Electromagnetic-Power-based Modal Classification, Modal Expansion, and Modal Decomposition for Perfect Electric Conductors

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1 LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 1 Eletromgneti-ower-bsed Modl Clssifition Modl Expnsion nd Modl Deomposition for erfet Eletri Condutors Renzun Lin Abstrt Trditionlly ll working modes of perfet eletri ondutor re lssified into onnt modes utive modes nd pitive modes nd the onnt modes re further lssified into internl onnt modes nd externl onnt modes. In this pper the onnt modes re lterntively lssified into intrinsilly onnt modes nd non-intrinsilly onnt modes nd the intrinsilly onnt modes re further lssified into non-rditive intrinsilly onnt modes nd rditive intrinsilly onnt modes. Bsed on the modl expnsion orponding to this new modl lssifition n lterntive modl deomposition method is proposed. In ddition it is lso proved tht: ll intrinsilly onnt modes nd ll non-rditive intrinsilly onnt modes onstitute liner spes petively but ll onnt modes nd ll rditive intrinsilly onnt modes nnot onstitute spes petively. Index Terms Chrteristi mode (CM) modl lssifition modl deomposition modl expnsion rdition onne. R I. INTRODUCTION ESONANCE is n importnt onept in eletromgnetis. Bsed on whether the onnt modes rdite they re lssified into internl onnt modes nd externl onnt modes [1] nd these two ks of onnt modes re widely pplied in eletromgneti (EM) vities [2] nd EM ntenns [3] petively. The most ommonly used mthemtil method for erhing internl onnt modes is norml eigen-mode theory (EMT) [2] [4] nd the norml EMT n onstrut bsis of internl onne spe (whih is onstituted by ll internl onnt modes [5]) nd the bsis re lled s norml eigen-modes. The most ommonly used mthemtil methods for erhing externl onnt modes re singulr EMT [6] nd hrteristi mode theory (CMT) [7] nd the bsis onstruted by singulr EMT nd CMT re petively lled s singulr eigen-modes nd rditive hrteristi modes (CMs) where the quottion mrk on bsis will be explined in Se. IV. Reently pper [8] generlized the trditionl CMT to internl onne problem nd proved tht: ll non-rditive per submitted Februry R. Z. Lin is with the Shool of Eletroni Siene nd Engineering University of Eletroni Siene nd Tehnology of Chin Chengdu Chin. (e-mil: rzlin@vip.163.om). modes onstitute liner spe lled s non-rdition spe nd this spe is the sme s internl onne spe; ll non-rditive CMs onstitute bsis of non-rdition spe nd internl onne spe nd then they re equivlent to the norml eigen-modes from the spet of modl expnsion. Bsed on bove observtions the bsis used to expnd onnt modes n be lssified into four tegories internl onnt norml eigen-modes externl onnt singulr eigen-modes rditive CMs nd non-rditive CMs nd the reltionships nd differenes mong the first three of these bsis re nlyzed in pper [1]. This pper lterntively lssifies ll onnt modes into three tegories non-rditive intrinsilly onnt modes rditive intrinsilly onnt modes nd non-intrinsilly onnt modes nd disusses the reltionships nd differenes mong them. By employing the modl expnsion orponding to this new modl lssifition n lterntive modl deomposition method is proposed in this pper nd t the sme time some further onlusions re obtined. II. MODAL CLASSIFICATION When field F inidents on perfet eletri ondutor (EC) urrent will be ued on the EC. All possible working modes onstitute liner spe lled s modl spe [4]-[8]. If the is expnded in terms of ependent nd omplete bsis funtions there exists one-to-one orpondene between the nd its expnsion vetor [5] [7] [8] nd the liner spe onstituted by ll possible is lled s expnsion vetor spe (where the is the vetor onstituted by ll expnsion oeffiients). The following prts of this pper re disussed in expnsion vetor spe nd frequeny domin. In expnsion vetor spe the omplex power done by E on hs the mtrix form nd then the rdited rd power Re nd the retively red power Im n be orpondingly expsed s the mtrix rd rd forms nd [8]. ere the supersript repents the trnspose onjugte of mtrix or vetor nd the method to obtin the mtrix n be found in ppers [7] nd [8] nd rd ( ) 2 nd ( ) 2 j [8].

