ECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 7. Waveguides Part 4: Rectangular and Circular Waveguide

Transcription

1 ECE Mirowve Engineering Fll 01 Prof. Dvid R. Jkson Dept. of ECE Notes 7 Wveguides Prt 4: Retngulr nd Cirulr Wveguide 1

2 Retngulr Wveguide One of the erliest wveguides. Still ommon for high power nd high mirowve / millimeter-wve pplitions.,, It is essentilly n eletromgneti pipe with retngulr ross-setion. Single ondutor No TEM mode For onveniene b. the long dimension lies long x.

3 TE Modes Rell H x, y, h x, y e jk where x y k h x, y 0 1/ k k k,, Subjet to B.C. s: E x H y y 0, b E nd 0 y H x 0, 3

4 TE Modes (ont.) h,, x y k h x y x y (eigenvlue problem) Using seprtion of vribles, let h x, y X xy y d X d Y Y X k XY dx dy Must be onstnt 1 d X 1 d Y k X dx Y dy 1 d X 1 d Y X dx Y dy k x nd k y where k k k x y dispersion reltionship 4

5 Hene, TE Modes (ont.) X ( x) Y ( y) h x, y ( Aos k x Bsin k x)( C os k y Dsin k y) x x y y Boundry Conditions: h 0 y h 0 y x 0, b A B A B D 0 nd n ky b n 0,1,,... B 0 nd m kx m 0,1,,... m x n y h x, y Amn osos nd k b m n b 5

6 TE Modes (ont.) Therefore, m n H Amn os xos ye b jk k k k m n k b From the previous field-representtion equtions, we n show jn m n Ex Amn os xsin ye k b b jm m n Ey Amn sin xos ye k b jk m m n Hx Amn sin xos ye k b jk n m n H y Amn os xsin ye k b b jk jk jk jk Note: m = 0,1,, n = 0,1,, But m = n = 0 is not llowed! (non-physil solution) jk H ˆ A e ; H

7 TE Modes (ont.) Lossless Cse mn mn m n k k k k b TE mn mode is t utoff when mn k k f mn 1 m n b Lowest utoff frequeny is for TE 10 mode ( > b) We will revisit this mode f Dominnt TE mode (lowest f ) 7

8 TE Modes (ont.) At the utoff frequeny of the TE 10 mode (lossless wveguide): 10 1 f f d d d 10 f 1 f f / d For given operting wvelength (orresponding to f > f ), the dimension must be t lest this big in order for the TE 10 mode to propgte. Exmple: Air-filled wveguide, f = 10 GH. We hve tht > 3.0 m/ = 1.5 m. 8

9 TM Modes Rell E x, y, e x, y e jk,, where e,, x y k e x y x y 1/ k k k (eigenvlue problem) Subjet to B.C. s: 0 x y 0, b Thus, following sme proedure s before, we hve the following result: 9

10 TM Modes (ont.) X ( x) Y ( y) e x, y ( Aos k x Bsin k x)( Cos k y Dsin k y) x x y y Boundry Conditions: 0 y x 0, b A B A B n C 0 nd ky n 0,1,,... b m A 0 nd kx m 0,1,,... m n e B sin x sin y b mn nd k m n b 10

11 TM Modes (ont.) Therefore m n E Bmn sin xsin ye b From the previous field-representtion equtions, we n show jk j n m n Hx Bmn sin xos ye k b b j m m n H y Bmn os xsin ye k b jk m m n Ex Bmn os xsin ye k b jk n m n Ey Bmn sin xos ye k b b jk jk jk jk k k k m k n b m=1,,3, n =1,,3, Note: If either m or n is ero, the field beomes trivil one in the TM se. 11

12 TM Modes (ont.) Lossless Cse mn mn m n k k k b (sme s for TE modes) f mn 1 m n b Lowest utoff frequeny is for the TM 11 mode f b Dominnt TM mode (lowest f ) 1

13 Mode Chrt Two ses re onsidered:,, b < / Single mode opertion TE 10 TE 0 TE01 TE11 TM 11 b > / f > b The mximum bnd for single mode opertion is f 10. b / Single mode opertion TE 10 TE01 TE0 TE 11 TM 11 f f mn 1 m n b 13

14 Dominnt Mode: TE 10 Mode For this mode we hve m 1, n 0, k Hene we hve os H A10 xe jk k sin H x j A10 xe 10 jk j sin Ey A10 x e E Ex E H y 0 jk k k A,, sin Ey E10 xe j E jk 14

