Chapter 6 Stability of Finite-Difference Time-Domain (FDTD) Method with Nonlinear Lumped Elements

Transcription

1 hap 6 Sabl of F-ff Tm-oma FT Mhod wh Nola Lumpd lms 6. Ioduo Tadoall o su umal sabl of h algohm fo a la modl h oua-fdh-lw FL o 3.5. has o b fulflld Taflov 995. I a usabl algohm h ompud ad fld ompos wll as whou lm as h smulao pogsss. Th FL o has b dvd wh h assumpo of homogous la dl ad uboudd mdum usg s Fou Tasfom FT h o Numa appoah s Appd. Fo a paal mowav u modl h FL o svs as a ul-of-humb a bs. A fw amps l dd h sabl aalss o lud la dspsv mda Pda.al 00 ad la lumpd lms Thl ad Kah 00. Th s also a amp o as h FT mhod o av ma quao Rms 000. Th mhods pod sll l o mahmaal ools fo la ssms.. supposo ppl FT. ad wll fal fo ola modls. To-da o h bs of h auho s owldg h s sll o sasfao ho o pla h sabl of FT fomulao oag o-homogous dls bouda odo ola dl ad also la ad ola lumpd lms aagd a aba ma. I hs hap w sabl homs basd o h g mhod a dvd o addss h ssu. Ths wo s spd b sabl ho of damal ssms Shma 996 oabl h Sod Lapuov Mhod lad 000 M 997 Khall 996. Alhough o show du o la of spa h FT fomulao s auall a ds damal ssm. Th poofs of h w homs a show Appd 3 o Appd 6 ad h applao s dmosad h ma. Th homs a usd o pov wha hav b ow houghou h as va smulao ha h opoao of a lumpd ompos suh as sso apao dod ad bpola uo asso h FT famwo s foud o b sabl. Th homs also show wh as how a modl oag o-homogous ad ola dl s sabl. Th homs a dd o omplm h FL 97

2 98 o ad h suls of Pda.al 00 Thl ad Kah 00 ad Rms 000. I So 6.3 s show how h homs a usd as a s o dm whh h luso of w lumpd lm FT s pop whou pfomg lgh smulao. A pop fomulao wll psv h sabl of h FT modl. Ths hap bgs b osdg a 3 FT modl fo P o mowav u whou a sou volag o u sou wh pf l oduo P as h modl boudas. Th w homs a sad ad a usd o pov h sabl of h soulss modl. Th h homs a dd o h h sabl of a 3 FT modl wh a ssv volag sou. A smpl smulao ampl svs o subsaa h suls. Fall so of h homs o a hdmsoal P modl wh absobg bouda odos As s show Appd Th Nw Sabl Thoms Fgu 6. shows a pal 3 FT modl fo mowav u P whou a volag o u sou wh pf l oduo P as h modl boudas. Th modl has ad lls alog ad as spvl. Th modl s assumd o b o-mag.. µ µ o fo all flds. Also h lls a h sam s. All h ad fld ompos wh h modl hav upda quaos gv b h followg fom as a b s fom So 4.3 ad So 4.4: µ 6..a µ 6..b µ 6.. J 6..d J 6..

3 99 J 6..f Wh 6..a-6..f 6..g 6..h ad so foh fo ad ms. No ha h so fo has b movd allowg o va aodg o loao ad oao. Though o dad a also b fuos of fld ompos. Smlal h u ds m J also dpds ol o fld ompos a al msps. Fgu 6. A pal mowav u modul. Tmals RF/Mowav oo asg ossg of P Spal duo Av/Passv ds ompos ad gad us Rsoag suus P

4 00 quaos 6..a-6..f wll b ow as h aoal FT Fom fo ad fld ompos. Upda quaos fo ma applaos a usuall b w hs fom. L us ow odu wo w quas as dfd b: [ ] o µ 6..a d J P 6..b wh. I quao 6..a s ow as h umal g of h 3 FT modl a squ s aalogous o h sod lomag g a phsal ssm. I ompss all h fld ompos a m-sp ad fld ompos a m-sp. Smlal h umal g ompss all ad fld ompos a m-sps ad spvl. I 6..b ah m J wll b ow as lmal dsspao s s h appoma pow ds dsspad b lumpd lm odg wh fld. Th gav sum of all lmal dsspao mulpld b s h oal dsspao P d of h modl. S all h boudas a P h umal g wh h modl ao sap fom h bouda. Th followg homs gv h laoshp bw umal g ad oal dsspao ad h sabl sul.

