Prof. David R. Jackson ECE Dept. Spring 2014 Notes 8 ECE 6341 1.
Prof. David R. Jackson ECE Dept. Spring 2014 Notes 35 ECE 6341 1.
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Transcript of Prof. David R. Jackson ECE Dept. Spring 2014 Notes 35 ECE 6341 1.
1
Prof. David R. JacksonECE Dept.
Spring 2014
Notes 35
ECE 6341
2
Higher-order Steepest-Descent Method
Assume
g z
C
I f z e dz
0 0f z
This important special case arises when asymptotically evaluating the field along an interface (discussed later in the notes).
r
3
Higher-order SDM (cont.)
Along SDP:
g z
C
I f z e dz
00 g z g zg z
SDP
I e f z e dz
20 ( )s g z g z real
20 ( )g z sdz
I e f z s e dsds
Hence
Note: The variable s is taken as positive after we leave the SP, and negative before we enter the SP.
4
Higher-order SDM (cont.)
Define
Assume
20 ( )g z sdz
I e f z s e dsds
( )dz
h s f z sds
20g z sI e h s e ds
210 0 0
2h s h h s h s
0 0f z
Then
5
Higher-order SDM (cont.)
Note: (s is odd)2
0ss e ds
Hence 01
~ 02
g zsI h e I
2 22 2
02s s
sI s e ds s e ds
where
Use2
2
t s
dt s ds
6
Higher-order SDM (cont.)
0
1
23 02
3 3
2 2
3
2
12
2
1
1 3 1 1 1
2 2 2
1
2
ts
t
tI e dt
t
t e dt
1
0
x tx t e dt
1/ 2
1 ! 1 2 !
1 1
x x x x
x x
Recall:
7
Higher-order SDM (cont.)
Hence
0
3
2
1 1~ 0
4g zI h e
We now we need to calculate 0h
Note: The leading term of the expansion now behaves as 1 / 3/2.
8
Higher-order SDM (cont.)
2
3( ) 2
( )
h s f z s z s
h s f z s z s f z s z s
h s f z s z s f z s z s z s
f z s z s z s f z s z s
3
0 0 00 0 3 0 0 0h f z z f z z z f z z
9
Higher-order SDM (cont.)
Hence 3
0 00 0 3 0 0h f z z f z z z
We next take two derivatives with respect to s in order to calculate 0 , 0z z
2
2
2
s g z z s
g z z s g z z s
(1st)
(2nd)
20s g z g z
10
Higher-order SDM (cont.)
From (2nd):
or
1
2
0
20z
g z
0
20 SDPjz e
g z
2
0 02 0 0g z z g z z
We then have
0
2 2
arg
SDP
g z
11
Higher-order SDM (cont.)
3
3
0 2
3
g z z s g z z s z s g z z s z s g z z s
g z z s g z z s z s g z z s
(3rd)
22 g z z s g z z s (2nd)
3
0 0 00 0 3 0 0 0g z z g z z z g z z
2
0 00 0 3 0g z z g z z
or
12
Higher-order SDM (cont.)
Hence,
02
0
20
3
g zz
g z
2
0
0
010
3
g z zz
g z
1
2
0
20z
g z
Therefore,
where
13
Higher-order SDM (cont.)
We then have
3
23
00
00 2
0 0
20
2 23
3
SDP
SDP
j
j
h f z eg z
g zf z e
g z g z
14
Higher-order SDM (cont.)Summary
3
23
00
00 2
0 0
20
2 23
3
SDP
SDP
j
j
h f z eg z
g zf z e
g z g z
g z
C
I f z e dz 0 0f z
0
3
2
1 1~ 0
4g zI h e
0
2 2
arg
SDP
g z
15
Example
A line source is located on the interface.
y
0Ix
0
1Semi-infinite lossy earth
16
Example
zA
The field is TMz and also TEy. ˆ zE zE
y
0Ix
0
1Semi-infinite lossy earth
h
Consider the line source at a height h above the interface.
E j A
TE
17
Example
y
0Ix
0
1Semi-infinite lossy earth
h
TE
0 0 00 0
0 0
1 1
4y y y xjk y h jk h jk y jk xTE
xy y
Ie e e e dk
j k k
1 0
1 0
TE TETE
TE TE
Z Z
Z Z
00
0
01
1
TE
y
TE
y
Zk
Zk
TEy:
18
Example
00 0 1 0
0 1 0
11
4y x
TE TEjk y jk x
xTE TEy
I Z Ze e dk
j k Z Z
1/22 20 0y xk k k
y
0Ix
0
1Semi-infinite lossy earth
The line source is now at the interface (h = 0).
zA
19
Example (cont.)
1 0 1
1 0 0 1
21
TE TE TE
TE TE TE TE
Z Z Z
Z Z Z Z
00
0
01
1
TE
y
TE
y
Zk
Zk
1/22 20 0y xk k k
1/22 21 1y xk k k
Simplifying,
20
Example (cont.)
