EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N)
Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for wveguide mode. y The wveguide mode is no M mode, u i cn e modeled s wve on rnsmission line. I + - V
Wveguide Trnsmission Line Model (con.) For wveguide mode, volge nd curren re no uniquely defined. y A B Ey ( ) Mode jωµπ π Ey A e j sin c y B jωµπ π j π V ( ) VAB ( ) E dr Ey dy A sin e V sin e A c The volge depends on! j 3
Wveguide Trnsmission Line Model (con.) For wveguide mode, volge nd curren re no uniquely defined. y y Mode H ( ) j π π H A e j sin c urren on op wll: Noe: If we inegre round he enire oundry, we ge ero curren (he op nd oom wlls hve opposie curren, s do he lef nd righ wlls). jπ π s sin c op ( ) ( ) ( ) j I J d H d A e d jπ π π A cos cos e c π I π π e cos cos j j The curren depends on he lengh of he inervl! 4
Wveguide Trnsmission Line Model (con.) Emine he rnsverse (, y) fields: Modl mpliudes ( + j + j ) E( y,, ) e( y, ) A e + A e ( + j + j ) H( y,, ) h( y, ) A e A e Noe: The minus sign ove rises from: ± ± H ( ˆ ± E ) w Wve impednce w or TM ωµ TM ωε c 5
Wveguide Trnsmission Line Model (con.) Inroduce defined volge ino he field equions: V A V A + + We my use whever definiion of volge we wish here. (In oher words, is rirry.) We hen hve: E( y,, ) e( y, ) V e + V e ( + j + j ) H( y,, ) h( y, ) V e V e + j + j ( ) 6
Wveguide Trnsmission Line Model (con.) Inroduce chrcerisic impednce (hving n rirry vlue) ino he equions: H y h y V e V e + j + j (,, ) (, ) H y h V y V e + j j (,, ) (, ) e + where 7
Wveguide Trnsmission Line Model (con.) Summry V( ) E( y,, ) e( y, ) Ve + Ve ( + j + j ) I ( ) + V j V + j (,, ) (, ) H y h y e e The dependence of he rnsverse fields ehves lie volge nd curren on rnsmission line. 8
Wveguide Trnsmission Line Model (con.) The rnsmission-line model is clled he Trnsverse Equivlen Newor (N) model of he wveguide. I( ) N + V ( ) -, E H V I Wveguide 9
Wveguide Trnsmission Line Model (con.) Power flow down he wveguide (comple power): ( ) ( ) WG * P E H ds S ˆ * * V( I ) ( ) * ( e( y, ) h( y, )) ds ˆ S ( ) ( ) * ( (, ) (, )) WG N * P P e y h y ds S ˆ omple power flowing down he N rnsmission line.
Wveguide Trnsmission Line Model (con.) Assume we choose o hve: N ( ) ( ) WG P P Then we hve he following consrin: ( (, ) (, )) e y h y ds * * ˆ S I is no necessry o me his ssumpion of equl powers, u i is useful choice h cn e mde.
Wveguide Trnsmission Line Model (con.) Summry of onsns (ssuming equl powers) ( (, ) (, )) e y h y ds * * ˆ S Once we pic, he consns re deermined. The mos common choice: w
Mode of Recngulr Wveguide We me he following choices: y hoose Assume power equliy ( ) ωµ e y h y ds * * (, ) (, ) ˆ S π ω µε c 3
Mode (con.) y ( (, ) (, )) e y h y ds * * ˆ S so π sin ds * * S π sin * * dyd π e ˆ ysin π h ˆ sin ωµ 4
Mode (con.) * * y Te he conjuge of he second one nd hen muliply he wo equions ogeher. Noe: The soluion is unique o wihin phse erm (we choose he phse o e ero here). Soluion: π e ˆ ysin π h ˆ sin ωµ 5
Mode (con.) π e ˆ ysin π h ˆ sin ωµ y Recll: V( ) E( y,, ) e( y, ) Ve + Ve ( + j + j ) I ( ) + V j V + j (,, ) (, ) H y h y e e 6
ωµ Emple: Mode (con.) π y Hence we hve for our finl modeling equions: V( ) π E(,, ) ˆ y ysin Ve Ve + ( + j + j ) I( ) + V j V + j π H(,, ) ˆ y sin e e 7
Emple: Wveguide Disconinuiy For [V/m] (field he cener of he guide) inciden mode in guide A, find he mode fields in oh guides, nd he refleced nd rnsmied powers. B y.856 cm.6 cm ε r.54 f GH A ε ε r µ µ µ π 58. rd / m [ ] π ε r 34. rd / m [ ] 8
, V + ΓV + N, Emple (con.) TV + π e ˆ ysin π h ˆ sin ωµ onvenion: hoose Assume power equliy j ( A + ) inc E ( y,, ) e( y, ) e ωµ 499.7 Ω [ ] ωµ 59.6 Ω [ ] ( ) V/m A + ( since e y, lredy hs [ ]) V A + + 9
Emple (con.) + j + j ( ) ( +Γ ) V V e e ( ) V V Te + + j + j ( ) ( Γ ) j V I e e VT + j I ( ) e Equivlen reflecion prolem:, V + ΓV + o o Γ.36 + o N, o TV + T +Γ.684 Noe: The ove N enforces he coninuiy of volge nd curren he juncion, nd hence he ngenil elecric nd mgneic fields re coninuous in he WG prolem.
