwok sulatos Thus teal pots ca be used as teface betwee EM sulatos ad etwok sulatos The detals of ths HEMT stuctue as well as the sulato esults ae peseted Chapte 44 The pleetato of teal pot FF esh ca be caed out two ways the fst oe two luped te vayg chages of equal agtudes but of opposte polates ae seted at the two odes of the teal pot ad the secod oe a ectagula shaped cuet cayg we s assued betwee two etal plates/les wth the stuctue ude cosdeato The pevous woks o the pleetato of teal pots/luped eleets FT ethod ae peseted [8-3] The pupose of the wok peseted hee s to pove the capabltes of the Fte-ffeece ethod fequecy doa FF, whe used fo stuctues of coplex geoety ad wth actve eleets The a cotbutos ae the devato of PML paaetes ad the cluso of teal pot the FF schee The paastc effects of PML as well as teal pots ae exploed ad the ways how to eove the ae dscussed A 4-quadat slot atea wth actve ccuts s teated as a good exaple whee the use of both the PML walls ad teal pot s a ust the followg sub-chaptes, the FF ethod, asotopc veso of PML bouday codto ad teal pot ae dscussed befly The thess s ogazed as follows: Chapte descbes the PML elated covegece pobles ad how they ca be ccuveted Chapte 3 addesses the fluece of the PML o the ode spectu of a wavegude wth a PML lateal bouday The ode spectu cossts of both PML ad physcal odes coespodg to the egevalues of the syste atx The teal pot pleetato ad the coespodg paastc effects ae gve Chapte 4 Fally, the esults ae vefed fo the exaple of a 4-quadat slot atea wth tegated electocs The desg ssues of the slot atea ae descbed Chapte 5 eployg the ewly developed PML walls as well as teal pots the F3 sulatos - 7 -

16 Chapte toducto a Top vew b : Z -Z coss-secto c : A, B, C zoo : Fg Tape shaped tassto HEMT stuctue [5] half above the syety plae: a ogasato of the gate, da fges ad souce pads togethe wth the co-stp les coectg the gate ad da fges though the tapes b coss-sectoal vew of the stuctue at Z -Z of a The souce pads ae coected wth each othe though a bdge ad to the goud though a va The substate used s SC ε c the eccled egos of A, B ad C ae sae The zooed ego of eccled A o B o C shows the thee teal pots betwee gate fge ad souce bdge, ad da fge ad the souce bdge, espectvely - 8 -

17 Chapte toducto Fte ffeece Fequecy oa FF Method The FF ethod [4, 5] s the fequecy doa coutepat of the FT [3] ethod whch s based o the dscete appoxato of the Maxwell's equatos Ulke the FT whch deals wth a tal value poble, the FF ethod te tes the equatos ae elated leavg the coplex atue ad a bouday value poble s teated ts splest fo, the bouday ego s a ectagula box as show Fg 3 Ths lage box s subdvded to a ube of eleetay cells by a thee-desoal o-equdstat Catesa gd accodg to Yee [] Each eleetay cell s flled wth hoogeeous ateal descbed by ts pettvty ad peeablty Hece a chage ateal popetes ca oly be located o the sufaces of the eleetay cells The electc feld copoets ae defed at the cetes of the cell edges whee as the agetc feld copoets ae defed at the cetes of the cell sufaces Up to ths pot, the settg s detcal to the coo FT appoach ode to descbe the EM felds sde the eclosue of the ectagula box, fequecy doa, oe has to solve a lage syste of lea equatos stead of applyg the leapfog algoth [8] the FT ethod The soluto of the lage lea equato syste s caed out teatvely A bef descpto of the FF foulatos s gve Appedx 7 The atx equato s solved by eas of a teatve uecal algoth of Kylovsubspace type [, 5] the F3 sulato [] whch s used to cay out all these FF vestgatos ths thess wok ths pocedue the ukows ae updated as log as the values of the ukows covege to a pedefed toleace The ube of teatos s a easue to the uecal effots ad dectly popotoal to the CPU te So the fast covegece s the key ssue of the teatve soluto FF ethod The FF ethod s patculaly advatageous copaed to FT ethod to sulate esoatg stuctue lke cavtes whee hgh possbltes of eegy tappg exst eleetay cell FF esh Fg 3 Meshg of the FF bouday value poble ad descpto of the eleetay cells wth the assgets of the feld copoets - 9 -

18 Chapte toducto 3 Asotopc PML Absobg Bouday Codto The PML absobg bouday codto povdes a eas of sulatg the ope boudaes t has the followg tasks dug the EM sulatos: educg the sze of the fte calculato doa suppessg the fluece of the boudaes o the electcal behavou of a stuctue 3 absobg adato fo a atea o paastc effects lke leaky waves ove the sgal les ad adato due to the dscotutes, thus allowg to opeate stuctues vtually ope space the F3 EM sulato [, ], based o FF ethod, the PML has bee pleeted accodg to the asotopc ateal appoach [8] ths pleetato the PML half space s sply a asotopc absobg edu defed by a coplex dagoal teso The pettvty ad peeablty of the gve abtay half space whch has the teface wth the PML half space ae ultpled by the PML teso to calculate the pettvty ad peeablty of the PML layes Fg 4 shows the teface of a abtay ateal ad a PML half space y x z ε ε ε µ µ µ teface [ ε] ε ε [ η] [ µ ] µ µ [ η] PML teso [ η] η η η η σ E j ϖε abtay half space PML half space Fg 4 efto of the PML edu accodg to the asotopc ateal appoach σ E ad σ M ae the atfcal electc ad agetc coductvtes, espectvely, ad oalzed to ε The PML teso s so chose thus the atteuato takes place the z decto σ E σ M ε µ The PML teso eleet of the decto of desed atteuato, ths case the z-decto, s ade vese to the othe two eleets Ude ths aageet, whe waves tavel fo the sulato ego of teest abtay half space to the asotopc PML half space, thee s o eflecto at the teface of the two eda whle the atteuato takes place the z-decto As a cosequece, the PML pat ca be teated by ay bouday wall at z costat wthout affectg the chaactestcs at z the fte dffeece F esh the PML half space s usually teated by electc o agetc walls ad the teso eleets η ae calculated fo a atfcal coductvty σ E pedefed fo a patcula PML wall, whch s estated by the esdual eflectos at the PML teface ue to the dscetzed descpto of the F ethod, the teface of the PML / o - -

19 Chapte toducto PML egos esults spuous eflectos [9] Ths satch poble depeds o the dffeece coductvtes of the eghbog egos ad thus ca be allevated by subdvdg the PML ego to a ube of sub layes wth vayg coductvtes, statg wth elatvely low values at the teface ad ceasg towads the tucato of the esh The oal eflecto coeffcet th, e the esdual eflecto at the PML / o PML teface s gve by equato, whee d s the thckess of the PML wall z decto ad p s the ode of the polyoal vaato fo the vayg coductvtes at dffeet PML sub layes Fo costat, lea ad paabolc vaato of the PML coductvtes the value of p s,, ad, espectvely th dσ ε E p+ ε c e 4 teal Pots teal pots ae a potat featue of ay EM sulatos They allow to feed o excte stuctue fo sde the stuctue as show Fg ulke ts coutepat of the wavegude pots, whch excte a stuctue wth ceta ode cofguato o spectu geeal, teal pots ca be pleeted two ways: the fst oe uses a coducto cuet souce ad the secod oe uses elaxato cuet souce The wave equato of the electcal felds deved fo the Maxwell s equatos s gve by ε ad µ ae the pettvty ad peeablty of the edu J e s the coducto cuet desty wheeas q v s the space-chage desty A o-zeo value of J e eas the edu cotas a coducto cuet souce O the othe had, a te vayg o-zeo value of q v eas the edu has a souce of o-statc suface chages E + ω εµ E jωµ J e + q v ε J e da 3 q Q qv dv jω 4 The coducto cuet teal pot s defed geeal as setg a atfcally cuet gve by equato 3 betwee two ceta pots odes wth two coductos ad as show Fg 5 ue to the exctato by the cuet a agetc feld s excted ad, depedg o the stuctue, a voltage U s developed betwee the coductos Ths pot ca be copaed wth a cuet souce except that a cuet souce the cuet does ot essetally eed a decto whle the teal pot cuet ust have a decto ode to coect the th we betwee the coductos The le cuet coes up wth a ductace due to the agetc felds suoudg the we ad ths s to be accouted fo f the pot fluece eeds to be elated - -

20 Chapte toducto coducto ode E delectc H Hdl U Edl coducto ode Fg 5 Coducto cuet teal pot : le cuet souce q v, J e t s also possble to set two te vayg chages -Q Q Q ad Q defed by equato 4 of equal agtudes but of opposte polates at two odes as show Fg 6 q s the put cuet ad elated to the te vayg chages Q ad Q - q etes the stuctue at ode ad coes out of the stuctue at ode ue to the pesece of the chages electc felds develop betwee the coductos esultg a potetal dffeece of U Ths pot utlzes the elaxato cuet betwee the coductos ad ca be thought of a voltage souce povdg chages of equal but opposte sg at the two odes Aothe pot s that ths pot does ot eed a decto whch s dffeet to the le cuet souce E Q -Q ode delectc Q +Q ode Q q jω U Edl Fg 6 Relaxato cuet teal pot : voltage souce q v, J e The FF ethod solves the dscetzed veso of the wave equato The coducto cuet teal pot s cluded FF by defg a path betwee two coductos ad the toducg a le cuet souce at the edges of the cells alog the path Wth the F appoxato, t ca be assued that the le cuet s hoogeeously dstbuted alog the coss secto of the dual cells [8] of the path so t has a ectagula coss-sectoal aea See Appedx 73 fo the FF foulato of ths pocedue O the othe had, accodg to [8], the pleetato of the elaxato cuet teal pot the FF schee see Appedx 7 by setg the te vayg chages at the cetes of the dual cells defg the odes of Fg 8 at the cetes of the dual cells s ot feasble due to the evolvg o physcal solutos - -

21 PML Paaetes ad Covegece Motvato ad Geeal Cosdeatos Accodg to the FF ethod the aalytcal wave equato of electoagetc EM felds, gve by, s tasfoed to a dscetzed fo of wave equato followg the copoato of gad dv te [6] ad syetcal foulato [7], gve by, that descbes the EM felds sde the eclosue of a ectagula box cotag the stuctue ude cosdeato ad theeby defg a bouday value poble see Appedx 7 ths atx fo of the dscetzed wave equato, e s a vecto that cotas the electc feld copoets fo all esh cells The syste atx M cotas all the foato about the ateal popetes, desos ad fequecy b epesets geeal the souce fo the exctato though wavegude pots o teal pots case of the wavegude pots, b coespods to the su ove all tasvesal electc felds of the avalable odes at the wavegude pots E k E M e b t s woth to look to the costucto of the atx wave equato closely The F3 solve based o the FF ethod s developed accodg to a Catesa coodate syste whee all the foato of the ateals, desos etc as well as felds ad souces ae assged to the atx ad vecto eleets, espectvely, cell by cell, fst of all the x decto, the the y decto ad lastly the z decto as gve by the equato of 3 M j s, j x, y, z ae the sub atces wth deso of, whee s the sze of the esh, ad e s as well as b s ae the vectos wth deso A oe copehesve epesetato of 3 s gve by 4 whee a,j s ae the eleets of atx M cosstg of ateal ad desoal foato, e ad b 3 epeset the feld ad souce copoets coespodg to the vectos e ad b, espectvely t should be oted that M s syetc but ot postve defte ad cossts oly of vey few dagoals wth ozeo eleets M M M xx yx zx M M M xy yy zy M M M xz yz zz e e e x y z b b b x y z 3 a, a3, a, j a a, 3 3, 3 e e3 b b

22 Chapte PML Paaetes ad Covegece The atx equato of 4 s solved fo the ukows of e wth a teatve uecal algoth, aely a Kylov-subspace ethod [, 5] wth the depedet Set Odeg ad SSOR-pecodtog pocedue see the Appedx 77 ths pocedue the values of the ukows ae updated as log as the ukows covege to a pedefed toleace As the asotopc PML satsfes Maxwell s equatos, the PML cells becoe pat of the bouded ego autoatcally dug the dscetzato The fluece of these asotopc PML cells o the covegece of the teatve soluto ca be explaed wth the help of the Geschgo ccles [45] The uo of the Geschgo ccles of adus cotas all the egevalues of the syste atx M, whee the adus s calculated as the suato of the agtudes of all the o-dagoal eleets of the th ow M gve by 5 A quatty ρ s defed as the ato betwee the suato of the agtudes of all the o-dagoal eleets ad the dagoal eleet at the th ow M accodg to equato 6 Aga ρ ax s the axu value of ρ cosdeg all the ows of the syste atx M gve by the equato 7 Soe exaples of Geschgo ccles ae show Fg 3 a, j 5 j j ρ 6 a, ρ ax < < 3 { ρ } ax 7 a, egevalues Rea, Geschgo ccles Fg Geschgo ccles wth cetes at a,, Rea, Fo M to be a postve defte atx, ρ ust be salle tha oe, all the egevalues ust le wth the ght had sde of the agay axs ode to ake the teatve soluto povde a stable covegece But the case of FF ethod, M s usually ot a postve - 4 -

23 Chapte PML Paaetes ad Covegece defte atx, e the pephey of the ccles wth soe of the egevalues exteds to the left had sde of the agay axs whch ay lead to a bad covegece ode to descbe the potace ad cosequeces of the paaetes of, a ad ρ fo the uecal covegece of the FF ethod wth PML the patch atea[9] Fg 6a s exaed t s assued that the atea has oly oe PML X-PML bouday followed by a agetc wall the decto pepedcula to the patch Othe ope boudaes ae teated by agetc walls Ude these codtos the atea s sulated at 6 GHz ad the paaetes, a ad ρ ae calculated out of the syste atx at each ow The sae s doe fo the case wth the PML edu eplaced by a whle keepg the dscetzato schee the sae, ths s teed as the case wthout PML o PML The atos betwee the cases of wth PML ad wthout PML fo all these thee paaetes ae calculated each case the atos whch ae equal to oe, coespodg to the e esh cells ad ows of the syste atx ot affected by the PML edu, ae excluded The eag atos ae sapled lealy couted stepped ages ad daw @3 values : o : a a pl o pl pl : o pl ρ Fg estes of the values gve by the The total ube of ows M s ad the ube of the PML affected ows whee the atos ae ot equal to oe s 73 Lookg at Fg, t s evdet that toducg the PML esults a ceased ube of ows M whee the values of the atos ae lage tha addto, the values ae lage tha those Ths helps to state that the addto of the PML wall ceases the values of ρ ad coespodgly ρ ax, whch eas that the ad of the Geschgo ccles becoe lage ad the possblty of the eal pats of the egevalues to be oe egatve ceases The outcoe s that the codto ube gows lage ad the covegece of the teatve pocess s oe ustable whe the PML wall s cluded ths way the cluso of the PML wall esults a chage the uecal popetes of the syste atx ad theeby affects the covegece of the solve, e, the ube of teatos The ube of teatos s take hee as a easue fo the uecal effots ad t s dectly popotoal to the CPU te eeded to obta the soluto - 5 -

