ECE Microwave Engineering

Transcription

1 ECE Mirowve Engineering Adpted from notes y Prof. Jeffery T. Willims Fll 018 Prof. Dvid R. Jkson Dept. of ECE Notes 9 Wveguiding Strutures Prt 4: Retngulr nd Cirulr Wveguide 1

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

3 TE Modes For + propgtion: ( ) ( ) H y,, h ye, jk y PEC where y + + k h( y, ) 0 ( ) 1/ k k k ε, µσ, From previous field tle: Sujet to B.C. s: E E y j E H ωµ k y k + j E H ωµ k y k + E E y 0 H 0 y 0 H y 0, 3

4 TE Modes (ont.) + h,, y k h y y ( ) ( ) (eigenvlue prolem) Using seprtion of vriles, let h( y, ) X( Y ) ( y) d X dy Y X k XY d + dy Must e onstnt This is the seprtion eqution. 1 d X 1 dy k X d + Y dy (If we tke one term ross the equl sign, we hve funtion of equl to funtion of y.) 1 d X 1 dy X d Y dy k nd k y where k + k k y seprtion eqution 4

5 TE Modes (ont.) Hene, ( ) X( ) Y( y) h y, ( Aos k+ Bsin k)( Cos ky+ Dsin ky) y y Boundry Conditions: y h h 0 y 0, A B A B D 0 nd nπ ky n 0,1,,... B 0 nd mπ k m 0,1,,... (, ) mn os m π h y A os n π y k m π n π nd + 5

6 TE Modes (ont.) Therefore, mπ nπ H ( y,, ) Amn os os y e jk k k k mπ nπ k From the field tle, we otin the following: k mπ nπ + jωµ nπ mπ nπ E Amn os sin y e k jωµ mπ mπ nπ Ey Amn sin os y e k jk mπ mπ nπ H Amn sin os y e k jk nπ mπ nπ Hy Amn os sin y e k jk jk jk jk Note: m 0,1,, n 0,1,, But m n 0 is not llowed! (non-physil solution) jk H A ˆ e ; H

7 TE Modes (ont.) Reson for non-physil solution Strt with the vetor wve eqution: H k H 0 Vetor wve eqution: from Mwell s equtions. ( ) Tke divergene of oth sides. ( ) ( ) H k H 0 The divergene of url is ero. H 0 Mgneti Guss lw 7

8 TE Modes (ont.) Reson for non-physil solution Revisit how we otined the vetor Helmholt eqution: H k H ( ) 0 Vetor wve eqution: from Mwell s equtions. ( ) H H k H H 0 From definition of vetor Lplin Now use: H 0 Mgneti Guss lw A needed ssumption! H + k H 0 Vetor Helmholt eqution (wht we hve solved) 8

9 TE Modes (ont.) Reson for non-physil solution Vetor wve eqution mgneti Guss lw Vetor Helmholt eqution mgneti Guss lw The vetor Helmholt eqution does not gurntee tht the mgneti Guss lw is stisfied. In the mthemtil derivtion, we need to ssume the mgneti Guss lw in order to rrive t the vetor Helmholt eqution. Note: The TE 00 mode is the only one tht violtes the mgneti Guss lw. 9

10 TE Modes (ont.) Lossless se ( ε ε ε ) mn mn mπ nπ k k ( k ) k TE mn mode is t utoff when mn k k ( k ω mn ) µε f mn 1 m n + µε Lowest utoff frequeny is for TE 10 mode ( > ) We will revisit this mode lter f Dominnt TE mode µε (lowest f ) 10

11 TE Modes (ont.) At the utoff frequeny of the TE 10 mode (lossless wveguide): λ d f µε d d d 10 f 1 so λ d / f f For given frequeny (with f > f ), the dimension must e t lest λ d / in order for the TE 10 mode to propgte. Emple: Air-filled wveguide, f 10 GH. We hve tht > 3.0 m / 1.5 m. 11

