Auhos name Giuliano Beini* Albeo Bicci** Tile Equivalen waveguide epesenaion fo Diac plane waves Absac Ideas abou he elecon as a so of a bound elecomagneic wave and/o he elecon as elecomagneic field apped in (equivalen waveguide can be found moe o less eplicil in man papes, fo eample b Zhi-Yong Wang, Roald Ekhold, David Hesenes, V.A.Induchoodan Menon, J. G. Williamson, M. B. van de Mak. Wha we wan o show hee is ha he Diac equaion fo elecon and posion plane waves admis an equivalen elecical cicui, consising of an equivalen ansmission line. The same ansmission line is epesenaive of a mode in waveguide, so ou can also sa ha he Diac equaion fo plane waves includes an implici epesenaion in ems of an equivalen waveguide. All he calculaion will be done in elemena fom, wih he usual noaions of cicui heo and elecomagneism, wihou he need o eso o Cliffod algeba as in pevious papes. * Reied. Ealie: Selenia SpA, Rome and IDS SpA, Pisa. Also Adjunc Pofesso a he Univesi of Pisa, Adjunc Pofesso a Naval Academ, Leghon (Ialian Nav. E-mail: maiaeesacapa@iscali.i **IDS SpA, Pisa E-mail: a.bicci@ids-spa.i
Equivalen waveguide epesenaion fo Diac plane waves Inoducion and summa In his pape an equivalence beween ansmission line and Diac plane waves is inoduced. The same ansmission line is epesenaive of TE, TM modes in waveguide, so ou can also sa ha he Diac equaion fo plane waves includes an implici analog wih an equivalen waveguide. PART ONE: Saing fom Mawell equaions, equaions in a waveguide fo he ansvese componens ae deived. PART TWO: in hese equaions we decouple he dependence on, inoducing an analogue volage and cuen V and I equivalen o a waveguide mode (a TE mode. This pemis o define an equivalen ansmission line fo he mode. PART THREE: hee is a degee of feedom in he definiion of a scale faco fo V and I. Wih a pope choice of he scale faco fo V, I (and he impedance Z he equaions fo V, I ae educed o he fom of he Diac equaions fo plane waves. Thus he plane wave Diac equaions admis he pope equivalen cicui in ems of equivalen ansmission line and/o equivalen waveguide. Fo simplici he calculaion will be done in eended fom onl fo a TE mode, and shol fo TM. All he calculaion will be done in he classical fomalism, wih he usual noaions of cicui heo and elecomagneism, wihou he need o eso o Cliffod algeba as in [].
PART ONE: Mawell's equaions in a waveguide fo he ansvese componens In his secion we deive he equaions saisfied b he ansvese componen of he E, H guided fields. In paicula we conside a clindical waveguide (of whaeve coss-secion wih he ais paallel o he ais. The non-evanescen E, H fields ae heefoe assumed o have a dependence on ime and coodinaes descibed b i ik e. Fo ansvese componen of E, H we mean he ( E and componen, ansvese o he -ais. We sa fom Mawell's equaions in naual unis (c: H E oe, oh, dive, divh τ τ and in paicula fom hese wo equaions: whee H oe, dive τ E E iˆ E ˆ j Ekˆ H H iˆ H ˆj H kˆ which, in ems of he individual componens, ae: ( E E H ( E E H (3 E E H (4 E E E Foming (i( i.e. summing i imes he equaion ( o minus equaion ( we ge: ( E ( i E i Similal, foming (4 i(3, we ge: ( i ( E E i H We can epea he pocedue fo he ohe wo Mawell equaions, i.e. divh. The esuling 4 equaions ae: oh E τ and ( E ( i E i ( i ( E E i H 3
( i H i ( E ( i H i E Now we specialise o he waveguide case and we eamine fis he TE mode ( E. The above equaions become: (5 ( E i (6 ( i ( E i H (7 ( i H i ( E (8 ( i H Suppose now a popagaion wih an eponenial Replace evewhee : i e i ik (IEEE convenion. (5 ( E (6 ( i ( E H (7 ( i H ( E (8 ( i H We wan equaions epessed in ems of he ansvese componens ( E and H onl. Take equaion (7 and use equaion (6 o eliminae he componen ( H as follows. Fom (6 we ge: ( i H ( i ( i ( E and hen, as i is well know fom he heo of waveguides, being: ( we aive a: ( i ( i ( E ( ( E k ( E i H kc ( E which can be subsiued in (7, obaining: c o: kc ( E ( E 4
( kc ( E Bu (in naual unis c : k c k so ha: k ( ( c Fom (5 and (7 we hen have: (9 E ( ( ( ( E To esablish a moe diec coespondence wih he ansmission line equaions: dv d ( iµ I di ( i ε V d we ewie he equaions (9 and (9 as: (3 i E i ( (4 i ( i( E Noe: his means ha equaions simila o hose of he ansmission involve he quaniies i ( E and and no ( E and, ie hee is an imagina i beween. 5
