chinaxiv: v1

Transcription

1 Matheatcal pcple of essto etwos Zh-Zhog Ta * Zhe Ta. Depatet of physcs, Natog Uvesty, Natog, 69, ha. School of Ifoato Scece ad Techology, Natog Uvesty, Natog, 69, ha (9-3-) Abstact The ufed pocessg ad eseach of ultple etwo odels ae pleeted, ad a ew theoetcal beathough s ade, whch sets up two ew theoes o evaluatg the eact electcal chaactestcs (potetal ad esstace) of the cople essto etwos by the Recuso-Tasfo ethod wth potetal paaetes (RT-V), apples to a vaety of dffeet types of lattce stuctue wth abtay boudaes such as the oegula ectagula etwos ad the oegula cyldcal etwos. Ou eseach gves the aalytcal solutos of electcal chaactestcs of the cople etwos (fte, se-fte ad fte), whch has ot bee solved befoe. As applcatos of the theoes, a sees of aalytcal solutos of potetal ad esstace of the cople essto etwos ae dscoveed. I patcula, thee ovel atheatcal popostos ae dscoveed whe copag the esstace two essto etwos, ad ay teestg tgooetc dettes ae dscoveed as well. Key wods: cople etwo, RT-V ethod, electcal popetes, bouday codtos, tgooetc detty, Laplace equato PAS : 5.5.+q, 84.3.Bv, 89..Ff,..Y,.55.+b Subject Aeas: Itedscplay Physcs, Matheatcal Physcs, odesed Matte Physcs, ople Netwos * taz@tu.edu.c, tazzh@63.co zhzhta@hotal.co

2 Ⅰ. Itoducto May cople scetfc pobles ca be sulated by the essto etwo odel, such as ay electcal ad o-electcal pobles the feld of physcs, egeeg ad atheatcs. The pogess of ccut theoy ot oly pootes the developet of tegated ccut ad electfcato scece but also pootes the tedscplay developet of atual scece. Ressto etwo odels ae so potat that the ssues of vaous dscples ca be studed by sulatg essto etwo, such as coducto asotopc dsodeed systes [], pecolato ad coducto [], Asotopy electcal coductvty [3], Nolea localzed odes two-desoal electcal lattces [4], Electc ccut etwos equvalet to chaotc quatu bllads [5], photoc cystal ccuts [6], Mafestg the evoluto of egestates fo quatu bllads [7], dyacal sgatue of factoalzato [8], the pocessg of heagoally sapled two desoal sgals [9], topologcal sulatos ad supecoductos [], topologcal popetes of lea ccut lattces [], topologcal sp ectatos [], thee-desoal pted eshes [3], topologcal sulato suface [4], factoal-ode ccut etwo [5], the ea feld theoy [6, 7], fte-sze coectos of the de odel [8], lattce Gee s fuctos [9-], esstace dstace [3], ad so o. I patcula, two potat equatos of Posso equato ad Laplace equato [4, 5] ca be sulated by essto etwo odel [6]. I addto, a eal plae etwo of gaphee ests the eal atue. It s well ow that calculatg the equvalet esstace betwee two abtay lattce stes a essto etwo s always a potat but dffcult poble sce t eques ot oly the ccut theoy but also the ovatve algeba. Fo eaple, whe the bouday of essto etwo s abtay, t s usually vey dffcult to obta the eact potetal ad esstace of the cople etwos wth abtay boudaes. I fact, the bouday s le a wall o tap, whch affects the soluto of the poble. Theefoe, the ealty eques us to ceate ew theoes to accuately calculate the electcal chaactestcs (voltage ad esstace) of the cople ccut etwo. Let's evew the eseach hstoy of essto etwos. I 845 Kchhoff establshed the basc ccut theoy (the ode cuet law ad the ccut voltage law). Afte 5 yeas, set [7] calculated the two-pot esstace of the fte etwo by Gee s fucto techque, whch s aly focused o fte lattces, ad soe applcatos of Gee s fucto techque wee obtaed late lteatue [8, 9]. I 4 Wu [3] foulated a dffeet appoach (call the Laplaca at ethod) ad deved the eplct esstace abtay fte ad fte lattces wth oatve bouday (such as fee, peodc bouday etc.) tes of the egevalues ad egevectos of the Laplaca at, whch eles o two atces alog two vetcal dectos. Late, the Laplaca at aalyss has also bee appled to pedace etwos [3], afte soe

3 poveets, seveal ew essto etwo pobles have bee esolved [3-34]. Howeve, the Laplaca appoach caot apply to the etwo wth abtay bouday sce t s possble to gve the eplct egevalues fo the abtay at eleets (assocatg abtay boudaes). But the bouday codto s potat sce t s eal case occug eal lfe. I Ta poeeed a ew techque fo studyg cople essto etwos [35], whch ow s called Recuso-Tasfo (RT) theoy of essto etwos [6]. The RT ethod depeds o oe at cotag oe dectos, whch s obvously dffeet fo the Laplaca ethod whch depeds o two atces alog two dectos. Wth the developet of the RT techque, ay pobles of o-egula etwo wth zeo essto edges have bee esolved [36-45]. I addto, the advatage of the RT ethod s that all esstace esults ae a sgle suato dffes fo the Laplaca appoach gave esstace esults ae the fo of a double suato. Recetly, the RT ethod have bee subdvded to two fos: oe fo s the at equato epessed by cuet paaetes [38-44], whch s sply called the RT-I ethod; aothe fo s the at equato epessed by potetal paaetes [6, 45], whch s sply called the RT-V ethod. Suazg the pevous applcatos of the RT (cludg RT-I ad RT-V) ethod, t s ot had to see that the pevous eseaches of the RT ethod all ely o the zeo essto bouday, such as the globe etwo[36, 44] belogs to cyldcal etwo wth two zeo essto boudaes, the cobweb etwo[6, 4] belogs to cyldcal etwo wth oe zeo essto bouday, the fa etwo[37, 45] belogs to ectagula etwo wth oe zeo essto bouday, ad the haoc etwo[34, 43] belogs to ectagula etwo wth two zeo essto boudaes. Obvously, how to study the cople etwo wthout zeo essto bouday by the RT ethod s a questo. Y (,) (,) d d A(,) (,) X Fg.. A abtay essto etwo wth two abtay bouday esstos, whee ad ae the au coodate values of (, ). Bods the hozotal ad vetcal dectos ae esstos ad ecept fo two abtay bouday esstos of ad. Ths pape developed a ew techque ad poved the RT theoy to allow us to study abtay essto etwos wthout elyg o zeo essto bouday, whch ca deve the electcal popetes (potetal ad esstace) of the abtay cople etwos wth cople 3

4 boudaes. Hee we buld two ew theoes lead to lage pobles to be esolved. Ou study shows the uvesal RT ethod s vey teestg ad useful to solve the cople etwo. We focus o eseachg the electcal popetes (potetal ad esstace) of Fg. ad Fg. o two cople essto etwos wth two abtay boudaes by the advaced RT-V ethod, whch have ot bee esolved befoe. It s woth ephaszg that the o-egula cople etwos wth two abtay boudaes ae the ult-pupose etwo odel because t ca poduce vaous geoetcal stuctue as show Fgs.5-. Thus a lage ube of pobles of essto etwos wll be esolved by ths pape. d A (,) Fg. A oegula cyldcal essto etwo, whee ad ae the au coodate value of (, ), wth the ut esstaces ad the espectve hozotal ad vetcal (loop) dectos ecept fo two abtay bouday esstos of ad. Fo the above aalyss, pofesso Wu [3] was the fst to gve seveal accuate equvalet esstace foulas fo the egula essto etwos by the Laplaca at ethod, fo the sae of copaatve study, hee we toduce two a esults of essto etwos fo Ref.[3]. ase-. osde Fg. wth s a egula ectagle etwo, whee ad ae the au coodate values of (, ), esstos ad ae boded espectvely the hozotal ad vetcal dectos, the esstace foula fo Fg. s d (,) R d d y y (, ) [ cos( y ) cos( y ) ] ( )( ), () ( cos ) ( cos ), j, j j j (,) X whee, cos( ) j, j ( ), j j ( ), d(, y ) ad d(, y ) ae abtay two odes the etwo. ase-. osde Fg. wth s a cyldcal essto etwo, whee ad ae the au coodate values of (, ), esstos ad ae boded espectvely the hozotal ad vetcal dectos, the esstace foula fo Fg. s 4