12 TM Modes Rell: ( ) ( ) E y,, e ye, where jk + e,, y ke y y ( ) ( ) Sujet to B.C. s: 0 0, y PEC ε, µσ, (eigenvlue prolem) ( ) 1/ k k y 0, Thus, following sme proedure s efore, we hve the following result: 1

13 TM Modes (ont.) ( ) X( ) Y( y) e y, ( Aos k+ Bsin k)( Cos ky+ Dsin ky) y y Boundry Conditions: 0 y 0, A B A B nπ C 0 nd ky n 0,1,,... mπ A 0 nd k m 0,1,,... mn sin m π e B sin n π y k m π n π nd + 13

14 TM Modes (ont.) Therefore mπ nπ E ( y,, ) Bmn sin sin y e From the field tle, we otin the following: jωε nπ mπ nπ H Bmn sin os y e k jk jωε mπ mπ nπ Hy Bmn os sin y e k jk mπ mπ nπ E Bmn os sin y e k jk nπ mπ nπ Ey Bmn sin os y e k jk jk jk jk k k k mπ k k nπ mπ nπ + m 1,,3 n 1,,3 Note: If either m or n is ero, the field eomes trivil one in the TM se. 14

15 TM Modes (ont.) Lossless se ( ε ε ε ) mn mn mπ nπ k k ( k ) k (sme s for TE modes) f mn 1 m n + µε The lowest utoff frequeny is otined for the TM 11 mode f µε Dominnt TM mode (lowest f ) 15

16 Mode Chrt Lossless se ( ε ε ε ) Two ses re onsidered: < / Single mode opertion TE 10 TE 0 TE01 TE11 TM 11 > / Single mode opertion TE 10 TE01 TE0 TE 11 TM 11 f BW f f f 1 f f 0 enter frequeny 0 y The mimum ndwidth for single-mode opertion is 67%. f mn PEC ε, µσ, ( /) 1 m n + µε 16

17 Dominnt Mode: TE 10 Mode For this mode we hve π m 1, n 0, k Hene we hve os π H A10 e jk y PEC ε, µσ, k sin π H j A10 e π 10 jk jωµ sin π Ey A10 e π E E E Hy 0 jk 10 π k k k sin π Ey E10 e A π jωµ E jk 17

18 Dominnt Mode: TE 10 Mode (ont.) The fields n e put in terms of E 10 : sin π Ey E10 e 1 π H E sin e jk 10 ZTE jk y PEC ε, µσ, π os π H E10 e jωµ jk 10 π k k k E E Hy 0 Z TE ωµ k 18

19 Dispersion Digrm for TE 10 Mode 10 ω ω Lossless se ( ε ε ε ) v g slope v p slope f > f 1 µε k k π β k ω µε ( Light line ) β Phse veloity: Group veloity: v p v g ω β dω d β Veloities re slopes on the dispersion plot. 19

20 Field Plots for TE 10 Mode Top view y PEC ε, µσ, y y E H End view Side view 0

21 Field Plots for TE 10 Mode (ont.) Top view y PEC ε, µσ, y y J s H End view Side view 1

22 Power Flow for TE 10 Mode Time-verge power flow in the diretion for + mode: 1 + * P10 Re ( ) ˆ E H dyd Re * EyH dyd k Re E 10 e α ωµ Simplifying, we hve + P10 Re{ k} E10 e α 4ωµ 00 sin Note: π dyd sin π jk Ey E10 e 1 π H E10 sin e ZTE ωµ ZTE k jk Note: For given mimum eletri field level (e.g., the rekdown field), the power is inresed y inresing the ross-setionl re ().