PART TWO: decouple he dependence on, In his secion we esablish a clea coespondence of equaions (3 and (4 wih he ansmission line equaions. The meaning of he imagina uni i which muliplies he second membe of he equaions (3 and (4 is well epessed b equaion (5: (5 E i ( which shows ha ( E and ae each ohe a 9 in he, plane; i s i and H which ae "paallel. Thei quoien, as well as V / I in a ( E ie ( ansmission line, i is puel ohmic (o bee puel eal such ha V ZI wih Z eal, jus as in a lossless ansmission line- Moe pecisel, as he, dependence is given b he eponenial: e i ik making he wo deivaives and : ik i we ge fom (5: ik ( E This shows again and eplicil ha i ( E and ae "paallel", and hei quoien is eal: i( E k o: i ( E k Wie now he ansvese fields ie ( E and ih ( H in he fom: (5 E e(, V ( H h(, I( 6
Noe ha if hee ae no phsical condiions which deemine V and I, he ampliudes o be assigned individuall o e (,, V ( as well as o h (,, I( ae abia, povided hei poduc emains consan and equal o he ampliude of E and especivel H. We can ewie he pevious equaion: (6 i E as: ( k (7 ie H k o even: (8 ìe(, V ( h(, I( k The (8 shows wha we need igh now, a paallelism beween e(, Epess he paallelism in he fom: (9 ìe(, Ah(, This allows o eliminae he dependence on,, as shown below. Thanks o he definiion (5, equaions (3 (4 become: o ie H ih i( ie iev ihi hi i( iev bu being: ìe Ah ì and h (,. we obain: I V i A I i( AV 7
if hee ae no phsical condiions ha uniquel deemine V and I (as i happens fo eample fo TE and TM in waveguide ou migh make fo A he choice which is mos convenien, e.g. A. Wih his choice he above equaions ae wien in he final fom: ( V ii ( I i( V Compae i wih he usual equaions of he ansmission lines. Since V and I depend onl on we can wie he equaions ( and ( in he usual fom of ansmission line equivalen o a TE mode in MKSA unis (see fo eample Ramo Whinne []: ( dv d di d iµ I i ε V The equaions of a ansmission line ae: (3 dv d di d ZI YV whee Z and Y depend on he ansmission line and Z is he chaaceisic Y impedance of he line. Equaions ( hen implicil assume as he chaaceisic line impedance: (4 Z iµ Y Noe also ha iε µ Z (5 ε Y is equal o he mode impedance ("Schelkunoff choice": 8
(6 Z Z Z Y TE The equivalen line has inducance and capaciance as in he following figue: L µ L ε C ε The ansmission line is dispesive because he chaaceisic impedance (9 is fequenc dependen and i esonaes when: LC (7 In naual unis (see (, ( we have insead: L L C Remains, fo he waveguide, all he emaining abiainess in he definiion of impedance and hus V, I discussed [], which we summaie hee in he following. 9
PART THREE: educion o he fom of he Diac equaion As we have seen in he pevious secion, in he heo of waveguides, we can inoduce an equivalen volage and cuen, V and I. Fo all modes bu he TEM one he definiion of V and I leaves he feedom in he choice of a scale faco, as shown below. We emembe he definiion of ansvese fields in ems of V and I: (8 E H (,, V ( e(, (,, I( h(, wih he condiion: * (9 P Re E H nds ˆ Re( VI S The phsical meaning of (8 is ha V and I delibeael ignoe he deailed configuaion of E and H on he ansvese plane. Accoding o (9 V and I coecl epoduce he value of he oal eneg ha popagaes. The impedance Z is given of couse b: V (3 Z I Equaion (8 leaves a degee of feedom in he definiion of V and I: we can ale V and I and simulaneousl e h, as follows: (3 V ' αv, e' e α I ' I, h' αh α which leaves condiion (9 invaian: * (3 P Re( VI Re( V ' I'* Accodingl he value of he impedance Z becomes: V ' I ' V I (33 Z' α α Z
This feedom does no change he value of quaniies elaed o eneg soage and popagaion, such as: *,, VI ZI Z V We can now deive an eplici fom of he equivalen ansmission line which is implied b he Diac equaion fo plane wave. Selec he scale faco in (3 as: (34 α Subsiuing in (, ( we ge: ( ' ' I i i d dv (35 ( ' ' V i i d di and he new Z is: (36 TE Z I V Z ' ' ' I is immediae now o see ha he equaions (35 fo volage and cuen ae acuall he Diac equaion fo 3 and. To see his we efe o he Diac equaion wien in eended fom, as can be found fo eample in Schiff [3]: 3 4 τ im i 4 3 τ im i 3 τ im i