2 LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 2 A. Trditionl modl lssifition Mtrix is positive semi-definite [8] so rd 0 for ny nd the modes orponding to rd 0 nd rd 0 re lled s rditive modes nd non-rditive modes petively. In ddition the semi-definiteness of mtrix implies tht: rd 0 if nd only if [9] i.e. rd rd rd 0 rd rd 0 0 Mode is non- rditive (1) Thus ll non-rditive modes onstitute liner spe (i.e. the null spe of [9]) lled s non-rdition spe (whih is identil to the internl onne spe [8]) nd ny stisfies the following orthogonlity: non rd rd non rd rd rd 0 (2) non rd non rd for ny working mode. The mtrix is efinite [7] [8] so n be zero or positive or negtive nd the modes orponding to 0 0 nd 0 re lled s onnt modes utive modes nd pitive modes petively [3] [4] [7] [8]. Aording to whether the onnt modes rdite the onnt modes re further lssified into internl onnt modes (whih don t rdite so this pper lls them s non-rditive onnt modes) nd externl onnt modes (whih rdite so this pper lls them s rditive onnt modes) [1] [5] [8]. As demonstrted in [8] the non-rditive modes must be onnt so ll utive nd pitive modes must be rditive nd then this pper lls them s rditive utive modes nd rditive pitive modes petively. B. New modl lssifition (An lterntive lssifition for onnt modes) Besides lssifying ll modes into onnt modes (inluding non-rditive onnt modes nd rditive onnt modes) rditive utive modes nd rditive pitive modes trditionlly n lterntive lssifition for the onnt modes is proposed in this sub-setion. Mtrix is efinite so 0 doesn t imply tht 0 [9] though 0 lwys implies tht 0. This is equivlent to sying tht 0 0 Mode is onnt (3) i.e. the ondition 0 is stronger ondition thn the ondition 0 to gurntee onne. Bsed on this the 0 n be prtiulrly lled s intrinsi onne ondition if the 0 is viewed s onne ondition. Corpondingly the modes stisfying 0 re lled s intrinsilly onnt modes nd the onnt modes not stisfying 0 re lled s non-intrinsilly onnt modes. Obviously ll intrinsilly onnt modes onstitute liner spe i.e. the null spe of nd this spe is lled s intrinsi onne spe. Similrly to (2) ny intrinsilly onnt mode stisfies the following (4) for ny : 0 (4) When intrinsilly onnt mode stisfies ondition rd ( ) 0 it is lled s non-rditive intrinsilly onnt mode nd orpondingly denoted s. When intrinsilly onnt mode stisfies ondition rd ( ) 0 it is lled s rditive intrinsilly onnt mode nd orpondingly denoted s. As demonstrted in pper [8] if. This implies tht the intrinsi onne spe ontins the whole non-rdition spe. Then the set onstituted by ll must be liner spe nd this spe is just the non-rdition spe; ll non-intrinsilly onnt modes re rditive nd they re prtiulrly denoted s ; for ny mode the stisfies the following orthogonlity: non rd 0 rd 0 non non rd rd non rd non rd rd rd 0 (5.1) non rd non rd 0 (5.2) non rd non rd In summry by introduing the onepts of intrinsi onne nd non-intrinsi onne this sub-setion lterntively lssifies ll onnt modes into non-rditive intrinsilly onnt modes non rd rditive intrinsilly onnt modes rd nd rditive non-intrinsilly onnt modes non rd. Beuse the non-rditive intrinsilly onnt modes non rd re just the trditionl internl onnt modes the introdution of rditive intrinsilly onnt modes rd non nd rditive non-intrinsilly onnt modes rd is essentilly subdivision for the trditionl externl onnt modes. C. Clssifition for hrteristi modes Beuse both bove trditionl nd new modl lssifitions re suitble for whole modl spe they re lso vlid for CM set. ere the symbol is used to repent the expnsion vetor of CM in order to be distinguished from the expnsion vetor of generl mode. Trditionl lssifition for CMs Trditionlly CM set re divided into four sub-sets [1] [7] [8]: non-rditive onnt CM set non rd ; rditive onnt CM set rd ; rditive utive CM set rd ; p nd rditive pitive CM set rd ;. For the onveniene of the following prts of this sub-setion the non-rditive nd rditive onnt CMs re olletively referred to s onnt CMs nd the union of sets non rd ; nd rd ; is orpondingly denoted s i.e. ; nonrd rd ;. An lterntive lssifition for onnt CMs As illustrted in ppers [1] [7] nd [8] ll stisfy the hrteristi eqution 0. In ft this eqution is just the intrinsi onne ondition introdued in Se. II-B so ll the re intrinsilly onnt nd then they re prtiulrly denoted s. Corpondingly the non rd ; nd rd ; re prtiulrly denoted s non rd ; nd rd ;