23 Condutor Attenution for TE 10 Mode Rell α P (0) l P 0 P0 P (lulted on previous slide) y R s Rs Pl(0) Js d C J nˆ H on ondutor s Lossless left right ot top C C + C + C + C 3

24 Condutor Attenution for TE 10 Mode Side wlls y R 0: J ˆ H yh ˆ ya ˆ e left s right s : J ˆ H yh ˆ ya ˆ e jk jk Lossless Hene: J J A e left sy right sy 10 jk π jk H A10 os e k π H j A10 sin e π jk 4

25 Condutor Attenution for TE 10 Mode (ont.) Top nd ottom y 0: J yˆ H ot y : J yˆ H top s y 0 y y R s J top s J ot s (The fields of this mode re independent of y.) Lossless Hene: ot π jk Js A10 os e ot k π Js j A10 sin e π jk π jk H A10 os e k π H j A10 sin e π jk 5

26 Condutor Attenution for TE 10 Mode (ont.) y We then hve: R s R left ot s R s Pl(0) J s dy J s d left ot ot s sy s s s 0 0 ( ) R J dy + R J + J d π k π Rs A10 dy + Rs A10 os + j A10 sin d π 0 0 π R A dy d d π k s 10 + os + sin π k Rs A π Lossless 6

27 Attenution for TE 10 Mode (ont.) Assume f > f y k β (The wvenumer is tken s tht of guide with perfet wlls.) R s β Pl(0) Rs A π P β E 4ωµ A π E jωµ Lossless Simplify, using β k k 10 k π α Pl (0) P 0 α Finl result: R [np/m] 3 β + s ( ) ( 3 π k kη ) 7

28 Attenution for TE 10 Mode (ont.) y Finl Formuls R s Two lterntive forms for the finl result: Lossless α α R [np/m] 3 β + s ( ) ( 3 π k kη ) R 1 f 1 + [np/m] s η 1 ( f / f ) f 8

29 Attenution for TE 10 Mode (ont.) Brss X-nd ir-filled wveguide 7 ( σ.6 10 [S/m]) X nd : 8 1 [GH] (See the tle on the net slide.).0 m (from the Por ook) 9

30 Attenution for TE 10 Mode (ont.) Mirowve Frequeny Bnds Letter Designtion Frequeny rnge L nd 1 to GH S nd to 4 GH C nd 4 to 8 GH X nd 8 to 1 GH Ku nd 1 to 18 GH K nd 18 to 6.5 GH K nd 6.5 to 40 GH Q nd 33 to 50 GH U nd 40 to 60 GH V nd 50 to 75 GH E nd 60 to 90 GH W nd 75 to 110 GH F nd 90 to 140 GH D nd 110 to 170 GH (from Wikipedi) 30

31 Modes in n X-Bnd Wveguide.9m (0.90in) 1.0m (0.40in) Stndrd X-nd wveguide (WR90) Mode TE TE TE TE TM TE TE TM f [GH] X nd : 8 1 [ GH] 1" 0.5" 50 mil (0.05 ) thikness 31

32 Emple: X-Bnd Wveguide Determine β, α, nd λ g (s pproprite) t 10 GH nd 6 GH for the TE 10 mode in lossless ir-filled X-nd wveguide..9m ε 0, µ 0 10 GH β ω µε 10 π π10 π β [rd/m] λ g π π β λ g 3.97 [m] 3

33 Emple: X-Bnd Wveguide GH k 9 π π6 10 π ω µε j [1/m] α [np/m] λ g π β [db/m] Evnesent mode: β 0; λ g is not defined! 33

34 Fields of Guided Wve Fields Equtions in Cylindril Coordintes H H E j E ρ k k ρ φ ρ φ ρ k ± + k H j E k H ωε ± k ρ ρ y j ωε E ωµ H ρ ρ φ These equtions give the trnsverse field omponents in terms of the longitudinl omponents, E nd H. ( ) e F k ω µε jk E φ j k E H ωµ + k ρ φ ρ k k k 34

35 Cirulr Wveguide ε, µσ, PEC TM mode: ( ρφ, ) ke( ρφ, ) e (eigenvlue prolem) k k k The solution in ylindril oordintes is: e ( ρφ, ) Jn( kρ) sin( nφ ) Yn( kρ) os( nφ) Note: The vlue n must e n integer to hve unique fields. 35

