EE 537-635 Microwve Engineering Adped from noes y Prof. Jeffery T. Willims Fll 8 Prof. Dvid R. Jcson Dep. of EE Noes Wveguiding Srucures Pr 7: Trnsverse Equivlen Newor (N)
Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for wveguide mode. y x The wveguide mode is no M mode, u i cn e modeled s wve on rnsmission line. I + - V Trnsverse Equivlen Newor model of wveguide mode
Wveguide Trnsmission Line Model (con.) For wveguide mode, volge nd curren re no uniquely defined. y V + - A B Ey ( x) Mode x π j jωµπ π H( x, y, ) A cos x e Ey A sin x e c j B jωµπ π j π V ( ) VAB ( ) E dr Ey dy A sin x e V sin x e A c y j x The volge depends on x! jωµπ V A c 3
Wveguide Trnsmission Line Model (con.) For wveguide mode, volge nd curren re no uniquely defined. y y x I x Mode Hx ( x) x j π π Hx A x e j sin c x urren on op wll: Noe: If we inegre round he enire oundry, we ge ero curren (he op nd oom wlls hve opposie curren, nd he lef nd righ wlls hve no curren in he direcion). x x x jπ π s x sin x x x c op ( ) ( ) ( ) j I J x dx H x dx A x e dx jπ π π A cos x cos x e c π π π I cos x cos x e j The curren depends on he lengh of he inervl! j I j π A π c 4
Wveguide Trnsmission Line Model (con.) Exmine he rnsverse (x, y) fields of wveguide mode: Modl mpliudes ( + j + j ) E( xy,, ) e( xy, ) A e + A e ( + j + j ) H( xy,, ) h( xy, ) A e A e h( xy, ) ( ˆ e) Noe: The minus sign ove rises from: ± ± H ( ˆ ± E ) w w Wve impednce w or TM Noe: The shpe funcions hve n rirry mpliude normliion. 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( xy,, ) e( xy, ) V e + V e ( + j + j ) H( xy,, ) h( xy, ) V e V e + j + j ( ) 6
Wveguide Trnsmission Line Model (con.) Nex, inroduce chrcerisic impednce (hving n rirry vlue) ino he equions: H xy h xy V e V e + j + j (,, ) (, ) H xy h V xy V e + j j (,, ) (, ) e + where 7
Wveguide Trnsmission Line Model (con.) Summry of Fields V( ) E( xy,, ) e( xy, ) Ve + Ve ( + j + j ) I ( ) + V j V + j (,, ) (, ) H xy h xy 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 sme Wveguide 9
Wveguide Trnsmission Line Model (con.) Power flow down he wveguide (complex power): ( ) ( ) WG * P E H ds S ˆ * * V( I ) ( ) * ( e( xy, ) h( xy, )) ds ˆ S ( ) ( ) * ( (, ) (, )) WG N * P P e xy h xy ds S ˆ omplex 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 xy h xy ds * * ˆ S I is no necessry o me his ssumpion of equl powers, u i is useful choice h cn e mde (we will dop his choice).
Wveguide Trnsmission Line Model (con.) Summry of onsns (ssuming equl powers) ( (, ) (, )) e xy h xy 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 x ( ) ωµ e xy h xy ds * * (, ) (, ) ˆ S π ω µε c 3
Mode (con.) lcule his erm: y ( (, ) (, )) e xy h xy ds * * ˆ so S π x sin ds * * S π sin * * x dydx π e ˆ ysin x π h ˆ x sin x ωµ Noe: The rnsverse shpe funcion e hs een chosen (rirrily) o hve uni mpliude he cener of he wveguide. x 4
Mode (con.) We hve: y * * Te he conjuge of he second one nd hen muliply he wo equions ogeher. x Noe: The soluion is unique o wihin phse erm (we choose he phse o e ero here). Soluion: π e ˆ ysin x π h ˆ x sin x ωµ 5
Mode (con.) π e ˆ ysin x π h ˆ x sin x ωµ y x Recll: V( ) E( xy,, ) e( xy, ) Ve + Ve ( + j + j ) I ( ) + V j V + j (,, ) (, ) H xy h xy e e 6
Mode (con.) y Hence we hve for our finl modeling equions: ωµ π x V( ) π E(,, ) ˆ xy ysin x Ve Ve + ( + j + j ) I( ) + V j V + j π H(,, ) ˆ xy x sin x e e 7
Exmple: 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 x.856 cm A ε ε r µ µ µ π 58. rd / m [ ].6 cm ε.54 r f GH π ε r 34. rd / m [ ] 8
, V + ΓV + N, onvenion: hoose Assume power equliy Exmple (con.) TV + π e ˆ ysin x π h ˆ x sin x ωµ Noe: is he sme for oh guides, u is differen. ωµ 499.7 Ω [ ] ωµ 59.6 Ω [ ] ( ) V/m A + ( since e xy, lredy hs [ ]) V A + + j ( A + ) inc E ( xy,, ) e( xy, ) e 9
Exmple (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: ΓV + o o Γ.36 + o N,, V + 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 uomiclly coninuous in he WG prolem.