24 Chapte PML Paaetes ad Covegece Whe dscussg the uecal cosequeces of toducg the PML, t s potat to have a close look o the esultg pettvty ad peeablty tesos, whch ae wtte hee as the poducts of the sotopc physcal ateal costats ad a teso [η] epesetg the PML popetes accodg to [3, 4] see equatos 8 ad 9 [η] s a dagoal teso wth the coplex paaete η, the agay pat of whch defes the atteuato ad thus the esdual eflectos at the PML see Fg 3 The oal eflecto coeffcet th of equato ca be wtte tes of the agay pat of η η as As atteuato ad eflecto ae vesely popotoal, atteuato α th ca be wtte as Fo low eflecto, e, hgh atteuato the PML, the poduct of η ad d ust be lage O the othe had, η should ot be too lage, as the atx codto depeds o η So fo hghe atteuato as well as bette atx codto a elatvely sall η ad lage d should be chose Whe usg PML walls fo seveal oute boudaes, these walls ovelap at edges ad coes as llustated Fg 3 case of the ovelappg at the edges eg ovelappg & ad coes ovelappg, & 3 the esultg PML teso s the poduct of the PML tesos of the dvdual PML walls that fo the edges ad coes, espectvely [ ] ε ε [ η] 8 ε [ ] µ µ [ η] 9 µ th ε ω η p + c exp d th d α ω η Fg 3 PML Tesos [η] see equatos 8 ad 9 fo PML walls thee dectos detalg the ovelappg egos σ deotes the atfcal coductvtes electcal of the espectve PML wall as deved fo a gve oal eflecto facto equato - 6 -

25 Chapte PML Paaetes ad Covegece The specto of the tesos accodg to Fg 3 eveals that, because always both η ad ts vese appea, the PML ceases the agtude age of the ateal tesos ad thus that of the esultg etes the syste atx, whch tu ceases the ad of the Geschgo ccles as etoed above As these ccles cota all the egevalues of the syste atx, lage ad of those ccles eas hghe possbltes of the eal pats of the egevalues to assue salle values ad theeby wose covegece Ths stuato becoes patculaly ctcal fo the ovelappg egos sce thee seveal factos of the η ae ultpled ode to deostate ths, Tab shows typcal values fo the PML teso eleets based o a exaple at GHz wth 6dB oal eflecto Ths s the basc effect why PML causes the uecal popetes to deteoate but oe eeds a -depth vestgato to bette udestad the udelyg effects ad to fd the solutos zg the odds Ths s peseted the followg sectos η /η η /η /η η η η /η η η 3 η /η η Tab agoal eleets of PML tesos accodg to Fg 3 ε of PML wall ad ae ad espectvely; σ of PML wall ad ae 46 ad 4 espectvely; µ s all cases; fo calculato of σ see [8, 9] Havg kow that PML geeal woses the covegece of the teatve soluto t s vestgated quattatvely whch paaetes of the PML wall affect the covegece ostly ad how that ca be tepeed by odfyg the PML egos ad paaetes The ost vtal odfcatos ad paaetes wll be dscussed sepaately followg soe geeal cosdeatos egadg covegece as gve below: As aleady etoed chapte, the teface of the PML/o PML egos the dscetzed descpto of FF ethod esults spuous eflectos Ths satch poble ca be allevated by subdvdg the PML ego to a ube of layes wth vayg coductvty [9] ths pot t s potat to kow the pact of the ube of PML layes o the covegece Moe layes pove of couse the accuacy but cease the uecal expeses t s foud that a 5-laye cofguato povdes a good copose ad ths ube s used all the followg vestgatos The covegece of the teatve solve s flueced also by the choce of the PML backg Cooly, PML walls ae followed by electc o agetc walls We foud that fo the stuctues cosdeed ths wok, electc walls lead to a less ube of teatos tha agetc walls 3 Adjacet PML ad o-pml cells should have szes of copaable desos to have the esults less eoeous ad teatos educed ube 4 The ube of teatos ceases wth the cease ε ax -ε ad fequecy, whee ε ax ad ε ae lagest ad sallest elatve delectc costats wth the stuctue These ae geeal chaactestcs of the FF sulato, depedetly whethe PML s cluded o ot - 7 -

26 Chapte PML Paaetes ad Covegece Ovelappg PML Walls Whle fo a sgle PML wall the teso eleets ae the age /η η, ths becoes wose f ovelappg egos of two PML walls ae cluded Assug that both PML walls have the sae popetes, the teso eleets wth the stuctue ow cove the age /η η Poceedg wth 3 ovelappg PML egos the oveall stuato does ot chage fo ths pot of vew because the addtoal eleets ae of sae ode f oe assues detcal PML popetes To gve ubes, Tab shows that fo a sgle PML wall eg PML wall the age of teso eleets s - 83 ad whe the PML wall s cluded the age of atx eleets exteds to 65 Gog beyod these elatvely sple aguets, extesve vestgatos wee pefoed sulatg typcal cowave stuctues wth dffeet PML cofguatos t s foud that the PML dsposto deed has a dastc fluece o the effots equed fo solvg the syste of equatos toducg a sgle PML wall aleady leads to a sgfcat cease the ube of teatos of the solve typcally a facto of cludg ovelappg egos wth PML esults futhe dastc deteoatos a addtoal facto >5, whch ay cases ede ths appoach useless fo pactcal desg wok Addg oe PML walls wth ovelappg egos, covegece deteoates futhe t s foud that ths deteoato of covegece ca be allevated by odfyg the PML defto the edge ad coe egos ffeet deas have bee tested cludg a agle-depedet descpto ad the seto of addtoal physcal coductvtes the PML foulato fo uecal stablzato All these appoaches dd ot yeld the desed poveets The best soluto tued out s to avod ay ovelappg Avodg the ovelappg PMLs s caed out by dsjot PMLs called o-ovelappg PML whee the edges ad coes ae eplaced by the coplete PML walls Sce the edges ad coes fo oly a vey sall facto of the oveall PML suface, the eflectos occug thee ea eglgble ost cases ad oe ca use a PML wth atteuato a sgle decto stead Fg 4 llustates ths dea of o-ovelappg PML fo the edge case Whe PML ovedes PML wall, t s efeed to as oetato the followg ad vce vesa Ths appoach to ccuvet the poble wth ovelappg PMLs poves ts effectveess fo vaous stuctues lke patch atea, costp, flp-chp o LTCC stuctues The vestgatos of how the cluso of PML ovelappg ad o-ovelappg PML flueces the atx eleets, ad theeby the covegece of the teatve soluto wll be peseted the followg dealg a co-stp le ad a patch atea Fg 4 Edge of the stuctue accodg to Fg 3 wth o-ovelappg dsjot PML walls llustatg possbltes of edge descpto - 8 -

27 Chapte PML Paaetes ad Covegece The co-stp stuctue of Fg 5a s sulated at GHz fst of all wthout ay PML wall ode to keep the esh the sae fo all the sulatos, the PML walls ae ot just excluded fo the stuctue, athe the teso eleets of the PML walls ae ade uty η ode to epeset the case wthout PML ths case the PML cells ae thee but ot flled wth PML ateal The desty dstbuto of the o-zeo atx eleets as a fucto of agtudes s show Fg 5b a b PML edge PML coe 5 wthout PML pot PML wall ube of eleets a j c d ube of eleets 5 ovelappg PML 4 3 ube of eleets wthout PML o ove ove PML a j a j Fg 5 a The co-stp MS suouded by PML walls o all sdes othe tha the pot The MS s µ thck ad 5 µ wde The substate ε 98 s µ thck The PML walls cosstg of 5 layes wth vaable thckesses have a oveall eflecto eo of -7dB ad paabolc spatal vaato of the coductvtes The desty of the atx eleets a j elato to the agtudes fo the case wthout PML walls b, wth PML walls cludg the ovelappg egos c ad wth o-ovelappg PML walls d d gaphs b ad c ae edaw fo copaso The lowest ad lagest values of the agtudes ae b 74E- ad 8456, c 754E-5 ad , ad, d 74E-5 ad 9946 espectvely - 9 -

28 Chapte PML Paaetes ad Covegece the sae way, the MS stuctue s sulated wth all the PML walls cludg the ovelappg egos of the coes ad edges at GHz The desty of the agtudes of the o-zeo atx eleets s peseted Fg 5c The stuctue s sulated also wth o-ovelappg PML walls ad aga the desty of the agtudes of the o-zeo atx eleets ae plotted togethe wth those of the cases wthout PML ad ovelappg PML walls Fg 5d Fo Fgs 5b ad 5c t s clea that the PML geeal wdes the age of agtudes of the atx eleets The vey sall ad lage atx eleets, few ube but vey dvese fo each othe, ae due to the ovelappg PML egos ecall the age: /η η ad esult a wose codto of the syste atx ode to pove the codto of the atx, we ust check whch of the lagest ad lowest values we ca exclude that do ot have sgfcat fluece o the eflecto level Fg 5d shows that the uppe lt of the age of agtudes asg due to the ovelappg egos ca be educed by usg the o-ovelappg PML That s why o-ovelappg PML walls povde bette codto of the syste atx ad also lowe ube of teatos The above-etoed esults obtaed the case of the MS le ae also tue fo the patch atea stuctue of Fg 6a We shall see futhe wth ths patch atea exaple how ovelappg ad o-ovelappg PML fluece the Geschgo ccles ad evetually the ube of teatos the pctue we see the aageet of the PML walls oovelappg case, whch s foed by eplacg the ovelappg coes ad edges the x decto the PML s copletely x dected X-PML The yz colus below the x dected PML have bee ade y dected Y-PML ad the z dected PML Z-PML s suouded by the x ad y dected PML walls Ths s teed as xyz oetato of PML walls ode to copae the o-ovelappg PML agast the ovelappg oe, the atea was sulated wth all the PML walls cludg all the ovelappg egos The outcoe was useless, as the PML fully cludg all the ovelappg descptos esults a teato cout above 3, aleady at 3 GHz ad stll wthout ay covegece So the ovelappg egos wee checked oe by oe Sce the outcoe was the sae all these cases, oly the ovelappg ego foed by the X-PML ad Z-PML wll be descbed the followg Magetc walls ae used stead of PML walls the y dectos Now the atea s sulated wthout ay PML as well as wth ovelappg ad oovelappg PML walls x ad z dectos The the ρ s fo each ow of the syste atx M all these thee cases ae calculated at 6 GHz Clealy a sgfcat ube of ρ s ae sae fo all those thee cases, aely fo the cells coespodg to the o-pml egos So these ρ s ae excluded The est of the ρ s whch ae affected by PML walls ae daw as a ato wth espect to the o-pml case, sepaately fo the o-ovelappg ad ovelappg cases Fg 6b ad c, espectvely As oe ca see, fo the o-ovelappg case we have aleady a odeate ube of ows M, whee the ρ s ae lage tha the o PML case Now the cluso of ovelappg PML wall the xz edge exteds the ube ad values of the ρ s exteely as evdet the lowe chat Ths eas that fo ths hgh ube of ows, the ad of Geschgo ccles becoe lage extedg the pephees the left sde of the agay axs, whch fst esults obvously eal pat of egevalues to be oe egatve ad theeby bad covegece The sae s tue also fo othe fequeces - -

29 Chapte PML Paaetes ad Covegece a b ubes 6 GHz c ρ o ovelappg PML / ρ o PML hoo-ovelappg PML / hoo PML ubes 6 GHz ρ ovelappg PML / ρ o PML hoovelappg PML / hoo PML Fg 6 a : The patch atea[9] suouded by PML walls o X ad Z Y-PML eplaced by agetc wall The PML walls cosst of 5 layes wth vaable thckesses ad have a oveall eflecto eo of -7dB ad a paabolc spatal vaato of the coductvtes The desty of PML affected ρ ae show fo case of o-ovelappg PML b ad ovelappg PML c, espectvely, elato wth the case wthout PML - -

30 Chapte PML Paaetes ad Covegece Sce the ovelappg egos of coes ad edges fo a vey sall facto of the whole PML suface, the eflecto occug thee f o-ovelappg PML walls ae used ca be expected to be eglgble Fo exaple, we ca see Fg 7a that thee s o sgfcat chage the esultg S-paaetes due to the splfcatos But we have vey sgfcat poveet the covegece of the teatve soluto whe o-ovelappg PML walls ae used Fo o-ovelappg walls the ube of teatos eas below up to 6 GHz whee as fo ovelappg PML case, the ube ses fo 7 to above, Fg 7b a b S / db ovelappg o ovelappg fequecy / GHz teatos ovelappg o ovelappg fequecy / GHz Fg 7 a : Reflecto coeffcets of the atea usg ovelappg ad o-ovelappg PML walls b : The ube of teato fo the sulato of the atea usg ovelappg ad o-ovelappg PML walls both cases oly X- ad Z-PML walls ae used Oe should ote that the choce of PML decto the edge ad coe egos teed as oetato Fg 4 affects the covegece as well Oe of the easos s the costucto of the syste atx whee the ode of eleets flueces the ube of teatos ths s solely due to the chaactestc of the solve, ad has othg to do wth PML geeal ode to see the fluece of the oetato of the stuctue as well as the PML walls, we have to look to the costucto of the atx equatos 3 ad 4 closely As etoed above, ou FF ethod s developed accodg to a Catesa coodate syste whee all the foato of ateals, desos etc as well as felds ad souces ae assged cell by cell fst of all the x decto, the the y decto ad lastly the z decto So the costucto of the atx equato s totally dffeet fo dffeet oetatos of the stuctue ad the sae way those of the PML walls Let us cosde two exaples of Fg 8 the fst case, oe half of a patch atea top s calculated wth the decto of adato the z decto Two o-ovelappg PML walls the z ad x dectos ae used Hee we see that thee s a lage dffeece the ube of teatos fo dffeet oetatos of xz ad zx o-ovelappg PML walls By choosg zx PML the ube of teatos s educed by 5% - -

31 Chapte PML Paaetes ad Covegece teatos XZ ZX fequecy / GHz 5% YZ ZY 45% X Z Z Y Y X Fg 8 teatos fo dffeet oetato ad otato wth espect to the Catesa co-odate syste of a half patch atea The substate of the atea s Teflo ε The deso of the patch s 8 µ 5 µ 6 µ ad that of the feedg costp s 6 µ 5 µ 8 µ Now the atea s otated as gve by botto, whee the adato takes place the y decto ad two PML walls ae used the y ad z dectos Hee we ca see that wth ths oetato the ube of teato s educed by 45% eve though thee s o sgfcat chage the ube of teatos fo yz ad zy oetatos Howeve the depedece of covegece o the oetato of a stuctue the x, y o z dectos ca be tepeed by povg the solve puttg pocedue eaagg the atx eleets 3 PML Cell Sze The cell sze wth the PML layes tued out to be a vtal quatty, whch geatly affects the uecal covegece Fo equato t s clea that thckess of the PML wall d has a doat ule ove the atteuato Fo a costat atteuato, a cease d eas a decease η ad theeby a poveet uecal covegece Extesve vestgatos wee caed out ode to check the fluece of PML cell sze o both the accuacy ad uecal covegece wth the help of dffeet stuctues wth sgle PML walls oly oe PML wall oe decto ad ultple PML walls oe tha oe PML wall dffeet dectos ad o-ovelappg The stuctues vestgated clude MS, CPW, flp-chp, spal ducto, couple, atea etc The ost potat esults wll be peseted the followg sub-chaptes A sple MS stuctue wth a sgle PML at the ed wll be vestgated detal wth dffeet lateal ad logtudal dscetzato schees to check the absobg capablty of the PML as well as the pact of ts thckess cell szes o covegece ths case the PML wall woks as a atched laye The depedecy of the covegece o ρ ax wll be explaed - 3 -