1 01
0 1 0 1
0 1
12
221 1
TEy y
TE TEy y
y y
k kZ
Z Z k kk k
Hence
Therefore,
00 0
0 1
1
2y x
jk y jk xx
y y
Ie e dk
j k k
Now use
0
0
0 0
sin
cos
cos
x
x
y
k k
dk k d
k k
sin
cos
x
y
21
Example (cont.)
or
0 cos0 0 0
12 2 2 2
0 1 0
cos
2cos sin
j k
C
I ke d
jk k k
0 cos0 0
12 2 21
cos
2cos sin
j k
C
Ie d
jn
1 1 0/n k k
Note: 2 2 21
0 1 0 0 12 2 2 21 1 0
2 2 20 0 1 0 1
sin , sinsin
sin , siny
k k k k kk k k
jk k k k k
22
Example (cont.)
12 2 21
cos
cos sinf
n
0 Saddle point:
We then identify
Note: There are branch points in the complex plane arising from ky1, but we are ignoring these for a lossy earth (n1 is complex). (There are no branch points in the plane arising from ky0, since the steepest-descent transformation has removed them.)
cosg j
k
1/22 20 0 cosy xk k k k
23
Example (cont.)
As
090 , 90
0 0f (unless 1 = 2 )
12 2 21
cos
cos sinf
n
Hence
24
Example (cont.)
0 12 2 21
cos
cos sinf
n
Far-field (antenna) radiation pattern:
0g j
k
0 / 2
0g j
0 /40 0
12 2 021
cos 2
2cos sin
j k jIe e
j k jn
0
00
2~ SDPg z jI f z e e
g z
25
Example (cont.)
Final far-field radiation pattern:
0 /40 0
12 2 021
cos 2
2cos sin
j k jz
IE j e e
j kn
y
0Ix
0
1Semi-infinite lossy earth
0 / 2
Note: The pattern has a null at the horizon.
26
Example (cont.)
Interface field: / 2
0 cos0 0
12 2 21
cos
2cos sin
j k
C
Ie d
jn
0
3
2
1 1~ 0
4g zI h e
0 0
2
II
j
From the higher-order steepest-descent method, we have:
27
Example (cont.)
00 0
3/2
0
1 10
2 4j k
z
IE j h e
j k
We then have
28
Example (cont.)We have
cosg j
with
3
23
00
00 2
0 0
20
2 23
3
SDP
SDP
j
j
h f eg
gf e
g g
0 / 2
12 2 21
cos
cos sinf
n
4SDP
29
Example (cont.)We have
cosg j
0g j
0 0g
0g j
0 0g
0 / 2
30
Example (cont.)
We therefore have
with
3
3 /4200 2 jh f e
12 2 21
cos
cos sinf
n
31
Example (cont.)We then have
12 2 21
cos
cos sinf
n
1 1
2 2 2 22 21 1sin cos sin sin cos 1 sin cosN n n
2
D N N Df
D
1 12 2 2 22 21 1
212 2 21
sin cos sin sin cos 1 sin cos
cos sin
n n
f
n
21
2 2 21cos sinD n
32
Example (cont.)We then have
2
D N N Df
D
1 12 2 2 22 2
1 1
1 1 32 2 2 2 2 2 22 2 2
1 1 1
cos cos sin sin sin 1 sin cos
cos sin 1 sin cos sin cos sin sin cos sin sin cos
N n n
n n n
1 1
2 2 2 22 21 12 cos sin sin sin sin cosD n n
where
33
Example (cont.)At the saddle point we have
0 0N
20 12 1D n
0 21
1
1f
n
20 1 1D n
0 21
2
1f
n
20 1 1N n
34
Example (cont.)We then have
3
3 /4221
20 2
1jh e
n
0
33 /40 0 2
3/221 0
1 2 12
2 4 1j kj
z
IE j e e
j n k
The field along the interface is then
35
Example (cont.)
The field in space ( < / 2) is
0
33 /40 0 2
3/221 0
1 2 12
2 4 1j kj
z
IE j e e
j n k
The field along the interface ( = / 2) is
0 /40 0
12 2 021
cos 2
2cos sin
j k jz
IE j e e
j kn
Summary
36
Example (cont.)
Space wave Line Source
Lateral wave
01 jke
0
3/2
1 jke
Lossy earth
37
Extension to Dipole
Dipole SourceSpace wave
0
2
1 jk rer
01 jk rer
Lateral wave
Lossy earth
38
Important Geophysical Problem
Lateral wave
TX line source
RX line sourceSpace wave
c c
R
1R 2RrR
Lossy earth
The field is asymptotically evaluated for R
Two types of wave fields are important for large distances:
Space wave Lateral wave
Note: More will be said about these waves in the next chapter on “Radiation Physics of Layered Media.”
39
Important Geophysical Problem (cont.)
1 11 1
rjk R jk RTE
r
e eR R
1 1 1 2
0
3/ 21 2
1 jk R jk Rjk e e
eR R
Space wave:
Lateral wave:This will be the dominant
field for a lossy earth (k1 is complex).
Note: Amplitude constants have been suppressed here.
Lateral wave
TX line source
RX line sourceSpace wave
c c
R
1R 2RrR
Lossy earth
h1h2