ECE 6340 Intermediate EM Waves. Fall 2016 Prof. David R. Jackson Dept. of ECE. Notes 18

Transcription

1 C 6340 Intermediate M Waves Fall 206 Prof. David R. Jacson Dept. of C Notes 8

2 T - Plane Waves φˆ θˆ T φˆ θˆ A homogeneous plane wave is shown for simplicit (but the principle is general). 2

3 Arbitrar Polariation: Decomposition Assume that and are real vectors in the figure for simplicit. 0 α 0 φˆ and T parts: θˆ 0 cos α φˆ 0 sin α φˆ 0 cos α 0 sin α T θˆ θˆ 3

4 Plane Wave or or = jωε j = jωε =ωε c c c Tae component: ωε ˆ ( ˆ ˆ ˆ ) ( ˆ ˆ c = ) = so = ωε c Assume that the wave is propagating upward (+ direction) so that is positive (or has a positive real part for a loss medium). 4

5 Plane Wave (cont.) Tae component: ωε ˆ ( ˆ ˆ ˆ ) ( ˆ ˆ c = ) = = ωε c Define: = ωε c 5

6 Plane Wave (cont.) Both results are summaried in a vector equation: or = ( ˆ ) t t = ( ˆ t) t Note: t stands for transverse, meaning the and components. Consider replacing Recall: = ωε c 6

7 Plane Wave (cont.) so t = ( ˆ t) (ere is the same as before, defined for a + wave.) Summar for both cases: t = ± ( ˆ t) = ( ˆ ) t t 7

8 T Plane Wave = jωµ or j = jωµ or =ωµ Tae component: ωµ ˆ ( ˆ ˆ ˆ ) ( ˆ ˆ = ) = so ωµ = 8

9 T Plane Wave (cont.) Tae component: ωµ ˆ ( ˆ ˆ ˆ ) ( ˆ ˆ = ) = so ωµ = = η Define: T ωµ = 9

10 T Plane Wave (cont.) Then T = ( ˆ ) t t = ( ˆ t) T t Allowing for both directions, + and -, we have: t =± ( ˆ t) T T = ( ˆ ) t t 0

11 Transverse quivalent Networ Denote Assume a plane wave going upward (the propagation is in the + direction). (,, ) = eˆ V ( ) ψ (, ) t t (,, ) = hˆ I ( ) ψ (, ) t t where j ( + ) ψ t (, ) = e eˆ hˆ = ˆ ρ = ˆ φ t t As we will see, V () and I () behave as voltage and current on a TL

12 TN (cont.) Assume a + wave: V ( ) = Ae I ( ) = Be j j The form is the same as the waves on a TL. Also Therefore t = ( ˆ t) ˆ ψ (, ) ( ) (, ) ( )( ˆ ˆ t I h = ψt V e ) We choose: hˆ = ˆ eˆ 2

13 TN (cont.) We then have I ( ) = V ( ) If we assume a - (downward) wave: I ( ) = V ( ) ence, in summar: I ( ) = ± V ( ) This proves that the transverse fields behave as voltage and current on a TL. 3

14 TN (cont.) I () + V () - t t T I T () + V T () - T t t 4

15 TN (cont.) Note: V() and I() model onl the transverse fields, but we can obtain the component of the fields from these. ample: Find for a plane wave: = jωε c ˆ = ( ) jωε c = jωεc = ( j ) + ( j ) jωε c 5

16 Reflection From Interface Incident Reflected θ i θ r # θ t #2 Transmitted We want to find the directions of the reflected and transmitted waves, and the reflection and transmission coefficients. Note: The plane is the plane of incidence (the plane containing the incident wave vector and the unit normal to the boundar). 6

17 Reflection From Interface Incident Reflected θ i θ r # θ t #2 Transmitted = sinθcosφ = sinθsinφ = cosθ φ = π /2 = 0 = sinθ = cosθ 7

18 Reflection From Interface (cont.) θ i θ r # θ t #2 Phase matching condition: ence = = θ i r t i = θ r (law of reflection) sinθ = i sinθ r 8

19 Reflection From Interface (cont.) θ i θ r # θ t #2 Phase matching condition: i = r = t ence sinθ = i 2 sinθ t (Snell s law) 9

20 Reflection From Interface (cont.) Determine reflection and transmission coefficients. (Assume a T wave) i i θ i θ r T Γ # θ t #2 20

21 Reflection From Interface (cont.) Modeling equations: V I i i = V ( ) e r r = V ( ) e t t = V ( ) e j j j i V ( ) = e r V ( ) =Γe t V ( ) = Te j + j j 2 sin θi 2 2 = cos = = i θ θ 2 2 cos 2= 2 = 2 t 2

22 Reflection From Interface (cont.) Practical note: When dealing with loss media, the wave in region 2 will be inhomogeneous. Therefore the transmitted angle will be comple. In this case it is usuall easier to wor with the separation equation (the square-root formula for 2 ) rather than the transmitted-angle formula. = sinθ = = sinθ = real 2 2 t i comple It is difficult to solve for θ t θ 2 2 cos 2= 2 2 = 2 t as ard (needs comple angle) 22

23 Reflection From Interface (cont.) Γ T # #2 T T 0 02 Γ= T T 02 0 T T T = +Γ= 2 T 02 T T

24 Reflection From Interface (cont.) ence Γ= ωµ 2 ωµ 2 ωµ 2 ωµ + 2 or Γ= µ µ µ + µ T = 2µ 2 µ + µ

25 Reflection From Interface (cont.) Percent power reflected: P = 00 Γ % r 2 Percent power transmitted: P = 00 P % % t r 25