32 Chapte PML Paaetes ad Covegece PML walls ae used ostly to absob adato Theefoe, a adatg atea stuctue wll be studed to optze the PML cell szes Multple PML walls ae used ad they ae assued to be o-ovelappg at the coes ad edges Evetually the slot atea of Fg 5 wll be exaed as a exaple of pactcal teest 3 Sgle PML Wall : Mco-stp Stuctue The co-stp stuctue Fg 9 s vestgated fo dffeet lateal ad logtudal dscetzato schees ode to optze the PML cell sze wth egad to covegece Two dffeet lateal dscetzato schees ae used as gve Fg, whee the uppe pctue a the sallest ad lagest cell szes the x decto ae of 3 ad 84 µ espectvely, the y decto of 6 ad 6 µ, espectvely O the othe had, the lowe pctue b the sallest cell szes ae sae as the uppe oes but the lagest cell szes the x ad y dectos ae educed to 4 µ both cases the deso of the xy coss-secto s detcal, oly the sze ad ube of cells ae vaed Alog wth these two lateal dscetzato schees, the cell szes the logtudal z- decto both the PML ad o-pml egos ae vaed abtaly ode to fd out the PML cell szes copaso to the o-pml cell szes fo whch the ube of teatos educes to a u all cases, PML s chose such that eflecto at the ed of the MS le ust appoach to the oal eflecto th of the PML wall Thee logtudal dscetzato schees ae show Fg, whee the legth ad the ube of cells ae kept costat each case veso a the sallest cells ae the PML cells, a the stuctue s hoogeeous ad a3 the lagest cells ae the PML cells Sulatos ae caed out cosdeg each cobato of the lateal ad logtudal dscetzato schees of Fg ad espectvely The esultg eflecto coeffcets ad the ube of teatos ae plotted Fgs ad 3 espectvely X Z Y Fg 9 The co-stp stuctue wth a PML wall at the ed The stp s µ thck ad 5 µ wde The substate s µ thck ad the delectc costat of the substate s 98 The PML wall cosstg of 5 layes wth vaable thckesses has a oveall eflecto eo of 6 db ad paabolc spatal vaato of the coductvtes - 4 -

33 Chapte PML Paaetes ad Covegece a b Fg scetzato schees the xy plae of the MS stuctue gve Fg 9 The ube of cells the x ad y dectos ae 4 ad 7, espectvely, a, ad 35 ad 4, espectvely, b Fo Fg t s clea that the eflecto coeffcets both the cases of dffeet lateal dscetzatos ea the sae, whee the hoogeeous dscetzato povdes the lowest eflecto eo As the oal eflecto s take as 6dB, deally the stuctue should povde a eflecto of 6 db, fo the whole fequecy age We see that at the lowe fequecy age up to GHz the eflecto levels all cases stay costat at 6 db The eflecto coeffcets ae below 45 db up to GHz The a s the S s ae due to - 5 -

34 Chapte PML Paaetes ad Covegece the cacelg of each othe of the eflected waves at the PML laye tefaces O the othe had, lage eflecto > -6 db the hghe fequecy age s due to the o-quasy statc behavou of the felds Fg scetzato schees the logtudal z decto The legth of the stuctue s µ The last cell s a electcal wall ad the PML wall cossts of 5 cells adjacet to the electcal wall PML cell szes ae vaed fo µ to 4 µ ad gve the backets The sallest ad lagest cells fo a ad a3 ae ad 7 µ, ad 3 ad 4 µ, espectvely a b S / db a µ a µ a3 4 µ S / db a µ a µ a3 4 µ fequecy / GHz fequecy / GHz Fg Reflecto coeffcets fo the dffeet logtudal dscetzatos accodg to Fg The gaphs a ad b coespod to the lateal dscetzato schees of Fga ad b espectvely The PML cell szes ae gve backet - 6 -

35 Chapte PML Paaetes ad Covegece Regadg covegece, a3 wth lagest cells the PML ego shows the lowest ube of teatos, whch s below 5 up to GHz, both the cases of the lateal dscetzatos Fg 3 The ube of teatos of the sallest PML cells the case of lateal dscetzato of Fg a s below 3 Fg a wheeas the case of lateal dscetzato of Fg b t s above 5 Fg 3b The easo behd ths cease teatos s that the case of Fg b we have hghe ube of cells 35, 4 ad 6 tha the case of Fg a 4, 7 ad 6 a b 5 5 teatos 4 3 a µ a µ a3 4 µ teatos 4 3 a µ a µ a3 4 µ 5 5 fequecy / GHz 5 5 fequecy / GHz Fg 3 Nube of teatos fo the dffeet logtudal dscetzatos accodg to Fg The gaphs a ad b coespod to the lateal dscetzato of Fg a ad b espectvely The PML cell szes ae gve backet ode to bette copae the ube of teatos fo the two cases of coss-sectoal dscetzatos, the teatos pe cell ae plotted Fg 4 ths fgue t s clea that all cases a, a ad a3 the ube of teatos b ae lowe tha a both cases whe the PML cells the logtudal decto ae the lagest oes we have the lowest ube of teatos Regadg the left pctues the lagest PML cells a3, 4 µ ae salle tha the lagest cells the x 67 µ ad the y 36 µ dectos Regadg the ght pctue the lagest PML cells a3, 4 µ ae lage tha the cells the x ad y dectos We coclude that the PML cells ust ot oly be the lagest cells the decto of PML but also the cell szes othe dectos ths case x ad y ust be salle tha the PML cells oe deftely, PML cells ust be the lagest oes oveall ode to educe the teatos to the u - 7 -

36 Chapte PML Paaetes ad Covegece a b Fg 4 The ube of teatos pe cell The gaphs a ad b coespod to the xy dscetzatos Fg a ad b, espectvely PML cell szes ae the backets The paaete, ρ ax,as defed by equato 7, the axu value of the atos of the su of the agtudes of all the o-dagoal eleets to the agtude of the dagoal eleet of the syste atx, ae daw wth espect to fequecy Fg 5 We see hee a coelato wth the ube of teatos, whch, howeve, povdes o pot-to-pot elato, athe a oveall appg s evdet The gaphs a ad b coespod to the ube of teatos gve left- ad ght-had gaphs Fg 4, espectvely each case t s evdet that ρ ax deceases wth the decease the ube of teatos f we copae betwee the left ad ght pctues, we see that ρ ax s of the ght pctue ae lowe tha that of the left oe Ths gves the easos why the ube of teatos pe cells gve Fg 4b ae lowe tha those gve Fg 4a a ρ ax a µ a µ a3 4 µ 5 5 fequecy / GHz b ρ ax a µ a µ a3 4 µ 5 5 fequecy / GHz Fg 5 Vaato ρ ax The left ad ght pctues coespod to the xy dscetzatos Fg a ad b espectvely PML cell szes ae the backets - 8 -

37 Chapte PML Paaetes ad Covegece The geeal ted ca be explaed also aothe way: all the calculatos peseted above, a costat oal eflecto coeffcet th s assued, whch efes to a costat atteuato the PML ego Accodg to equato, ceasg the PML cell szes e d deceases η thus povg the uecal codto fo a costat atteuato 3 Multple PML Walls : Patch Atea A sgle PML wall at the ed of a MS as descbed the pevous secto woks as a atched teato of wavegude Now we shall study a adatg stuctue, the patch atea Fg 6a Fst, t s sulated wth a sgle PML wall x decto ad the wth ultple PML walls all ope dectos x, y ad z dectos ode to estate the optu PML cell sze egadg covegece each case PML walls cosst of 5 layes followed by a electc wall The PML walls ae o-ovelappg case of a sgle PML wall the x decto, the othe ope boudaes the y ad z dectos ae teated by agetc walls Thee ae fou, thee ad two dffeet dscetzato schees used the x Fg 6, y Fg 7a ad z Fg 7b dectos, espectvely Fg 6 The dscetzatos of the patch atea the x decto The ube of cells the x decto s 45 each case The PML cell szes ae the lagest cell szes each case ad gve the backets - 9 -

38 Chapte PML Paaetes ad Covegece a b Fg 7 The dscetzatos of the patch atea y a ad z b dectos The ubes of cells the y ad z dectos ae 94 ad 97, espectvely The lagest cell szes each case ae gve backets Fst of all, a total of 4 sulatos cobato of the dscetzato schees the x, y ad z dectos wth each othe ae caed out wth a sgle PML the x decto whee the cells wth dces fo 4 to 44 as depcted Fg 6 ae defed as the PML cells Cosdeg all the sulato esults wth espect to the ube of teatos, t s foud that the ube of teatos deceases as PML cell szes x decto cease The ost potat sulato esults ae suazed Fg 8 a the ube of teatos goes dow at 5 GHz fo 54 to 5 whe the PML cell szes cease fo 5 µ to µ whee the lagest cells y ad z dectos ae kept at 5µ b the ube of teatos deceases at 5 GHz fo 7 to 375 as the PML cell szes goes up fo 5 µ to µ whle keepg the lagest cells y ad z dectos at 9µ These two sets of esults clealy suggest that the PML cells should be the lagest oes copaso to the o PML cells all dectos ode to obta fast covegece t should be oted that f the PML cell szes ae ceased futhe, the decease ube of teatos eaches satuato t eas futhe cease the PML cell sze wll poduce vey lttle decease teatos Aga the PML cell szes ust ot be lage tha oe teth of the wave legth ode to keep the dspeso eo wth a acceptable lt Now t ust be checked what wll happe egadg the ube of teatos f PML walls ae seted y ad z dectos as well Aga the 4 sulatos as aleady used fo the sgle x PML wee caed out ad the ost potat esults ae lsted Fg 9 The cells wth dces fo to 6 ad fo 88 to 9 y decto Fg 7a, ad the cells wth dces fo 9 to 96 z decto Fg 7b ae defed as PML cells alog wth x PML cells defed as befoe Copag Fg 8 ad 9 t s otceable that the cluso of y ad z PML walls ases the ube of teatos the lowe fequecy ages akg the teatos level alost costat fo 3 to 5GHz Ths s because the y ad z PML cell szes of 5 µ dscetzato schees: y ad z ae aleady the satuato, e, the y ad z PML cells ae aleady lage eough ad f they ae ceased to 9 µ dscetzato schees: y 3 ad z the teato levels do ot go dow sgfcatly Ths ca be see by copag left ad ght-had gaphs of Fg 9 Fo ths obsevato t ca be stated that the PML cell szes ust be lage tha the o-pml cell szes all dectos, espectve of PML dectos - 3 -

39 Chapte PML Paaetes ad Covegece So optzg the PML cell szes x 4 : µ, y 3 : 9 µ, z : 9 µ the ube of teatos goes dow below 8 Copag the eflecto coeffcets of ths set of dscetzatos wth that of the sae stuctue sulated wth CST Mcowave Studo povdes a vey good atchg as gve Fg a b teatos x 5 µ x 75 µ x 3 µ x 4 µ fequecy / GHz teatos x 5 µ x 75 µ x 3 µ x 4 µ fequecy / GHz Fg 8 : The ube of teatos fo a sgle PML the x decto fo dffeet PML cell szes wth the dscetzato schees accodg to Fg 6 The gaph a coespods to the dscetzatos the y ad z dectos labelled by y ad z Fg 7 espectvely The gaph b coespod to the dscetzatos the y ad z dectos labelled by y 3 ad z Fg 7 espectvely a b Fg 9 : The ube of teatos fo ultple PML walls the x, y ad z dectos fo dffeet PML cell szes the x decto wth dffeet x dscetzatos accodg to Fg 6 The gaph a coespods to the dscetzatos the y ad z dectos labelled by y ad z Fg 7, espectvely The gaph b coespods to the dscetzatos the y ad z dectos labelled by y 3 ad z Fg 7, espectvely - 3 -

40 Chapte PML Paaetes ad Covegece S / db CST MWS F fequecy / GHz Fg : The eflecto coeffcets of the patch atea calculated usg F3 ad CST Mcowave Studo MWS F3, calculato the dscetzatos used ae x 4, y 3, z of Fg 6 ad 7, espectvely the above vestgatos of the patch atea, the PML cell szes all x, y ad z dectos wee aleady lage tha the o-pml cells ad the the values wee ceased We shall check the ube of teatos whe soe of the PML cells ae salle tha o- PML cells fo ths stuctue wth ultple o ovelappg PML walls The esh sze of the patch atea Fg 6 ad 7 was whch was ecessay to keep the esh sze sae fo all the dscetzato schees whle vayg the PML cell szes x, y ad z dectos ealty we do ot eed such a hgh esoluto fo ths sple patch atea Afte kowg that we eed PML cell szes lage tha o PML cell szes all decto to ake the teato level low, we ca ow dscetze the stuctue a oe coveet way to keep the PML cells the lagest oes The ew esh of the sae atea stuctue s ow The a of the followg vestgatos wth salle esh sze s to show that esh optzato by usg lage PML cell szes s tue ot oly fo a patcula esh sze wth dffeet cell szes, but also fo dffeet esh szes of a patcula stuctue Fst of all, the stuctue of the ew esh s sulated wth dffeet PML cell szes the x decto, vayg fo 4 µ to 5 µ accodg to Fg whle the PML cell szes both the y ad z dectos ae kept costat at 5 ad µ espectvely The o PML cells the y ad z dectos ae salle tha the PML cells Fg a shows the copaso the ube of teatos fo these thee types of dscetzato schees wth dffeet PML cell szes x decto The ube of teatos ca be educed by 86% at 7 GHz by ceasg the PML cell szes fo 4 µ to 5 µ Fg b shows that the esultg S-paaetes do ot vay due to the chage PML cell szes t should be oted hee that the pevous vestgatos of the patch atea the cteo to teate the teatve solve was e-8 ad the vestgatos of Fg the value s take e- Howeve both the cases holds the aguet that f the PML cells lage tha the o-pml oes, ths esults a sgfcatly lowe ube of teatos The sae calculatos wee pefoed fo PML cell szes the y ad z dectos whle keepg the PML cell szes othe dectos lage tha the o-pml cell szes The esults suppoted the pevous fdgs - 3 -

41 Chapte PML Paaetes ad Covegece Fg : The dscetzato schees x decto of the patch atea gve Fg 6a The PML cells ae wth dces fo to 6 The PML cells szes ae 4 µ top, 7 µ ddle ad 5 µ botto The sze of the lagest o-pml cells x decto ae 63 µ top, 43 µ ddle ad 343 µ botto a b teatos µ 75 µ 5 µ fequecy / GHz S / db fequecy / GHz 4 µ 7 µ 5 µ Fg : The ube of teatos la ad eflecto coeffcets b fo the dffeet PML cell szes the x decto accodg to Fg

42 Chapte PML Paaetes ad Covegece 33 Slot Atea The 4-quadat slot atea [36, 37] as show Fg 5 was developed fo ea-age seso coucatos The atea povdes lateal adato ad atches wth the coo flp-chp type of tegato The esoat popetes of the atea eable t to be used as a put / output flte A actve chp ca be seted sde the slot ad ca ake the atea self-depedet The challeges of desgg the slot atea ad sulato FF wth the F3 sulato as well as easueet esults wll be deostated Chapte 5 The atea s excted wth a teal pot The popetes of the teal pots FF ethod wll be dscussed Chapte 4 Hee the atea wll be used as a exaple to check the PML cell sze ad ts fluece o the covegece All the ope boudaes ae teated by PML bouday codtos Thee sulatos ae caed out wth half of the stuctue see Fg 54 the fst oe, the PML cell szes ae so chose that the ato betwee the PML cell szes ad the lagest o- PML cell szes each decto vaes betwee 8 ad 9 The atos vay betwee 3 ad 4 the secod sulato ad betwee 34 ad 44 the thd sulato The o PML cell szes the e esh eas sae fo all the thee cases So, the fst case the PML cell szes ae salle tha the o-pml cell szes ad fo the othe two cases t s vce vesa The sulato esults ae plotted Fg 3 Whe the PML cells ae salle tha the o- PML cells, teato cout exceeds 65 up to 47, whee as whe PML cells ae lage tha the o-pml cells, the teato cout s lowe tha The esultg S-paaetes ae copaed wth each othe of the thee sulatos, togethe wth a copaso wth CST MWS te doa Thee exsts a vey good fttg the eflecto coeffcet cuves a b teatos * atos fequecy / GHz S / db - -4 atos CST MWS fequecy / GHz Fg 3: The ube of teatos a equed to sulate the slot atea of Fg 54 fo dffeet atos betwee PML cell sze ad the lagest o PML cell sze The ube of esh cells s 944 all cases The eflecto coeffcets b fo dffeet atos betwee PML cell sze ad the lagest o-pml cell sze The copaso wth CST MWS FT ethod fo the sae stuctue s also cluded