Hence we hve: Exmple (con.) N 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: E( xy,, ) e( xy, ) V H ( xy,, ) h( x, y) I ( ) ( ) [ ] [ ] 58. rd / m 34. rd / m,, V + ΓV + π e ˆ ysin x π h ˆ x sin x ωµ TV +
Exmple (con.) y B ε r Hence, for he wveguide prolem we hve he fields s: E( xy,, ) e( xy, ) + j + j ( e (.36) e ) x A ε H ( xy,, ) h( xy, ) 6 j + j ( e (.3 ) e ) j E ( xy,, ) e( xy, ) (. 684) e H( xy,, ) h( xy, ).684 ( ) e j
Exmple (con.) Susiuing in for nd e, we hve (guide A): E( xy,, ) e( xy, ) + j + j ( e (.36) e ) π + (,, ) ˆ E xy ysin x e. 3 6 j j ( + ( ) e ) π e ˆ ysin x π h ˆ x sin x ωµ 3
Exmple (con.) Susiuing in for nd h, we hve (guide A): H ( xy,, ) h( xy, ) 6 j + j ( e (.3 ) e ) H (,, ) ˆ xy x sin x π + j j ( e (.36) e ) 4
Exmple (con.) Susiuing in for nd e, we hve (guide B): j E ( xy,, ) e( xy, ) (. 684) e (,, ) ˆ π E xy ysin x (.684) e j 5
Exmple (con.) Susiuing in for nd h, we hve (guide B): H( xy,, ) h( xy, ) (.684) e j π H (,, ) ˆ xy x sin x (.684) j e 6
Exmple (con.) Summry of Fields E(,, ) ˆ xy ysin x e +.36 e π + j j ( ( ) ) H (,, ) ˆ xy x sin x e.36 e π + j j ( ( ) ) [ ] [ ] 499.7 Ω 59.6 Ω π j E(,, ) ˆ xy ysin x (.684) e π j H (,, ) ˆ xy x sin x (.684 ) e [ ] [ ] 58. rd / m 34. rd / m 7
Exmple (con.) Power lculions: Noe: In his exmple, nd Γ re rel. Recll: ( ) V A + + + + * + Re * inc P Re V I V P P ref rns Γ Γ + + * Re V I + + * Re V I Γ Γ ( ) ( ) 8
Exmple (con.) Finl Resuls: P P P inc refl rns [ ] [ ] [ ].6 mw.6 mw.45 mw.856 cm.6 cm ε.54 r f GH B ε r A x ε Noe: 9.% of he inciden power is rnsmied. y 9
Qurer-Wve Trnsformer in Wveguide A qurer-wve rnsformer is shown here. T d λ gt /4 Gol: Deermine : d, εt Top view x i i ε rt ε r T d Now % of he inciden power is now rnsmied. 3
Qurer-Wve Trnsformer in Wveguide (con.) Wveguide prolem Top view x ε rt ε r T d N, T T d 3
Qurer-Wve Trnsformer in Wveguide (con.) Design recipe: Top view x ) Deermine : T T ε rt ε r ) Deermine : ωµ / T T T π 3) Deermine T : T T 4) Deermineε : 5) Deermine β : rt T rt T T T 6) Deermineλ : λ π / β gt gt T 7) Deermine d: d λ /4 β gt ε T d T d λ gt /4, T T d 3
Qurer-Wve Trnsformer in Wveguide (con.) Exmple:.856 cm.6 cm ε.54 r f GH ε.53 rt Resuls: d.76 cm Top view x ε rt ε r T d 33
Mching Elemens in Wveguide 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. 34
Mching Elemens in Wveguide (con.) Top view Γ Inducive iris in ir-filled wveguide x 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 35
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 36
Using Irises for Mching An iris is shown here eing used for mching o lod (illusred for horn nenn lod). Top view Wveguide x Iris Horn w (,) d An inducive iris is shown eing used for mching. 37
Using Irises for Mching (con.) The N is shown Physicl Prolem Iris Wveguide Horn w (,) d jx w s L N Model 38
Using Irises for Mching (con.) A field proing cn e used o deermine he unnown lod impednce L. Proe Horn +ΓL SWR Γ L Γ L SWR - SWR + + Γ L E E y inc y ( ) ( ) - ΓL d min 39
Using Irises for Mching (con.) A field proing cn e used o deermine he unnown lod impednce L. L +ΓL SWR Γ Γ L L SWR - SWR + N Model + Γ L V( ) inc V ( ) - ΓL d min 4
Using Irises for Mching (con.) A Smih chr cn e used o find he unnown lod impednce. Γplne V min Γ L d min N L V mx N ( SWR R in ) 4
Using Irises for Mching (con.) A Smih chr cn e used o find he locion of he shun suscepnce nd is vlue. d jx / Y s L Y s s jx jb s s B N s B Y s Noe : B s X s 4
Noe: We mus use his upper poin for n inducive iris. Using Irises for Mching (con.) N Y L + d N jb in Γ plne Γ Γ Γ L hoose: G in N G in B N s B N ( Bs < ) N in 43
Using Irises for Mching (con.) Anoher mching scheme is shown here. d is rirry (I gives us sfey uffer from he horn disconinuiy, o llow he higher-order modes o decy.) x Top view Wveguide (,) w T Iris Horn λ /4 / g λ g d Qurer-wve rnsformer Sfey uffers The iris is used o cncel he inpu suscepnce -d. 44