ECE Microwave Engineering

Transcription

1 ECE Mirowve Engineering Apte from notes Prof. Jeffer T. Willims Fll 18 Prof. Dvi R. Jkson Dept. of ECE Notes 1 Wveguiing Strutures Prt 5: Coil Cle 1

2 TEM Solution Proess A) Solve Lple s eqution sujet to pproprite B.C.s.: ( ) Φ, B) Fin the trnsverse eletri fiel: t (, ) Φ (, ) e C) Fin the totl eletri fiel: ( ) ( ) E,, e e, jk, k k t D) Fin the mgneti fiel: 1 H ( ± ˆ E) ; ± propgting η Note: The onl frequen epenene is in the wvenumer k k.

3 Coil Line: TEM Moe To fin the TEM moe fiels, we nee to solve: ( ρφ) Φ, ; Φ ( ) V Φ ( ) Φ ρ ρ ρ Φ ( ρ) Cln ρ + D ε, µσ, φ PEC or ρ Φ ( ρ) C ln ρ V Cln Zero volt potentil referene lotion (ρ ). C V ln 3

4 Coil Line: TEM Moe (ont.) Hene V Φ ( ρ) ln ρ ln Thus, (,, ) (, ) E e e t jk jk Φ ( (, )) ˆ tφ e ρ e ρ jk ε, µσ, TEM PEC k k ω µε k jk : V 1 (,, ) ˆ ρ E ln 1 H ˆ E η e ρ ( ) jk H ˆ V φ η ln 1 e ρ jk ε ε j σ ω µ η ε 4

5 Coil Line: TEM Moe (ont.) B B ( ) ( ˆ ) ( ˆ ˆ ρ ˆ ) V ( ) V E r ρe ρρ+ φρφ+ E ρ AB A A ρ V( ) Ve V ln jk 1 e ρ π π s φ I( ) J H φ π V 1 jk e φ Z η ln πv I( ) e η ln η µ ε jk jk ρ φ ρ A B point on inner onutor point on outer onutor V I + + Hene ( ) ( ) η Z ln π ε, µσ, PEC Note: This oes not ount for onutor loss. 5

6 Coil Line: TEM Moe (ont.) Attenution: α α + α Dieletri ttenution: ε, µσ, PEC TEM: α k Geometr for ieletri ttenution TEM k β jα k ω µε k jk : ε ε j σ ω 6

7 Coil Line: TEM Moe (ont.) Attenution: α α + α α Conutor ttenution: P Pl () P 1 Z I R s ε ε ε R s Geometr for onutor ttenution (We ssume Z is rel here.) 7

8 Coil Line: TEM Moe (ont.) Conutor ttenution: 1 Pl() Rs Js C + C 1 π π Rs s φ s Rs J + J φ π π s Rs I R I φ + π π π π Rs 1 Rs 1 φ π π Rs 1 Rs 1 I + I π π 1 Rs Rs I + 4π φ I + I φ Geometr for onutor ttenution R s R s ε ε ε ωµ m σ (Here σ m enote the onutivit of the metl.) m R s 8

9 Coil Line: TEM Moe (ont.) Conutor ttenution: α P P () P 1 Z I l 1 Rs Rs Pl () I + 4π R s ε ε ε R s Geometr for onutor ttenution Hene we hve α I 1 Rs Rs 4 + π 1 Z I or α 1 1 R R + s s Z 4π 9

10 Coil Line: TEM Moe (ont.) Let s reo the lultion of onutor ttenution using the Wheeler inrementl inutne formul. Wheeler s formul: α on R Z s Zη R s ε ε ε R s Geometr for onutor ttenution The formul is pplie for eh onutor n the onutor ttenution from eh of the two onutors is then e. In this formul, l (for given onutor) is the istne whih the onuting ounr is reee w from the fiel region. η µ ε 1

11 α Rs Z Zη Rs Z α + Zη Hene Coil Line: TEM Moe (ont.) α R s Z α Zη η Z ln π ( ) ( ) Rs η 1 Zη π Rs η 1 α Zη π or so 1 η R R α + Geometr for onutor ttenution s s Zη π 1 1 R R α + R s ε ε ε R s s s Z 4π 11

12 Coil Line: TEM Moe (ont.) We n lso lulte the funmentl per-unit-length prmeters of the loss oil line. From previous lultions: (From Notes 3) lossless L Z µε C µε / Z lossless ε, µσ, R s R s µ µ (From Notes 7) G R α ( ωc) tn δ ( Z lossless ) Z lossless where 1 π η ln ε r ε εε r The lossless supersript hs een e to here to emphsie tht these vlues re rel. 1