36 Referenes for Bessel Funtions M. R. Spiegel, Shum s Outline Mthemtil Hndook, MGrw-Hill, M. Armowit nd I. E. Stegun, Hndook of Mthemtil Funtions with Formuls, Grphs, nd Mthemtil Tles, Ntionl Bureu of Stndrds, Government Printing Offie, Tenth Printing, 197. N. N. Leedev, Speil Funtions & Their Applitions, Dover Pulitions, New York,

37 Plot of Bessel Funtions n 0 J (0) n is finite J n () J0( ) J1( ) n 1 n Jn(, ) nπ π Jn( ) ~ os, π 4 n 1 Jn( ) ~ n 0,1,,..., 0 n n! 37

38 Plot of Bessel Funtions (ont.) n 0 n 1 n Y n () Y0( ) Y1( ) 3 Y (0) n is infinite Yn(, ) nπ π Yn ( ) ~ sin, π 4 Y0 ( ) ~ ln γ, γ , 0 π + n 1 Yn ( ) ~ ( n 1)!, n 1,,3,..., 0 π 38 10

39 Cirulr Wveguide (ont.) Choose (somewht ritrrily) os( nφ ) Jn( kρ) e ( ρφ,, ) os( nφ) Yn( kρ) The field should e finite on the is Y ( k ρ) n is not llowed Hene, we hve ( ) e ρφ, os( nφ) J ( k ρ) n ( ) E ρφ,, os( nφ) J ( k ρ) e n jk 39

40 Cirulr Wveguide (ont.), φ, 0 B.C. s: E ( ) Hene J ( k ) 0 n J ( ) n Sketh for typil vlue of n (n 0). Note: Por uses the nottion p mn. n1 n n3 k np k np Note: The vlue n0 0 is not inluded ρ sine this would yield trivil solution: Jn n Jn( ) (This is true unless n 0, in whih se we nnot hve p 0.) 40

41 Cirulr Wveguide (ont.) TM np mode: ρ E (,, ) os( ) jk ρφ nφ Jn np e n 0,1, np k k p 1,,3,... 41

42 Cutoff Frequeny: TM k k k Assume k is rel here. At f f : k 0 k k TM π f µε np np f TM d π np d ε r 4

43 Cutoff Frequeny: TM (ont.) np vlues p \ n TM 01, TM 11, TM 1, TM 0,.. 43

44 TE Modes Proeeding s efore, we now hve tht H ρφ,, os( nφ) J ( k ρ) e ( ) n jk Set E ( ) φ, φ, 0 E φ H jωε ( ) H ρ 0 ρ 1 ρ H ρ (From Ampere s lw) H ρ 0 ρ Hene J ( k) 0 n The prime denotes derivtive with respet to the rgument. 44

45 TE Modes (ont.) J ( k) 0 n J ( ) n Sketh for typil vlue of n (n 1). n1 n n3 k k np np p 1,,3,... We don t need to onsider p 0; this is eplined on the net slide. 45

46 TE Modes (ont.) ρ H (,, ) os( ) jk ρφ nφ Jn np e p 1,, We then hve, for p 0: Note: If p 0, then np 0 n 0 ρ Jn np Jn ( 0) 0 ρ n 0 J0 np J0( 0) 1 (unless n 1) (trivil solution) jk jk jk H e H e ˆ H e ˆ (nonphysil solution) (violtes the mgneti Guss lw) The TE 00 mode is not physil. 46

47 Cirulr Wveguide (ont.) TE np mode: ρ H (,, ) os( ) jk ρφ nφ Jn np e n 0,1, np k k p 1,,3,... 47

48 Cutoff Frequeny: TE k k k Assume k is rel here. k 0 k k np TE π f µε np Hene f TE d π np d ε r 48

49 Cutoff Frequeny: TE np vlues p \ n TE 11, TE 1, TE 01, TE 31,.. 49

50 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 e thought of s n evolution of the TE 10 mode of retngulr wveguide s the oundry hnges shpe. 50

