ECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 8. Waveguides Part 5: Coaxial Cable

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1 ECE Mirowve Engineering Fll 17 Prof. Dvid R. Jkson Dept. of ECE Notes 8 Wveguides Prt 5: Coil Cle 1

2 Coil Line: TEM Mode To find the TEM mode fields, we need 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 Hene Thus, Coil Line: TEM Mode (ont.) V Φ ( ρ) ln ρ ln jk Φ E(,, ) ( (, )) ˆ tφ e ρ e ρ V ρ ln (,, ) ˆ ρ E 1 H ˆ E η ( ) e jk jk ˆ V H φ ηρ ln TEM e ε, µσ, k k ω µε k jk : jk PEC ε ε η j σ ω µ ε 3

4 Coil Line: TEM Mode (ont.) ( ) ( ˆ ) ( ˆ ˆ ˆ ρ ) V ( ) V E dr ρe ρd ρ+ φρdφ+ d E dρ AB V( ) Ve V ρ ln jk π π s φ I( ) J d H dφ V η ln πv I( ) e η ln B B A A e jk π jk e dφ jk dρ φ ρ A B point on inner ondutor point on outer ondutor Z Hene ρ V I + + ( ) ( ) η Z ln π ε, µσ, PEC Note: This does not ount for ondutor loss. 4

5 Coil Line: TEM Mode (ont.) Attenution: α α + α d Dieletri ttenution: TEM: α d k ε, µσ, PEC Geometr for dieletri ttenution TEM k β jα k ω µε k jk : d ε ε j σ ω 5

6 Coil Line: TEM Mode (ont.) Attenution: α α + α α d Condutor ttenution: P Pl () P 1 Z I R s ε ε ε R s Geometr for ondutor ttenution (We ssume Z is rel here.) 6

7 Coil Line: TEM Mode (ont.) Condutor ttenution: 1 Pl() Rs Js d C + C 1 π π Rs s φ s Rs J d + J dφ π π s Rs I R I dφ + π π π π Rs 1 Rs 1 φ π π Rs 1 Rs 1 I + I π π 1 Rs Rs I + 4π dφ I d + I dφ Geometr for ondutor ttenution R s R s ε ε ε ωµ σ R s (Here σ denote the ondutivit of the metl.) 7

8 Coil Line: TEM Mode (ont.) Condutor ttenution: α P P () P 1 Z I l 1 Rs Rs Pl () I + 4π R s ε ε ε R s Geometr for ondutor ttenution Hene we hve α I 1 Rs Rs 4 + π 1 Z I or α 1 1 R R + s s Z 4π 8

9 Coil Line: TEM Mode (ont.) Let s redo the lultion of ondutor ttenution using the Wheeler inrementl indutne formul. Wheeler s formul: R dz α s Zη d R s ε ε ε R s Geometr for ondutor ttenution The formul is pplied for eh ondutor nd the ondutor ttenution from eh of the two ondutors is then dded. In this formul, dl (for given ondutor) is the distne whih the onduting oundr is reeded w from the field region. η µ ε 9

10 α Rs dz Zη d Rs dz α + Zη d Hene Coil Line: TEM Mode (ont.) α R s dz α Zη d η Z ln π ( d d) ( d d) Rs η 1 Zη π Rs η 1 α Zη π or so 1 η R R α + Geometr for ondutor ttenution s s Zη π 1 1 R R α + R s ε ε ε R s s s Z 4π 1

11 Coil Line: TEM Mode (ont.) We n lso lulte the fundmentl per-unit-length prmeters of the loss oil line. From previous lultions: R s R s (From Notes 1) (From Notes 5) lossless L Z µε C G R α µε / Z ( ωc) tn lossless δ ( Z lossless ) ε, µσ, where lossless lossless η Z ln π η lossless µ ε The lossless supersript hs een dded to here to emphsie tht these vlues re rel. 11

12 Attenution for RG59 Co Approimte ttenution in db/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 overmoded 9.7 GH (TE wveguide mode) 11 Z 75Ω r.9 mm 1.85mm ε.5 (from Wikipedi) 1

13 Coil Line: Higher-Order Modes We look t the higher-order modes* of oil line. The lowest wveguide mode is the TE 11 mode. PEC ε, µσ, Sketh of field lines for TE 11 mode *Here the term higher-order modes mens the wveguide modes tht eist in ddition to the desired TEM mode. 13

14 Coil Line: Higher-Order Modes (ont.) TE : ( ρφ) kh( ρφ) h, +, eigenvlue prolem k k k ε, µσ, PEC The solution in lindril oordintes is: ( ρφ ) ( ρφ) H,, h, e jk h ( ρφ, ) Jn( kρ) sin( nφ ) Yn( kρ) os( nφ) Note: The vlue n must e n integer to hve unique fields. 14

15 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 15

16 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 16

17 Coil Line: Higher-Order Modes (ont.) We hoose (somewht ritrril) the osine funtion for the ngle vrition. Wve trveling in + diretion: ( ρφ ) ( ρφ) h,, h, e jk ε, µσ, PEC ( ρφ, ) os( φ) ( ( ρ) + ( ρ) ) h n AJ k BY k n n The osine hoie orresponds to hving the trnsverse eletri field E ρ eing n even funtion ofφ, whih is the field tht would e eited proe loted t φ. 17

18 Coil Line: Higher-Order Modes (ont.) Boundr Conditions: E φ Eφ (, φ ) H jωε ( ) H ρ ρ 1 ρ H Eφ (, φ ) 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 denotes derivtive with respet to the rgument. 18

19 Coil Line: Higher-Order Modes (ont.) AJ ( k ) + BY ( k ) n n AJ ( k ) + BY ( k ) n n In order for this homogenous sstem of equtions for the unknowns A nd B to hve non-trivil solution, we require the determinnt 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 19

20 Coil Line: Higher-Order Modes (ont.) J ( ky ) ( k) J ( ky ) ( k) n n n n Denote k Then we hve: ε, µ, σ ( ) ( ( )) ( ) F( ; n, / ) J ( ) Y / J / Y ( ) n n n n For given hoie of n nd given vlue of /, we n solve the ove eqution for to find the eros.

21 Coil Line: Higher-Order Modes (ont.) A grph of the determinnt revels the eros of the determinnt. F( n ;, / ) th np p ero Note: These vlues re not the sme s those of the irulr wveguide, lthough the sme nottion for the eros is eing used. n3 n1 n k np k np 1

22 Coil Line: Higher-Order Modes (ont.) Approimte solution: k 1 + / n 1 p 1 The TE 11 mode is the dominnt higher-order mode of the o (i.e., the wveguide mode with the lowest utoff frequen). Et solution Figure 3.16 from the Por ook

23 Coil Line: Lossless Cse Wvenumer: k ( ) k k k is rel here k f f π f µε k k Use formul on previous slide k f k k π µε π µε ε π r [m/s] TE 11 mode of o: f 1 ε π 1 + / r 3

24 Coil Line: Lossless Cse (ont.) f 1 1 ε π 1 + / r At the utoff frequen, the wvelength (in the dieletri) is: λ d f d f π + ε r ( 1 / ) Compre with the utoff frequen ondition of the TE 1 mode of RWG: d λ so ( ) λd π + ε r or λd + π / ( ) 4

25 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] 5