5 R ( d, d ) y y ( y y) cos( y y ) ( )( ), () ( cos ) ( cos ), j, j, j, j j j whee, cos( ) j, j ( ), j j ( ). The above esults wee foud fo the fst te by Wu. Late Refs.[3-34] poved the Laplaca at ethod to ae t applcable to egula cobweb ad haoc etwos. Howeve, the poved Wu ethod stll caot esolve the essto etwo wth abtay bouday, such as the etwos wth two abtay boudaes of Fg. ad Fg.. I addto, the equvalet esstace Eqs.() ad () ae the double suato ot a sgle su. ( ) V V ( ) V ( ) V V ( ) V V Fg.3 The essto sub-etwo wth the potetal paaetes. Ⅱ. RT-V theoy ad Posso equato ( ) V ( ) V osde two ds of cople essto etwos of Fg. ad Fg., whee ad ae the au coodate values of (, ). Assue A (,) s the og of the ectagula coodate syste, ad deotg odes of the etwo by coodate {, y}. Assue the electc cuet J goes fo the put d(, y ) to the output d(, y ). Deote the odal potetal of the sub-etwo s show Fg.3, ad epessg the odal potetal at d(, y ) by We wll study the cople essto etwos fou steps. U y V ( y) (, ). The fst step, settg up dscete Posso equato based o the sub-etwo of Fg.3. By Kchhoff law ( V ) to set up the odal potetal equatos alog the vetcal decto, we acheve a dscete statc feld equato (o call Posso equato) fo ay etwo, 5

6 ( h ) V I, (3) ( y) ( y) y, whee h, ad I J( ) cotas the put ad output codtos of the cuet, ( y) y, y y, y ( y) ( y ) ( y) ( y ) V V V V ad V V V V deote secod ode dscete equato, ad ( y) ( y) ( y) ( y) y whe ( y), Eq.(3) educes to the dscete Laplace equato ( h ) V. Fo the y abtay etwo togethe wth the uppe ad lowe bouday codtos, by Eq.(3) we ae led to V A V V I, (4), whee V ad I ae espectvely ( ) colu ates, ad eads T V () () () V V V V, (5) ( y) ( y) ( y) ( y) T I I I I I, (6) A s the at bult alog the vetcal decto. Fo Fg. ad Fg., the A s ad h bh h bh h ( h) h A, (7) h ( h) h bh h h bh whee b 3, ad 3 s the essto betwee (,) ad (, ) Fg., whe b ( ), 3 the A belogs to Fg.; whe b ( 3 ), the A belogs to Fg.. The pupose of toducg 3 s to epess two dffeet essto etwos ufoly. The secod step, cosde the bouday codtos of the left ad ght edges the etwo of Fg. ad Fg.. Applyg Kchhoff's law ( V we obta two at equatos of bouday codtos, ) to each of the left ad ght boudaes, hv [ A ( h) E] V, (8) h V [ A ( h ) E] V, (9) whee h, h, E s the ( ) ( ) detty at, at A s gve by Eq.(7). Equatos (4)-(9) ae all the equatos we eed to copute the ode potetal. Howeve, t s possble fo us to get the soluto of the above equatos dectly. Thas to the RT theoy of Ta that gave the at tasfo ethod [6, 38-4] ad we ceate the ew techque hee. I the followg we ae gog to gve the tasfoato techology based o RT-V theoy. The thd step, ceatg at tasfo. Fstly, we wo out the egevalue t of at 6

7 A, whch s gve by solvg deteatal equato of det A t E (just b ad b ), yelds t ( h) hcos, (,,, ) () whee ( b) ( ), ad b fo Fg., b fo Fg.. Net to tasfo Eqs.(4)-(9) by the followg appoaches P A dag{ t, t,, t } P, () X P V o V ( P ) X, () () () ( ) T whee X X X X. Assug P s the ow vectos of at P+, such as P,,,,. (3) Thus, we ultply Eq.(4) fo the left-had sde by P+, we get whee Eqs.() ad () ae used. X t X X J ( ), (4) ( ) ( ) ( ), y,, y, Slaly, applyg P to Eqs.(8) ad (9), we ae led to h X ( t h ) X, (5) ( ) ( ) ( ) ( ) h X ( t h ) X. (6) The above Eqs.()-(6) ae all essetal equatos fo evaluatg the ode potetal. ( The fouth step, solvg the at equatos (3)-(6). Selectg ) V ( ) J as the efeece potetal, by Eqs.(4)-(6), we obta afte soe algeba ad educto the soluto, () X J, (7) b whee b fo Fg., b fo Fg., ad s defed Eq.(6) below, ad have X ( ) ( ) y, y, ( t ) G whee,,, G ae, espectvely, defed Eqs.(9)-(5) below. s, J, ( ) (8) The RT-V theoy. The above ethod of establshg ecusve at equatos wth voltage paaetes, pleetg at tasfo ad obtag the solutos of at equatos s called RT-V theoy. The detaled cotet of the RT-V theoy (Recuso- Tasfo theoy wth potetal paaetes) ca be foud by the above fou steps Eq.(3)-(8). Hee we gve a suay of the RT-V theoy o the cossts of fou steps: The fst step ceates a a at equato of potetal dstbutos alog the Y as (such as the buld of Eqs.(3)-(7)); The secod step bulds the 7

8 costat equatos (cludg bouday codtos) of odal potetals (such as settg up Eqs.(8) ad (9)); The thd step dagoalzes the at elato to educe the equatos fo two deso to oe deso(such as covetg Eq.(4) to Eq.(4) et al.); The fouth step fgues out the aalytc soluto of the equatos (such as the esults of Eqs.(7) ad (8)), the aes the vese at tasfo by Eq.() to deve the aalytcal solutos of odal potetal ad gves the esstace foula by oh's law. Ⅲ. Two theoes of essto etwos A. Seveal deftos I ode to facltate ad splfy the epesso of the solutos of at equatos, we defe seveal vaables of, ad, fo late uses, wth Ad defe vaables F, y, cos( ) y, y cos( ) y y y, (9), h hcos ( h hcos ). () h hcos ( h hcos ) b fo Fg. ( b) ( ), () b fo Fg., s, ad s, G fo late uses by F ( ) ( ), ( ) ( ) ( ) s, s F F F, () ( ) ( ) ( ) F ( h ) F, hs s. (3) f, (4) ( ) ( ) ( ), s,, s s s ( ) ( ) ( ),,, f s s s G F ( h h ) F ( h )( h ) F. (5) ( ) ( ) ( ) ( ) The above deftos ae applcable to the ete atcle. All of these deftos ae eat to llustate the followg two fudaetal theoes, ad we always assue that the electc cuet J goes fo the put d(, y ) to the output d(, y ) ou ete pape. Theoe-. osde the abtay B. Two fudaetal theoes essto etwos of Fg. ad Fg. whose au coodate value s (, ). The the potetal of ode d(, y ) the essto 8