5 Lmma 6. Rlaoshp bw umal g ad oal dsspao osd a 3 FT modl wh P boudas of Fgu 6.. Gv ha all fld upda quaos a of h fom 6..a-6..f wh ad P d as dfd 6..a ad 6..b h h followg lao s u: Pd 6..3 Lmma 6. Posv sm-dfss of osd a 3 FT modl wh P boudas of Fgu 6.. Gv ha all fld upda quaos a of h fom 6..a-6..f h s posv df f ad ol f: a > 0 0 ad µ µ > a > 0 > b Fo m{ } ad o m l: µ o < m 6..4b m m wh { L } { L } { L }. m A fuo f s posv df wh 0 mpls f > 0 ad f 0 wh 0 Khall 996. No ha a b a vo o a sala. Th poofs fo Lmma 6. ad Lmma 6. a show Appd 3. I hs o sabl mpls h FT algohm fo h 3 modl s boh umall ad damall sabl. Ths mas ha f w w o du o o o as h m-sp o f h soluo fo ad fld ompos would alwas ma boudd. Th dfo fo sabl of h 3 FT modl s qu smla o fo.. I wll b dfd aga h o of h FT algohm ad fld ompos. 0

6 fo 6. Sabl of FT algohm Suppos w osu a vo ompos of h modl: X whos lms oss of all ad fld X M M 6..5a X M 6 3 dm 6..5b Th h FT algohm s sabl wh X T 3 K N 6..5 Th smbol mas ag h om of a vo Oga 987 whh s a masu of h dsa bw ad h og. T s a posv al valu whh dpds ol o T h mamum ompuao m. N T / s h mamum squ. As 0 N wll appoah f. owv 6..5 das ha T mus b f as N appoahs f as log as T s boudd. W allow o b a fuo of T as a soluo of h FT a as wh m. Fo sa wh h s a sou h modl ha ass wh m s h asoabl o p h fld ompos o gaduall as oo. Wha 6..5 mas s ha h soluo mas boudd fo f m val. If 6..5 s o fulflld as ass h h algohm s o sabl. Fall w df h sabl hom: 0

7 Thom 6.3 Sabl hom fo 3 FT modl osd a 3 FT modl wh P boudas of Fgu 6. f h modl fulflls all h odos Lmma 6. h a suff odo fo o b sabl s P 0. d Th poof s gv Appd 4. Th so wll shows how Lmma 6. Lmma 6. ad Thom 6.3 a b usd o dm h sabl of a gal 3 FT modl fo mowav us. 6.3 Applao ampl sablshg h Sabl of a Soulss 3 Pd u oad P Modl Assum a P modl of Fgu 6.. Th P modl oas o-homogous dl lumpd ssos lumpd apaos ad ola ompos suh as PN uos ad bpola uo assos JTs. Th FT upda quaos fo ad flds of all hs lms a b foud So 4.3 ad So 4.4. Iall ol a soulss 3 FT modl wll b osdd fo asos o b plad So 6.4. Ths mas h modl wll o oa a sou suh as h lumpd ssv volag sou quao Th aalss bgs b ompug h lmal dsspao J of ah lm ad show ha hs s alwas ga o qual o o ud omal ad fld valus. Fuhmo wll also b show ha s alwas posv ud omal fld valus. lm mg hs wo haass s alld pop. A lmal dsspao > 0 mas ha h lm s absobg umal g fom h modl. Wh all h lms a pop h oal dsspao P d wll b qual o o o gav. If s posv df h odo of Thom 6.3 wll b fulflld. Th m dsao s as dad b 6..4b of Lmma 6.. Wh h fal odo m Thom 6.3 lls us ha h modl wll b sabl. Fo smpl s assumd h lms a od h do. Ths a alwas b gald o lms od alog oh dos. 03