43 Chapte PML Paaetes ad Covegece 4 PML Layeed Meda ths sub-secto, the questo of how to estate the atfcal electc ad agetc coductvtes of the PML edu the pesece of layeed eda s addessed ode to calculate these coductvtes fo a gve oal eflecto th eo a thub ule s developed to estate the delectc costat of the PML ego These vestgatos ae potat, because o oe had, the gve oal eflecto eo whch depeds o the delectc costat of the PML ego ad the calculated absopto level of the PML afte the FF sulato ust be sae, o the othe had, the gve oal eflecto eo has sgfcat fluece o covegece The PML edu s defed by the atfcal electc ad agetc coductvtes ode to show how they ust be chose the FF ethod, the PML teso s peseted hee aga by σ E ad σ M ae the electc ad agetc PML coductvtes [5] t s show that f σ E ad σ M ae assged at cell edges ad cell cetes, espectvely, bette absopto by the PML wall s acheved tha f both σ E ad σ M ae cosdeed to be located at the cell cetes Fo the esultg S-paaetes of that wok t s evdet that the eflecto coeffcet at the lowest fequecy does ot appoach the oal eflecto coeffcet th Ths s due to the ssg te of the elatve effectve delectc costat fo the PML edu ε pl the calculato of σ E fo gve a th Now afte cludg ε pl [9] the equato of calculatg σ E becoes as gve by 3 Hee p deotes the ode of polyoal vaato ad d the thckess of the PML wall The te ε pl should ot be cofused wth the effectve delectc costat ε eff of a ceta ode; ε eff vaes wth the fequecy betwee the lowest ad hghest delectc costats of the avalable eda the wavegude coss secto ε pl s a appoxate value selected fo the age of ε eff fo the whole fequecy age ad has a sgfcat fluece o the absopto level η jσ E ε [ η] η ; η ; σ E σ M ωε µ η σ E ε c p + l th d ε pl 3 Accodg to the ew foulato, the MS stuctue of Fg 9 wth lateal ad logtudal dscetzato schees gve by Fg b ad a3 espectvely s sulated aga wth a th of 6dB fo dffeet assgets of σ E ad σ M The esultg S-paaetes ad the ube of teatos ae daw Fg 4 t s clea fo Fg 3 that locatg σ E ad σ M at the cell edge ad cell cete, espectvely, esults a bette absopto Futheoe, the eflecto coeffcet at the lowest fequecy appoaches the th value of 6dB t should be oted that the esultg eflecto coeffcet S should appoach th, at least at the lowest fequecy, because the PML seves hee as atched absobg laye O the othe had, thee s ot ay chage the teato level fo the dffeet assget of σ E ad σ M

44 Chapte PML Paaetes ad Covegece S / db σ E & σ M : cell edges σ E : cell edge & σ M : cell cete Fequecy / GHz teatos σ E & σ M : cell edges σ E : cell edge & σ M : cell cete Fequecy / GHz Fg 4 Reflecto coeffcets left ad the ube of teatos ght of the MS stuctue of Fg 9 fo the dffeet deftos of electc ad agetc PML coductvtes t wll be show the followg how to estate the value of ε pl fo a layeed edu wth the help of the MS stuctues of Fg 5 All the desoal foato of the stuctues ae gve Tab all cases a PML wall at the ed of the stuctue s added ad the absopto by the PML wall s vefed whle vayg the paaetes of substate thckess ad ε pl We shall stat ou vestgato wth the stuctue of Fg 5a the PML we have thee dffeet eda othe tha the PML edu fo ths stuctue: a, PEC stp ad ceac 98 ε pl s vaed fo to 98 fo the calculato of σ E yeldg th fo db to 6dB The esultg S-paaetes ae daw Fg 6 We see both that the eflecto coeffcets appoach the oal value th f ε pl s equal to 98 The explaato s that ost of the felds ae cocetated the ceac substate ad ts pephey to a aoud the stp Statg wth ths sple layeed stuctue, we tetoally add aothe th-fl laye of LTaO 3 ε 4, Fg 5b Fo ths stuctue ε pl s vaed fo to 4 wth a th value of db The esultg S-paaetes fo two dffeet thckesses of the th fl ae plotted Fg 7 A aveage value of ε pl s calculated accodg to equato of 4, whee t s ae the thckesses of the dffeet layes except stp ad ε s ae the elatve delectc costats of the layes t ε aveage ε pl 4 t

45 Chapte PML Paaetes ad Covegece a b c Fg 5 a MS le o ceac substate b Mcostp le o th-fl of LTaO 3 ad ceac substate Thckess of th fl, t TF, s vaed fo to 5 µ c Mcostp le o a sadwch stuctue of GaAs, th-fl of LTaO 3 ad ceac All the detals of the stuctues ae lsted Tab The delectc costats of ceac, GaAs ad LTaO 3 ae 98, 9 ad 4, espectvely thckess µ dscetzato a b c decto a b c t s d µ, u of cells d µ, u of cells d µ, u of cells t ceac X 7, 35 7 & 754, 7, 43 4 & 43 t TF - & 5 Y 5, 4 5, 4 5, 4 t GaAs Z, 6, 6, 6 Tab Thckesses ad desos of the stuctues gve Fg 5 The wdth of the costp s 5 µ x, y ad z efe to the dectos of thckess, wdth ad legth of the stuctues Total ube of cells x, y ad z dectos s also attached each case a b S / db ε pl fequecy / GHz S / db ε pl fequecy / GHz Fg 6 The Reflecto coeffcets of the stuctue Fg 5a fo vayg ε pl a : th db ad b : th 6 db

46 Chapte PML Paaetes ad Covegece Fg 7 shows that oe of the values of ε pl esults a eflecto coeffcet equal to the oal eflecto coeffcet th of db at the lowest fequeces The easo fo such devato s aga as stated befoe that ost of the felds ae cocetated the ceac, TF ad ea the substate-a bouday The sae calculatos ae caed out except excludg the a space above the substates the calculato of aveage ε pl The Fg 8 shows the esultg S-paaetes wth dffeet th of db ad 6dB The aveage values of ε pl, 5 ad ae calculated cosdeg dffeet thckesses of th fl t TF :, ad 5 µ espectvely We see that each case the eflecto coeffcets appoach the value of th a b S / db ε pl aveage 5 5 fequecy / GHz S / db ε pl aveage 5 5 fequecy / GHz Fg 7 Reflecto coeffcets fo the stuctue Fg 6b fo vayg ε pl th db a : t TF µ ad b : th t TF 5 µ a S / db t TF : µ, ε pl : -8 t TF : µ, ε pl : 5 t TF : 5 µ, ε pl : fequecy / GHz b S / db t TF : µ, ε pl : t TF : µ, ε pl : 5 t TF : 5 µ, ε pl : 5 5 fequecy / GHz Fg 8 Reflecto coeffcets fo the stuctue Fg 6b vayg t TF The aveage ε pl s calculated usg 4 a : th db ad b : th 6 db

47 Chapte PML Paaetes ad Covegece Fally the above etoed expeece s checked wth the stuctue of Fg 5c Hee aothe laye of GaAs ε 9 s added Now ε pl appoxates 65 accodg to equato 4 wthout takg a to accout ad the esultg S-paaetes ae show Fg 9a We see a vey good atchg wth the th value at the lowest fequecy left pctue Oe obseves that a thub ule fo ε pl would be the aveage of the elatve delectc costats of dffeet substates accodg to 4, excludg the egos e a Fg 5 fo the calculato whee the feld cocetato s eglgble copaed to the othe egos e substates Fg 5 As becoes clea fo Fg 9b, th has a fluece o the ube of teatos, teestgly lowe th gves lowe ube of teatos at the hghe fequeces a S / db th - db th -6 db fequecy / GHz b teatos th - db th -6 db 5 5 fequecy / GHz Fg 9 Reflecto coeffcets a ad the ube of teatos b of the stuctue Fg 5c fo dffeet th

48 Chapte PML Paaetes ad Covegece 5 poved Nuecal Solve the couse of ths wok, the Weestass-sttute WAS, Bel optzed the F3 solve [-3] ode to tackle the covegece by the PML walls the poved veso, the solve s odfed by eaagg the syste atx M gve by equato the old veso uodfed veso, called the old solve, the syste atx s pecodtoed dectly by the SSOR pocedue followed by the teatve soluto usg the Kylov-Subspace ethod [4, 5] As aleady etoed see the esults of Fg 8, the posto of the ows as well as atx eleets M has a geat fluece o the covegece ad the eshufflg of the atx eleets ca esult a poved covegece Takg that d the solve was odfed by usg dffeet sotg ules fo depedet Set Odeg to accoplsh the eaageet of the syste atx ths way the syste atx s educed to a salle deso Schu-copleet, S whch ca be solved faste afte the usual SSOR pecodtog Ths odfed veso wll be called hee ew solve The detals of the odfcatos ad pocedue of the F3 solve ae gve the Appedx 77 ode to check the beefts of the ew solve veso ove the old oe the copaatvely sple patch atea stuctue of Fg 3 wll be deostated fst Both the old ad ew solves ae used to sulate the atea sepaately wth x ad y PML as well as x ad z PML walls cludg the ovelappg egos both the cases the teatve pocedue fo a ceta fequecy s teated ethe f covegece s eached o f the ube of teatos exceeds 3, wthout covegece The esultg ubes of teatos ae gve Fg 3 Fo the ovelappg PML walls x ad y dectos, the old solve does ot covege aleady at GHz wheeas the ew solve eaches the teato wthout covegece oly at 65 GHz Fg 3a Fo the ovelappg PML walls x ad z dectos, aga the old solve does ot covege aleady at GHz whee as the ew solve coveges the whole fequecy age wth a teato cout of 36, at 8GHz Fg 3b So the ew solve woks uch bette tha the old oe egadg the ube of teatos whe appled to the PML walls wth ovelappg egos Howeve, eve though the ew solve educes the ube of teatos sgfcatly the cases of ovelappg PML walls, the ube of teatos s stll too hgh to be used ost pactcal cases So we eed o-ovelappg PML ad the othe cocepts descbed the pevous sub-sectos Fg 3 Half of a ectagula patch atea The desos of the half patch, feedg costp le ad coputatoal doa ae 5µ 6µ 6µ, 5µ 3µ 8µ ad 4844µ 8µ 3µ x, y ad z dectos espectvely The PML cell sze ad ube of cells x, y ad z dectos ae 8µ, µ, µ ad 3, 5, 4, espectvely - 4 -

49 Chapte PML Paaetes ad Covegece a b teatos ew solve old solve teatos ew solve old solve fequecy / GHz fequecy / GHz Fg 3 teatos of the patch atea Fg 3 wth ovelappg PML walls x ad y dectos a, ad, x ad z dectos b The ew solve allows dffeet sotg of the syste atx ad theeby the eaageet of the eleets of the syste atx The ube of teatos depeds sgfcatly o the dffeet sotg ules The sulatos coespodg to the esults Fg 3 wee caed out wth the optzed sotg of the syste atx The optzato of the sotg wll be aalyzed the followg wth help of the patch atea Fg 6a dscetzato schees : x 4, y 3, z 4 ; Fg ad Fg 3a shows the copaso the teato cout wth the o ovelappg PML walls x, y ad z dectos fo thee dffeet sotg ules of alteate odeg, evese odeg ad SOEG odeg [4] epedg o the sotg, the ube of teatos ca be educed by 7% at 6 GHz Also, the covegece s geatly depedet o aothe paaete called the elaxato facto, ω [5] The value of ω les betwee ad Fg 3b shows the depedecy of the ube of teatos o ω at dffeet fequeces Fo all fequeces, the ube of teatos ceases abuptly fo values of ω above 6 8 t has bee foud that 5 s a good choce fo ω fo the stuctues studed hee, cludg patch atea, MS le, CPW le, LTCC, Flp-chp etc a b teatos alteate evesed teatos GHz 6 GHz 5 SOEG fequecy / GHz GHz ω Fg 3 The ube of teatos fo the dffeet sotg of syste atx usg the ew solve a ad fo dffeet values of ω b appled to the patch atea Fg 6a wth dscetzato schees of x 4, y 3 ad z Fg 6 ad 7-4 -

50 Chapte PML Paaetes ad Covegece Fg 33 shows the fluece of the ew solve copaso to the old solve fo the case wthout PML ad the case wth o-ovelappg PML Now the sae atea Fg 6a wth dscetzato schees of x 4, y 3 ad z Fg 6 ad 7 s sulated ode to check the fluece of the ew solve copaso to the old solve fo the cases wthout PML ad o-ovelappg PML We see that thee s a slght poveet the hghe fequecy ages the case wthout PML whe usg the ew solve O the othe had, we see a 6 to % decease the ube of teatos the lowe fequecy ages wth the ew solve the o-ovelappg PML case As the atx eleets ae fequecy depedet, the sotg ules wok bette fo dffeet fequecy ages egadg the poveet covegece The SOEG sotg s used fo eaagg the syste atx the ew solve case wheeas the old solve case o sotg s used o eaageet of the syste atx We see that SOEG sotg ths case deceases the ube of teatos the hghe fequecy ages fo the o-pml case slghtly, ad the lowe fequecy ages fo the o-ovelappg case a teatos ew solve old solve fequecy / GHz b teatos ew solve old solve fequecy / GHz Fg 33 The ube of teatos of the patch atea Fg 6a wth dscetzato schees of x 4, y 3 ad z Fg 6 ad 7 fo the case wthout PML a ad fo the case o ovelappg PML walls x, y ad z dectos b elato wth the old ad ew solves The patch atea stuctues studed befoe ae elatvely sple ad ght ot povde the full pctue wth egad to the solve s potetalty Theefoe, the copaatvely coplex flpchp stuctue wth o ovelappg PML walls Fg 34 s aalyzed the followg Noovelappg PML walls ae used the lateal decto to eplace the ope boudaes The esultg teato ubes ae plotted Fg 35 We see that the SOEG sotg of the ew solve povdes a oveall poveet covegece copaso to the old solve the whole fequecy age wth a decease the ube of teatos by % By choosg the evese sotg, the ube of teatos ca be educed by %, but oly the lowe fequecy ages - 4 -

51 Chapte PML Paaetes ad Covegece Fg 34 MS to CPW flp-chp tecoect teatos ew SOEG old atual ew evese 3 4 fequecy / GHz Fg 35 Copaso of the ew ad old solve: ube of teatos agast fequecy fo the flp-chp stuctue of Fg

