ECE 3317
Prof. D. R. Wilton
Notes 19
Waveguiding Structures
[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
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)

 R  j L  G  jC 
k z   j   j
 R  j L  G  jC 
   j
Lossless: kz   LC  
  k
(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
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).
Multi-Mode Fiber
Higher index core region
http://en.wikipedia.org/wiki/Optical_fiber
Waveguide
Properties
 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)
http://en.wikipedia.org/wiki/Waveguide_(electromagnetism)
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 determined
by guide geometry
(wavenumber of material inside waveguide)
(wavenumber of material at cutoff frequency c )
  c : k z  k 2  kc2  real
(propagation)
  c : k z   j kc2  k 2  imaginary
(evanescent decay)
Field Expressions of a Guided Wave
Statement:
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.
"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.
Note for such fields,    =  jk z   )
z
A proof of the “statement” above is given next.
Field Expressions (cont.)
Proof (illustrated for Ey)
H  j E
or
1  H z H x 
Ey 



j  x
z 
1  H z

Ey 


jk
H
z
x

j  x

Now solve for Hx :
 E   j  H
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  E z

1  H z
Ey 
 jk z  
 jk z E y  


j  x
 j  y

Next, multiply by
 j  j   k 2
Field Expressions (cont.)
This gives us
H z
E z
k E y  j
 jk z
 k z2 E y
x
y
2
Solving for Ey, we have:
 j  H z  jk z  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.
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  jk z  E z
Ey   2
 2
2 
2 
k

k

x
k

k

z 

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
Field Expressions (cont.)
TMz Fields
TEz Fields
 jk z  E z
Ex    2
2 
 k  k z  x
  j  H z
Ex   2
2 
k

k
z  y

 jk z  E z
Ey   2
2 
k

k
z  y

 j  H z
Ey   2
2 
k

k
z  x

 j  E z
Hx   2
2 
k

k
z  y

 jk z  H z
Hx    2
2 
k

k
z  x

  j  E z
Hy   2
2 
k

k
z  x

 jk z  H z
Hy   2
2 
k

k
z  y

TEMz Wave
Assume a TEMz wave:
Ez  0
Hz  0
To avoid having a completely zero field,
k 2  kz2  0
Hence,
TEMz
kz  k
TEMz Wave (cont.)
Examples of TEMz waves:
 A wave in a transmission line
 A plane wave
In each case the fields do not have a z component!
z



S




E
H

H
x
coax
E
plane wave
y
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Hx
Therefore, we have





 
Hx
k

 
Ey

TEMz Wave (cont.)
 E   j H
Now take the y component of both sides:
Hence,
  jk  Ex   jHy
Therefore, we have
Hence,
E z E x


  j H y
x
z
Ex

Hy
E x 






Hy
k

 
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 one
another, and their amplitudes are related by .
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
z







E = ˆ
V0
e jkz
 ln b a
S
V0
H = ˆ
e jkz
 ln b a

coax
plane wave
y
H
x
E
z
The fields look like a plane
wave in the central region.
d
V0  jkz x
V0  jkz microstrip
H  yˆ
e
E  xˆ
e
d
d
y
Waveguide
In a waveguide, the fields cannot be TEMz.
y
PEC boundary
A
Proof:
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)
Waveguide (cont.)
In a waveguide, there are two types of fields:
TMz: Hz = 0, Ez  0
TEz: Ez = 0,
Hz  0