9 etwo ca be wtte as V y X, (6) (, ) y, (,,, ) whee (,,, ),,, ad y, s defed Eq.(3), y, s the cojugate cople of y,, ad X s the soluto of the at equato (4) togethe wth the bouday codto equatos. Foula (6) s a geeal foula whch s sutable fo ay essto etwo odel. ( ) I patcula, whe selectg V ( ) J as the efeece potetal, the potetal of ode d(, y ) the essto etwos ca be wtte as b V y J X, (7) (, ) y, whee ( ), s the cojugate cople of y, (thee be y, y, f y, y, s just a eal ube), ad s a pecewse fucto,,, (8), ad X s gve by (8) whch s the soluto of equatos (3)-(6). Theoe-. osde the abtay essto etwos of Fg. ad Fg. whose au coodate value s (, ). The the esstace betwee ay two odes d(, y ) ad d(, y ) the etwo s gve by R ( ) ( ) X y, X y, ( d, d). (9) (, ) J y, y, whee X s the soluto of the at equato (4) togethe wth the bouday codto equatos, Foula (9) s a geeal foula whch s sutable fo ay essto etwo odel. I patcula, fo the etwos of Fg. ad Fg., the esstace betwee two odes d(, y ) ad d(, y ) ca be wtte as ( ) ( ) b y, y, R ( d, d) J 9 X X. (3) whee b s the case of Fg., ad b s the case of Fg., ad X s gve by (8) whch s the soluto of equatos (3)-(6). The above two ew theoes cota a wde vaety of geoetc stuctue of the etwo odel, whch ca poduce ay ew esults of potetal ad esstace, ad ca ceate ew

10 atheatcal detty (see the secto 6). I the followg, we ae gog to pove the coectess of two theoes.. Poof of theoes osde the essto etwo wth two abtay boudaes show Fg. ad Fg., the toducto, we have bult the ey Eqs.(4)-(9) by the RT-V theoy, ad coveted the equatos to Eqs.(4)-(6), ad deved Eqs.(7) ad (8). Now we wll wo out the eact egevalues of at A Eq.(7). Eq.() ca be deved by solvg equato det A te, ad the we eed to cosde two cases below. Oe s Fo Fg., substtutg Eq.to () wth b A, we get the egevectos cosv cosv cosv cos v cos v cos v P, (3) whee v, ad ( ). By caeful calculato, the vese at ca be easly obtaed whee [ ] T deote at taspose. T P P, (3) Thus, the te, appeaed Eq.(3) ca be specfcally ewtte as, ( y ) (33) y, y, y, y, cos( y ), ( ) (34) Aothe s Fo fg., substtutg Eq.to () wth b A, the egevecto s obtaed afte soe algeba ad devato, ep( ) ep( ) ep( ) ep( ) ep( ) ep( ) P, (35) whee ( ) (,,, ). Accodg to stct calculatos, the vese at eads ep( ) ep( ) P. (36) ep( ) ep( ) Thus, the te appeaed Eq.(3) ca be specfcally ewtte as,

11 , ( y ) (37) y, y, y, ep( y ), y, ep( y ) (38) We fd that Eq.(3) ad Eq.(36) ca be ewtte as a ufed fo below,,,,,, P, (39) (,,, ),,, whee (,,, ),,, ad s the cojugate cople of y,. y, Usg Eq.(), we have V ( P ) X, epadg ths at equato, the we get ( y) () ( ) V X y, X y, (,,, ), (4) Equato (4) agees wth the foula (6) that we eed to vefy. Futhe, we get (,,, ) by copag equato (39) wth equatos (3) ad (36). b ( Ad whe selectg ) V ( ) J, we have Eq.(7). Substtutg Eqs.(7), (33) ad (37) to Eq.(4), the Eq.(7) ca be vefed edately. Net, we vefy Eqs.(9) ad (3), by oh's law, we have R d, d [ V (, y) V (, y)]. (4) J Substtutg Eq.(6) wth {, ad y { y, y to Eq.(4), we theefoe obta Eq.(9). Substtutg Eq.(7) wth {, ad y { y, y to Eq.(4), we edately obta Eq.(3). Thus, two theoes ae vefed. I subsequet secto we cosde applcatos of theoes to abtay lattces. I all applcatos, we stpulate all paaetes Eqs.(8)-(39) apply to all essto etwos, ad deote the esstos alog the two pcpal dectos by ad ecept fo esstos o the left-ght boudaes, ad the put ad output odes of cuet ae espectvely at d(, y ) ad d(, y ). Ⅳ. Electcal popetes of cople ectagula etwo osde the o-egula A. Nodal potetal of cople ectagula etwo essto etwo show Fg., whee the au

12 ( ) coodate s (, ), selectg V ( ) J as the efeece potetal, the potetal of ay ode d(, y ) the fte ad se-fte etwos ca be wtte as ( ) ( ) y, y, y, U (, y) J, (4) ( cos ) G U (, y) J h h y, y, ( cos ) y,, (43) whee ( ), ad,,, s, G ae, espectvely, defed Eqs.(9)-(5). Fo Eq.(43), thee be,, wth fte. I patcula, whe (eas the put ad output odes of cuets ae at the sae vetcal as), foulae (4) ad (43) educe to U (, y) ( ), (44) y, y, y, J ( cos ) G U (, ) ( ) y J h h y, y, y, ( cos ). (45) Poof of Eq.(4). Fo Fg., substtutg Eq.(34) wth y { y, y} to (8), we acheve X ( ) ( ) y, y, ( t ) G J, ( ) (46) Substtutg Eq.(46) ad (34) to (7) wth b, we theefoe acheve Eq.(4). Fo povg Eq.(43), whe,, wth fte, t ca be got a lt by usg Eqs.()-(5) l ( ) G t. (47) So, substtutg Eq.(47) to (4) wth, we theefoe vefed Eq.(43). Foula (4) s a eagful esult because the etwo of Fg. s vey cople ad has ot bee esolved befoe, whch cotas a lot of dffeet etwo odels sce the dffeet bouday esstos ca poduce dffeet geoetc stuctues. Hee seveal specal applcatos of foula (4) ae gve below. Applcato. Whe, Fg. degades to a egula ectagula etwo, the potetal of a ode d(, y ) the etwo s U( y, ) J ( cos ) F ( ) ( ) y, y, y,, (48)

13 whee educes to, s F F. ( ) ( ) ( ), s s I patcula, whe, potetal foula (48) educes futhe to U (, y) ( ). (49) y, y, y, J ( cos ) F Applcato. Whe h ( ), Fg. degades to a fa etwo as show Fg.4(a), whee ad ae the espectve esstos alog logtude (adus) ad lattude (ac) dectos, ad the essto eleet o the oute ac s (a abtay bouday essto). The potetal of a ode d(, y ) the fa etwo ca be wtte as U(, y) J F ( h ) F ( ) ( ) y, y, ( ) ( ) y,, (5) whee we edefe F, f ) ad ( ) ( ) ( ) s s s F, f ). ( ) ( ) ( ) s s s Please ote that a o-egula fa etwo (the oute ac essto s abtay) s a scetfc coudu, whch has ot bee solved befoe. Ref.[6] has eseached just the egula fa etwo (the oute ac essto s ), but ou foula (5) wth s dffeet fo the esult Ref.[6] because two esults depeds o the dffeet atces alog the othogoal decto. (a) Fg.4. Two essto etwo odels. (a) s a Fa etwo wth a abtay bouday essto ; (b) s a abtay haoc etwo. (b) Applcato 3. Whe, Fg. degades to a haoc etwo as show Fg.4(b), the potetal of a ode d(, y ) the U(, y) J F haoc etwo ca be wtte as ( ) ( ) y, y, y,, (5) whee we edefe ( ) ( ) ( ) F s F s s f ) ad ( ) ( ) ( ) F s F s s f ). I patcula, whe d(, y ) ad d(, y ) ae espectvely at the left ad ght poles, the potetal of Eq.(5) educes to 3

