prof. david r. jackson notes 19 waveguiding structures waveguiding structures ece 3317 1 spring 2013
TRANSCRIPT
2
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
3
Transmission lines
j 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)
4
Fiber-Optic Guide
Properties
Has a single dielectric rod Can propagate a signal at any frequency (in theory) 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 (“hybrid mode”)
5
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).
7
Multi-Mode Fiber (cont.)
maxsin sin2air rod cn n
maxsin cosrod cn
c
/ 2 c maxrodn
At left end of rod:
Assume cladding is air
Higher index core region
max
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Multi-Mode Fiber (cont.)
max
2
2
sin cos
1 sin
1 1 /
rod c
rod c
rod rod
n
n
n n
osin sin 90 1rod c airn n
2sin 1max rodn
sin 1 /c rodn
At top boundary with air:
Higher index core region
max
9
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)
Inside microwave oven
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Waveguides (cont.)
Cutoff frequency property (derived later)
1/22 2z ck k k
k (wavenumber of material inside waveguide)
c ck (definition of cutoff frequency)
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
We can write
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Field Expressions of a Guided Wave
All six field components of a guided wave 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.)
A proof of this statement is given next.
Statement:
"Guided-wave theorem"
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Field Expressions (cont.)
Proof (illustrated for Ey)
or
H E
HH1E xz
y
j
j x z
Now solve for Hx :
E Hj
H1E Hz
y z xjkj x
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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
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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.
2 2 H EE z z
z y zk k j jkx y
or
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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
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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,
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TEMz Wave (cont.)
Examples of TEMz waves:
In each case the fields do not have a z component!
A wave in a transmission line (no conductor loss) A plane wave
E
H
Coax x
y
z
EH
S Plane wave
zk k
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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
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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
20
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
each other, and the amplitudes of them are related by .
Summary:E
Hy
x
E
Hx
y
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TEMz Wave (cont.)
Examples
E
H
Coax x
y
z
EH
S Plane wave
Microstrip
The fields look like a plane wave in the central region.
EH
“Quasi-TEM” (TEM-like at low frequency)
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Waveguide
In a waveguide, the fields cannot be TEMz.
0B
A
E dr (property of flux line)
ˆ 0z
C S S
BBE dr n dS dS
t t
(Faraday's law in integral form)
0C
E dr contradiction!
Proof:
Assume a TEMz field
y
x
waveguide
E
PEC boundary
A
B
Cˆ ˆn z