13 Attenution for RG59 Co Approimte ttenution in B/m f Frequen RG59 Co 1 [MH].1 1 [MH].3 1 [MH].11 1 [GH].4 5 [GH] 1. 1 [GH] 1.5 [GH].3 5 [GH] OM* 1 [GH] OM* *OM overmoe 9.7 GH (TE wveguie moe) 11 Z 75Ω r.9 mm 1.85mm ε.5 (from Wikipei) 13

14 Coil Line: Higher-Orer Moes We look t the higher-orer moes* of oil line. The lowest wveguie moe is the TE 11 moe. ε, µσ, PEC Sketh of fiel lines for TE 11 moe *Here the term higher-orer moes mens the wveguie moes tht eist in ition to the esire TEM moe. 14

15 Coil Line: Higher-Orer Moes (ont.) TE : ( ρφ, ) kh( ρφ, ) h The solution in linril oorintes is: h ( ρφ, ) eigenvlue prolem k k k ε, µσ, Jn( kρ) sin( nφ ) Yn( kρ) os( nφ) ( ρφ ) ( ρφ) H,, h, e PEC jk Note: The vlue n must e n integer to hve unique fiels. 15

16 Plot of Bessel Funtions n n 1 J () n is finite J n () J( ) J1( ) Jn(, ).4. n n 1 Jn( ) ~ n,1,,..., n n! nπ π Jn( ) ~ os, π 4 16

17 Plot of Bessel Funtions (ont.) n n 1 n Y n () Y( ) Y1( ) Yn(, ) 3 Y () n is infinite nπ π Yn ( ) ~ sin, π 4 Y ( ) ~ ln γ, γ , π + 1 Yn ( ) ~ ( n 1)!, n 1,,3,..., π n 17

18 Coil Line: Higher-Orer Moes (ont.) We hoose (somewht ritrril) the osine funtion for the ngle vrition. Wve trveling in + iretion: ( ρφ ) ( ρφ) h,, h, e jk ( ρφ, ) os( φ) ( ( ρ) + ( ρ) ) h n AJ k BY k n n ε, µσ, PEC The osine hoie orrespons to hving the trnsverse eletri fiel E ρ eing n even funtion ofφ, whih is the fiel tht woul e eite proe lote t φ. 18

19 Coil Line: Higher-Orer Moes (ont.) Bounr Conitions: Eφ (, φ ) Eφ (, φ ) E φ H jωε ( ) H ρ ρ 1 ρ H ρ (From Ampere s lw) ε, µσ, PEC Hene H ρ ρ, ( ) ( ) k AJ ( k ) + BY ( k ) n n k AJ ( k ) + BY ( k ) n n Note: The prime enotes erivtive with respet to the rgument. 19

20 Coil Line: Higher-Orer Moes (ont.) AJ ( k ) + BY ( k ) n n AJ ( k ) + BY ( k ) n n In orer for this homogenous sstem of equtions for the unknowns A n B to hve non-trivil solution, we require the eterminnt to e ero. ε, µσ, PEC Hene J n( k ) Y n( k ) Det ( k ) J ( k) Y ( k) n n J ( ky ) ( k) J ( ky ) ( k) n n n n

21 Coil Line: Higher-Orer Moes (ont.) J ( ky ) ( k) J ( ky ) ( k) n n n n Denote k Then we hve: ε, µσ, PEC ( ) ( ( )) ( ) F( ; n, / ) J ( ) Y / J / Y ( ) n n n n For given hoie of n n given vlue of /, we n solve the ove eqution for to fin the eros. 1

22 Coil Line: Higher-Orer Moes (ont.) A grph of the eterminnt revels the eros of the eterminnt. F( n ;, / ) th np p ero n1 n n3 Note: These vlues re not the sme s those of the irulr wveguie, lthough the sme nottion for the eros is eing use. k np k np TE 11 moe: k 11

23 Coil Line: Higher-Orer Moes (ont.) Approimte solution: k 1 + / n 1 p 1 The TE 11 moe is the ominnt higher-orer moe of the o (i.e., the wveguie moe with the lowest utoff frequen). Et solution Figure 3.16 from the Por ook 3

24 Cutoff Frequen of TE 11 Moe Wvenumer: k ( ) k k k is rel here k f f π f µε k k Use formul on previous slie k f k k π µε π µε π ε r [m/s] TE 11 moe of o: f 1 π ε 1 + / r 4

25 Coil Line: Lossless Cse (ont.) f 1 1 ε π 1 + / r At the utoff frequen, the wvelength (in the ieletri) is then: so λ f f r ( 1 / ) ( ) π + π + π ε Compre with the utoff frequen onition of the TE 1 moe of RWG: ε r λ λ π / 5

26 r Emple Emple 3.3, p. 133 of the Por ook: RG 14 o: 4.35 inhes [m] inhes [m] ε. / 3.31 f 1 1 ε π 1 + / r f 16.8 [GH] 6