3 LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 3 petively. All re ependent of eh other [7] nd the rnk of set equls to the rnk of the null spe of so they onstitute bsis of intrinsi onne spe [9] i.e. ny intrinsilly onnt mode n be uniquely expnded in terms of. In ddition the onstitute bsis of non-rdition spe [8] i.e. ny non-rditive mode n be uniquely expnded in terms of. non rd ; non rd ; III. MODAL EXANSION non rd In this setion further disussion on the CM-bsed modl expnsions for vrious modes is provided bsed on the new modl lssifition proposed in bove Se. II. A. Modl expnsion for generl modes Bsed on the ependene property nd ompleteness of CM set [7] [8] ny mode n be uniquely expnded in terms of some non-rditive onnt CMs some rditive onnt CMs some rditive utive CMs nd some rditive pitive CMs s follows: rd ; rd ; p rd ; non rd ; p p non rd ; nonrd ; rd ; rd ; rd ; rd ; rd ; rd ; (6) where the on to use insted of will be explined in Se. IV. Bsed on expnsion (6) some vluble onlusions shown in Fig. 1 n be derived nd they re proved s below. The proof for 1 is obvious. The proof for 2 : It is obvious tht 0 0 so mode p p is intrinsilly onnt. Thus if rd ; rd ; rd ; rd ; 0 p p then rd ; rd ; rd ; rd ; is intrinsilly onnt. The proofs for 3 nd non rd ; nonrd ; rd ; rd ; is intrinsilly onnt. Thus the is intrinsilly onnt if nd only if the term p p rd ; rd ; rd ; rd ; is intrinsilly onnt bsed on the intrinsi onne ondition introdued in Se. II-B. The proof for 4 is obvious beuse of (3). The proofs for 5 nd 6 : Beuse the term non rd ; nonrd ; rd ; rd ; is intrinsilly onnt the retively red power of equls to the retively red p p power of term rd ; rd ; rd ; rd ; due to the orthogonlity (4). Thus both the 5 nd 6 hold. p p The proof for 8 : If rd ; rd ; rd ; rd ; is intrinsilly onnt then the is intrinsilly onnt due to 3. This implies tht the n be expnded in terms of rd ; non rd ; s onluded in Se. II-C. Beuse of the uniqueness of the CM-bsed modl expnsion for ertin the oeffiients rd ; nd p rd ; in (6) must be zeros nd then both the terms p p rd ; rd ; nd rd ; rd ; must be zeros. The proof for 9 is obvious beuse of 8 nd : It is obvious tht the term B. Modl expnsion for generl onnt modes n be expnded s fol- Obviously ny onnt mode lows: p p non rd ; nonrd ; rd ; rd ; rd ; rd ; rd ; rd ; (7) where the on to use insted of will be explined in Se. IV. C. Modl expnsion for intrinsilly onnt modes The onlusions given in Se. II-C nd Fig. 1 imply tht ny intrinsilly onnt mode n be expnded s follows: (8) non rd ; non rd ; rd ; rd ; i.e. there doesn t exist ny utive CMs nd pitive CMs in the CM-bsed modl expnsion formultion for n intrinsilly onnt mode. As pointed out in Se. II-C nd pper [8] ny non-rditive intrinsilly onnt mode n be expnded s follows: p rd ; p p rd ; rd ; rd ; rd ; p p rd ; rd ; rd ; rd ; p p rd ; rd ; rd ; rd ; non rd rd ; (9) nonrd non rd ; nonrd ; owever it nnot be gurnteed tht the non-rditive term nonrd ; nonrd ; in the modl expnsion of rditive intrinsilly onnt mode is zero beuse of the (5) i.e. rd rd non rd ; non rd ; rd ; rd ; (10) where the on to use insted of will be explined in Se. IV. D. Modl expnsion for non-intrinsilly onnt modes If non-intrinsilly onnt mode follows: non rd is expnded s non p p rd non rd ; nonrd ; rd ; rd ; rd ; rd ; rd ; rd ; (11) it n be onluded tht p p rd ; rd ; rd ; rd ; 0 (12) bsed on Fig. 1. In ft it n be further onluded tht 0 8 is intrinsilly onnt 37 Mode is intrinsilly onnt 4 Mode is onnt 5 6 p p rd ; rd ; rd ; rd ; is onnt Fig. 1. Some equivlene reltionships relted to onne.

