ECE Spring Prof. David R. Jackson ECE Dept. Notes 7

Transcription

1 ECE 6341 Spring 216 Prof. David R. Jackson ECE Dept. Notes 7 1

2 Two-ayer Stripline Structure h 2 h 1 ε, µ r2 r2 ε, µ r1 r1 Goal: Derive a transcendental equation for the wavenumber k of the TM modes of propagation. Assumption: There is no variation of the fields in the y direction, and propagation is along the direction. 2

3 Two-ayer Stripline Structure (cont.) TEN: 1 R 2 h 1 h 2 TM impedances: 1 k 1 ωε 1 2 k 2 ωε 2 Note: Because this is a lossless closed structure, k is either real (above cutoff) or imaginary (below cutoff). ( 2 2 ) 1/2 ( 2 2 k ) 1/2 2 k2 k k k k 1 1 Transverse Resonance Equation (TRE): in in 3

4 Two-ayer Stripline Structure (cont.) in in 1 2 h 1 h 2 in 1 tan ( 1 1) j tan ( k h ) j k h in TRE: or ( ) tan ( ) tan k h k h k ωε tan ( k h ) tan ( k h ) ωε2 k 4

5 TEN (cont.) Hence ε ( ) ε tan ( ) k tan k h k k h r r Assuming nonmagnetic materials, ( 2 2 ε ) 1/2 ( 2 2 k ) 1/2 2 kεr2 k k k k 1 r1 A similar analysis could be applied for the TE modes. 5

6 Parallel-Plate Waveguide Special case: parallel-plate waveguide h h1+ h2 h ε, µ r r k k k 1 2 6

7 Parallel-Plate Waveguide (cont.) ( ) ε tan ( ) ε k tan kh k kh r 1 r 2 or ( kh) ( kh) tan + tan 1 2 Helpful identity: We then have tan tan( a) + tan( b) + 1 tan( a) tan( b) ( a b) ( kh kh) ( kh) ( kh) tan tan 1 tan 2 This will be satisfied if ( kh kh) tan

8 Parallel-Plate Waveguide (cont.) Hence we have so ( ) tan kh kh mπ, m,1, 2 We have (from the separation equation) Hence k k ε k 2 2 r Note: TM and TE modes have the same wavenumber, but only the TM mode can eist for m. (the electric field is perpendicular to the metal walls). k 2 2 mπ kε r h 8

9 Waveguide With Slab y TE mn modes TM mn modes TEN : b w w a ( a w) /2 1 9

10 Waveguide With Slab (cont.) TE TE 1 ωµ k ωµ 1 k 1 TM TM 1 k ωε k 1 ωε 1 Wavenumbers: 2 2 nπ k k k b 2 2 nπ k1 kε r k b 2 1/2 2 1/2 Note: A F nπ y sin F( ) e b nπ y cos G( ) e b jk jk Note: In the BCs, A acts like E while F acts like H. 1

11 Waveguide With Slab (cont.) First try reference plane at : R 1 11

12 Waveguide With Slab (cont.) (2) (1) in in 1 w General formula: + j tan βl in j + tan β l Apply this twice: j tan( k ) (1) in + j tan( k w) j tan( k w) (1) (2) in 1 1 in 1 (1) 1 + in 1 12

13 Waveguide With Slab (cont.) (2) (1) in in 1 w Apply again: ( 2) in + j tan( k) ( 2) + jin tan( k) A mess! 13

14 Waveguide With Slab (cont.) Now try a reference plane at the center of the structure (the origin is now re-defined here). w R 1 TRE: But (from symmetry): Hence (This is the only finite number that will work.) 14

15 Waveguide With Slab (cont.) Now use an admittance formulation w 1 Y Y TRE: Y Y But (from symmetry): Y Y Hence Y (only finite number that will work) Hence 15

16 Waveguide With Slab (cont.) Hence, there are two valid solutions: PEC wall PMC wall PEC wall PMC wall 16

17 Waveguide With Slab (cont.) From symmetry: ( ) ( ) E AE Odd mode: A -1 Even mode: A +1 E E Plots of E Odd mode: PEC wall Even mode: PMC wall 17

18 Waveguide With Slab (cont.) PEC wall: odd mode PMC wall: even mode PEC : PMC : Ht Et, Hn, n Et Ht, En, n Note: even/odd is classification is based on the E field. E E Plots of E Odd mode: PEC wall Even mode: PMC wall 18

19 Eample: TE Even Modes (1) w in + j1 tan k1 2 1 (1) w 1 + jin tan k ( ) R w 1 in Set (1) w 1 + jin tan k1 2 w 1 + j[ j tan( k) ] tan k1 2 w 1 tan( k) tan k1 2 19

20 TE Even Modes (cont.) ωµ ωµ w tan ( k) tan k1 k k 2 1 or w k k 1tan( k) tan k1 2 TE even where 2 1/2 2 1/2 2 2 nπ 2 2 nπ 1 ε r k k k k k k b b Hence, the transcendental equation is in the form Fk ( ) 2

21 Four Possible Cases (We have only done one of them: TE even.) TE even TM even TE odd TM odd 21

22 Cutoff Frequency (TE Even) Then Set k (closed structure) nπ 2 2 nπ 1 ε r k k k k k k b b Hence, the transcendental equation is now in the form Fk ( ) where k f 2π fc µε c cutoff frequency 22

23 SE/SM Terminology y This terminology is often used in the microwave community. SE: ongitudinal Section Electric. This means the same thing as TE. The electric field vector of the mode has no component, and hence it lies within the y plane (This is called the longitudinal plane, which means the plane parallel to the slab face. SM: ongitudinal Section Magnetic. This means the same thing as TM. The magnetic field vector of the mode has no component, and hence it lies within the y plane (the longitudinal plane). 23