52 Chapte PML Paaetes ad Covegece 6 Cocluso The fluece of the PML boudaes o the covegece of the teatve soluto ca be suazed as follows: PML elages the age of the agtudes of the eleets of the syste atx t s show that due to the PML, seveal ows of the syste atx ae affected whee the o-dagoal eleets becoe sgfcatly lage agtude tha the a dagoal oes The outcoe s that the codto ube of the syste atx gows ad, cosequetly, the ube of teatos theeby the CPU te whe solvg the syste of equatos ceases dastcally Ths effect ca be tgated, of couse, by povg the atheatcal solve appled, as t s descbed Sec 5 Howeve, the esultg poveets ae ot sgfcat eough to solve the poble Two a ssues have bee foud: Ovelappg PML egos at edges ad coes ake the usage of the PML bouday codto alost possble fo pactcal cowave stuctues t coes out that by avodg ay ovelappg, the hgh cout of the ube of teatos ca be loweed vey sgfcatly wthout sacfcg accuacy too uch PML cell sze s the ost potat paaete egadg the covegece of the teatve soluto The vestgatos o seveal stuctues lke co-stp, patch atea, flp-chp tecoect, spal ducto, couple, eveal that by akg the PML cell szes the lagest oes the whole esh, the teato level ca be loweed dastcally Fally, a thub ule s developed fo estatg the elatve delectc costat the PML ego, f the PML wall s seted a layeed eda, thus esug that the absopto level of the PML appoaches the gve oal eflecto eo

53 3 Wavegude Pot Sulato wth PML Mostly, thee-desoal 3 sulatos of passve stuctues ae caed out though wavegude pot exctatos Eve though oly the desed physcal odes ae to be excted at the wave gude pot, paastc atfcal odes ae also excted due to the fte bouday of the calculato doa cluso of PML as ope bouday akes the stuato eve wose, eg because t exctes odes at the wavegude pot wth coplex popagato costats γ eve f the vestgated stuctue s loss fee The goal of ths chapte s to vestgate the chages ode spectu of a logtudally hoogeeous stuctue due to PML A sple MS stuctue s used to descbe the effects of PML o ode spectu at the wavegude pots 3 Physcal ad atfcal odes As the FF ethod deals wth a bouday value poble, the wavegude pots suppot two types of odes The fst oes ae the physcal odes that ae elated to the physcal wavegude MS, CPW les fo exaple The secod oes ae atfcal odes, also called box odes, whch exst due to the bouday whch s toduced oly fo the sulato puposes ad s ot peset ealty Coespodgly, these odes deped o the sze ad type of the boudaes, electc o agetc walls, fo exaple The box odes have usually ozeo cutoff fequeces whee as the physcal odes ay have zeo cutoff fequeces eg MS, CPW odes ad ozeo cutoff fequeces eg hghe ode substate odes The box odes lke the physcal odes ae able to popagate alog the wavegude ad theeby ay tefee wth each othe wheeve thee s a dscotuty alog the wavegude f thee ae p ubes of coductg les at the wave gude pot, the ube of physcal odes wth zeo cutoff fequecy s equal to p assug that the desos of the les ae suffcetly sall copaed wth the wavelegth Oe should ote that ths dstcto betwee physcal ad atfcal odes does ot apply stctly ove the ete spectu Ths s because of the F dscetzato Fo the dscetzed stuctue, oe has a fxed ube of odes, whch solely deped o the ube of esh cells of the wavegude Moe pecsely, the ube of egevalues whch s equal to twce the ube of esh cells N at the pot Theefoe, the age fo hgh γ α + jβ, thee ae a lot of odes whch ae oe o less deteed by the dscetzato oly ad thus ethe physcal o box odes PML bouday codtos ae used ode to sulate ope space So the box odes exstg due to the atfcal boudaes should ot appea whe PML walls ae used Sce the total ube of odes s fxed see above, these odes ca ot be elated but oly shfted the spectu, desably to a age fa away fo the physcal odes Howeve PML s desged to absob waves pgg oto the PML t s ot equally sutable as a lateal bouday of a wavegude Theefoe, ot all the box odes ae shfted popely ad they ay appea betwee the physcal oes Oe has to fd out how to dstgush aog physcal, box ad est of the atfcal odes usg PML Oe possblty s the cteo PPP Powe Pat PML whch s defed [5] as the ato betwee the powe of a patcula ode the PML ego ad the total powe of

54 Chapte 3 Wavegude Pot Sulato wth PML that ode calculated at ceta wavegude pot Accodgly, the odes whch have copaatvely lage PPP values wll be defed hee as PML odes Obvously the PML odes belog to the goup of atfcal odes The PPP values of the physcal ad box odes deped o the aout of felds the PML ego Howeve, t tus out that oly based o the PPP values the PML odes ca ot be sepaated fo the desed physcal odes The PML odes ad the effect o physcal ad box odes wll be descbed the ext sub-chapte A MS stuctue Fg 3a s used to show the appeaace of all these physcal ad atfcal odes A dscotuty of two MS les wth dffeet wdths s used devotedly to check the couplg of MS ode to othe odes Except fo stp wdth, the wavegude pots pot ad ae the sae at both eds of the stuctue The coss secto of pot s depcted Fg 3b The stuctue s sulated wth ad wthout PML Whe sulated wth PML, two PML walls desgated by X- ad Y-PML ae used ad whe sulated wthout PML, the PML egos ae just eplaced by the coespodg eghboug ateals eg a ad ceac So both cases the oute desos of the stuctue ea the sae The PML walls ae followed by agetc walls MW, wheeas aothe agetc wall syety plae s used to sulate oly half of the stuctue ad the ceac substate s gouded by electc wall PEC PML walls ae used such a way that PML cells ae the lagest oes A ovelappg PML wall s used at the cossg of X- ad Y- PMLs the vestgatos of ths sub-chapte ode to stck to the basc PML defto [8], howeve, t s foud that XY o ovelappg PML walls show the sae effects as the ovelappg PML does a b Fg 3 a: scotuty of two MS les o ceac substate The wdths of the MS les at pot ad ae µ ad µ espectvely The thckess of the etallzato s 5 µ b: Coss-sectoal vew at pot The thckesses of the X-PML a x ad Y-PML a y walls as well as the heght b x ad wdth of the stuctue b y ae vaed The cosssecto at pot s sae as at pot except fo the dffeet wdth of the stp The stuctue s teated by a electc wall PEC at the botto ad othewse by agetc walls MW the wavegude sulato, efeed to as the followg, all the odes ae soted fst by the ascedg ode of α α beg the agay pat of the popagato costat ad the by the descedg ode of β f α s of the two odes ae sae [] So the odes wth salle α s appea fst ths way the evaescet odes, whch ae ot of teest

55 Chapte 3 Wavegude Pot Sulato wth PML ost cases, ae shfted at the ed of the ode lst O the othe had, the PPP cteo ca be used to flte out the odes The value of PPP vaes betwee ad So f PPP equal s used all the odes ae cluded ad f PPP s 3, fo exaple, oly the odes whose powe poto the PML ego s less the 3% ae cosdeed The MS stuctue of Fg 3 s sulated wthout PML o-pml ad wth PML cosdeg the fst odes each pot both cases PPP equal to s used wth PML case The atteuato costats α ad effectve delectc costats ε _effectve β/β of these odes at pot the cases of o-pml ad wth PML ae show Fg 3a ad 3b espectvely Out of these odes, ode s the MS ode [4] Mode s the TM ode ad ode 6 6 s the TE ode, both of these ae suface wave odes, tasvesal agetc ad electc, espectvely, to the pepedcula decto to the substate x decto The est of the odes ae box odes The feld pattes of TM ad TE odes ae show Fg 33 a α / Np/ wthout PML fequecy / GHz b ε -effectve wthout PML fequecy / GHz Fg 3 Wthout PML: atteuato a ad effectve delectc costats b at pot of Fg 3 agast fequecy fo dffeet odes b x 9 µ ad b y 4 µ TM TE 6 Fg 33 Wthout PML: The electc felds of TM ad TE odes appeag at pot of Fg 3 TM has the feld copoets of E x, E z ad H y TE has the feld copoets of E y, H x ad H y [4,46]

56 Chapte 3 Wavegude Pot Sulato wth PML t s wothwhle to dscuss these suface wave odes oe detal Not oly a gouded delectc wavegude but also othe teestg wavegudes lke ugouded delectc slab, delectc od, cougated coducto etc suppot suface waves [4-44, 46] Suface waves ca also be excted o soe types of plaa tassso les, such as MS le eg Fg 3, CPW le eg Fg 35 ad slot le Suface waves have felds whch decay away fo the delectc substate whle ost of the felds ea o ea the a/delectc teface The hghe the fequeces, the oe tghtly boud ae the suface wave felds the delectc, so the suface waves gudg wavegudes ae ostly of pactcal teest at hghe fequeces Aothe teestg chaactestc of suface waves s that the phase velocty s less tha the velocty of lght vacuu due to the pesece of the delectc The pcpal types of suface wave odes suppoted by a gouded delectc ae the tasvesal agetc TM ad tasvesal electc TE odes The cutoff fequeces of these odes ae gve by 3 ad 3, espectvely, assug that the gouded delectc s ftely exteded the y ad z decto whle the vacuu above the substate s also ubouded x decto Fg 3 would be the case whe MS s ot peset ad all the lateal boudaes ae shfted to fty h s the heght of the delectc ad c s the velocty of lght vacuu f f TM TE c,,, h ε c,, 3 4h ε 3 3 Ay ozeo thckess h > delectc slab wth elatve delectc costat ε lage tha uty ust suppot at least oe popagatg TM ode, the TM ode, whch s the doat ode of the delectc slab wavegude wth zeo cutoff fequecy The gudace equato of the TM odes s gve by 33, whee k d ad k a ae the cutoff wave ubes the two ego of delectc ad a, espectvely The phase velocty of TM odes β s elated to k d ad k a by 34 ad 35, espectvely Though soe atheatcal apulatos aog 33, 34 ad 35, k d, k a ad β ca be deteed The TM odes allow oly E z, E x ad H y feld copoets whch ca be calculated eadly oce the values of β ae kow k d ta k h ε k 33 d a kd ε k β 34 k a k β 35 TE odes ulke TM odes suppot oly ozeo cutoff fequeces The gudace equato fo TE odes s gve by 36 The TE odes allow oly H z, H x ad E y feld copoets whch ca be calculated eadly oce the values of k d ad k a ae kow Accodg to 3 ad 3 the ode of the popagato of the TM ad TE suface wave odes ae TM, TE, TM, TE, etc

57 Chapte 3 Wavegude Pot Sulato wth PML k cot k h k 36 d d a Now the stuctue of Fg 3 s copaed wth the deal case Ths stuctue suppots oly two popagatg suface wave odes, TM ad TE f TE GHz up to the fequecy of 5 GHz Accodg to [4] the effectve pettvty of the TM ode stats to devate fo the value of at the fequecy f of 6 GHz, calculated by 37 c f 37 h addto to the pesece of the MS le, the stuctue s bouded by agetc walls y ad postve x dectos Ths toduces a ozeo cutoff fo all odes except the MS oe TM, Fg 3b athe stats at GHz wth ε _effectve equal to Ths depedecy of cutoff fequecy of TM o the pesece of the agetc bouday x decto s checked by vayg the legth of b x Fg 34 t s obvous that shftg the agetc wall fa away fo the delectc ca shft the cutoff fequecy of the TM ode at lowe fequeces ad suppot the dspeso fequecy f gve by 37 ε -effectve TM ε + / f TM, h/λ b x fequecy / GHz Fg 34 Wthout PML : epedece of the cutoff fequecy of TM ode o the dstace of the agetc bouday x decto of Fg 3 e b x s vaed f s 6 GHz ad f TM, s 3 GHz Aga oe has to cosde that the gudace equato gve by 33 fo TM ode s tue oly fo upetubed slab, eg f copaed wth Fg 3, the MS le s ot peset ad b x exteds to fty Fo the sake of claty the stuctue of Fg 3 s calculated ad the effectve delectc costats fo both the cases of wth ad wthout MS ae lsted Fg 35a The electc felds of TM ode fo the cases of wth ad wthout MS le ae gve Fg 35b ad 35c, espectvely, clealy showg that the E x felds of the TM ode ae petubed by the MS le t s checked that the gudace equato of 33 s vald oly fo the wthout MS case f ε _effectve The dffeece ε _effectve values fo both the cases vaes betwee % ad 8% wth espect to the case wth MS

58 Chapte 3 Wavegude Pot Sulato wth PML The TE ode s suppoted by the stuctue of Fg 3 as well Ths ode s affected two ways ths stuctue pesece of the MS le ad the agetc wall the y decto ue to the agetc wall, the E y copoet of the ode ust vash at the bouday That s why the TE 6, Fg, 3b stats wth ε _effectve equal to athe tha at GHz The gudace equato of 36 s fulflled the fequecy age of to 5 GHz whee the values of ε _effectve stat to be lage tha [4] suggests that ode to avod ay couplg betwee the suface wave odes ad the MS ode, the uppe fequecy level should be kept below the fequecy f TM, gve by 38 whee the ε +/ le eets the TM dspeso cuve, as show Fg 34 a ε -effectve TM wthout MS wth MS fequecy / GHz b TM ode wth MS c TM ode wthout MS Fg 35 Wthout PML: fluece of MS le o TM ode b x of Fg 3 s chose to f TM, c ta ε πh ε 38 Apat fo the physcal odes of MS, TM ad TE, the othe odes Fg 3 ae due to the atfcal agetc boudaes ad, theefoe, ae ot of teest - 5 -

59 Chapte 3 Wavegude Pot Sulato wth PML 3 PML odes ad the couplg to othe odes Afte kowg what type of physcal odes ad box odes ca be suppoted by the stuctue of Fg 3 t wll be ease to detect the PML odes Sce we kow ow that sae PML odes ay be close to physcal oes the spectu ad ot always ealy to sepaate, oe has to clafy whethe ad how they affect the behavou of the physcal odes a 3 stuctue, e, the S-atx of the physcal odes Ths s doe the followg PML walls ae used such a way that PML cells ae the lagest oes A ovelappg PML wall s used at the cossg of X- ad Y- PMLs the vestgatos of ths sub-chapte ode to stck to the basc PML defto [8], howeve t s foud that XY o-ovelappg PML walls show the sae effects as the ovelappg PML does The oal eflecto coeffcets used fo the PML walls s 8dB PML odes ae typfed geeal by the hghe PPP values togethe wth ozeo values of α ad β Aga α values of PML odes ae also lage copaed to the physcal ad box odes How PML odes ca be sepaated fo the physcal oes based o PPP cteo ad α values ae show [5] Howeve pobles ase whe suffcet aout of felds of physcal odes ae the PML ego ths case, the PPP values ae also vey hgh fo the physcal odes ad copaable to the PPP values of PML odes The MS stuctue Fg 3 s sulated wth PML as etoed above ad the PML odes ae flteed out accodg to the PPP ad α values The PPP, α ad ε _effectve values of the physcal MS, TM, TE ad box odes 3 ad 7 excted at pot ae plotted Fg 36 The PPP of MS ode s ealy zeo as ost of the felds cocetate ea the MS ad alost o felds the PML ego TM ode has vey hgh values of PPP at the lowe fequeces ad PPP deceases at hghe fequeces Ths s because, at lowe fequeces the felds E x copoet exst alog the whole Y-PML ego ad at hghe fequeces the felds cocetate ea the delectc Nevetheless, a lage poto of the felds ae stll the Y PML ego ad that s why ths ode s hghly affected by PML Eve at 5 GHz ths ode has a PPP value of above 4 f a PPP value of 4 s assued to flte out the PML odes, the TM ode wll be flteed out as well Ths eas the effect of ths physcal ode o MS ode s eglected the 3 calculato t s also tue fo TE ode ad all box odes Fo TE ode the PPP vaes fo 9 to 3 Nea the cutoff fequeces of these odes PPP values ae the lagest oes Oly odes of 3 ad 7 ae show hee because of the low PPP values ad lowe atteuato costats The physcal as well as the box odes ae also atteuated o aplfed due to the PML, eve though the stuctue s loss fee Whle α of the MS ode eas ea to zeo, α of TM ode fluctuate betwee 5 to 5 Ths ode s atteuated betwee 6 ad 9 GHz ad at othe fequeces s aplfed TE ode s also atteuated ad the atteuato stats to decease at GHz fo 6 to 36 at 5 GHz teestgly both TM ad TE have a axu atteuato costat at 8 ad GHz, espectvely The box odes of 3 ad 7 ae also hghly atteuated The dspeso chaactestcs of the MS ode do ot chage whe copaed wth the o- PML case Howeve the TM ode ow stats to popagate at lowe fequeces wth ε _effectve values of ad eas at ths value up to 6 GHz Lke TM TE ode also stats to popagate wth ε _effectve values of at GHz Because of vey hgh α ad low β values copaso to soe PML odes at cutoff fequeces the box odes ea at the lowe ed of the ode spectu ad do ot appea f a defte ube of odes eg ae excted at the pot 3 ad 7 ae show Fg