8 04 Losslss La l Fo h fld of a losslss la dl 3.3.4f: o 6.3.a wh s as dfd 6..h. ompag 6.3.a wh h aoal FT Fom fo ompo: 0 J ad o 6.3.b Thus lmal dsspao 0 J. S 0 > o h losslss la dl fomulao s pop. Pf l oduo P Th l fld a P s alwas 0. I a b w as: 0 P 6.3. wh P assumg 0 0. Aga ompaso wh aoal FT Fom shows ha 0 J mplg lmal dsspao s 0. Th P fomulao s pop. apao Fo a lumpd apao od as aodg o 4.3.8: o o o o 6.3.3a ompag 6.3.3a wh aoal FT Fom:

9 05 0 J ad o o 6.3.3b Fom 6.3.3b h ffv pmv s alwas posv ad h lmal dsspao s o. Th apao fomulao s hfo pop. La l wh Loss Fo h fld of a la dl wh oduv aodg o 3.3.4f: 6.3.4a ompag 6.3.4a wh aoal FT Fom: 6.3.4b J J Fom 6.3.4b s alwas posv bu o h lmal dsspao. Usg d ad h lmal dsspao s: [ ] J d d d d Ths psso s o posv df a ombaos of ad wll aus o bom gav. To su ha s alwas posv o o a odos d o b odud. Rqug ha 0 d : Fo 0: 6.3.6a Fo < 0: < < 6.3.6b Mos of h m h odos of 6.3.6a o 6.3.6b a m spall wh s small low loss a < 0. sv al-m amaos of lmal

10 dsspao dug FT smulao show ha h valu s alwas posv fo low o mdum loss. Usuall h odos 6.3.6a ad 6.3.6b do o hav o b pll mposd dug smulao. quao 6.3.6a ad 6.3.6b mpl ha u flowg hough h lm s alwas lmd. Ths s smla o a lal u wh a fw pahs havg low ssa. v hough h low ssa pah a suppo lag u h u ofguao wll d o lm h u hough h pahs sug posv pow dsspao. I h as of FT smulao h ssm modl wll usuall lm h mag fld ompos suoudg h l fld so ha pow dsspao s posv. owv s f-dff s ol a appomao o h aual Mawll s quaos s pd ha h lmal dsspao a bom gav o a whl. sv smulaos show ha wh s subsaall ga ha 0 h lmal dsspao of suls gav valus o a whl. Mos loss dl maal wll hav muh small ha. Wh h lmal dsspao s gav h followg qual a b mposd omplg wh 6.3.6a ad 6.3.6b o fo : 6.3.7a Applg 6.3.7a o h upda quao of 6.3.4a would sul : 6.3.7b I s foud ha usg 6.3.7b fo h as wh lmal dsspao s gav dos o aus a oabl hag h FT smulao suls. A pal flow of updag h fld fo loss dl wh lmal dsspao hg ad oo s show Fgu 6.. Thfo b modfg h upda ou fo la dl wh loss aodg o h flow of Fgu 6. w ould aga olud ha h fomulao s pop. A smla podu a b usd o show ha poal m-sppg shm Taflov 995 fo hgh loss maal s also pop h dals a omd du o la of spa. Fomulao suh as hs wh w lm gv wll b ow as odoall pop. 06

11 07 Fgu 6. - Modfd upda ou fo la dl wh oo oduv. Rsso Th upda quao fo a lumpd ssv lm of ssa R s v smla o h fom fo la dl wh loss. W us pla h m wh R o R o o R or Usg smla modfd upda ou as Fgu 6. h fomulao fo sso s also pop. Aga smulao vd shows ha hs s o qud mos of h m p fo low ssa. od o PN Juo osd a PN uo paalll o h -as wh h upda quao fo h ospodg l fld gv b usg h fs od appomao 4.4.8b: Sa Upda usg:? 0 d Ys No lmal dsspao:

12 08 q T T s d di N N I I I T o o o p η N s h oao osa s f h lm s od do.. posv u flows do as plad So 4.4. L N I o a b w as: o ompag hs wh h aoal FT Fom: o d di o o N I T T s η η p 6.3.0a o J J o 6.3.0b Fom 6.3.0a h ffv pmv s a fuo of ad s alwas posv s h uo apaa ad h poal m alwas gvs posv valus. L us fom h followg podu o am h sg of lmal dsspao: J o 6.3. s a fuo of ad. Usg ad a plo of vsus ad fo a pal sufa-mou Sho dod SMS-80 Agl Thologs 000 s show Fgu 6.3. Th dod s od do N - ad h ag of ad a: [ ] [ ]

