ECE 1100 Introduction to Electrical and Computer Engineering

1
ECE 6340 Intermediate EM Waves
Fall 2005
Prof. David R. Jackson ECE Dept.
Notes 17
2
General Plane Waves
Assume
where
Helmholtz Eq.
so
which gives
(separation equation)
or
3
General Plane Waves (cont.)
Denote
Then
(separation equation)
and
4
General Plane Waves (cont.)
We can also write
The ? vector gives the direction of most rapid
phase change. The ? vector gives the direction
of most rapid attenuation.
Now look at Maxwells equations
5
General Plane Waves (cont.)
Hence
(1)
Similarly
Note ??c in general.
(2)
6
General Plane Waves (cont.)
(1)
(2)
Note ??c in general.
7
General Plane Waves (cont.)
Equations (1) and (2) imply that
since
8
Power Flow
9
Power Flow (cont.)
so
Use
so
10
Power Flow (cont.)
Assume E0 real vector (or a real vector times a
constant)
Hence
11
Power Flow (cont.)
so
Denote
12
Power Flow (cont.)
Then
Power flows in the direction of ?
Note The conclusion is also valid if we assume
that k real vector (times a complex constant).
13
Direction Angles
First, assume k real vector
The direction angles (?, f) are defined by
14
Direction Angles (cont.)
Even when (kx , ky , kz) become complex, these
equations define the direction angles, which may
be complex.
15
Homogeneous Plane Wave
(?, f) are real angles
In this case
Hence
where
16
Example
Plane wave
An infinite current sheet at z 0 launches a
plane wave.
17
Example (cont.)
Part (a)
Choose
Then
18
Example (cont.)
Part (b)
Choose
Then
19
Example
Power flow (in xy plane)
y
so
Note the inverse cosine should be chosen so that
the sin is correct (to give the correct kx).
so
Wave is evanescent in the z direction
20
Example (cont.)
Free space acts as a low-pass filter.
21
Radiation from Waveguide
22
Radiation from Waveguide (cont.)
Fourier transform pair
23
Radiation from Waveguide (cont.)
Hence
24
Radiation from Waveguide (cont.)
Hence
25
Theorems
Theorem 1
Theorem 2
If PW is homogeneous
(lossy)
(lossless)
Theorem 3
If medium is lossless
26
Example
Note It can be seen that
Find H and compare its magnitude with that of E.
27
Example (cont.)
28
Example (cont.)
Note The field magnitudes are not related by ?0 !