14 U(, y) J (. (5) ) Applcato 4. osde the put cuet J s at d(, y ) o the left edge, ad the output cuet J s at d(, y ) o the ght edge, the potetal of a ode d(, y ) the o-egula essto etwo of Fg. s U(, y) J ( ) ( cos ) G ( ) ( ), y,, y, y,, (53) whee s defed Eq.(3)., I patcula, whe ad h ( ), the potetal (53) educes to U(, y) F J ( ) F y, y,, (54) Applcato 5. osde a egula ectagula etwo of Fg. wth, whe d(, y ) s o the left edge, ad d(, y ) s o the ght edge, the potetal of a ode d(, y ) the ectagula etwo s F F, (55) ( ) ( ) U(, y) y, y, y, J ( ) ( cos ) F I patcula, whe y y, the potetal equato (55) educes to ( ) ( ) U(, y) F F y, y, J ( ) cos F, (56) Applcato 6. osde d(,) s at the botto edge, ad d(, ) s o the top edge, the potetal of a ode d(, y ) the o-egula essto etwo of Fg. s ( ) ( ) ( ) (),, y, (57) ( cos ) U(, y) J G whee, s defed Eq.(4), ad, cos( ). I patcula, whe, the potetal equato (57) educes to U(, y) [ ( ) ] J ( cos ) G whe ad, the potetal equato (57) educes to U(, y) [ ( ) ] F J ( cos ) F (), y,, (58) (), y,, (59) Applcato 7. osde d (,), ad d (, ) ae o two dagoal odes, the potetal of 4

15 a ode d(, y ) the o-egula essto etwo s U(, y) ( ) J ( ) ( cos ) G ( ) ( ),,, y,, (6) whee s defed Eq.(3)., I patcula, whe, the potetal of Eq.(6) educes to F ( ) F, (6) ( ) ( ) U(, y), y, J ( ) ( cos ) F Applcato 8. Assue Fg. s a se-fte etwo, ad but, ad y ae fte. osde d(, y ) s o the left edge, ad d(, y ) s o the ght edge, whe, the potetal of a ode d(, y ) the se-fte ectagula etwo s U (, y) J whee cos( y y ), F ( ) ( ) wth ( F F ) y y y d, (6) ( cos ) F h hcos ( h hcos ). Please ote that these deftos apply to all such ssues as appea below. Eq.(6) ca be deved by tag the lt of Eq.(55). Applcato 9. Assue Fg. s a se-fte etwo, whee but, ad y ae fte. Whe ad h ( ), d (, y) ad d (, y), tag the lt Eq.(54), we acheve the potetal a se-fte U (, y) F J etwo y yd, (63) F Applcato. Assue Fg. s a fte etwo, but ad y y ae fte, tag the lt Eq.(43), we have the potetal the fte ectagula etwo y y yy ( cos ) U(, y) d J, (64) h h Notce that Eqs.(6) ad (63) belog to the case of a se-fte etwo, whle Eq.(64) belogs to the case of a fte etwo. B. Resstace of cople ectagula etwo osde a ectagula etwo wth two abtay boudaes show Fg., whee the au coodate s (, ). Defg, the esstace betwee two odes d(, y ) ( ) ( ), s, s ad d(, y ) the fte ad se-fte etwos ae espectvely 5

16 ( ) ( ) ( ), y,, y, y,, y, ( cos ) G R ( d, d ), (65) R ( d, d ) y, y, y, y, ( h hcos ). (66) whee ( ), ad,, G ae, espectvely, defed Eqs.(9)-(5). Fo Eq.(66), thee be,, wth fte. Eq.(66) ca be deved by tag the lt Eq.(65). Poof of Eq.(65). Fo Fg., substtutg Eq.(4) wth, to (4), the Eq.(65) s vefed. Eq.(65) s a eact epesso whch stll cotas a vaety of esstace esults wth all ds of bouday codtos because the left ad ght boudaes ae the abtay esstos. Fo clealy udestadg foula (65), we set h ad h, o as specal values, ad gve seveal specal cases to udestad ts applcato ad eag. ase. Whe h, the etwo of Fg. degades to a ectagula etwo wth a abtay ght bouday, the foula (65) educes to whee educes to s, ( ) ( ) ( ), y,, y, y,, y, (, ) ( ) ( ) ( cos )[ F ( h ) F ] R d d ( ) ( ) ( ) ( ), s F F h s F s [ ( ) ]., (67) ase. Whe h h, the etwo of Fg. degades to a oal ectagula etwo, the foula (65) educes to ( ) ( ) ( ), y,, y, y,, y, ( cos ) F R ( d, d ), (68) whee educes to s, F F. Ths poble has bee eseached Ref.[3], ad gave ( ) ( ) ( ), s s Eq.() wth a double sus. lealy, ou esult (68) s dffeet fo Eq.(). Two dffeet esults wll be copaed the last secto. Ths also shows that the equvalet esstace ca be epessed dffeet fos. ase 3. Whe h, the etwo of Fg. degades to a o-egula fa etwo wth a abtay bouday as show Fg.4(a), by Eq.(65), we obta the esstace of a fa etwo ( ) ( ) ( ) -,,,,,,, (, ) y y y y ( ) ( ) F ( h ) F R d d, (69) whee s e-defed as s, ( ) ( ) ( ) ( ), s F F h s F s [ ( ) ]. I patcula, whe h, h, the etwo of Fg. degades to a oal fa 6

17 etwo, the foula (69) educes to ( ) ( ) ( ), y,, y, y,, y, F R ( d, d ), (7) whee s e-defed as s, F F. Ths case has bee eseached [37], but the esult ( ) ( ) ( ), s s s dffeet fo Eq.(7), howeve they ae equvalet to each othe. The easo s that they choce the dffeet at alog dffeet decto. Ths also shows that the equvalet esstace ca be epessed dffeet fos. ase 4. Whe h, h, the etwo degades to a fa etwo wth double essto edge, the foula (65) educes to ( ) ( ) ( ), y,, y, y,, y, (, ) R d d, (7) whee s e-defed as s,, F ( ). ( ) ( ) s s s ase 5. Whe h h, the etwo of Fg. degades to a haoc etwo, so, foula (65) educes to whee s, ( ) ( ) ( ), y,, y, y,, y, F R ( d, d ) s e-defed as F F ( ) ( ) ( ), s s, (7). Ths poble has bee eseached Ref.[34], but the esult s dffeet fo Eq.(7)], the easo s that they choce the dffeet at alog dffeet decto. I patcula, whe d(, y ) ad d(, y ) ae at the left ad ght poles, Eq.(7) educes to R ({, },{, }) y y ( ). (73) ase 6. Whe two odes ae o the sae vetcal as, fo (65) we have the esstace betwee two odes d(, y) ad d(, y ), ( ) R ({, y},{, y }) G cos y, y,. (74) Whee ad, G ae defed (4) ad (5). I patclla, whe, foula (74) educes to ( ) ( ) F ( ) F R ({, y},{, y }) F cos y, y,. (75) ase 7. Whe both d(, y ) ad d(, y ) ae at the left edge, foula (65) educes to ( ) ( ) F ( ( ) h ) F R ({, y},{, y }) G cos 7 y, y,. (76)

18 I patclla, whe h, d (,) ad d (, ), foula (76) educes to R ( ) ( ) F ( h ) F ({,},{, }) [ ( ) ]cot ( ) ( ) ( ) F ( h ) F. (77) It ca be foud that Eq.(77) agees wth the esult Ref.[4], clealy both of these esults ae vefed dectly by each othe. ase 8. Whe both d(, y ) ad d(, y ) ae at the sae hozotal as, fo (65), we have the esstace ( ) ( ) ( ),,, R ({, y},{, y}) G cos cos [( y ) ], (78) whee ad s, G ae defed (4) ad (5). Especally, whe both d(,) ad d(,) ae at the botto edge, Eq.(78) educes to ( ) ( ) ( ),,, R ({,},{,}) cot ( ). (79) G Whe, d (,) ad d (,) ae two coe pots at the botto edge, Eq.(79) educes to a eat esult, aely R ({,},{,}) F + cot ( ) F. (8) ase 9. Whe d(, y ) s o the left edge ad d(, y ) s o the ght edge, foula (65) educes to whee h h h h ( ) ( ), y, y, y,, y, ( cos ) G R ({, y },{, y }) F ( h ) F. ( ) ( ) ( ),, (8) I patcula, whe h h, Eq.(8) educes to ( ) F ( ) y, y, y, y, ( cos ) F R ({, y },{, y }). (8) ase. Whe d(,) s at the botto edge ad d(, ) s o the top edge, the foula (65) educes to ( ) R ({,},{, }) cot ( ), (83) ( ) ( ) ( ),,, G whee ad s, G ae defed (4) ad (5). ase. Whe d (,) s at the coodate og, ad d (, ) y s a abtay pot, by Eq.(65), the equvalet esstace s 8