15 Dispersion Digrm for TE 10 Mode 10 Lossless Cse vg slope f f vp slope 1 k g ( Light line ) k Phse veloity: Group veloity: v p v g d d Veloities re slopes on the dispersion plot. 15

16 Field Plots for TE 10 Mode Top view,, y x E H b End view x Side view 16

17 Field Plots for TE 10 Mode (ont.) Top view,, y x J s H b End view x Side view 17

18 Power Flow for TE 10 Mode Time-verge power flow in the diretion: b 1 * P10 Re E H ˆ dydx Re 4 b A10 * EyH x dydx b Re In terms of mplitude of the field mplitude, we hve b P Rek E k A j E b 0 0 sin Note: x dydx b For given mximum eletri field level (e.g., the brekdown field), the power is inresed by inresing the ross-setionl re (b). 18

19 Attenution for TE 10 Mode Rell Pl (0) P 0 P 0 P 10 (lulted on previous slide) Rs Pl (0) Js d C J nˆ H on ondutor s,, Side wlls x 0: J xˆ H yh ˆ ya ˆ e s x0 x : J xˆ H yh ˆ ya ˆ e s x jk jk os jk H A10 xe k sin H x j A10 xe jk side Jsy A e 10 jk 19

20 Attenution for TE 10 Mode (ont.) Top nd bottom wlls y 0: J yˆ H y b : J yˆ H s s y0 yb,, J top s J bot s (sine fields of this mode re independent of y) b R s side R s top Pl (0) J s dy J s dx 0 0 b side top top s sy s sx s 0 0 R J dy R J J dx 3 k Rs A 10 b jk H A10 os xe k Hx j A10 sin xe J J top s top sx jk J H J H top top s x sx k j A10 sin xe jk A10 os xe jk 0

21 Attenution for TE 10 Mode (ont.) Assume f > f k (The wvenumber is tken s tht of guide with perfet wlls.) 3 Pl (0) Rs A10 b,, b P E E 10 ja 10 Simplify, using k k 10 k P(0) l P 10 Finl result: R [np/m] 3 b s 3 b k k 1

22 Attenution in db/m Let = distne down the guide in meters. Attenution [db/m] 10 e db/m 0log / 0log ( e) / ,, Hene db/m = [np/m]

23 Attenution for TE 10 Mode (ont.) Brss X-bnd ir-filled wveguide [S/m] X bnd: 8 1 [GH] (See the tble on the next slide.) 3

24 Attenution for TE 10 Mode (ont.) Letter Designtion Mirowve Frequeny Bnds Frequeny rnge L bnd S bnd C bnd X bnd K u bnd K bnd K bnd Q bnd U bnd V bnd E bnd W bnd F bnd D bnd 1 to GH to 4 GH 4 to 8 GH 8 to 1 GH 1 to 18 GH 18 to 6.5 GH 6.5 to 40 GH 33 to 50 GH 40 to 60 GH 50 to 75 GH 60 to 90 GH 75 to 110 GH 90 to 140 GH 110 to 170 GH (from Wikipedi) 4

25 Modes in n X-Bnd Wveguide b.9m (0.90") 1.0m (0.40") Stndrd X-bnd wveguide (WR90) Mode f [GH] X bnd: 8 1 [ GH] TE TE TE TE TM TE TE TM mil (0.05 ) thikness 5

26 Exmple: X-Bnd Wveguide Determine nd g t 10 GH nd 6 GH for the TE 10 mode in n irfilled wveguide., 0 10 GH [rd/m] g g 3.97 [m] 6

27 Exmple: X-Bnd Wveguide 6 GH k j [1/ m] [np/m] g [db/m] Evnesent mode: = 0; g is not defined! 7

28 Cirulr Wveguide TM mode: k E E,, k k k The solution in ylindril oordintes is: E 0, Jn ( k ) sin( n ) Yn ( k) os( n ) Note: The vlue n must be n integer to hve unique fields. 8

29 Plot of Bessel Funtions n = 0 J (0) n is finite J n (x) J0( x) J1( x) n = 1 n = Jn( x) x 10 n Jn ( x) ~ os x, x x 4 n 1 J n( x) ~ x n 0,1,,..., x 0 n n! 9

30 Plot of Bessel Funtions (ont.) n = 0 n = 1 Y n (x) Y0( x) Y1( x) 1 3 n = Y (0) n is infinite Yn( x) x n Yn ( x) ~ sin x, x x 4 x x Y0 ( x) ~ ln, , x 0 n 1 Yn ( x) ~ ( n 1)!, n 1,,3,..., x 0 x 30 10