13 Usg N h ag fo ospods o 0.55 o 4.5 ols fom had fowad basd o had vs basd. Ths ag of ad s pall oud smulao fo low volag RF us. Th ag ould b lagd o as h ovag f wshd wh smla sul obad. I s s fom Fgu 6.3 ha s wal flud b. Fgu 6.3 ofms ha h lmal dsspao s alwas posv o 0 fo omal valus of fld. I gal a b show ha hs s also u fo all paal dod modls. Thus fom h abov agums h dod o PN uo fomulao s also pop. Rag of : -000 o 7500 Rag of : o 5000 m 0.0 N Paams usd fo gag h plo: 0.75mm 0.80mm 0.55mm.0pS o q T300 P I s.-8 η.08 o 0.7pF τ m 0.5 F 0.5 N Fgu 6.3 vsus ad. 09

14 0 pola Juo Tasso JT A podu smla o aalg h haass of h PN uo a b mplod o aal h JT. Th JT s fomulad aodg o So 4.4 wh osss of wo PN uos h m ad h ollo uos. Th lmal dsspao of ad uos a summd up smulaousl o gv h oal lmal dsspao of h dv. Th dvdual dsspao a b gav bu h oal lmal dsspao s alwas posv o o as log as h asso s suabl basd. Assumg h ad uos a od alog -as as show Fgu 6.4. I a b show ha wh h JT s o pushd o m sauao o u-off h JT fomulao s also odoall pop. Ths s do b vfg h oal dv dsspao ad ffv pmv of uos a posv df wh h fou opag gos of a JT.. av sauao vs ad u-off gos. Th JT fomulao s hus odoall pop. Fom So 4.4: 6.3.a 6.3.b f w vaabls ' ad : S N I ' 6.3.3a 6.3.3b Fom a b w as: ' S N I Subsug hs o 6.3.a: ' ompa wh h aoal FT Upda Fom. Th followg pssos wll b obad.

15 6.3.6a ' J 6.3.6b Followg smla podus 6.3.b a b w as: ' 6.3.7a S N I ' 6.3.7b ' J 6.3.7d As h PN uo fomulao h haass of a aual NPN asso s sd. Th assos usd a FR9A ad FG50 fom Phllps Smoduo Phllps Smoduo 995 ad 997. FR9A s a wdbad NPN asso wh aso fqu f T Ga ad M 993 up o 5.0G. FG50 s a ula-wdbad NPN asso wh 0 9. T f G. Th oao of ad uos fo boh assos s assumd as show Fgu 6.4 whh amou o N N. Fgu 6.4 Oao of ad uos fo asso FR9A ad FG50.

16 g b vfg ha h oal dv dsspao s posv df wh suabl opag odos. Th show ha wh hs suabl odos ad a alwas posv. Fs df h oal dv dsspao as: J J Ths a b w a mo ompa fom as: wh ad I s mpossbl o plo as a fuo of hs vaabls as ffh dmsoal spa s qud. Isad a hausv sah s ad ou usg a ompu b alulag h valu of wh a doma G as follows fo asso FR9A ad FG50: FR9A : [ ] [ ] [ ] [ ] G 6.3.9a FG50 : [ ] [ ] [ ] [ ] G 6.3.9b Th sah s ad ou ds sps wh Ths doma ompasss h fou opag gos of a JT fo vaous valus of ollo ad m u. Th lms fo ad allow ad uos o udgo had fowad-basd ad had vs-basd. Th lms fo ad. a hos pall oud h smulao of low pow RF us. Fo boh assos s obsvd ha s wal flud b ad. Allowg fo h gos fo ad wh s posv s alld h JT sabl go. Ths s show Fgu 6.5a ad Fgu 6.5b fo FR9A ad FG50 spvl. Th da go ospods o usabl go of h assos whl h lgh go s h sabl go wh

17 0.77 N Ivs Sauao N 0 Usabl go Nomal u-off Sabl go sao:.0ps 0.7mm 0.8mm 0.5mm Gumml-Poo Modl fo FR9A I ss β F 0.64 β R 8. F R A 6.67 I KF 3.0 I S I KR.855 I S J 0.60 m J τ F J m J τ R 0 F0.5 0 N N -. Fgu 6.5a Plo of vsus ad fo FR9A. 3