19 R ({,},{, y}) h h ( cos ) G ( ) ( ) ( ), y,, y, y,, y,, (84) whee ad, ae defed Eqs.(4) ad (5). s, I patcula, whe h h ad y, Eq.(84) educes to R ( ) ( ) ( ) ( ) ( ) ({,},{, }) F F F F cot ( ) F, (85) ase. Whe d (,) ad d (, ) ae two dagoal odes, by (65) we have the esstace betwee two aally sepaated odes R h ( ) h h h ( ) ( ),, ({,},{, }) cot ( ) G, (86) whee F ( h ) F, ( ). ( ) ( ) ( ), I patcula, whe, Eq.(86) educes to R ( ) ({,},{, }) F + cot ( ) F. (87) Please ote that Eq.(87) s a desed equvalet esstace betwee two au sepaated odes a abtay essto etwo. Ths s a teestg esult because t s sple ad easy to eseach the asyptotc epaso fo the au esstace. Refs.[46, 47] studed the asyptotc epaso by ag use of the esult (). Obvously, the cocse Eq.(87) s oe coducve to the study of the asyptotc epesso of the au esstace. I addto, thee ae slates betwee equatos (8) ad (87), but oly o dffeeces, whee Eq.(8) s the esstace betwee two coe pots at the botto edge, Eq.(87) s the esstace betwee two au sepaated odes o the dagoal le. ase 3. Whe, wth, ad y, y ae fte. Tag the lt of Eq.(76), we have the esstace the se-fte etwo R ({, y },{, y }) y, y, h cos 9 ( ). (88) Obvously, case 3 s a se-fte etwo poble. The easo why the equvalet esstace s depedet of the ght bouday s that the etwo s fte o the ght. ase 4. Whe,, h, ad y, y ae fte. It eas that the etwo s fte at the botto ad left but fte at the top ad ght. Tag the lt of Eq.(88), we have [cos( y ) cos( y ) ] R ({, y},{, y}) ( ) d cos, (89) whee h hcos ( h hcos ). Ths defto apples to all such ssues as appea

20 below. I patclla, whe,, ad y, y, h ( ), but y y s fte, tag the lt of Eq.(88), we have the equvalet esstace o the left edge cos( y y) R ({, y},{, y}) ( ) d. (9) cos Eqs.(89) ad (9) belog to the poble of se-fte etwo, whch has ot bee solved befoe. ase 5. Whe,, but ad y y ae fte, we have the esstace betwee two abtay odes d(, y) ad d(, y ) cos( y y) ( h hcos ) R ( d, d) d, (9) Foula (9) ca be poved by tag the lt, Eq.(66). Eq.(9) shows that the equvalet esstace of a fte etwo s depedet of the bouday codtos. Notce that Eqs.(88)-(9) belog to the case of a se-fte etwo, whle Eq.(9) belogs to the case of a fte etwo. Sce Eq.(65) s a pofoud ad vesatle foula that t cotas ultple outcoes, fo eades to bette udestad the eag of Eqs.(65) ad (9), we ae gog to aga llustate the eag ad usefuless by fou sple eaples below. B A B A Fg.5. A -step essto etwo wth two abtay boudaes, the esstos the hozotal ad vetcal dectos ae, espectvely, ad ecept fo two abtay boudaes. B A ase 6. Whe, the essto etwo of Fg. degades to a essto etwo as show Fg.5, the equvalet esstace betwee ay two odes the essto etwo ca be wtte as R A B,,, (, ) 4 G R A A,,, (, ) 4 G whee s,, G ae, espectvely, defed Eq.(4) ad (5), ad, (9), (93) h h h, h h h. (94)

21 B B B A A A Fg.6. A abtay cople etwo of esstos wth fou abtay eleets whch ae, espectvely,,, ad. ase 7. Whe, the essto etwo of Fg. degades to a abtay essto etwo as show Fg.6, whee thee be fou abtay essto eleets (,,, ), the equvalet esstace betwee ay two odes d(, y ) ad d(, y ) ca be wtte as () () () () () (),,,,,, (, ) () () 3 G 9G R A A () () () () () (),,,,,, (, ) () () 3 G 9G R A B whee, h ad (5). Ad () () () (),, 4, 4, R ( A, ) () () 3 G 9G ( =, ), ad F ( ) ( ), wth, s,. (95). (96). (97) G ae, espectvely, defed Eqs.(4) ( ) h h 4h, ( 9 3h h h). (98) Fo the above devato, we fd that foula (65) s a geealzed esult, whch s applcable to ay etwo pobles ad suazed a vaety of cople etwo odels sce t cotas s abtay eleets (,,,,, ). ase 8. osde a egula ectagula etwo, whe (the, etwo s fte at the left but fte at the botto, top ad ght.), by Eq.(9), the esstace o the left edge s R ({, y},{, y }) h ( h)acs h h h. (99) I patcula, whe h ( ), Eq.(99) educes to R y y ({, },{, }). () ase 9. osde a egula ectagula etwo, whe (the, etwo s fte at

22 the left but fte at the botto, top ad ght.), by Eq.(9) we have the esstace o the left edge R ({, y},{, y }) ( h) h h h acs ( ) h h h. () I patcula, whe h, fo Eq.() we get R ({,},{,}). () Eqs.(99)-() ae two ovel esults whch ae gve fo the fst te. Ⅴ. Electcal popetes of cople cyldcal etwo osde the o-egula A. Nodal potetal of cople cyldcal etwo cyldcal etwo show Fg., whee the au ( coodate value s (, ), selectg ) V ( ) J as the efeece potetal, defg ( ), y y cos( y y ), the potetal of ay ode d(, y ) the fte ad se-fte etwos ca be wtte as whee ad ( ) ( ) y y y y U (, y) J ( ) ( cos ) G yy yy ( ) ( cos ), (3) U (, y) J h h, (4) G ae, espectvely, defed Eqs.()-(5). Fo Eq.(96), thee be,, wth fte. Eq.(4) ca be deved by tag the lt Eq.(3). I patcula, whe, foula (3) ad (4) educe to U (, y) yy yy ( ) J ( cos ) G, (5) U (, y) J h h yy yy ( ) ( cos ), (6) Poof of Eq.(3). Fo Fg., substtutg Eq.(38) to Eq.(8), we acheve ( ) X The substtuto of (7) to Eq.(7) yelds ep( y ) ep( y ) J. (7) ( ) ( ) ( t ) G ( ) ( ) y y y y U(, y) J ( cos ) G

23 s[( y y) ] s[( y y) ], (8) ( cos ) ( ) ( ) G Because the eleets the etwo s eal ube, the potetal U(, y ) ust be eal ube. Thus, etactg the eal pat of Eq.(8) to poduce Eq.(3). Foula (3) s a eagful esult because the etwo of Fg. s vey cople ad has ot bee esolved befoe, cotas a lot of essto etwo odels, whee each of the dffeet bouday essto epesets a dffeet etwo stuctue. So Foula (3) ca ceate ay teestg esults. I the followg applcatos we always assue that the cuet J goes fo d(, y ) to d(, y ) ecept fo specal stuctos. Applcato. osde a abtay cyldcal etwo of Fg. wth, by (3) we have the odal potetal whee educes to, ( ) ( ) y y y y U(, y) J ( ) ( cos ) F F F. ( ) ( ) ( ), s s I patcula, whe Eq.(9) educes to U(, y) yy yy ( ) J ( cos ) F, (9), () (a) (b) Fg.7. Two essto etwo odels. (a) s a cobweb etwo wth a abtay bouday essto ; (b) s a abtay globe etwo. Applcato. osde a cyldcal etwo of Fg.. Whe, Fg. degades to a cobweb etwo as show Fg.7(a), by (3) we have the odal potetal U(, y) J F ( h ) F ( ) ( ) y y y y ( ) ( ), () ( ) ( ) ( ) whee s edefed as F, f s) ad s s s F, f ). ( ) ( ) ( ) s s s I patcula, whe d(, y ) s at left edge, ad d(, y ) s at ght edge. Eq.() educes to 3

