ECE Spring Prof. David R. Jackson ECE Dept. Notes 6

Transcription

1 ECE 6345 Spring 2015 Prof. David R. Jackon ECE Dpt. Not 6 1

2 Ovrviw In thi t of not w look at two diffrnt modl for calculating th radiation pattrn of a microtrip antnna: Elctric currnt modl Magntic currnt modl W alo look at two diffrnt ubtrat aumption: Infinit ubtrat Truncatd ubtrat (truncatd at th dg of th patch). 2

3 Rviw of Equivalnc Principl Original problm: Nw problm: J M ( EH, ) ( EH, ) + - ε r Arbitrary mdia out ε, µ out (0,0) out ε, µ out S J nˆ H S ˆn M nˆ E 3

4 Rviw of Equivalnc Principl A common choic (PEC inid): J M Th lctric urfac currnt itting on th PEC objct do not radiat, and can b ignord. ( EH, ) PEC J nˆ H M nˆ E out ε, µ out S ˆn 4

5 Modl of Patch and Fd Patch h Prob ε r Infinit ubtrat S A (aprtur) Coa fd 5

6 Modl of Patch and Fd h ε r ( EH, ) S A S Put ro fild Put ground plan Magntic frill modl: Aprtur: M ˆ E h M ε r 6

7 Elctric Currnt Modl: Infinit Subtrat h Th urfac S hug th PEC mtal. S ε r ( EH, ) Infinit ubtrat Put ro fild Rmov patch and prob Not: Th frill i ignord. M nˆ E nˆ E 0 J nˆ H J t top J bot J h prob J ε r 7

8 Elctric Currnt Modl: Infinit Subtrat (cont.) Lt J J + J patch top bot patch J h prob J ε r Top viw patch J prob J 8

9 Magntic Currnt Modl: Infinit Subtrat h S h S S t S b M ε r ( EH, ) Put ro fild Rmov patch, prob, and frill currnt Put ubtrat and ground plan M nˆ E 0, r S, S t b J + nˆ H 0, r S J 0, r S J 0, r S b t h (wak fild) (approimat PMC) 9

10 Magntic Currnt Modl: Infinit Subtrat (cont.) Eact modl: J h M J J M ε r Approimat modl: h M M ε r M nˆ E Not: Th magntic currnt radiat inid an infinit ubtrat abov a ground plan. 10

11 Magntic Currnt Modl: Truncatd Subtrat ε r S b Th ubtrat i truncatd at th dg of th patch. Approimat modl: Put ro fild Rmov th ubtrat M Not: Th magntic currnt radiat in fr pac abov a ground plan. 11

12 Elctric Currnt Modl: Truncatd Subtrat ε r S Th patch and prob ar rplacd by urfac currnt, a bfor. Th ubtrat i truncatd at th dg of th patch. J J + J patch top bot patch J Nt, w rplac th dilctric with polariation currnt. ε r 12

13 Elctric Currnt Modl: Truncatd Subtrat (cont.) H jωε E ( ) ( 1) jω ε ε E+ jωε E 0 0 jωε ε E+ jωε E 0 r 0 ( ε ) pol J jωε 0 r 1 E ε 0 patch J prob J J pol In thi modl w hav thr parat lctric currnt. ( ) J patch prob pol J : J, J, J 13

14 Commnt on Modl Infinit Subtrat Th lctric currnt modl i act (if w nglct th frill), but it rquir knowldg of th act patch and prob currnt. Th magntic currnt modl i approimat, but fairly impl. For a rctangular patch, both modl ar fairly impl if only th (1,0) mod i aumd. For a circular patch, th magntic currnt modl i much implr (it do not involv Bl function). 14

15 Commnt on Modl (cont.) Truncatd Subtrat Th lctric currnt modl i act (if w nglct th frill), but it rquir knowldg of th act patch and prob currnt, a wll a th fild inid th patch cavity (to gt th polariation currnt). It i a complicatd modl. Th magntic currnt modl i approimat, but vry impl. Thi i th rcommndd modl. For th magntic currnt modl th am formulation appli a for th infinit ubtrat th ubtrat i imply takn to b air. 15

16 Thorm Th lctric and magntic modl yild idntical rult at th ronanc frquncy of th cavity mod. Aumption: 1) Th lctric and magntic currnt modl ar bad on th fild of a ingl cavity mod corrponding to an idal lol cavity with PMC wall. 2) Th prob currnt i nglctd in th lctric currnt modl. Not: Thi thorm i tru for ithr infinit or truncatd ubtrat. 16

17 Thorm (cont.) Elctric-currnt modl: J J ˆ H h ε r Magntic-currnt modl: h M M ε r M nˆ E (E, H) fild of ronant cavity mod with PMC id wall 17

18 Thorm (cont.) Proof: W tart with an idal cavity having PMC wall on th id. Thi cavity will upport a valid non-ro t of fild at th ronanc frquncy f 0 of th mod. Idal cavity PEC ( EH, ) ε r PMC At ( ) ( ) f f 0 : EH, 0,0 18

19 Proof Proof for infinit ubtrat PEC S ( EH, ) PMC Equivalnc principl: Put (0, 0) outid S Kp (E, H) inid S Th PEC and PMC wall hav bn rmovd in th ro fild (outid) rgion. W kp th ubtrat and ground plan in th outid rgion. S ( EH, ) ( 0,0) 19

20 Proof (cont.) J nˆ H M nˆ E i i Not th inward pointing normal nˆi ( ) J 0,0 ( EH, ) nˆi M Not: Th lctric currnt on th ground i nglctd (it do not radiat). 20