18 N Ivs Sauao N Usabl go Nomal u-off Sabl go sao:.0ps 0.7mm 0.8mm 0.5mm Gumml-Poo Modl fo FG50 I ss β F 0.8 β R F R A I KF I S I KR I S J 0.60 m J τ F J m J τ R 0 F0.5 0 N N -. Fgu 6.5b Plo of vsus ad fo FG50. Fom boh Fgu 6.5a ad Fgu 6.5b s obsvd ha h JT modl aodg o So 4.4 wll hav gav dv dsspao wh h uo of h asso s havl fowad-basd ad havl vs-basd. Thfo h JT fomulao s odoall sabl. As log as dug h smulao w a pv h opag po of h JT fom dfg o h da gos of Fgu 6.5a ad Fgu 6.5b h JT dvs FR9A ad FG50 wll b sabl. Th s a small go h v of 00 wh h dv dsspao s gav fo boh 4

19 FR9A ad FG50. Upo los su of hs go b usg hausv sah h mmum valu of hs go s v small fo FR9A ous appomal a wh. Fo FG50 hs ous appomal a wh S s v small ad los o o s ff s glgbl h JT opag odo wll o lg hs go fo log h mom h ad uo volag s o o h dv dsspao wll bom posv aga. Th sam a b sad fo FG50. To fuh sgh h ofd ha h s dd o oh po h sabl go ha suls gav h Mhod of Sps s Wsm ad hag 978 s ad ou a fw adoml hos pos h fou opag gos of h JT. Aga hs dos o ld a po wh gav s h ushadd gos. Th fal sp s o vf ha boh ad a posv alwas. padg 6.3.6a ad 6.3.7: 6.3.0a 6.3.0b Fom h dfo of ad So 4.4 hs vaabls ol dpd o ad. A plo of ad vsus ad s show Fgu 6.6. Ths ofms ha h ffv pmv s alwas posv h sabl gos of FR9A ad FG50. Th paams fo boh assos a smla o h paams usd gag Fgu 6.5a ad Fgu 6.5b. Aoh wa o show hs sam s o o ha: I S I S 6.3.a 6.3.b 5

20 Th dffao of h u I S ad I S a omall posv fo pop asso modl s us a poal fom ad h uo apaa ad a also alwas posv. Thus ad a posv. plog vsus ad f s alwas posv h go of s h ad wll also b posv as hs a us mulplg ad dvdd wh ad spvl. 0 0 FR9A ffv s gav ffv s posv 0 0 FG50 Fgu ad vsus ad fo FR9A ad FG50 asso. Th ag fo ad s smla o h ags Fgu 6.5a ad Fgu 6.5b. 6

21 Now suppos h modl osss of ma ubs of smla s wh 0.75mm 0.8mm 0.5mm. Th modl s o-mag bu h dl osa s o ufom wh vaao of aoss h modl ad a poo of h modl s f spa. Also fom 6.3.b 6.3.3b 6.3.4b 6.3.0b 6.3.0a ad 6.3.0b h ffv pmv s alwas b ga ha o. Fom hs fomao w olud ha h smalls pmv quals o o as a dl whh s o a wll hav ga ha u. Usg odo 6..4b of Lmma 6.: m µ o o 7 4π < m m m < m ompa hs wh FL a: < m Th w sabl o has a as so b 9.84%. Fall a b oludd ha f all lms usd hav upda quaos of h abov ad < h aodg o Lmma 6. ad Thom 6.3 h modl of Fgu 6. wll b sabl. Ths mas ha f h al ad fld ompos a m-sp 0 s o o all fld ompos wll ma boudd as w adva h m-sp. No ha houghou hs so h ol qum fo h ffv pmv s ha s posv. Thus h pmv a hag wh loao b a ola fuo of fld ompos ad h soulss 3 modl s sll sabl. So h mhod poposd h ould also b usd o pov h sabl of modl wh o-homogous dl ad ola dl. I h so h as wh h s a ssv volag sou h modl wll b osdd. 7