24 U(, y) h F cos( y y) J F h F ( ) ( ) ( ) ( ), () Please ote that the cobweb etwo wth a abtay bouday has ot bee esolved befoe, the pevous wo oly studed the oal cobweb etwo (the bouday essto s ) [6], Eq.s a ogal esult. Applcato 3. osde a abtay globe etwo show Fg.7(b). That s to say that Fg. degades to a globe etwo whe, fo (3) we have the odal potetal U y ( ) ( ) (, ) y y y y J F, (3) whee we edefe ( ) ( ) ( ) F s F s s f ) ad ( ) ( ) ( ) F s F s s f ). I patcula, whe d(, y ) s at left pole, ad d(, y ) s at ght pole, Eq.(3) educes to U(, y) J (. (4) ) Foula (4) s vey sple ad vey teestg because the potetal dstbuto s oly elated to the ad has othg to do wth y, whch shows the odal potetal s equal the sae lattude. Applcato 4. osde a o-egula cyldcal etwo of Fg.. Assue d(, y ) s o the left edge, ad d(, y ) s o the ght edge. by (3) we have the odal potetal whee ( ) ( ) ( ), U(, y) h h J G ( ) ( ), y y, yy ( ) ( ) ( cos ) F ( h ) F s defed Eq.(3), ad y y cos( y y ). I patcula, whe y y, Eq.(5) educes to h ( ) ( ) U(, y),, y ( ) ( ) y J ( cos ) G h, (5), (6) whe h h, Eq.(5) educes to U(, y) F F J F whe h h, y y, Eq.(5) educes to ( ) ( ) y y y y ( ) ( ) ( cos ) F F ( ) ( ) U(, y) () ( ) ( ) ( cos ) y J y F, (7). (8) Applcato 5. osde a o-egula cyldcal etwo of Fg.. Whe the ode d(, y) ad d(, y ) ae located o the left edge, Eq.(3) educes to 4

25 U(, y) yy yy, J ( ) cos G, (9) whee y y cos( y y ), ad (), s defed Eq.(3). Applcato 6. osde a o-egula cyldcal etwo of Fg.. Whe the ode d(, y ) ad d(, y ) ae located o the ght edge, Eq.(3) educes to U(, y) yy yy, J ( ) cos G, () Oe ow the potetal fucto have potat applcato value fo solvg the Laplace equato. I ths pape, the aalytcal solutos of ode potetal fuctos ude vaous codtos ae gve, whch povdes a ew theoy fo pactcal applcato. I patcula, these sple equatos of Eqs.(9) ad () ae vey teestg ad eagful fo applcatos. B. Resstace of cople cyldcal etwo osde a cople cyldcal etwo wth two abtay boudaes show Fg., ( ) ( ) whee the au coodate value s (, ), ad defg, s, s, The equvalet esstace betwee two odes d(, y ) ad d(, y ) the fte ad se-fte cyldcal etwos ae espectvely R ( d, d ) whee ( ), y y y, cos( y ), () ( ) ( ) ( ),,, ( ) ( cos ) G y, () cos( ) (, ) R d d ( h hcos ) ad s, G ae, espectvely, defed Eqs.(4)-(5). Fo Eq.(), thee be,, wth fte. Eq.() ca be deved by tag the lt Eq.(). Poof of Eq.(). Fo Fg., substtutg Eq.(3) wth, to Eq.(4), we theefoe acheve (). Foula s a eact ad ectg esult because the etwo of Fg. s vey cople ad has ot bee esolved befoe, ad cotas a lot of essto etwo odels, whee each of the dffeet bouday essto epesets a dffeet etwo stuctue. I patcula, whe tag soe specfc value fo ad, Eq.() gves se to a sees of specal cases below. ase. osde a o-egula cyldcal etwo of Fg.. Whe, the esstace of Eq.() educes to 5

26 cos( y ) R d d ( ), (3) ( cos )[ ( ) ] ( ) ( ) ( ),,, (, ) ( ) ( ) F h F whee educes to s, ( ) ( ) ( ) ( ), s F F h s F s [ ( ) ]. ase. osde a oal cyldcal etwo of Fg. wth, the esstace of Eq.() educes to R ( d, d ) cos( y ), (4) ( ) ( ) ( ),,, ( ) ( cos ) F whee educes to s, F F. ( ) ( ) ( ), s s ase 3. osde a o-egula cyldcal etwo of Fg., whe h, the left bouday collapses to a pole, the etwo of Fg. degades to a cobweb etwo wth a abtay bouday essto as show Fg.7(a), we have the equvalet esstace whee s e-defed as s, R d d cos( y ), (5) ( ) ( ) ( ),,, (, ) ( ) ( ) F ( h ) F ( ) ( ) ( ) ( ), s F F h s F s [ ( ) ]. I patcula, whe h, h, the etwo of Fg.7(a) degades to a egula cobweb etwo, the esstace of Eq.(5) educes to whee s, R ( d, d ) ( ) ( ) ( ) s edefed as F F. cos( y ), (6) ( ) ( ) ( ),,, F, s s Please ote that case 3 has bee eseached Ref.[37], but the esult s dffeet fo Eq.(6), howeve they ae equvalet to each othe. The easo s that they choce the dffeet at alog dffeet decto, whee Ref.[37] set up at alog the logtude, but ths pape set up at alog the lattude. ase 4. Whe h h, the left ad ght bouday collapse espectvely to two poles, the etwo of Fg. degades to a globe etwo as show Fg.7(b), we have cos( y ) R ( d, d ), (7) ( ) ( ) ( ),,, F whee s e-defed as s, F F. ( ) ( ) ( ), s s Please ote that case 4 has bee eseached Ref.[36], but the esult s dffeet fo Eq.(7), howeve they ae equvalet to each othe. The easo s that they choce the dffeet at alog dffeet aes. Ths also shows that the equvalet esstace ca be epessed dffeet fos. 6

27 ase 5. osde a o-egula cyldcal etwo of Fg., whe d (,) ad d (, ) y ae o the sae lattude, the esstace of Eq.() educes to R ({,},{, y}), cos( y ) G cos, (8) ( ) ( ) ( ) whee,,,, ad s defed Eq.(3). s, I patcula, whe h h, the etwo of Fg. degades to a egula cyldcal etwo, the esstace of Eq.(8) educes to R ({,},{, y}) F F cos( y ), (9) ( ) ( ) F cos Especally, whe d (,) ad d (, ) y ae o the left edge, Eq.(9) educes to R ({,},{, y}) F cos( y) F, (3) cos ase 6. osde a o-egula cyldcal etwo of Fg., whe both d(,) ad d(,) ae o the sae hozotal as, we have R ({,},{,}) ( ) ( ) ( ),,, ( ) ( cos ) G. (3) ase 7. Whe d (,) s at the coodate og, ad d (, ) y s a abtay pot, foula () educes to whee ad, ( ) ( ) ( ),,, cos( ) ({,},{, }) h h y R y ( ), (3) ( cos ) G ae defed Eqs.(4) ad (5). s, I patcula, whe h h ad y, Eq.(3) educes to cos( ) ({,},{, }) F F y F F R y ( ), (33) ( cos ) G ( ) ( ) ( ) ( ) ase 8. osde a o-egula cyldcal etwo of Fg., whe d (,) s o the left edge ad d (, ) y s o the ght edge, the esstace betwee two edges s ( ) ( ),, cos( ) ({,},{, }) h h h h y R y ( ), (34) ( cos ) G I patcula, whe h h, Eq.(34) educes to cos( ) ({,},{, }) F y R y. (35) ( cos ) F 7

