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Prof. D. R. Wilton
Notes 19Notes 19
Waveguiding Structures Waveguiding Structures
ECE 3317
[Chapter 5]
Waveguiding Structures
A waveguiding structure is one that carries a signal (or power) from one point to another.
There are three common types:
Transmission lines
Fiber-optic guides
Waveguides
Transmission lines
R j L G j C
zk j j R j L G j C
j
zk LC k Lossless:
Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz)
Properties
(always real: = 0)
Fiber-Optic Guide
Properties
Has a single dielectric rod Can propagate a signal only at high frequencies Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields
Fiber-Optic Guide (cont.)
Two types of fiber-optic guides:
1) Single-mode fiber
2) Multi-mode fiber
Carries a single mode, as with the mode on a waveguide. Requires the fiber diameter to be small relative to a wavelength.
Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).
Multi-Mode Fiber
http://en.wikipedia.org/wiki/Optical_fiber
Higher index core region
Waveguide
Has a single hollow metal pipe Can propagate a signal only at high frequency: > c
The width must be at least one-half of a wavelength Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)
Properties
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)
Waveguides (cont.)
Cutoff frequency property (derived later)
1/22 2z ck k k
k (wavenumber of material inside waveguide)
c ck (wavenumber of material at cutoff frequency c )
2 2:c z ck k k real
2 2:c z ck j k k imaginary
(propagation)
(evanescent decay)
In a waveguide:ck
constant determined
by guide geometry
We can write
Field Expressions of a Guided Wave
All six field components of a wave guided along the z-axis can be expressed in terms of the two fundamental field components Ez and Hz.
Assumption:
0
0
E , , E ,
H , , H ,
z
z
jk z
jk z
x y z x y e
x y z x y e
(This is the definition of a guided wave.Note for such fields, )
A proof of the “statement” above is given next.
Statement:
"Guided-wave theorem"
= zjkz
Field Expressions (cont.)
Proof (illustrated for Ey)
or
H E
HH1E
H1E H
xzy
zy z x
j
j x z
jkj x
Now solve for Hx :
E Hj
Field Expressions (cont.)
E Hj
EE1H
E1E
yzx
zz y
j y z
jkj y
Substituting this into the equation for Ey yields the result
Next, multiply by
H E1 1E Ez z
y z z yjk jkj x j y
2j j k
Field Expressions (cont.)
2 2H EE Ez z
y z z yk j jk kx y
2 2 2 2
H EE z z z
yz z
jkj
k k x k k y
Solving for Ey, we have:
This gives us
The other three components Ex, Hx, Hy may be found similarly.
Field Expressions (cont.)
2 2 2 2
H EE z z z
xz z
jkj
k k y k k x
2 2 2 2
H EE z z z
yz z
jkj
k k x k k y
2 2 2 2
E HH z z z
xz z
jkj
k k y k k x
2 2 2 2
E HH z z z
yz z
jkj
k k x k k y
Summary of Fields
Field Expressions (cont.)
2 2
EE z z
xz
jk
k k x
2 2
EE z z
yz
jk
k k y
2 2
EH z
xz
j
k k y
2 2
EH z
yz
j
k k x
TMz Fields
2 2
HE z
xz
j
k k y
2 2
HE z
yz
j
k k x
2 2
HH z z
xz
jk
k k x
2 2
HH z z
yz
jk
k k y
TEz Fields
TEMz Wave
E 0
H 0z
z
2 2 0zk k To avoid having a completely zero field,
Assume a TEMz wave:
zk kTEMz
Hence,
TEMz Wave (cont.)
Examples of TEMz waves:
In each case the fields do not have a z component!
A wave in a transmission line A plane wave
E
H
coaxx
y
z
EH
S plane wave
TEMz Wave (cont.)
Wave Impedance Property of TEMz Mode
E Hj
EEHyz
xjy z
Faraday's Law:
Take the x component of both sides:
E Hy xjk j
0E , , E , jkzy yx y z x y eThe field varies as
Hence,
Therefore, we haveE
Hy
x k
TEMz Wave (cont.)
Now take the y component of both sides:
Hence,
Therefore, we have
E Hj
EEHxz
yjx z
E Hx yjk j
E
Hx
y k
Hence,E
Hx
y
TEMz Wave (cont.)
These two equations may be written as a single vector equation:
ˆE Hz
The electric and magnetic fields of a TEMz wave are perpendicular to one
another, and their amplitudes are related by .
Summary:E
Hy
x
E
Hx
y
The electric and magnetic fields of a TEMz wave are transverse to z, and their transverse variation as the same as electro- and magneto-static fields, rsp.
TEMz Wave (cont.)
Examples
coax x
y
z
EH
S plane wave
microstrip
The fields look like a plane wave in the central region.
0ˆE =ln
jkzVe
b a
0ˆH =ln
jkzVe
b a
x
y
0ˆE x jkzVe
d 0ˆH y jkzV
ed
d
z
Waveguide
In a waveguide, the fields cannot be TEMz.
Proof:
0B
A
E dr (property of flux line)
ˆ 0z
C S S
BBE dr n dS dS
t t
Assume a TEMz field
(Faraday's law in integral form)
y
x
waveguide
E
PEC boundary
A
B
C
0C
E dr contradiction!