51 TE 11 Mode (ont.) The ttenution due to ondutor loss for the TE 11 mode is: α R 1 f s η 1 ( f / ) f 11 f k 11 The derivtion is in the Por ook (see Eq ). 51

52 TE 01 Mode The TE 01 mode of irulr wveguide hs the unusul property tht the ondutor ttenution dereses with frequeny. (With most wveguide modes, the ondutor ttenution inreses with frequeny.) α R s ( f / f ) ( ) η 1 f / f Reson: This mode hs urrent only in the φ diretion, nd this omponent of urrent (orresponding to H ) dereses s the frequeny inreses (for fied power flow down the guide). (Plese see the equtions on the net slide.) The TE 01 mode ws studied etensively s ndidte for long-rnge ommunitions ut eventully fier-opti les eme ville with even lower loss. It is still useful for some high-power pplitions. Note: This mode is not the dominnt mode! 5

53 TE 01 Mode (ont.) The fields of the TE 01 mode re: ρ jk H J0 01 e 1 ρ ωµ 01 Eφ j J 0 01 e k H E / Z ρ φ ( 0,1) TE jk Z ( 0,1) TE ωµ k ( 0,1) 53

54 TE 01 Mode (ont.) α TE11 TM01 TE1 TM 11 α P (0) l P 0 Note: P 0 0 t utoff TE 01 f TM01 TE1 TE01 f f f TE 11 f 54

55 TE 01 Mode (ont.) Prtil Note: The TE 01 mode hs only n imuthl (φ - direted) surfe urrent on the wll of the wveguide. Therefore, it n e supported y 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.) TE 01 mode: E H 55

56 TE 01 Mode (ont.) VerteRSI's Torrne Fility is leding supplier of ntenn feed omponents for the vrious ommeril nd militry nds. A ptented irulr polried 4-port dipleer meeting ll Intelst speifitions leds full rry of produts. Produts inlude: 4-Port Dipleers, CP or Liner; 3-Port Dipleers, R & 1T; -Port Dipleers, RT, X-Pol or Co-Pol, CP or Liner; TE1 Monopulse Trking Couplers; TE01 Mode Components; Trnsitions; Filters; Fle Wveguides; Wveguide Bends; Twists; Runs; et. Mny of the items re "off the shelf produts". Produts n e ustom tilored to ustomer's pplition. Mny of the produts n e supplied with stndrd feed horns for prime or offset ntenns. 56

57 TE 01 Mode (ont.) From the eginning, the most ovious pplition of wveguides hd een s ommunitions medium. It hd een determined y oth Shelkunoff nd Med, independently, in July 1933, tht n illy symmetri eletri wve (TE 01 ) in irulr wveguide would hve n ttenution ftor tht deresed with inresing frequeny [44]. This unique hrteristi ws elieved to offer gret potentil for wide-nd, 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 e prtil, nd Southworth eventully egn to relie tht the role of wveguide would e somewht different thn originlly epeted. In memorndum dted Otoer 3, 1939, he onluded tht mirowve rdio with highly diretive ntenns ws to e preferred to long trnsmission lines. "Thus," he wrote, we ome to the onlusion tht the hollow, ylindril ondutor is to e vlued primrily s new iruit element, ut not yet s new type of toll le [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 oth prtil nd theoretil dvne. K. S. Pkrd, The origins of wveguide: A se of multiple redisovery, IEEE Trns. Mirowve Theory nd Tehniques, pp , Sept

58 TE 01 Mode (ont.) In memorndum dted Otoer 3, 1939, he onluded tht mirowve rdio with highly diretive ntenns ws to e preferred to long trnsmission lines. Rell the omprison of db ttenution: Wveguiding system: db 8.686( α) λ 4π Wireless system: db 10log ( GG ) 0log 0 + 0log ( r) 10 t r