60 Chapte 3 Wavegude Pot Sulato wth PML 8 wth PML PPP 6 4 TM 3 MS TE fequecy / GHz α / NP/ wth PML MS TM TE fequecy / GHz wth PML ε -effectve MS TM 3 TE fequecy / GHz Fg 36 Wth PML : PPP top, atteuato costats cete ad effectve elatve delectc costats botto of the MS, TM, TE ad box odes 3 ad 7 Fg 3 at pot of Fg 3 as a fucto of fequecy a x, a y, b x ad b y ae 95, 4, 9 ad 4 µ, espectvely - 5 -

61 Chapte 3 Wavegude Pot Sulato wth PML 3 box ode 7 box ode Fg 37 Wth ovelappg PML : ode spectus of box odes 3 ad 7, Fg 3 of patcula teest pot of Fg 3 at 5 GHz a x, a y, b x ad b y ae 95, 4, 9 ad 4 µ, espectvely Up to ths pot, all the esults ad dscusso have bee doe fo the pot of Fg 3 The sae s also tue fo the pot of Fg 3 Now a 3 sulato s caed out whle allowg odes at each pot ad the couplg of MS ode to all othe odes wll be checked Tab 3 these couplgs betwee the MS ode ad othe odes at pot of Fg 3 ae lsted togethe wth α, β ad PPP values at 5 ad 5 GHz fo exaple At 5 GHz the eflecto of MS s 7 db as the chaactestc pedaces of the two les ae 5 ad 7Ω espectvely At ths fequecy, except fo the MS ad TM ode, all ae PML odes ad these PML odes ae clafed by the hgh PPP ad α values The couplgs betwee MS ad PML odes ae vey low < -4dB Obvously the couplg betwee the MS ad TM ode s also vey low -38dB At 5 GHz, o the othe had, TE ad box odes ae also peset The couplgs betwee the PML odes ad MS ode ae aga vey low < 4dB The TM ode shows also vey low couplg -47dB at 5 GHz TE ad 7 show elatvely hgh couplg of 6dB ad 8 db, espectvely Ths s because the felds of these two odes look vey detcal ad cocetate wth a lage aout ea the MS see Fg 33 ad 37 The eflecto coeffcet of the MS le at ths fequecy s db odes type ε -eff α Np/ 5 GHz 5 GHz PPP couplg type ε -eff α PPP to MS Np/ db couplg to MS db MS MS 99 - TM TE PML TM PML PML PML PML PML PML PML PML PML PML PML Tab 3 Couplg betwee the MS ode ubeed ad all othe odes ubeed to fo 5 ad 5 GHz at pot of Fg 3 ae lsted Type, effectve delectc costat ε -eff, atteuato α, PPP values of each ode ae also gve fo copleteess

62 Chapte 3 Wavegude Pot Sulato wth PML The couplg of the MS ode at pot to othe odes at pot shows sla behavou as suased Tab 3, ot oly fo 5 ad 5 GHz couplg of PML odes to MS ode eas vey low <-4dB at all fequeces As TE ad 7 shows elatvely stoge couplg above 3 GHz to the MS ode, oe ust caefully decde whethe to eglect o ot eglect these odes by eovg the though the PPP cteo Up to 3 GHz PPP tus out to be a vey good cteo to eove PML odes ad TM ode whch have vey low couplg to the MS ode fo ths stuctue As the appeaace of suface odes ad box odes depeds o the stuctue tself ad ts bouday, oe eeds a tag of wth the help of PPP ad 3 sulato wth the help of couplg ode to eove uecessay odes fo the calculato doa As the case of TM ode whee ost of the felds ae the PML ego ad theeby stogly affected by the PML atteuated o aplfed, the felds of MS stp ode should be cocetated the o-pml ego Fg 38 shows that f the PML walls ae placed vey ea to the MS stuctue, the MS felds exst the PML ego ad the MS ode s hghly atteuated o aplfed Ths ases the questo of how fa the PML walls should be placed, patculaly fo plaa stuctues lke MS ad CPW le t s foud that a dstace of 5 tes the axu betwee substate thckess ad MS wdth fo the MS as well CPW les s a good thub ule to place the PML walls ad theeby to ze the effect of PML ε -effectve α / Np/ fequecy / GHz -3 Fg 38 Wth PML: effectve delectc costats ad atteuatos of MS ode a at pot of Fg 3 as a fucto of fequecy a x, a y, b x ad b y ae 5, 5, 5 ad 5 µ, espectvely The coespodg MS ode spectu b at 5 GHz

63 Chapte 3 Wavegude Pot Sulato wth PML 33 Cocluso t was show that the FF sulato wth PML geeal gves se to atfcal PML odes at the wavegude pots These PML odes ay appea close to the physcal odes the spectu but wee foud to show vey low couplg to the physcal odes of teest eg MS, CPW, PPL etc The MS o CPW odes, whose ost felds patculaly cocetate the o-pml ego, ae ot affected by the PML, f PML walls ae suffcetly fa away fve tes the axu betwee substate heght ad stp wdth The suface wave odes as well as box odes ay be hghly affected by the PML The so-called PPP cteo defed as the powe pat of a patcula ode the PML ego s useful to flte out PML odes but ts tet paaetes ay dffe fo stuctue to stuctue A caeful obsevato followg the usg PPP cteo ad 3 sulatos checkg the couplg of uwated PML, box ad physcal odes fo couple of fst sulato u s eeded ode to eove uecessay odes fo the calculato doa case of doubt, a ode should be cluded ode to ata best accuacy Sce the fluece of PML odes o physcal oes s foud to be low, the esultg possble eos ae sall

64 - 56 -

65 Chapte 4 teal Pot 4 teal Pots The objectve of ths chapte s to descbe how teal pots ca be pleeted the FF ethod chapte, t was show that the teal pots ca be ealzed geeal tes of a elaxato cuet souce o a coducto cuet souce t wll be show hee that the elaxato cuet exctato FF ethod does ot povde physcal soluto As s kow, the coducto cuet teal pots possess paastc ductaces t wll be show how the values of the paastc ductaces ca be estated fo deebeddg teal pots ake the way of exctg ay stuctue fo ay posto sde the stuctue t becoes vey uch effcet fo a EM sulato f a MMC togethe wth ts housg othe boad/package s ted to be sulated As the desos of the luped ad actve devces MMCs ae vey uch sall copaed to the passve eleets, the EM sulatos eed a vey lage esh sze to put all the ccut eleets togethe teal pots povde oe soluto fo ths poble The ty ccut copoets luped eleets, actve eleets ca be eplaced by teal pots ad the esultg cuet/voltage elatoshp eg S- o Z-paaetes foud at the teal pots though the EM sulatos ca be used etwok sulatos ode to teface wth the eplaced copoets Ths way wth the help of teal pots the coplex MMC stuctues alog wth the housg ca be sulated easly pat by pat, eg passve package ad the actve copoets sepaately, ad the the pats ca be tefaced the etwok sulatos As a exaple, Fg 4a shows the aay of a GHz powe tassto The tassto fges ae fed by the MS wavegude feedg les ot show hee though bod wes ug the desg of the devce, t s teestg fo the ccut sulatos to kow the S-paaetes at the pots whee the bod wes ae coected wth the tassto fges Ths s ot possble wthout local exctatos at those pots Aga the sulato of the stuctue as a whole s cubesoe ad effcet fo EM sulatos As a eedy the tassto aay s take out of the stuctue by setg teal pots at the tassto fges The stuctue wthout the tassto aay ca be sulated wth wavegude pots at the feedg les ad teal pots at the tassto fges The esultg S-paaetes ca be used to teface wth the tassto aay etwok sulatos As show Fg 4b, the teal pots ae placed at the feedg les fo the bod wes ad the esultg voltage ad cuets ae calculated to obta the esultg S-paaetes a b tassto fges feedg le fo bod wes space fo tassto teal pots aay feedg le fo bod wes bod we substates etal Fg 4 Powe tassto aay a Sulato stuctue fo the powe tassto s housg b

66 Chapte 4 teal Pot 4 Relaxato Cuet teal Pot The elaxato cuet teal pot s ealzed by setg two te vayg chages of equal agtudes but opposte polates at two odes of two coductos whch ae pat of a ceta ccuty ths way a te vayg cuet s defed to ete ad coe out of the ccuty at those two odes The basc defto of ths type of teal pot exctato show Fg 6 s epeated hee Fg 4 E Q -Q ode delectc Q +Q ode q Q jω U Edl Fg 4 Relaxato cuet teal pot : voltage souce q v, J e The pleetato of the elaxato cuet teal pot pocedue FF ethod s gve Appedx 7 The FF ethod deals wth a dscetzed wave equato gve by 4 Equato 4 s solved fo the vecto e whch cotas all the electc feld copoets of each cell of the FF esh M, M ad M 3 ae the atces that cota the foato o ateal popetes ad desos of the stuctue ude cosdeato M atx s defed to coespod to the opeato, o the othe had M 3 s defed to coespod the opeato Vecto efes to the elaxato cuet teal pot souce tes of te vayg chages ad s elated to atx M 3 by 4 M 3 s bought to the equato 4 ode to elate statc chages sde the coputatoal doa ad theeby to pove covegece [6, 7] M e whee, M 4 M + M + M e 3 M ε, µ, κ,, d M M ε, κ, d, ad M ω M ε, µ, κ, 3 3 d M 3 e

67 Chapte 4 teal Pot a hoogeeous ateal, 4 s the dscetzed epesetato of the Helholz s equato as gve below by 43 E + k E 43 Because the F foulato allows fo a oe-to-oe coelato wth the vecto dffeetal equato, the followg cosdeatos wll be based o the latte oe see also [8] 4 ca be wtte as E E k E Thus, the hoogeeous equato wth has two types of solutos, oe wth E ad E, the othe wth E ad E The fst efes to the electodyac but dvegece-fee ad thus chage-fee E-feld The othe efes to a o-physcal feld, whch volves E ad thus space chages f the exctato felds, e, 43 ad 44, ae dvegece-fee, the o-physcal felds ae ot excted ad thus zeo Usg a te-vayg chage, howeve, as doe the exctato cuet veso, E holds ad thus the esultg soluto fo E s a xtue of physcal ad o-physcal felds ad thus useless fo the teded EM sulato pupose the F foulato, ε E efes to M e 3 ad oe has aalogy to 44 two types of solutos fo 4 The physcal oe wth M e 3 ad M e, ad the o-physcal oe wth M 3e ad M e Thus, toducg a elaxato cuet souce, e te vayg put chages, the esultg soluto of equato 4 wll copse solutos wth M e ad M 3 e ad theeby lead to o-physcal solutos Ths o-physcal soluto akes the exctato of a stuctue by elaxato cuet souces o-feasble The o-feasblty of the elaxato cuet teal pot souce s explaed usg a exaple of the stuctue gve Fg 43 a cube of agetc walls flled wth a two detcal coppe les ae defed At the pots of A 3 ad B 3 the two odes of a elaxato cuet teal pot ae defed, whee a te vayg postve chage Q ad a egatve chage - Q ae appled at ode A 3 ad B 3 espectvely The agtudes of the chages ae calculated by eas of equato 79 fo Ap cuet at GHz

68 Chapte 4 teal Pot MW A le A le B B MW A B MW Z A 3 B 3 X Y MW Fg 43 Two detcal coppe σ 56e7 les le A ad B ae cotaed a cube of agetc walls MW The sdes of the MW cube ae 5 µ The legth l, wdth ad thckess of each le ae 5, 4 ad µ The les ae 4 µ apat fo each othe A, A, A 3 ad B, B, B 3 ae soe pots of teest sde the le A ad B espectvely The cell szes sde the les ae less tha oe fouth of sk depth At pot A 3 a postve te vayg chage +Q 5969E- C ad at pot B 3 a egatve te vayg chage Q -5969E- C ae appled The wave legth λ at GHz fo ths le syste s 8 Afte the 3 soluto of equato 4, the chages at each cell ae calculated The chage dstbuto at each cell oalzed by the axu chage apltude o the x 7 µ plae s show Fg 44 t s foud that the total aout of chage sde the cube s zeo The dual cells whch cota the pots A 3 ad B 3 possess the sae aout of chages wth the sae polates of the put chages le A, the sae aout of put chages Q but egatve polaty s seted by the soluto, deceasg fo the suoudg cells of A 3 towads the ed of the le the sae way, le B, a postve chage of Q s dstbuted by the syste deceasg fo the suoudg cells of B 3 towads the ed of the le Ths suggests that the soluto tes to oppose the seto of te vayg chages by dstbutg the sae aout of chages wth opposte polates f oe ed of a two coductos le syste s coected to cuet souce of Ap ad the othe ed s ope, the cuet deceases lealy alog the le assug l<<λ ad eds wth zeo at the ope ed of the le syste case of the exaple of Fg 43, the chages ad cuets alog the le A o B ae show Fg 45 Obvously ths does ot show a physcal behavou of cuet fo such a tassso le syste Eve though the put cuet s Ap, alog the le the cuet s alost eglgble Ths ples that the soluto povdes o-physcal chage as well as feld dstbuto - 6 -

69 Chapte 4 teal Pot The ad 3 electc feld dstbuto o the plae of x 7 µ, ae gve Fg 46 ad 47 espectvely Because of postve ad egatve put chages, we see at pot A 3 ad B 3 the electc felds cog out off ad gog to the chages espectvely We also see the pesece of egatve chages alog the le eg pots A, A etc whee the felds go to the chages Sae way alog the le B thee ae postve chages, so that electc felds ae cog out off these chages The stegth of the felds s also deceasg alog the le All of these povde o-physcal behavou of the soluto of equato 4 whch suggests that such a elaxato cuet teal pot exctato s ot feasble Fg 44 3 vew of chage dstbuto o the cete plae x decto x7 µ Q/Q ax put chage * 4 / Ap alog the legth of le / µ Fg 45 Cuet ad chage dstbuto Q alog the le A The sae s tue fo le B as well Q ax 5969E- C - 6 -

70 Chapte 4 teal Pot Fg 46 electc feld dstbuto o the cete plae x decto x7 µ Fg 47 3 electc feld dstbuto o the cete plae x decto x7 µ - 6 -

71 Chapte 4 teal Pot 4 Coducto Cuet teal Pot Le Cuet The coducto cuet teal pot s ealzed FF ethod by defg a path alog the edges of cells betwee two coductve stuctues The path ca be chose abtaly Howeve, ode to obta useful esults the path legth should be sall copaed to the wavelegth ode to have a well-defed voltage pot Alog ths path, a fte le cuet s defed The basc defto of ths type of exctato show Fg 5 s epeated hee Fg 48 ad the pleetato FF ethod s gve Appedx 73 Assug o souces othe tha a coducto cuet souce sde the stuctue, equato 4 looks lke as gve below by 45 Hee vecto c stads fo the coducto cuet souce The M, M ad M 3 atces ae defed to cota foato o ateal popetes ad desos as stated the pevous chapte The elato betwee c ad the atces ae gve by 46 ad 47 Ulke the elaxato cuet souce, hee the excted feld s dvegece fee e εe So whe 45 s solved fo e, the vecto that cotas the electc feld copoets at each cell, t does ot esult ay o-physcal soluto coducto ode E H delectc Hdl U Edl coducto ode Fg 48 Coducto cuet teal pot : le cuet souce q v, J e M e whee, M 45 M + M + M e c 3 M ε, µ, κ,, d M M ε, κ, d, ω ad M M ε, µ, κ, 3 3 d 46 M + M e c M 3 e