22 8 6.4 so of h Sabl Thom o 3 Modl wh Sou Suppos addo o h lms mod So 6.3 h 3 modl of Fgu 6. also oas a lumpd ssv volag sou wh upda quao fo fld gv b 4.3.5: 6.4. Wh R s s h dpd volag sou as a fuo of msp ad R s bg h sou ssa. a ps a osa d.. sou a puls susodal fuo ad so foh. ovg 6.4. o h aoal FT Fom w obsv ha h quval u ds s: R J s 6.4. odug ad v h lmal dsspao a b w as: [ ] [ ]v v J v R R s s Aga hs quao s df a b posv o gav. Wh sou lm suh as 6.4. s ps P d a bom posv ad fom Lmma 6. h umal g of h modl a as wh m-sp. Assumg v s a osa posv valu w a osd o b a d.. volag sou a plo of h go A h - pla wh lmal dsspao boms gav s show Fgu 6.7. A smla bu vd go a b asl plod wh v s a osa gav valu.

23 A lss ha 0 hs go I s v 0 A v s R s Fgu 6.7 Ngav go A of h ssv volag sou wh v > 0 ad s osa. A p quso s whh h s a fuh osa apa fom go A? I fa h s. Th fs osa s h u I s s Fgu 6.7 mus o b mo ha / R whh pss h sou u wh h mals a shod. s s Th a ass wh sou u a d hs lm bu fo a popl dsgd u / R s usuall h hshold. Th sod osa s fo h sou of 6.4. s s o ouousl suppl pow o h modl lmal dsspao mus alwas b gav o o a all squ. Ths mas f of squ sul 0 3 h of squ mus also sul 0. Appd 5 shows ha hs qums a b pssd mahmaall as: v < 6.4.4a v 0 v v 6.4.4b Usg h a of 6.4.4a ad assumg a w gav go alld s show Fgu 6.8a fo < ad Fgu 6.8b fo >. Wh s ousd h shadd go h ssv volag sou wll boud o sop 9

24 supplg umal pow o h modl fuu squ ad h ad fld ompos wll sa o das P v P 0 Rgo wh 6.4.4a o a fulflld fo. v v Paams: 0.75mm 0.80mm 0.55mm.0ps 4. R s / v.88.8 ooda of mas: P v v P v0 P v v 3 v P Fgu 6.8a Ngav go fo ssv volag sou fulfllg 6.4.4a o <. P o P 3 a ma pos. 0

25 5 0 4 P Rgo wh 6.4.4a o a fulflld fo. v Paams: 0.75mm 0.80mm 0.55mm.0ps.0 R s / v v P 0 v P 3 ooda of mas: P v v P v0 P 3 v M Fgu 6.8b Ngav go fo ssv volag sou fulfllg 6.4.4a o >. Th gav gos dfd Fgu 6.8a ad Fgu 6.8b a sll qu osvav vhlss s suff fo h agum. I od o hav 0 fo all h aual go ould b small ha show. Fo a ssv volag sou o ouousl suppl umal g o h modl s sa mus alwas b ofd o. Fuhmo f h sou s o osa h w a osu h gav go basd o h lags magud of Ngav gos fo ssv volag sou wh. s gav a b osud a smla ma usg 6.4.4a Th mpoa of h gav gos a b summd up as follows. Suppos a FT modl as Fgu 6.

26 oas all pop o odoall pop fomulaos ad a fw ssv volag sous. Th all h umal g of h ssm wll as. S Appd 4 has b pov ha s adall uboudd o all fld ompos a dvgg wll als a las o o mo o fld ompos ha a asg. u o h popagag au of Mawll s quaos a fld ompo ha s asg whou lm wll subsqul aus h s of h modl s ad fld ompos o as oo. Th mpoa fau of h gav gos s ha s boudd as log as h sou volag s s boudd ad R s ga ha 0. If h m of s o lmd b oh dsspav lms h modl wll vuall oss h bouda of h gav go. Th h lmal dsspao of h ssv volag sou wll boud o bom posv aga ad h m of umal g hals. Thus h s a bul- mhasm du o R s h ssv volag sou fomulao o pv uoolld m of h fld ompos. Wh hs w a olud ha h gal FT modl wh ssv volag sous wll b umall sabl f ad ol f h followg h odos a m:. Thom 6.3 s fulflld wh h modl dos o oa a sou.. Th s o oh sou p ssv volag sous. 3. FT fomulaos of oh lms a pop o odoall pop. 6.5 Smulao ampl A smulao s ad ou o vf h ops dsussd. a sho odug a gd b a ssv volag sou s od o a paalll R load. Th P bouda s usd hs pm. Th shma ad h op vw of h FT modl s show Fgu 6.9. Th smulad volag aoss h ssv volag sou ad h load s show Fgu 6.0A whl h umal g s show Fgu 6.0. I hs modl h ssv volag sou s mmsd h dl ad s dd. Is volag fuo s a sgl puls of amplud 3.3 av dug 0 00ps ad h s o 0 bod 00ps. As s Fgu 6.0 wh h volag sou s av ass apdl ad h sauas as h pow suppld b h sou quals h pow dsspad b h ssos h modl. ug av

