ECE 1100 Introduction to Electrical and Computer Engineering

1
ECE 6340 Intermediate EM Waves
Fall 2005
Prof. Donald R. Wilton ECE Dept.
Notes 9 (Notes based on those of D. R. Jackson)
2
Fields of a Guided Wave
z
Assume
Then
3
Fields of a Guided Wave (cont.)
Proof (for Ey)
or
Now solve for Hx
4
Fields of a Guided Wave (cont.)
Substituting this into the equation for Ey
yields the result
Multiply by
5
Fields of a Guided Wave (cont.)
or
The other components may be found similarly.
6
TEM Wave
Wavenumber property
To avoid having a completely zero field,
so
Use
Note
7
TEM Wave (cont.)
Lossless TL
so
The phase velocity is equal to the speed of light
in the dielectric.
8
TEM Wave (cont.)
Static property
and are 2D static field
functions.
9
TEM Wave (cont.)
Proof
Therefore, only a z component of the curl
exists. We next prove that this must be zero.
10
TEM Wave (cont.)
Now use
Also,
11
TEM Wave (cont.)
Hence
Therefore,
12
TEM Wave (cont.)
( No charge density in the time-harmonic steady
state, for a homogeneous medium)
Also,
Therefore,
Hence,
13
TEM Wave (cont.)
transmission line
14
TEM Wave (cont.)
A
B
The potential function is therefore unique, and
is the same as the static potential function.
15
TEM Wave (cont.)
Similarly,
so
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TEM Mode Magnetic Field
so
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TEM Magnetic Field (cont.)
Also,
so
This can be written as
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TEM Mode Charge Density
TEM mode
y
x
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TEM Charge Density (cont.)
so
Hence
Note ?? ? ?c
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Example Microstrip Line
Ignore substrate and ground plane
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Example (cont.)
Line in free space with a static charge density
(This was first derived by Maxwell using
conformal mapping.)
Hence
In this result, I0 is the total current Amps
on the strip.
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Example Coaxial Cable
Find E, H
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Example (cont.)
Boundary conditions
so
Hence
Therefore
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Example (cont.)
25
Example (cont.)
This result is valid at any frequency.
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Example (cont.)
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TEM Mode Telegraphers Eqs.
TEM mode (lossless conductors)
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Telegraphers Eqs. (cont.)
Note v is path independent in the (x,y) plane
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Telegraphers Eqs. (cont.)
Use
So
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Telegraphers Eqs. (cont.)
-?
But
so
Note L is the magnetostatic (DC) value.
or
31
Telegraphers Eqs. (cont.)
If we add R into the equation
This is justifiable if the mode is approximately
a TEM mode (small conductor loss).
32
Telegraphers Eqs. (cont.)
Amperes law
so
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Telegraphers Eqs. (cont.)
Now use
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Telegraphers Eqs. (cont.)
Hence
35
Telegraphers Eqs. (cont.)
36
Telegraphers Eqs. (cont.)
Hence
or
37
Telegraphers Eqs. (alternate derivation)
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Telegraphers Eqs. (alternate derivation)