4 LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 4 p p rd ; rd ; rd ; rd ; 0 (13) beuse: if the utive/pitive term is zero nd the pitive/utive term is non-zero then the retively red power of is less/lrger thn zero due to the orthogonlity (4) nd this leds to ontrdition; if both the utive term nd pitive term re zeros then the term p p rd ; rd ; rd ; rd ; must be zero nd this leds to ontrdition with (12). non rd IV. MODAL DECOMOSITION If the terms nonrd ; nonrd ; nd in bove-mentioned modl expnsion formultions re denoted s nd petively then the CM-bsed modl expnsions (6)-(11) n be lterntively written s follows: nd p p rd ; rd ; non rd rd rd ; rd ; rd p non rd rd rd rd p rd rd ; rd ; (6') (7') p non rd rd rd rd (8') non rd rd (9') non rd non rd (10') rd nonrd rd (11') non p rd non rd rd rd rd where to utilize symbol is to emphsize tht these terms re the building blok terms in CM-bsed modl expnsions. The (6') nd (7')-(11') re petively lled s the eletromgneti-power-bsed (EM-bsed) modl deompositions for generl modes nd vrious onnt modes. In ft the EM-bsed modl deompositions for ny rditive utive mode nd ny rditive pitive mode n be similrly expsed s follows: rd p rd nonrd rd rd rd p rd (14) (15) p p rd nonrd rd rd rd As the ontinution of the onlusions given in Ses. II nd III the following further onlusions n be derived bsed on bove EM-bsed modl deompositions. In (6') (8') nd (9') ll the terms in the right-hnd sides of these expnsions n be zero or non-zero. In (7') the nd n be zero or non-zero nd the nd mrked by single underlines n be simultneously zero or simultneously non-zero. In (11') the nd n be zero or non-zero nd the nd mrked by double underlines must be simultneously non-zero. In (10') (14) nd (15) the terms mrked by double underlines must be non-zero. These bove re just the ons to use nd in (7)-(11) (7')-(11') (14) nd (15). Beuse the term in (10') n be non-zero then the set onstituted by ll rditive intrinsilly onnt modes is not losed for ddition so ll rditive intrinsilly onnt modes nnot onstitute liner spe [9]. Obviously similr onlusions hold for the sets onstituted by ll non-intrinsilly onnt modes rditive utive modes nd rditive pitive modes beuse of (11') (14) nd (15). In ddition ll onnt modes lso nnot onstitute liner spe. For exmple: If the retively red powers of CMs nd p re normlized to 1 nd then the modes Ard Ard j p nd Ard Ae rd must be onnt for ny due to the orthogonlity of CMs [7]. owever the mode might be non-onnt beuse of the rbitrriness of. This implies tht the set onstituted by ll onnt modes is not losed for ddition. These re just the ons to use some quottion mrks on the bsis in Se. I. The (7') implies tht the might ontin the non rd rd nd terms; the (8') nd (10') imply tht the nd rd might ontin the non rd term; the (11') implies tht the non p rd must ontin the rd nd rd terms. In ft these re just the ons to ll the 0 s intrinsi onne ondition nd to ll the modes stisfying 0 s intrinsilly onnt modes. rd p rd non rd 1 rd non rd p rd rd rd rd A non rd p rd p rd The reltionships of vrious modes re illustrted in Fig. 2 where the modl lsses in boxes re liner spes. V. CONCLUSIONS This pper lterntively proposes n EM-bsed modl lssifition. Bsed on the new modl lssifition nd orponding CM-bsed modl expnsion n lterntive modl deomposition method is obtined i.e. ny mode n be expsed s the superposition of non-rditive intrinsilly onnt mode rditive intrinsilly onnt mode rditive utive mode nd rditive pitive mode. In ddition some onlusions re obtined for exmple: ll intrinsilly onnt modes nd ll non-rditive modes onstitute liner spes petively but other ks of onnt modes nnot onstitute liner spes petively. Non- rditive intrinsilly onnt modes Internl onnt modes Non- rditive modes Intrinsilly onnt modes Resonnt modes Rditive intrinsilly onnt modes All modes Rditive onnt modes Externl onnt modes Non- intrinsilly onnt modes whih must be rditive Rditive modes Indutive modes whih must be rditive Rditive non- onnt modes Cpitive modes whih must be rditive Fig. 2. The EM-bsed modl lssifition for ll working modes nd the reltionships mong vrious modl lsses.

5 LIAN: EM-BASED MODAL CLASSIFICATION EXANSION AND DECOMOSITION FOR EC 5 REFERENCES [1] T. Srkr E. Mokole nd M. Slzr-lm An expose on internl onne externl onne nd hrteristi modes IEEE Trns. Antenns ropg. vol. 64 no. 11 pp Nov [2] R. Collin Field Theory of Guided Wves. New York: Wiley [3] C. Blnis Antenn Theory: Anlysis nd Design. NY: Wiley [4]. kson Clssil Eletrodynmis 3rd. New York: Wiley [5] A. eterson S. Ry nd R. Mittr Computtionl Methods for Eletromgnetis. New York: IEEE s [6] C. Bum On the singulrity expnsion method for the solution of eletromgneti intertion problems AFWL Inter. Notes 88 De [7] R. rrington nd. Mutz Theory of hrteristi modes for onduting bodies IEEE Trns. Antenns ropg. vol. A-19 no. 5 pp My [8] R. Lin (Ot. 2016) Eletromgneti-power-bsed hrteristi mode theory for perfet eletri ondutors [Online]. Avilble: [9] R. orn nd C. ohnson Mtrix Anlysis 2nd. New York: Cmbridge University s 2013.