28 Whe h, Eq. (34) educes to ({,},{, }) ( ) ( ) F ( h ) F R y h F, (36) ase 9. osde a o-egula cyldcal etwo of Fg., whe d (,) s at the botto edge, d (, ) y s a abtay ode, ad,, but y, ae fte, by(8) we have ( h hcos ) [ cos( y )]( q ) R ({,},{, y}) d, (37) whee q ( h ) ( h ), h hcos ( h hcos ). I patcula, whe h, Eq.(37) educes to [ cos( y )]( ) R ({,},{, y}) d. (38) Whe, but y s fte, Eq.(37) educes to ( h hcos ) cos( y ) R ({,},{, y}) d, (39) ( h hcos ) Notce that Eqs.(37) ad (38) belog to the case of a se-fte etwo, whle Eq.(39) belogs to the case of a fte etwo. Fo the above esults we ow foula () ad () ae two geeal esults whch, cotas ay esults a vaety of lattce stuctues, ca poduce ay ew esstace foulae, ad ca ceate ew detty (see Secto 6). It s essetal to tae to accout foula () aga ode to help the eade futhe to udestad ts eag, hee two sple eaples ae gve below. B B A A Fg.8 A 3D etwo wth esstos ad the espectve hozotal ad vetcal dectos ecept fo ad o the left ad ght edges. ase. Whe, Fg. degades to a 3D essto etwo as show Fg.8. By Eq.() we have 3 (, ), substtutg to Eq.() yelds 3 3 h ( h). (4) 8

29 () () So we have, ad 3, 4 3. By Eq.(), we have the equvalet esstace t, s t, s betwee ay two odes cos( y 3) R ( A, P ) () () (),,, () 3 9 G, (4) whee P epesets the odes of A, B,, ad, () s, () G ae, espectvely, defed Eqs.(4) ad (5). Eq.(4) s a geeal foula, whch ca poduce two specfc esult below. osde the esstace betwee two odes A ad A, thee be y, Eq.(4) educes to R ( A, A ) osde the esstace betwee two odes educes to () () (),,, () 3 9 G A ad B (o R A B R A () () (),,, (, ) (, ) () 3 9 G, (4) ), thee be y, Eq.(4), (43) Fg.8 s a sple ad coo etwo odel, but gettg the equvalet esstace has always bee a dffcult poble because of the coplety of the bouday codtos. Eq.(4) s gve fo the fst te, whch povdes a ew theoetcal bass fo pactcal applcato. B A D B A D Fg.9 A 3D etwo wth esstos ad the espectve hozotal ad vetcal dectos ecept fo ad o the left ad ght edges. ase. Whe 3, Fg. degades to a 3D essto etwo as show Fg.9. We ca get the equvalet esstace betwee ay two odes by Eq.(). By Eq.() we have, substtutg t to Eq.() yelds, 3 h ( h) 9. (44) h ( h) (3) () So we have, ad,. By Eq.(), we have t, s t, s cos( y ) cos( y ), (45) () () () () () (),,,,,, () () 4 4 G 6 G R ( A, P )

30 whee P epesets the odes of A, B,, D, ad, s, G ae, espectvely, defed Eqs.(4) ad (5). Eq.(45) s a geeal foula, whch ca poduce thee specfc esult below. osde the esstace betwee two odes A ad () () () () () (),,,,,, () () 4 4 G 6 G R ( A, A ) osde the esstace betwee two odes A ad A, thee be y, Eq.(45) educes to, (46) () () () () (),,,,, () () 4 4 G 6 G R ( A, B ) osde the esstace betwee two odes A ad B, thee be y, Eq.(45) educes to, (47), thee be y, Eq.(45) educes to. (48) () () () () () (),,,,,, () () 4 4 G 6 G R ( A, ) The case tells us aga that the geeal foula s a eagful ad ultpupose esult sce just a 3D essto etwo has ch cotets ad ay fuctos such as Eq.(45)-(48). Ⅵ. opaato ad Tgooetc Idettes A. Poposto-. A geeal tgooetc detty- Defg y, cos( ) y, ad, j j. Whe, ad, y ae atual ubes, ad,, y, y, we have the tgooetc detty cos( ) cos( ) y, j y, j j ( cos ) h ( cos j) y y, (49) ( cos ) ( ) ( ) ( ), y,, y, y,, y, F whee F F ad ( ) ( ) ( ), s s F ( ) ( ), F F F wth ( ) ( ) ( ), h hcos ( h hcos ). (5) h hcos ( h hcos ) Please ote that the detty (49) s foud fo the fst te by ths pape. Idetty (49) educes a double su to a sgle su, whch povdes a ew poposto ad eseach ethod fo atheatcas. Poof of the Poposto- 3

31 osde a egula ectagula etwo show Fg. wth, Ref.[3] gave a esstace foula () by the Laplaca at ethod, whch s the fo of double su. Howeve, ths pape gves Eq.(68) by the RT-V ethod, whee the codto ad etwo stuctue agee wth Ref.[3]. Obvously, the two esults wth dffeet fo two dffeet atcles ae ecessaly equvalece because they ae fo the sae etwo wth the sae coodates. opag foula (68) wth foula (), we edately obta detty (49). We fd Eq.(49) s a teestg detty fo splfyg the double su to be a sgle su. I patcula, whe tag patcula values of y,, ad,, we have the followg sple tgooetc dettes. Deducto. Whe y y y, Eq.(49) educes to cos( ) j cos( ) j cos ( y ) j ( cos ) h ( cos j) ( ) ( ) ( ) cos ( ),, y,. (5) cos F I patcula, whe,, Eq.(5) educes to j [ ( ) ] cos( () j)cos( y ) cos ( ) F y ( ) j ( cos ) ( cos j) h F cos. (5) Deducto. Whe y y, Eq.(49) educes to cos( ) cos( ) j j 4 j ( cos ) ( cos ) h j cos ( ) cot ( ). (53) ( ) ( ) ( ),,, F Whe, y, thee be, Eq.(53) educes to cos( () () () ) j cos( ) j,,, () j j, (54) h ( cos ) F whee F F, () () (), s s F ( ) ( ) wth h h( h). () Deducto 3. Whe, y, cos( ) y, Eq.(49) educes to ( ) cos ( ) y, y, j j ( cos ) h ( cos j) 3

32 I patcula, whe, Eq.(55) educes to ( ) y y F cos ( ) ( ) F F y, y, (55) ( (),, ) cos ( ) y y j ( ) F ( y, y, ) y y ( ) j ( cos ) ( cos ) cos h j F. (56) whee F F F F ad the followg Eq.(76) s used. ( ) ( ) ( ) ( ) Whe y, y, Eq.(55) educes to [ ( ) ] cos( )cos( ) j 4 j ( cos ) h ( cos j) whee F F [ ( ) ]cot ( ). (57) ( ) ( ) F Deducto 4. Whe, y, y, we have, Eq.(49) educes to () () () () [cos( ) j cos( ) j],,, () j h ( cos j) F, (58) F ( ) ( ) wth h h( h) ad h h( h). I patcula, whe ad, Eq.(58) educes to. (59) j ( cos ) cos ( ) ( ) ( ) j F F ( ) h j F The above dscovees ae teestg because they ae foud ot atheatcs but physcs. Obvously, accodg to the detty (49) we ca deve a sees of specal tgooetc equaltes whe the dffeet coodates (, y ) s ade. B. Poposto-. A geeal tgooetc detty- Defg, cos( ) j j, ad, j j. Whe, ad, y ae atual ubes, ad,, y, we have the tgooetc detty cos( y ) ( cos ) ( cos ), j, j, j, j j h j y cos( y ) ( ) F ( ) ( ) ( ),,, y ( ) ( cos ), (6) whee F F ad ( ) ( ) ( ), s s F ( ) ( ), F F F wth ( ) ( ) ( ) 3