21 Proof (cont.) Etrior Fild: + E J + E M + 0 J nˆ H ˆ H J J patch J i (Th quivalnt lctric currnt i th am a th lctric currnt in th lctric currnt modl.) M nˆ E i + nˆ E M ( nˆ E) M (Th quivalnt currnt i th ngativ of th magntic currnt in th magntic currnt modl.) 21

22 Proof (cont.) Hnc + J M E J + E M + 0 or + J M E J + E M 22

23 Thorm for Truncatd Subtrat Proof for truncatd modl PEC S ( EH, ) PMC Rplac th dilctric with polariation currnt: PEC S ( EH, ) PMC J pol 23

24 Proof (cont.) J J M J patch M M ( 0,0) ( EH, ) J pol M + patch + pol + M E J E J E M or patch pol M E + J E + J E + M + Hnc J M E + J E + M 24

25 Rctangular Patch Idal cavity modl: y PMC 2 E + k 2 E 0 W L C E n 0 C Lt E( y, ) X( Y ) ( y) X Y + XY + k XY 2 0 Divid by X()Y(y): X Y k 0 X Y o X 2 Y k + X Y 25

26 Rctangular Patch (cont.) Hnc X ( ) X( ) contant k 2 Gnral olution: X ( ) Ain k+ Bco k Boundary condition: X (0) k Aco( k 0) k Bin( k 0) k A 0 Choo B 1 X ( ) co( k) A 0 Boundary condition: ( ) X ( L) kin kl 0 k mπ L 26

27 Rctangular Patch (cont.) o X( ) co m π L Rturning to th Hlmholt quation, o 2 Y 2 k + + k 0 Y Y contant ( k 2 k 2 ) 2 ky Y Following th am procdur a for th X() function, w hav: Y( y) co n π y W Hnc ( mn, ) mπ nπy E ( y, ) co co L W 27

28 Rctangular Patch (cont.) Uing k 2 k 2 y + k 2 0 w hav k mn m π L 2 + nπ W 2 whr k mn ω mn µε Hnc ω mn 1 µε m π L 2 2 nπ + W ω mn r 2 2 c mπ nπ + ε L W 28

29 Rctangular Patch (cont.) Currnt: patch J nˆ H ˆ H H 1 E jωµ 1 jωµ ( E ˆ ) 1 ( ˆ) E ˆ E jωµ o 1 H ˆ E jωμ 29

30 Rctangular Patch (cont.) Hnc patch 1 J ˆ ˆ E jωµ ( ) J patch 1 jωμ E patch 1 ˆ mπ mπ nπy n m n y J in co yˆ π π π co in jωµ + L L W W L W Dominant (1,0) Mod: E π ( y, ) co L 1 (, ) π ˆ in π J y jωµ L L 30

31 Rctangular Patch (cont.) Static (0,0) mod: E J ( y, ) 1 ω 00 0 ( y, ) 0 Thi i a tatic capacitor mod. A patch oprating in thi mod do not radiat at ro frquncy, but it can b mad ronant at a highr frquncy if th patch i loadd by an inductiv prob (a good way to mak a miniaturid patch). 31

32 Radiation Modl for (1,0) Mod Elctric-currnt modl: y J patch π π ˆ in jωµ L L W L patch J h ε r 32

33 Radiation Modl for (1,0) Mod (cont.) Magntic-currnt modl: ˆ ˆ ˆ co M M n E n L π 0 ˆ ˆ 0 ˆ ˆ ˆ y y W y y L n y L W 33

34 Radiation Modl for (1,0) Mod (cont.) Hnc M M yˆco π yˆ L yˆco 0 yˆ 0 π ˆ co y W L π ˆ co y 0 L W y radiating dg M M Th non-radiating dg do not contribut to th far-fild pattrn in th principal plan. L 34

35 Circular Patch a PMC 2 E + k 2 E 0 E Jn( kρ ρ) co( nφ ) mπ co Yn ( kρ ρ) in( nφ ) h t m ( ) 1/2 kρ k k k k 2 mπ h 2 1/2 35

36 Circular Patch (cont.) Not: coφ and inφ mod ar dgnrat (am ronanc frquncy). Choo coφ : co( φ) ( ρ) E n J k n E 0 ρ ρ a J ( ) n J ( ) 0 n ka n1 n2 36

37 Circular Patch (cont.) Hnc ka np o ω np c ε r np Dominant mod (lowt frquncy) i TM 11 : E (1,1) ( np, ) (1,1) ( ρφ, ) co φj ( kρ) 1 37

38 Circular Patch (cont.) Elctric currnt modl: J 1 1 E ˆ 1 E E ˆ ρ φ jωµ jωµ + ρ ρ φ J 1 ˆ 1 ˆ ρkco nφ J n( kρ) φ ( n)in nφ Jn( kρ) jωµ + ρ y TM 11 mod: n 1, p 1 J 1 ˆ 1 J ˆ ρkco φ J 1 ( kρ) φ in φ J1( kρ) jωµ ρ Vry complicatd! 38

39 Circular Patch (cont.) Magntic currnt modl: o M M nˆ E ˆ ρ ˆ φ E ( E ˆ ) M M ˆ φ co nφ J ( ka ) n TM 11 : n 1, p 1 M M ˆco φ φ J ( ka ) 1 39

40 Circular Patch (cont.) Not: V( φ) he ( φ) ρ a h co φ J ( ka) 1 At φ 0 V (0) V h J ( ka) 0 1 M M ˆ ˆ V φ co φ J1( ka) φ co φ h Hnc 0 o M M ˆ V h 0 φ coφ 40

41 Ring approimation: Circular Patch (cont.) K ˆ φ K φ V φ φ φ φ coφ h h M M 0 K M co 0 d h M h V0 K φ ( φ) V0 coφ y h M ε r K( φ) h K ε r 41