27 sag h sa of h ssv volag sou s dmd fom h ospodg ad flds ad plod h - pla of Fgu 6.. Fom h sul w ould lal s ha h ooda v lavs h gav go. Af h volag s s davad h ssv volag sou vs o a omal sso modl. W obsv ha h umal g of h modl sas o dl as ow all h lms h modl a pop. Rssv volag sou 3.3 s 0 Rssv volag sou Z 50 Paalll R load ps P boudas Fgu 6.9 Th shma ad op vw of h s modl. 5 lls alog -as 7 lls alog -as lls alog -a. l hss 3 lls 0.75mm 0.8mm 0.5mm.0ps 3

28 A Numal g of h modl Fgu 6.0 A olag aoss h ssv volag sou ad h R load. Numal g of h ssm. 4

29 P Ngav go 0 P 000 P 3 Fgu 6. Loao of h sa fo 0 o 000 a a val of 0 msps. 6.6 d Nos Th homs So 6. a usful. Th ovom h lmaos of o- Numa appoah whh sul h FL o. Fs ad fomos h a b usd o dm ad su h sabl of Y s FT modl fo mowav u o hgh-spd P wh h followg odos: aabl ad ola dl osa. 3 oag la ad ola lumpd lms. Iludg h ff of P bouda. I applg Thom 6.3 h ol qum fo pmv s ha mus b posv fo all fld ompos whl h qum fo pmabl µ s ha mus b qual o µ o fo all fld ompos. Sodl h odos mposd o J ad P d Lmma 6. ad Thom 6.3 a b usd as a s o h whh h FT fomulao of a lumpd lm s pop. Fo a w lumpd lm w a w s fld upda quao 5

30 h aoal FT Fom dm h quval u ds ad ompu s lmal dsspao J agas all possbl ombaos of dpd fld ompos as show So 6.3. As log as h lmal dsspao s ga o qual o o ad > 0 w ow ha h fomulao s pop ad wll o obu o sabl of h modl. Th o-numa ad oh la appoahs ao b usd o aal sabl of h ssm ud h abov odos of -3 ad adoall ul-of-humbs a usd o su ha h FT modl s sabl hap Swda 989. I addo h appoah psd h a also b dd o lud: aabl ad ola pmabl. No-ufom ll s. spsv lms ad lumpd duos. Absobg bouda odo. Th fs ad sod so a b ad ou b wg µ ad µ fomulag h odo fo o b posv df. Th hd so a b ad ou b mplmg a moog algohm muh l Fgu 6.. Th dals of h fouh so a show Appd 6. I s show how plag h P wh absobg bouda odo mplog mahmaal polao suh as Mu s A would o aff h sabl of h modl. Th poof Appd 6 s v usful as a b appld o a absobg bouda odo ha uss mahmaal appoah. Th fouh so a also b ahvd b odug a fw las of lls wh l oduv bfo h P bouda. A b appoah would b o odu mag oduv ad mag u addo o l oduv. Th fomula a absobg bouda odo A basd o Pfl Mahd La PML mhod g 994 Taflov 995. Smla podus as h pvous sos a b usd o dv dd hom opoag mag loss of M wh M s h 6

31 ospodg mag u. Th mhod dos hav som dsadvaags. I wll fal wh h s upda quaos fo h o fld ha ao b w h aoal FT Fom. Also h odo fo s slghl mo gd ha FL o. 7