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ECE 3317
Prof. Ji Chen
Spring 2014
Notes 19
Waveguiding Structures
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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
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Transmission lines
Properties
 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)
    j 
 R  j L G  jC 
k z   j   j
 R  j L  G  jC 
   j
Lossless: kz   LC  
  k
(always real:  = 0)
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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”)
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Fiber-Optic Guide (cont.)
Two types of fiber-optic guides:
1) Single-mode fiber
Carries a single mode, as with the mode on a
waveguide. Requires the fiber diameter to be small
relative to a wavelength.
2) Multi-mode fiber
Has a fiber diameter that is large relative to a
wavelength. It operates on the principle of total
internal reflection (critical angle effect).
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Multi-Mode Fiber
Higher index core region
  max
  max
http://en.wikipedia.org/wiki/Optical_fiber
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Multi-Mode Fiber (cont.)
Higher index core region
  max
c
 max
 / 2  c
nrod
Assume cladding is air
At left end of rod:
nair sin  max


 nrod sin    c 
2

sin max  nrod cos c
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Multi-Mode Fiber (cont.)
Higher index core region
  max
At top boundary with air:
sin  max  nrod cos  c
nrod sin  c  nair sin  90o   1
 nrod 1  sin 2  c
 nrod 1  1 / nrod 
2
sin c  1 / nrod
2
sin max  nrod
1
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Properties
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)
Inside microwave oven
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)
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Waveguides (cont.)
Cutoff frequency property (derived later)
In a waveguide:
k   
We can write
kc  c 
kz   k  k
2

2 1/2
c
kc  constant
(wavenumber of material inside waveguide)
(definition of cutoff frequency)
  c : kz  k 2  kc2  real
(propagation)
  c : kz   j kc2  k 2  imaginary
(evanescent decay)
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Field Expressions of a Guided Wave
Statement:
All six field components of a guided wave can be expressed
in terms of the two fundamental field components Ez and Hz.
"Guided-wave theorem"
Assumption:
E  x, y, z   E 0  x, y  e  jkz z
H  x, y , z   H 0  x, y  e
 jk z z
(This is the definition of a guided wave.)
A proof of this statement is given next.
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Field Expressions (cont.)
Proof (illustrated for Ey)
H  j E
1  H z H x 
Ey 



j  x
z 
or
1  H z

Ey 


jk
H
z
x

j  x

Now solve for Hx :
 E   j  H
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Field Expressions (cont.)
 E   j  H
1  E z E y 
Hx  



j  y
z 

1  E z

 jk z E y 

j  y

Substituting this into the equation for Ey yields the result


1  H z
1  E z
Ey 
 jk z  
 jk z E y  


j  x
 j  y

Next, multiply by
 j  j   k 2
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Field Expressions (cont.)
This gives us
H z
E z
k E y  j
 jk z
 k z2 E y
x
y
2
or
H z
E z
 k  k  E y  j x  jkz y
2
2
z
Solving for Ey, we have:
 j  H z  jkz  E z
Ey   2
 2
2 
2 
 k  k z  x  k  k z  y
The other three components Ex, Hx, Hy may be found similarly.
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Field Expressions (cont.)
Summary of Fields
  j  H z  jk z  E z
Ex   2
 2
2 
2 
 k  k z  y  k  k z  x
 j  H z  jkz  E z
Ey   2
 2
2 
2 
 k  k z  x  k  k z  y
 j  E z  jk z  H z
Hx   2
 2
2 
2 
k

k

y
k

k

z 

z  x
  j  E z  jk z  H z
Hy   2
 2
2 
2 
k

k

x
k

k

z 

z  y
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TEMz Wave
Assume a TEMz wave:
Ez  0
Hz  0
To avoid having a completely zero field,
k k 0
2
2
z
Hence,
TEMz
kz  k
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TEMz Wave (cont.)
Examples of TEMz waves:
 A wave in a transmission line (no conductor loss)
 A plane wave
In each case the fields do not have a z component!
kz  k

z






S
E
H

Coax
H
x
Plane wave
y
E
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TEMz Wave (cont.)
Wave Impedance Property of TEMz Mode
Faraday's Law:
  E   j H
Take the x component of both sides:
The field varies as
E z E y

  j H x
y
z
E y  x, y, z   E y 0  x, y  e jkz
Hence,
   jk  E y   j H x
Therefore, we have





 
Hx
k

 
Ey

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TEMz Wave (cont.)
  E   j H
Now take the y component of both sides:
Hence,
  jk  Ex   jH y
Therefore, we have
Hence,
E z E x


  j H y
x
z
E x 






Hy
k

 
Ex

Hy
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TEMz Wave (cont.)
Summary:
Ey
Hx
 
Ex

Hy
These two equations may be written as a single vector equation:
E    zˆ  H 
The electric and magnetic fields of a TEMz wave are perpendicular to
each other, and the amplitudes of them are related by .
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TEMz Wave (cont.)
Examples

z






S
E
H
Plane wave
y

H
E
x
Coax
E
H
The fields look like a plane
wave in the central region.
Microstrip
“Quasi-TEM” (TEM-like at low frequency)
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Waveguide
In a waveguide, the fields cannot be TEMz.
y
PEC boundary
A
Proof:
nˆ   zˆ
C
E
Assume a TEMz field
x
B
waveguide
B
 E  dr  0
(property of flux line)
C
A
 E  dr   
C
S
 E  dr  0
B
B
 nˆ dS   z dS  0
t
t
S
contradiction!
(Faraday's law in integral form)
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Waveguide (cont.)
In a waveguide (hollow pipe of metal), there are two types of fields:
TMz: Hz = 0,
Ez  0
TEz: Ez = 0,
Hz  0
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