31 Cirulr Wveguide (ont.) Choose (somewht rbitrrily) os( n ) J ( k ) E,, os( n ) e Yn ( k) n jk The field should be finite on the xis Y ( k ) n is not llowed E,, os( n ) J ( k ) e n jk 31

32 Cirulr Wveguide (ont.) B.C. s: E,, 0 Hene J n ( k ) 0 J n (x) Sketh for typil vlue of n (n 0). x n1 x n x n3 Note: Por uses the nottion p mn. x k x np k x np Note: The vlue x n0 = 0 is not inluded sine this would yield trivil solution: Jn xn0 J n 0 0 This is true unless n = 0, in whih se we nnot hve p = 0. 3

33 Cirulr Wveguide (ont.) TM np mode: E,, os( ) jk n Jn xnp e n 0,1, np k k p x 1,,3,... 33

34 Cutoff Frequeny: TM k k k At f = f : k 0 k k x np f x np f TM d x np d r 34

35 Cutoff Frequeny: TM (ont.) x np vlues p \ n TM 01, TM 11, TM 1, TM 0,.. 35

36 TE Modes Proeeding s before, we now hve tht H,, os( n ) J ( k ) e n Set E,, 0 jk E j H k (From Ampere s lw) H 0 Hene J( k ) 0 n 36

37 TE Modes (ont.) J( k ) 0 n J n ' (x) Sketh for typil vlue of n (n 1). x' n1 x' n x' n3 x k k x x np np p 1,,3,... We don t need to onsider p = 0; this is explined on the next slide. 37

38 TE Modes (ont.) H,, os( ) jk n Jn x np e p 1,, Note: If p = 0 x np 0 We then hve, for p = 0: n 0 n 0 J n x np Jn 0 0 J0 x np J0 0 1 (trivil solution) H ˆ e jk ˆ e jk (nonphysil solution) (The TE 00 mode is not physil.) 38

39 Cutoff Frequeny: TE k k k k 0 k k x np f x np Hene f TE d x np d r 39

40 Cutoff Frequeny: TE x np vlues p \ n TE 11, TE 1, TE 01, TE 31,.. 40

41 TE 11 Mode The dominnt mode of irulr wveguide is the TE 11 mode. Eletri field Mgneti field (From Wikipedi) TE 10 mode of retngulr wveguide TE 11 mode of irulr wveguide The mode n be thought of s n evolution of the TE 10 mode of retngulr wveguide s the boundry hnges shpe. 41

42 TE 01 Mode The TE 01 mode hs the unusul property tht the ondutor ttenution dereses with frequeny. (With most wveguide modes, the ondutor ttenution inreses with frequeny.) The TE 01 mode ws studied extensively s ndidte for longrnge ommunitions but eventully fiber-opti bles beme vilble with even lower loss. It is still useful for some high-power pplitions. 4

43 TE 01 Mode (ont.) TE 11 TM 01 TE 1 TM 11 P(0) l P 0 P 0 = 0 t utoff TE 01 f, TE11 f, TM01 f, TE1 f, TE01 f 43

44 TE 01 Mode (ont.) Prtil Note: The TE 01 mode hs only n imuthl (-direted) surfe urrent on the wll of the wveguide. Therefore, it n be supported by set of onduting rings, while the lower modes (TE 11,TM 01, TE 1, TM 11 ) will not propgte on suh struture. (A helil spring will lso work fine.) 44

45 TE 01 Mode (ont.) From the beginning, the most obvious pplition of wveguides hd been s ommunitions medium. It hd been determined by both Shelkunoff nd Med, independently, in July 1933, tht n xilly symmetri eletri wve (TE 01 ) in irulr wveguide would hve n ttenution ftor tht deresed with inresing frequeny [44]. This unique hrteristi ws believed to offer gret potentil for wide-bnd, multihnnel systems, nd for mny yers to ome the development of suh system ws mjor fous of work within the wveguide group t BTL. It is importnt to note, however, tht the use of wveguide s long trnsmission line never did prove to be prtil, nd Southworth eventully begn to relie tht the role of wveguide would be somewht different thn originlly expeted. In memorndum dted Otober 3, 1939, he onluded tht mirowve rdio with highly diretive ntenns ws to be preferred to long trnsmission lines. "Thus," he wrote, we ome to the onlusion tht the hollow, ylindril ondutor is to be vlued primrily s new iruit element, but not yet s new type of toll ble [45]. It ws s iruit element in militry rdr tht wveguide tehnology ws to find its first mjor pplition nd to reeive n enormous stimulus to both prtil nd theoretil dvne. K. S. Pkrd, The Origins of Wveguide: A Cse of Multiple Redisovery, IEEE Trns. MTT, pp , Sept