i τ im 4 Hee,,, 4 ae comple funcions such as V, I in he usual cicui heo. B i seing 4 and assuming he e dependence on, as i is fo he Diac s plane wave soluion we have fo he wo componens diffeen fom eo: (37 3 ( i i ( i i 3 which coincide wih (35. Thus he Diac equaions (37 ae pefecl analogous o he waveguide-ansmission line equaions (35, once we selec he choice (34 fo he scale faco α. In paicula he chaaceisic impedance fo he Diac equaion is no he "Schelkunoff choice (6, bu i s ha deemined b (35, i.e.: (38 Z Y
The equivalen ansmission line has inducances and capaciance as shown in he following figue: L ( C ( The naual unis emploed o wie (35 and (37 ae convenien bu he ma mask he ue meaning of he elecical paamees of inducance, capaciance and impedance. Rewie he equaions (35 in MKSA unis: (39 dv d di d ' I ( i i µ I' iµ ( ' ' V ( i i εv ' iε ( ' Compaing (39 wih (3 we deduce he paamees of he ansmission line. The equivalen ansmission line has seial Z and paallel Y like his: L µ ( C ε ( The "Diac choice" fo he chaaceisic impedance is hen: (4 Z Y µ Z ε Fo his impedance ends o he impedance Z of emp space. The same holds fo, which holds fo a TEM-like popagaion (which means also no waveguide and neuino equaions. 3
The equivalen cicui shows ha fo hee is volage V while he cuen I is eo, as can be seen also b he impedance (4. This seems phsicall easonable b he fac ha he whole calculaion fom he beginning has been developed fo a TE. Fo eponeniall damped evanescen waves popagae. Unil now he calculaion was done in eended fom fo a TE, fom he Mawell equaions (5 and (7. This will be now biefl epeaed fo a TM wih he pai of equaions (6, (8. Wih simila pocedue, fo TM mode we ge he following equivalen cicui C µ L µ C ε and equaions simila o (: (4 dv d di d i µ iεv I whee we use he MKSA unis. 4
Taking advanage of he abiainess inheen in volage and cuen and poceeding as fo (34 (35, bu now wih: (34bis we aive a: α (4 dv ' I d ( i i µ ' di ' V d ( i i ε ' You can now see ha he equaions (4 fo volage and cuen ae now coesponding o he Diac equaion fo 4 and. These ae, in a fom simila o (37: (43 ( i i 4 ( i i 4 Idenif wih (4 ecep fo a comple conjugae opeaion, which is inepeed as i ik i ik wave popagaion e insead of e. The equivalen cicui, deducible fom (4, is he following L µ ( C ε ( The " Diac choice" fo he chaaceisic impedance is hen: (44 Z Y µ Z ε 5
The chaaceisic impedance assumes a highl smmeic fom beween he TE and TM cases, see (4 and (44. Also he equivalen cicui is ve smmeical: i is alwas he same apa fo a change of sign in. Fo he impedance ends o he impedance Z of emp space. The same holds fo, which sill means a TEM-like popagaion (which means also no waveguide and neuino equaions, bu now wih opposie polaiaion. The equivalen cicui shows now ha fo hee is cuen I, while volage V is eo, as can be seen also b he impedance (4. This seems phsicall easonable b he fac ha now he calculaion has been developed fo a TM. 6
CONCLUSIONS We have hus shown ha he Diac equaion fo plane waves can be pu in coespondence wih an elecical cicui, equivalen o a ansmission line. The same ansmission line is epesenaive of a mode in waveguide, so ou can also sa ha he Diac equaion fo plane waves includes an implici epesenaion of an equivalen waveguide. The equivalence is embedded in he usual V and I descipion. To quoe Hesenes we wan o emphasie ha his inepeaion is b no means a adical speculaion; i is a fac! The inepeaion has been implici in he Diac heo all he ime. All we have done is make i eplici. (Hesenes hee efes o he inepeaion of he imagina i. The calculaion was done in eended fom fo a TE, fom he Mawell equaions (5 and (7. This was biefl epeaed fo a TM wih he pai of equaions (6, (8. Doing so, he full se of plane wave Diac equaions can be inepeed in ems of appopiae equivalen ansmission line cicuis and/o equivalen waveguide. Obviousl soluions wih opposie spin ae epesened b opposie polaiaion in he waveguide. The equivalen ansmission line shaes all he usual popeies of he ansmission lines, including he dispesive chaace, and evanescen waves. The evanescen waves ma be he coesponden of elecons popagaing hough a poenial baie. 7
REFERENCES [] G. Beini, Cliffod Algeba and Diac equaion fo TE, TM in waveguide, hp://via.og/abs/9.59 [] S. Ramo, J. R. Whinne, T. van Due, Fields and Waves in Communicaion Eleconics, John Wile (994 [3] L. I. Schiff, Quanum Mechanics, McGaw-Hill (968 8