72 Chapte 4 teal Pot The sple MS stuctue of Fg 49 s used hee to check the feasblty of ths exctato pocedue Two teal pots wth cuets ad ae used to feed the stuctue As two teal pots ae used, the stuctue ust be excted twce ode to get the coplete voltage/cuet elatoshp at the odes of the pots Tab 4 shows the values of ad as a exaple fo the exctatos Exctato / Ap / Ap - 3 Tab 4 Values of ad to feed the stuctue of Fg 49 Fo all of these exctatos the cuets alog the MS le MS ae calculated at GHz ad show Fg 4 Fo the fst exctato, both ad ae assged wth Ap ad both of the flow the sae decto e towad the stp ths case MS ust be equal to Ap at the two eds of the le ad lealy decease to zeo at the ddle of the le Fo the secod exctato ad ae assged wth Ap ad Ap ths case both the cuets ae opposte decto - a steady Ap cuet ust flow alog the le Fo the thd exctato, the MS s fed by Ap at oe ed ad ope at the othe ed So the cuet ust decease lealy fo Ap to zeo alog the le All these behavous ae physcal fo ay le syste whch s fed by cuet souces at the two ed of the le Hee t should be oted that, the lea behavous of the cuets Fg 4 ae due to the vey sall le legth copaed to wavelegth l<<λ, othewse they ae ot lea alog the le, athe susodal because of stadg wave patte X MS Y Z pot pot ceac 98 Fg 49 PEC MS o ceac ε 98 substate The legth l, wdth ad heght of the stp ae 6µ, 3 µ ad 5 µ espectvely The substate s µ thck At the two eds of the MS, two coducto cuet teal pots ae seted Pot ad feed the le wth cuets ad espectvely The wavelegth λ at GHz fo ths le syste s

73 Chapte 4 teal Pot MS / Ap Ap Ap Ap - Ap Ap alog MS / µ Fg 4 Cuets alog the MS le Fg 49 fo the thee dffeet exctatos accodg to Tab 4 Wth ay two of the exctatos of Tab 4, fo exaple, the Z-atx at the two pots of the MS le ca be calculated The foulas to calculate Z-atx fo ths coducto cuet souce pocedue s gve Appedx 74 f the efeece pedaces of the teal pots ae specfed, the the S-paaetes of the le ca be calculated ode to povde a atched codto at the pots, the chaactestc pedace of the MS le at each fequecy s take as the efeece pedaces of the pots ths case thee should ot be ay eflecto at the teal pots Accodgly, oe expects hgh etu loss at the two pots As ca be see fo Fg 4, howeve, cosdeable eflecto occus Ths ca be patly explaed by the fgg felds at the eds of the le, whch cause a exta capactace Sce also a ductace s foud at the ed, ths ust be due to the souce Ths paastc ductace wll be addessed the ext secto eebeddg the L ad C, oe has the le chaactestcs oly Sce the stuctue s syetc at the ddle of the le, the Z- ad S-paaetes ae also syetc at the two pots The eflecto coeffcets at pot ae show Fg 4 fo both the cases of wth ad wthout deebeddg of paastc effects Wthout deebeddg, the eflecto s vey lage: > -5 db above 5 GHz up to 5 db at 4 GHz Ths hgh eflecto s solely due to the paastcs as etoed above As the pot pedaces ae assued to be atched wth the MS le pedace, the vey sall paastcs becoe doat eough to fluece the MS le chaactestc t s clea that wth a LC deebeddg the eflecto goes below 5 db ad the a at half wave legth e 37 GHz becoe poet We see that thee s a sgfcat phase shft at hghe fequeces betwee the cases of wth ad wthout deebeddg eg at 5 GHz

74 Chapte 4 teal Pot a S / db - wthout deebeddg wth LC deebeddg fequecy / GHz b θ S / wth LC deebeddg wthout deebeddg fequecy / GHz Fg 4 Reflecto coeffcet a ad the coespodg phase b at pot of MS le Fg 49 The le pedace at each fequecy s take as the pot efeece pedace LC deebeddg wth L 8 ph ad C 3 ff 43 Paastc Effects of Coducto Cuet teal Pots As etoed befoe, the coducto cuet teal pot possesses paastc ductace The fgg felds at the eds/edges of the tassso le/etal plate, o the othe had, esult a paastc capactace Wheeve these paastc ductace ad capactace ae kow they ca be deebedded The deebeddg foulas ae gve Appedx 75 A uecal pocedue s developed to extact the paastc ductace ad capactace ad a look-up table s ade fo the ductace as a fucto of teal pot legth ad teal pot cell szes The extacto of teal pot paastc ductace L ex ad the ed capactace C ed s descbed wth the help of the ccut odel gve Fg 4 Ay tassso le, eg MS of Fg 49, fed by two detcal teal pots at the ed of the le ca be defed accodg to ths odel L ex ca be extacted fo Z ex by 48 ad 49, whee Z FF s the Z-atx of the EM-sulated stuctue cludg the paastcs, Z le s the Z-atx of the le wthout paastcs ad Y c s the adttace atx descbg the ed effect

75 Chapte 4 teal Pot Fg 4 Two detcal teal pots ae coected at the eds of the tassso le The tassso le s defed by chaactestc pedace Z, phase costat β ad legth l L ex stads fo the paastc ductace of the pots, whee as C ed epesets the ed effects of the le Y + Y Z ex Z FF le C 48 whee, Y le Z le jz jz cot βl secβl jz cosecβl cot jz βl ad Y C jω C ed jω C ed Z ex jω L ex jω L ex 49 [4] t s show how the ed capactace ca be calculated gaphcally C ed s the ff age ad gaphcal extacto of C ed ay lead to a accuacy L ex extacto Hee C ed s extacted a alteatve way A wavegude pot s seted at the cete of the le the logtudal decto ad the eflecto coeffcet S at the wavegude pot s calculated though the FF EM sulato C ed s the calculated fo S by 4 The extacto of C ed of the MS le Fg 49 s explaed Fg 43 as a exaple, whee t s assued that both cases the dscetzatos ae the sae C ed + 4 jωz whee, jφ ope e ad φope ac S + β l

76 Chapte 4 teal Pot wavegude pot MS Y X Z ope ed C ed S ceac 98 l / Fg 43 Extacto of C ed by setg a wavegude pot at the cete of the MS le of Fg 49 The dscetzato schees of both the half ad full stuctues ae kept the sae The values of C ed fo the half MS stuctue of Fg 43 ad theeby the values of L ex fo the full stuctue of Fg 49 ae lsted Fg 44 fo dffeet substate thckesses h As the teal pots ae coected betwee the goud ad MS, the legths of the teal pots ae sae as the substate thckesses A cease substate thckess eas a cease fgg felds ad theeby a cease ed capactace Fo ceasg substate thckess the o-quas-statc effects stat eale tes of fequecy Fo 3 µ substate thckess, C ed s alost costat up to 5 GHz, whethe fo µ substate thckess, C ed stats to devate fo the costat value aleady at GHz a C ed / ff µ µ 5 µ µ fequecy / GHz b L ex / ph µ µ 5 µ µ 35 ph 7 ph 9 ph 8 ph fequecy / GHz Fg 44 C ed a ad L ex b fo dffeet substate thckesses e legth of the teal pot accodg to Fg 49 ad 43, espectvely The teal pot cell szes y decto Y ad z decto Z alog the path of the teal pot ae kept µ

77 Chapte 4 teal Pot At the fequeces coespodg to ultples of half wave legth, the dffeece the sapled values of the stadg wave patte the eghboug cells becoes salle tha the uecal eos, whch esults vey lage o-physcal values case of L ex we see that ea zeo ad 37 GHz such vey lage ductace values Cosdeg the values whee L ex eas costat wth fequecy, we see that L ex ceases lealy wth the legth of the teal pot e thckess of the substate agg fo 8 ph fo 3 µ legth to 35 ph fo µ legth The values of L ex suggest that by educg the substate thckess ad thus the legth of the teal pot, oe ca educe the paastc effects aleady sgfcatly t s foud that L ex s vey uch sestve to the coss secto of the teal pot cells the pepedcula decto to the path of the teal pot Fo the MS exaple of Fg 49, the teal pot path s dected x decto, the teal pot cell szes y decto Y ad z decto Z copse the coss secto Fo the esults show Fg 4, the values of Y ad Z wee 375 µ wheeas fo the esults Fg 44 the values of Y ad Z wee µ The extacto of L ex has bee caed out whle vayg Y, Z ad substate thckess fo the above etoed MS stuctue 44 Aalytcal Foulato fo the Paastc ductace Sce oe ca ot pefo the vaous sulatos ode to extact the paastc ductace L ex fo each stuctue ude cosdeato, t would be desable to have a aalytcal appoxato Ths s peseted the followg, statg fo the wok [6, 7] Accodg to [6-9], the ductace of a we L h wth ectagula coss secto A a b esults a ductace gve by 4, assug that the we possesses hoogeeous cuet desty ove the coss secto the FF pocedue, the assget of a put cuet of teal pot s equvalet to a cuet desty, dstbuted evely ove the dual cell suface as show Fg 45 So the teal pot ca be thought of as a ectagula we wth a hoogeeous cuet desty ove the coss secto whch s sae as the coss secto of the dual cell Of couse, cotast to a coductve we, we do ot have a codto o the tagetal E-feld µ 4l 3 L l l π h l + 4 π a + b 3 3 X X Y Y Thus, eplacg a ad b 4 by the espectve cell szes a + ; b + fo equdstat esh oe obtas a aalytcal appoxato fo L ex gve by 4 Fo the exaple Fg 49: X Y ad Y Z, espectvely µ 4l 3 L l l π h l + 4 π X + Y

78 Chapte 4 teal Pot Fg 45 Foally, the cuet s assged at the cete of a edge alog the cuet path the FF foulato ths cuet ca be descbed to be dstbuted wth a hoogeeous cuet desty ove the coss secto A ths exaple, the cuet path s assued to be the x decto The agetc felds ae defed at the cete of the edges of A See Appedx fo 7 fo FF dscetzato The values of L h calculated accodg to 4 ae copaed wth L ex extacted fo the sulatos Fg 46 coespodg to the teal pots of Fg 49 The teal pot legths h ae vaed to 3,, 5 ad µ Two dffeet coss-sectos of teal pot ae cosdeed µ µ ad 75 µ 75 µ all cases thee s a lea behavou of ductace elato wth the pot legth ad the two deftos povde the sae ductace values The salle coss sectos esult hghe ductaces eg the case of µ legth the ductaces ae 7 ad 5 ph fo the salle ad lage coss sectos espectvely - 7 -

79 Chapte 4 teal Pot 5 } Y Z µ L ex L h L / ph 5 } Y Z 75 µ L ex L h teal pot legth / µ Fg 46 Copaso of L ex ad L h as a fucto of the teal pot legth fo dffeet coss sectos of the teal pot cells Fg 47 shows the extacted paastc ductaces L ex of the teal pots of Fg 49 as a fucto of pot coss-sectos fo dffeet pot legths As L h each case ae sae to L ex, oly oe cuve L s daw Fo ths gaph, t s evdet that the ductace deceases wth the cease coss secto whle ceasg appoxately lealy wth the cease legth t s woth to eto hee that the paastc ductaces ae the sla age wth the le ductace Fo exaple, the case of µ pot legth ad µ µ pot coss secto, the le l6 µ possesses total ductace of 3 H 8 ph / µ wheeas the paastc ductace s 7 ph, whch s 8% of the le ductace Ths ples that atched codto at the teal pots, the paastc ductaces have vey sgfcat fluece o the le, e the eflecto coeffcets, especally whe the sulated stuctue s sall copaed to the teal pot legth - 7 -

80 Chapte 4 teal Pot L / ph h : µ h : 5 µ h : µ h : 3 µ Y Z / µ Fg 47 teal pot paastc ductace L Fg 49 as a fucto of coss sectoal cell desos of the cuet path Y Z of the cells fo dffeet teal pot legth h Followg the sae pocedue of calculatg teal pot paastc ductace the case of a MS le Fg 49, a CPW le Fg 48 s studed Aga the teal pots ae placed at the eds of the les The paths of the teal pots ae defed ths case the y decto ad the coss sectos ae the XZ plaes L ex ad L h ae calculated fo dffeet dscetzatos ad gaps g betwee the sgal ad goud les t s foud that both values of paastc ductaces agee well all cases of dffeet dscetzatos ad gaps The paastc ductaces L ex ae plotted Fg 49 Aga, t s evdet that the ductace deceases wth the cease coss secto whle ceasg lealy wth the cease legth Futheoe t s teestg to copae the MS ad CPW cases see Fg 47 ad 49 Both yeld detcal paastc ductaces fo exaple fo 3 µ teal pot legth, coss sectos of µ µ ad 5µ 5µ, shows the ductace values of 3 ph ad ph both the cases of MS ad CPW les - 7 -

81 Chapte 4 teal Pot Y X Z ceac 98 pot pot syety pla g Fg 48 CPW: The coducto backed ceac substate s 54 µ thck The thckess of the etallzatos s 5 µ The wdths of the sgal ad goud les ae 6 µ ad µ espectvely The goud to sgal gap g s vaed Oly the half stuctue ade by agetc wall syety pla s sulated The teal pots ae seted betwee the goud ad sgal les at the two eds of the le L / ph g : 7 µ g : 5 µ g : 3 µ X Z / µ Fg 49 teal pot paastc ductace L ex fo the CPW stuctue of Fg 48 as a fucto of coss sectoal cell desos of the cuet path X Z fo dffeet teal pot legth h

82 Chapte 4 teal Pot 45 Sulatos wth teal Pot the pevous subsectos feasblty ad paastc effects of the coducto cuet teal pots wee vefed usg MS ad CPW les Hee soe applcatos of teal pots wll be deostated Fst, the coveso e coecto of a teal pot to a luped essto wll be exaed by a T-jucto Secod, the cuet dstbuto o the gate ad da fges of the tape shaped HEMT stuctue of Fg wll be exaed usg teal pots 45 T-jucto The T-jucto s ealzed wth th-fl co-stp les o BCB substate Fg 4 All coductos ae lossless The chaactestc pedaces of the les ae 5Ω The goud pads ae coected to the goud PEC wth vas Thee teal pots ae seted betwee the cetes of sgal pads ad the goud udeeath the substate Afte calculatg the Z- atx fo the T-jucto, teal pot 3 s coected to a 5 Ω essto See Appedx 74 fo the coectg luped eleets to teal pots The esultg -pot Z- ad S-paaetes ae copaed wth the easueet esults Fg 4 ug easueet of such T-jucto the sae pocedue s appled oe pot was teated wth a kow essto usually 5Ω ad the S-paaetes wee easued betwee the othe two pots the case of sulatos, the eflecto coeffcets ae 33 ad tassso coeffcets ae 66 fo the whole fequecy age The easueet esults show good fttg wth the sulato esults, although the sulato s caed out wth a lossless stuctue goud pads pot sgal pads Y X Z pot pot 3 BCB substate 7 Fg 4 The MS les ad pads of the T-jucto ae located o 53 µ thck BCB substate ε 7 Each a of the MS le s 35 µ log, 8 µ wde The goud ad sgal pads ae of 9 µ µ ad 75 µ 6 µ coss sectos The top etallzato s 4 µ thck