33 Poof of the Poposto-, h hcos ( h hcos ). (6) h hcos ( h hcos ) osde a egula cyldcal etwo show Fg. wth, we obta a esstace foula (6) by the RT-V ethod. Howeve Ref.[3] gave aothe esstace foula () by eas of the Laplaca at ethod. opag foula (4) wth Eq.(), we edately obta detty (6). I patcula, whe settg specal ube values of y,, ad,, we have the followg dettes. Deducto. Whe y, fo (6), we have whee ( cos ) h ( cos ) ( cos ) F [cos( ( ) ( ) ( ) ) cos( ) ] j j,,, ( ) j j. (6) I patcula, whe, we have cos, the Eq.(6) educes to F F, () () (), s s cos( () () () ) j cos( ) j,,, () j j, (63) h ( cos ) 4F F ( ) ( ), wth () h h( h), h h( h). (64) Deducto. Whe, Eq. (6) educes to cos [( ) j]( cos y) j ( cos ) h ( cos j) ( ) ( ) y F F cos y y F cos. (65) Deducto 3. Whe y, we have cos, j j ( ), fo (6), we have ( ), (66) j h ( cos ) () () (), j, j,,, () j F whee F () ( ) ( ), ad ae defed Eq.(64). Please ote that Eq.(66) s dffeet fo Eq.(58) because the ae dffeet fo each othe. Deducto 4. Whe,, ( ), y, by (65), we have cos( y ) y( y). (67) cos 33

34 The above equatos ae gve fo the fst te.. Poposto-3. A geeal tgooetc detty-3 ( ) ( ) ( ) () Defg F F, F ( ) ( ),,, ad, s s ad defg ( ),, ad h h cos ( h h cos ), (68), (69) h h cos ( h h cos ) Assue atual ubes satsfy,, whe atual ube satsfes, ad the abtay eal ube h, we have Poof of the Poposto-3 [cos( ) cos( ) ] coth( l ) ( cos ) h h h. (7) F ( ) ( ) ( ),,, ( cos ) osde a oal cyldcal etwo, whee the au coodate value s (, -). Whe d(,) ad d(,) ae o the sae logtude, by Eq.(3), we have R({,},{,}) ( ) ( ) ( ),,, ( cos ) F. (7) Howeve, Ref.[48] gave aothe esstace foula (the paaetes Ref.[48] have bee coveted to be eactly the sae as those Fg.) cos( ) cos( ) R ({,},{,}) coth( l ), (7) ( h h cos ) Thus foula (7) s equal to Eq.(7) sce they have the sae paaetes the sae etwo. By Eq.(7) equals (7) to yeld Eq.(7). That eas, poposto-3 holds. Deducto. Whe,, ( ),, ad s gve by (69), Eq.(7) educes to s ( )cos coth( l ) h F ( h h cos ), (73) 4 ( cos ) F whee F ( ) ( ), ad s gve by Eq.(68). Deducto. Whe, ( ), s gve by (69), Eq.(7) educes to 34

35 cos( () () () ) cos( ) h,,, () ( h h cos ) F coth(l ) 8. (74) whee () F F, F ( ) ( ), ad s gve by Eq.(64). () () (), s s Deducto 3. Sce coth(l ) ( ) ( ), substtutg to (74) togethe wth (69), whch yelds h h h F () () () h cos,,, cos( ) cos( ) h () ( cos ) 8. (75) whee ( ), F F, ad s gve by Eq.(64). () () (), s s Deducto 4. Whe, ( ), Eq.(7) educes to whee the followg detty s used [cos( ) cos( ) ], (76) cos ( cos ) h h h ( cos ) coth(l ) 35. (77) We fd that detty (7) s teestg because the left-had sde of the detty s the su ove but the ght-had sde s the su ove, whch povdes a ew atheatcal detty fo the applcato of atheatcs. Ⅶ. ocluso ad oet Ths pape set up a uvesal RT-V theoy (Recuso-Tasfo theoy wth potetal paaetes) ad eveals the basc pcple of electcal chaactestcs of cople essto etwos fo the fst te, such as two theoes of theoe- ad theoe- ae poposed, ad the eplct electcal chaactestcs (potetal ad esstace) foulae of the cople etwos ae gve, whch cotas the esults of fte ad fte etwos. It ust be ephaszed that pevous theoes (Maly efes Gee s fucto techque ad Laplaca at ethod) caot solve essto etwos wth cople boudaes, because the Gee s fucto techque s usually used to solve fte etwo pobles, ad the Laplaca at ethod depeds o the soluto of two egevalues whch eles o two atces alog two othogoal dectos. Usg the RT-V ethod to study essto etwos just eles o oe at alog oe vetcal dectos, whch avods the cofuso of aothe at wth abtay eleets that caot be solved eplctly, ad also gves cocse esults a sgle suato, such as the all equatos gve by ths pape.

36 As applcatos of two theoes, the aalytcal solutos of the electcal chaactestcs (cludg potetal fucto ad equvalet esstace) the cople essto etwos wth abtay boudaes ae gve, ad ay teestg esults of the vaous types of essto etwos ae poduced. Please ote that a o-egula ectagula etwo (see Fg.) cotas abtay fa (see Fg.4(a)) ad haoc (see Fg.4(b)) etwos; a o-egula cyldcal etwo (see Fg.) cotas abtay cobweb (see Fg.7(a)) ad globe (see Fg.7(b)) etwos. Thus the aalytcal solutos of the electcal chaactestcs gve by ths pape have geeal sgfcace, whch apples to a wde vaety of lattce stuctues, ad eas the Laplace equato wth cople bouday codtos s esolved by odellg essto etwos. The tgooetc dettes gve show that ou eseach wo establshed ew eseach deas ad appoaches fo the study of atheatcal dettes. I addto, esstace foulae (65), (66), () ad () et al. ca be eteded to pedace etwos sce the gd eleets ca be ethe esstos o pedaces Fg. ad Fg., Fo eaple, assue Z R jl, Z j /, (78) L that we ca theefoe study the abtay RL etwo f we do a plual aalyss [3, 4] to the esstace esults obtaed ths pape. Ou esstace foulae ca also apply to factoal ccuts, fo eaple, the fequecy doa pedace of factoal capactace ad ductace ae espectvely [5] ZL Lcos( ) j Ls( ), (79) Z Whe we eplace ad wth cos( ) j s( ). (8) Z L ad Z, ad use the plual aalyss [3, 4], the equvalet pedaces of the factoal RL etwo wth abtay boudaes ca be obtaed. REFERENES [] J. Scasco, oducto asotoyc dsodeed systes: Effectve-edu theoy, Phys. Rev. B 9, (974). [] S. Kpatc, Pecolato ad oducto, Rev. Mod. Phys. 45, (973). [3] N. I. Lebova, Y. Yu. Taasevch, N. V. Vygots, A. V. Eseepov, ad R. K. Ahuzhaov, Asotopy electcal coductvty of fls of alged tesectg coductg ods, Phys. Rev. E 98, 4 (8). [4] L. Q. Eglsh, F. Paleo, J. F. Stoes, J. uevas, R. aeteo-gozález, ad P. G. Keveds, Nolea localzed odes two-desoal electcal lattces, Phys. Rev. E 88, 9 (3). [5] E. N. Bulgaov, D. N. Masov, ad A. F. Sadeev, Electc ccut etwos equvalet to chaotc quatu bllads, Phys. Rev. E 7, 465 (5). [6] A. R. McGu, Photoc cystal ccuts: A theoy fo two- ad thee-desoal etwos, Phys. Rev. B 6, 335(). 36

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