Emple (con.) V ( ) e + (.36) e V ( ) (.684) e j + j ( ) I ( ) e (.36) e j + j ( ) (.684) I ( ) e j j Recll h for he mode:, V( ) ΓV + + j + j E( y,, ) e( y, )( Ve + Ve ) I ( ) + V j V + j H( y,, ) h( y, ) e e [ ] [ ] 58. rd / m 34. rd / m V + N, TV + π e ˆ ysin π h ˆ sin ωµ
Emple (con.) y B ε r Hence, for he wveguide prolem we hve he fields s: E( y,, ) e( y, ) + j + j ( e (.36) e ) A ε H ( y,, ) h( y, ) 6 j + j ( e (.3 ) e ) j E ( y,, ) e( y, ) (. 684) e H( y,, ) h( y, ).684 ( ) e j
Emple (con.) Susiuing in, we hve (guide A): E( y,, ) e( y, ) + j + j ( e (.36) e ) π + (,, ) ˆ E y ysin e. 3 6 j j ( + ( ) e ) π e ˆ ysin π h ˆ sin ωµ 3
Emple (con.) Susiuing in, we hve (guide A): H ( y,, ) h( y, ) 6 j + j ( e (.3 ) e ) (,, ) ˆ H y sin π + j j ( e (.36) e ) 4
Emple (con.) Susiuing in, we hve (guide B): j E ( y,, ) e( y, ) (. 684) e (,, ) ˆ π E y ysin (.684) e j 5
Emple (con.) Susiuing in, we hve (guide B): H( y,, ) h( y, ) (.684) e j (,, ) ˆ π H y sin (.684) e j 6
Emple (con.) Summry of Fields E(,, ) ˆ y ysin e +.36 e π + j j ( ( ) ) H (,, ) ˆ y sin e.36 e π + j j ( ( ) ) π j E(,, ) ˆ y ysin (.684) e [ ] [ ] 499.7 Ω 59.6 Ω [ ] [ ] 58. rd / m 34. rd / m π j H (,, ) ˆ y sin (.684 ) e 7
Emple (con.) Power lculions: Noe: In his prolem, nd Γ re rel. inc + + * + Re Re * P V I V P ref Γ Γ + + * Re V I P rns + + * Re V I Γ Γ ( ) ( ) 8
Emple (con.) For [V/m] inciden mode in guide A (field he cener of he guide), find he mode fields in oh guide, nd he refleced nd rnsmied powers. Finl Resuls: P P P inc refl rns [ ] [ ] [ ].6 mw.6 mw.45 mw.856 cm.6 cm ε r.54 f GH B A ε Noe: 9.% of he inciden power is rnsmied. ε r y 9
Mching Elemens in Wveguide A qurer-wve rnsformer is shown here. d λ gt T /4 T λ gt T π β T ( ) β lossless Top view T π T ε rt ε T rt T d T ωµ T Now % of he inciden power is rnsmied. 3
Mching Elemens in Wveguide (con.) Recngulr Wveguide (end view) Noe: Plnr disconinuiies re modeled s purely shun elemens. Inducive iris pciive iris Resonn iris The equivlen circui gives us he correc reflecion nd rnsmission of he mode. 3
Mching Elemens in Wveguide (con.) Top view Γ Inducive iris in ir-filled wveguide T ωµ η π Noe: The shun inducor models he effecs of he iris nd gives he mpliudes of he mode correcly everywhere, u he N model does no ell us how srong he higher-order modes re. Higher-order mode region N Model Becuse he elemen is shun disconinuiy, we hve T +Γ Ls 3
Mching Elemens in Wveguide (con.) Much more informion cn e found in he following reference: N. Mrcuvi, Wveguide Hndoo, Peer Perigrinus, Ld. (on ehlf of he Insiue of Elecricl Engineers), 986. Equivlen circuis for mny ypes of disconinuiies Accure AD formuls for mny of he disconinuiies Grphicl resuls for mny of he cses Someimes, mesured resuls 33