83 Chapte 4 teal Pot S S S easueet sulato easueet sulato fequecy / GHz θ / θ θ easueet sulato easueet sulato Fequecy / GHz Fg 4 Copaso betwee sulato wth teal pot ad easueet fo the S- paaetes agtude the left ad phase the ght sulato a 5Ω essto s coected to the pot 3, whle easueet pot 3 s teated by 5Ω as well 45 HEMT Stuctue A tape shaped HEMT stuctue Fg was toduced Chapte ode to show the ecessty ad use of teal pots Such HEMT stuctues ae optzed [5] ode to povde a equal dstbuto of cuet to the gate ad da fges of the tasstos The stuctue hee s ot optzed yet, peseted just to show the applcato of teal pots Fo claty Fg s edaw hee as Fg 4 Half of the stuctue s ealzed by a syety plae agetc wall ad cosdeed fo the sulato The dffeet desos of the stuctue ae lsted Tab 4 Betwee the MS ad fges the tape tagula etallzato s 5 µ log the decto of the MS le ad 75 µ log the decto pepedcula to the MS le 6 gate fges as well as 3 da fges ae coected to the coespodg MS les though tapes a The souce pads ae coected to each othe though a bdge ad to the goud though a va b All of these etallzatos ae placed o SC substate teal pots ae placed betwee the gate fges ad souce bdge as well as betwee the da fges ad souce bdge The MS les ae also fed by teal pots at the eds of the les The eccled egos of A, B ad C b ae the sae e all the tasstos ae of the sae desos c shows the zooed ego of eccled A o B o C whee the thee teal pots ae seted betwee the fges ad souce bdge A total of teal pots s used fo ths stuctue - 6 teal pots ae used fo the 6 gate fges, 3 teal pots fo the 3 da fges ad teal pots at the oute eds of the MS les A 3 sulato wth all these teal pots povdes a Z-atx of deso The goal was to detee the cuet dstbuto though the gate ad da fges vayg the output pedaces of the gate ad da fges

84 Chapte 4 teal Pot a : Top vew b : Z -Z coss-secto c : A, B, C zoo : Fg 4 Tape shaped tassto HEMT stuctue half above the syety plae d ogasato of the gate, da fges ad souce pads togethe wth the MS le coectg the gate ad da fges though the tapes e coss-sectoal vew of the stuctue at Z -Z of a The souce pads ae coected wth each othe though a bdge ad to the goud though a va The substate used s SC ε f The eccled egos of A, B ad C ae detcal The zooed ego of eccled A o B o C shows the thee teal pots betwee gate fge ad souce bdge, ad betwee da fge ad souce bdge

85 Chapte 4 teal Pot SC MS gate fge da fge souce pad thckess µ wdth µ Legth µ * Tab 4 esos of the tapeed HEMT stuctue of Fg 4 take fo [5] Fg 43 shows the scheatc to calculate the cuets at each gate as well as da fges afte the EM-sulatos have bee pefoed The esultg Z-atx s fed by two cuets g ad d eplacg the teal pots at the oute eds of the MS les, whch coect the gate ad da fges, espectvely Each gate ad da fge s teated by load pedaces of Z out g ad Z out d, espectvely coected to the teal pots at the fges The cuets though the loads ae otoed whle vayg the load pedaces The pedace values ae take accodg to [5] Fg 43 The stuctue Fg 4 s sulated by eas of the FF ethod The esultg Z-atx s fed by two cuet souces g ad d placed at the oute eds of the MS les The teal pots at the fges ae teated by Z out g ad Z out d The esultg cuets though the gate ad da fges ae calculated Fg 44 shows the agtudes ad phases of the output cuets though the gate ad da teatos fo Z out g 43 Ω ad Z out d 5-j84Ω, espectvely The put cuets ae A Hee G deotes gate fge, da fge Fg 4, etc The agtudes of the cuets though the gate fges cease fo gate at the cete to gate 6 at the pephey The dffeeces cease wth the cease fequecy At 5 GHz, fo exaple, the cuet though gate 6 G6 s 5% lage tha the cuet though gate G wheeas at GHz the dffeece s 5% The cuet dstbutos aog the da fges ae also ot equal The e-ost da fge allows oe cuet tha the oute-ost da fge 3 Aga the dffeece ceases hghe fequeces At 5 ad GHz the dffeece cuet agtudes betwee the e ad oute da fges ae 6% ad %, espectvely tes of powe ths dffeece s just double eg at 5 GHz the powe level at gate 6 s 5% lage tha the powe level of gate

86 Chapte 4 teal Pot a b out / A da G G G3 G4 G5 3 G6 gate fequecy / GHz θ / da G G G3 G4 G5 3 G6 gate fequecy / GHz Fg 44 Magtudes a ad phases b of the output cuets though the gate ad da fges Z out g 43 Ω, Z out d 5-j84 Ω ad g d A Ths ueve dstbuto of cuets aog the gate ad da fges depeds geatly o the pedace values of the teatos f the gate teatos ae Ω, we do ot see sgfcat chage cuet agtude aog the gate fges Fg 45a Howeve f the teatos ae Ω, thee s a vey lage dffeece cuet agtudes aog the gate fges Fg 45b Aleady at GHz, the oute-ost gate G6 cuet s 6% lage tha the e ost gate G cuet The sae s also tue fo the cuet dstbuto aog the da fges Eve though detcal tasstos ae used, the output powes at the gate ad da fges ay vay cosdeably depedg o the pedaces at the gate ad da fges a b 4 G G G3 G4 G5 G6 36 G G G3 G4 G5 G6 3 3 out / A 9 out / A fequecy / GHz fequecy / GHz Fg 45 Magtudes of the output cuets though the gate fges g d A a : Z out g Ω ad b : Z out g Ω

87 Chapte 4 teal Pot 46 Cocluso teal pots ae used to feed a stuctue at ceta pots sde the stuctue, especally whe the stuctue s etely bouded by PML as well as electc/agetc walls They allow to teface etwok descptos of sub-stuctues o actve eleets wth EM-sulato, whch s patculaly potat f actve eleets ad/o sub-stuctues wth cosdeable salle desos ae to be cluded Two dffeet appoaches fo pleetg teal pots FF ethod ae exaed The fst oe s the elaxato cuet souce, whee two luped te vayg chages of equal agtudes but opposte polates ae seted at the two odes of the teal pot t s show that ths appoach exctes o-physcal solutos ad theefoe s ot feasble to wok as a teal pot The oe potat appoach s the coducto cuet o le-cuet souce, whee a path of cells s defed betwee two coductos ad a souce cuet s toduced alog the path Ths appoach poved to be a vey useful ad vesatle tool A oe detaled study eveals that the coducto cuet teal pot appoach coes up wth a paastc ductace Accodg to the FF dscetzato schee, ths ductace has bee estated as the ductace of a ectagula we L h assug a hoogeeous cuet desty ove the coss secto of the we Two le stuctues, MS ad CPW ae exaed ode to extact the paastc ductace uecally L ex, whee two teal pots ae placed at the eds of the les t s foud that the closed-fo appoxato fo the paastc ductace appoxates the actual values wth good accuacy Fally, a T-jucto ad a HEMT feedg stuctue ae teated deostatg usefuless of the teal-pot cocept

88 - 8 -

89 5 4-Quadat Slot Atea 5 toducto stbuted seso systes ae pesetly a hot topc of eseach ad developet, sce they offe vaous applcatos dusty, poducto les, evoetal cotollg ad futheoe, such seso systes copse ueous vey sall seso odes ode to cotol ad easue local popetes, eg tepeatue, ateal desty, vscosty etc These seso odes ae used to be self-depedet, e equpped wth full fuctoaltes so that they ca exchage foato wth each othe ad evetually sed the desed foato to the base ode Keepg d such ecesstes, a atea fo low powe oolthc 4 GHz fot-eds was developed wth the faewok of a BMBF poject Ths s peseted the followg The atea s desged fo lateal adato, whch akes t oe attactve to be tegated wth the plae ccut boads ad allows the sepaato of adato to fou quadats, each of whch woks lke a ho type slot atea t ca be eadly tegated wth the fot-ed ccuty fo the use of tassso chp The tegato of the fot-ed chp wth the atea educes the ube ad legth of the devce-foted tecoects ad coectg sgal les, theeby esults educed paastc effects/ losses The ateas ae copact ad copatble wth stadad packagg techques Fg 5a shows the basc atea syste wth each atea to be cotolled by the RF ccuty depedetly fo each othe a b Fg 5 The fou quadats atea stuctue ade up of tagula cavtes sepaated by dagoally placed bup feces betwee two ectagula etal plates b descbes the wokg pcple of oe of the tagula cavtes of a The bup feces ae hee eplaced by closed etal walls fo splcty - 8 -

90 Chapte 5 4-Quadat Slot Atea Oe quadat of the atea ca be thought of as a tagula cavty esoato, a half of a coo ectagula cavty esoato Fg 5b The cavty s foed by a th slot betwee two etal plates ad sepaated fo the othe cavtes by dagoally placed bup feces The slot has a low chaactestc pedace Whe the cavty s excted, adato takes place The th slot wth the electc felds ca be descbed by a equvalet le of agetc cuet souce [44], adatg the fee space ue to the th slot the adato takes place lateally ode to have the axu adato the cavty ust be tapped such a way that the exctato at a ceta posto the cavty esults esoace ad the cavty acts as a tasfoe addto, as the esoace takes place at the fequecy of opeato, the atea povdes autoatcally a flteg behavo The basc challeges sulatg such a atea stuctue ae the huge esh sze due to the dvese desos ad the ope boudaes to evaluate adatos Moeove, wavegude pots ae ot sutable ths case as the whole stuctue s to be bouded by PML absobg bouday codtos Theefoe, teal pots ae ecessay to excte the atea teally Chapte t has bee show how the bad covegece behavo due to the cluso of asotopc PML the FF ethod ca be allevated The exctato of a stuctue teally though teal pots has bee descbed Chapte 4 The slot atea wll be deostated ths chapte as a pactcal exaple whee both the PML ad teal pots ae exploted the sulatos based o the FF ethod the FF sulatos the PML walls used ae o-ovelappg ad PML cell szes ae the lagest oes the whole esh Fally, the sulato esults accodg to the FF ethod ae copaed wth those obtaed CST MWS FT ethod 5 Atea esg The basc stuctue of the atea s gve Fg 5 The atea s bult as a sadwch stuctue t cossts of a uppe ad a lowe etalzed plaa substate stacks whch ae flp-chp outed face-to-face thus fog a paallel-plate esoato ad ceatg a a gap as thck as the bup heght Ths esoato s pattoed to 4 quadats by the bup feces The etal blocks below ad above ths sadwch stuctue ae abtay blocks wth pope sheldg electc walls whch esue appopate bea-fog ealty, these blocks ay cota a futhe stack of substates wth low fequecy electoc odules o the battey, fo stace The oly codto s that these blocks eed va feces o sla aageets at the sdes to sheld the agast the RF adato fo the slot The aageet of the atea stuctue allows to tegate a RF foted togethe wth a coplete tasceve fucto to a cube of copact sze Fg 53 shows a oe detaled vew of the lowe half of the atea wth ts layes a heachcal vew Backsde-etalzed Roges 43 ateal ε 338 of 58 µ thckess s used fo the atea substates The slot etallzato s 35 µ thck ad the apetue heght, e, the dstace betwee the two etallzato of the paallel-plate wave gude, s 3 µ The exctato bup s coected to the uppe etallzato ad the cuet flows back though the bup feces ove the lowe etallzato - 8 -

91 Chapte 5 4-Quadat Slot Atea a b Fg 5 The basc stuctue of the 4-quadat slot atea a The dagoally placed bup feces sepaate the stuctue to fou tagula ho type atea sectos b povdes a coss-sectoal vew The slot s foed betwee the etalzatos The bup feces eflect the electoagetc waves geeated by the exctato bup a way sla to a ho atea ad fo a electcal sepaato betwee the 4 quadats So each quadat woks as a ho type slot atea, ca be udestood as a tagula cavty wth a ope slot at the oute bouday ad excted by the bup ea the cete of the cavty Because of the low cavty heght thee s a stog satch betwee the cavty pedace ad that of the oute space Theefoe the cavty s opeated ea the esoace, whch allows pedace tasfoato ad esults a esoat atea wth flteg popetes The bups also have the fucto of echacally coectg the uppe ad lowe half of the esultg set-up Wth the uppe ad the lowe substate, va feces at the edges fo a atfcal sheldg to block the RF adato ogatg out of the slot fo peetatg to the substate sde the substate vas ae eeded to pevet esoaces the substate excted at exctato bup Fo ths pupose, soe vas ae dstbuted ove each quaduple secto Regadg desg the ost essetal specfcato fo the atea s that ts esoat fequecy should eet the cete fequecy of the SM bad aoud 4 GHz The shape of the atea lobe s ot a equally potat ssue because the atea s to be appled ea-age coucatos wth copaatvely low costats o adato patte The bup ad va feces ust have a ceta desty to ake sue that thee s oly low couplg betwee the sectos ad that o esoaces occu the substates, espectvely O the othe had, the dstace betwee the bups caot be chose abtaly sall because the bup ptch has a lowe lt fo techologcal easos Sla estctos hold fo the u va ptch the substate Thee ae dffeet paaetes that cause lage toleaces, whch tu esult a shft of the esoace fequecy o chage the adato level Aog these, the ost potat paaetes ae the legth of the slot e the oute deso of the atea, the posto, heght ad daete of the exctato bup ad the heght of the etal block Electoagetc sulatos ae caed out to check the toleaces of the above paaetes, to fd out the paastc esoaces ad to optze the layout

92 Chapte 5 4-Quadat Slot Atea a b c d Fg 53 Heachcal vew of the lowe syetcal half of the atea a: the bup feces that ealze sepaato of the ectagula cavty to fou quadats b: the lowe etallzato wth the opeg fo the exctato bup c: The Roges substate wth vas d: the etal block whch seves hee as a epesetato fo the space whee futhe fuctoaltes ca be ealzed

93 Chapte 5 4-Quadat Slot Atea 53 Sulato esults FF The F3 sulato based o the FF ethod wth PML ad teal pots s used fo desg ssues ad to detee the fluece of toleaces of the physcal paaetes of the atea As the fou tagula ateas, e the fou quadats, ae electcally sepaated ad decoupled fo each othe by the dagoally placed bups, t s suffcet to sulate oly oe of those ateas Fg 54 shows oe half of the whole atea stuctue copaso to the basc stuctue gve Fg 5 ad Fg 53, ectagula-type vas ad bups of sae coss-sectoal aeas ae used The heght of the etal block h, see Fg 54b s vaed betwee 35 µ to 5 µ PML walls of fve layes ae used fo the ope boudaes At the begg, soe vestgatos wee caed out ode to estate the ube of vas ad bups the feces whch ae ecessay to keep cosstalk aog the quadats suffcetly low Fo vey coase feces, addtoal esoaces occu due to the elatvely stog couplg ceasg the bup desty, the desed sgle-esoace behavou s foud but the value of the esoat fequecy stll chages because the esultg ductace of the bup feces vaes Fo the tageted puposes, t s foud that 9 vas o bups at each a yeld good esults a b c Fg 54 a: the stuctue sulated: oe half of the basc atea stuctue gve Fg 5 b: coss-sectoal vew ot scaled bups ae exteded though the substates to wok as vas as well teal pot s placed betwee the two etallzatos c : top vew fo at the cete of the slot defto of the slot legth ad the posto of the exctato bup / teal pot x fo the cete of the 4-quadat atea