Fundamental Waveguide
Theory
UCF
Types of Waveguides
1. TEM and quasi-TEM waveguides
2. Metallic waveguides (TE and TM modes)
3. Dielectric waveguides (TE, TM, TEM or
hybrid modes)
UCF
TEM and Quasi-TEM Waveguides
TEM:
Strip Line
Rectangular Coaxial
r
r
r
Microstrip Line
Slot Line
Coplanar Line
Coaxial
Quasi TEM:
UCF
Metallic Waveguides
Rectangular waveguide
Circular waveguide
Dielectric Waveguides
UCF
r
Planar dielectric
waveguide
 r1
r2
Fiber
 r1
r2
Ridge waveguide
UCF
Modes in Waveguides (1)
Modes: certain field patterns that can
propagate independently
TEM mode: Transverse Electromagnetic
mode. All the fields are in the cross
section or there are no Ez and Hz
components
UCF
Modes in Waveguides (2)
TE modes: transverse electric modes
Electric fields are in the cross section (or
no Ez component).
Only Hz component in the longitudinal
direction. Also called H modes
TM modes: transverse magnetic modes
Magnetic fields are in the cross section (or
no Hz component).
Only Ez component in the longitudinal
direction. Also called E modes
Conditions for the Existence of
TEM Modes
UCF


At least two perfect electric conductors
Dielectric distribution in the cross
section is homogeneous
Yes
Yes
No
No
Note: TEM line can have higher order
TE and TM modes
Quasi-TEM Line
UCF

Some planar waveguide structure with
inhomogeneous dielectric distribution.
Ez  0 , H z  0 But E z  Et , H z  H t
r
r
r
Microstrip Line
Slot Line
Coplanar Line
UCF
Basic Waveguide Theory (1)
Equivalent
E  Et  a z E z
H  Ht  a z H z
with Et   etm ( x, y )Vm ( z )
m
H t   h tm ( x, y )I m ( z )
m
Generalized
Fourier Transform
UCF
Basic Waveguide Theory (2)
dVm ( z )
  jk zm Z 0 m I m ( z )
dz
dI m ( z )
  jk zmY0 mVm ( z )
dz
Z0m 
1
Y0 m
is the characteristic impedance of the mth mode
k zm
is the propagation constant of the mth mode
UCF
Basic Waveguide Theory (3)
Vm ( z )  Ame
 jk zm z
Forward Wave
 Bme
jk zm z
Backward Wave
Effective Relative Dielectric Constant of the mth mode
 eff
k zm 2
2
2
 ( )  k zm  k0  eff
k0
The mode with the largest is called the dominant mode.  eff
UCF
TEM Mode (1)
Maxwell’s Equations
   E   j  H
   H  j  E


  E  0
   H  0
TEM mode
Ez 0
H z 0
  E t   j  H t
  H  j E

t
t

  E t  0
  H t  0



Let   a x
 az
 ay
y 
z
x



 t   z

TEM Mode (2)
UCF

( t   z )  E t   j H t   t  E t  0

E t
(a z )  E t   j H t  a z  (
)   j H t
z
z
(  t   z )  H t   j  E t   t  H t  0

(a z
)  H t  j  E t 
z
 t  Et  0
t  Ht  0
H t
az  (
)  j E t
z
UCF
From
TEM Mode (3)
 t  Et  0
 t  Et  0
t  Ht  0
t  Ht  0
The fields in the cross-section are similar to
2-D electrostatic & 2-D magnetostatic fields
for TEM mode even if actual operating
frequency can be very high.
TEM Mode (4)
UCF
Let
E t   t 
 t   0
2
Laplace Equation
V12 

2
1
E  d l For voltage
I   H  dl
For current
TEM Mode (5)
UCF
E t
az (
)   j  H t
z
az (
H t
)  j  E t
z

E t
Ht  
az (
)
j 
z
1
E t

1 
(a z 
 a z  
z
 j   z

)   j  E t

 2Et
2
a z  (a z 
)


E t
2
z
 Et
 Et
2
a z (a z 
)

(
a

a
)


E t
z
z
2
2
z
z
2
2
 2 Et
2


k
Et  0
2
z
Likewise
 2Et
)0
where a z (a z 
2
z
where k 2   2 
 2Ht
2

k
Ht  0
2
z
TEM Mode (6)
UCF
For +z direction propagation mode, we have
E t  e t ( x , y ) e  jk z z


  jk z
z
H t  h t ( x , y ) e  jk z z
From
 2E t
2

k
Et  0
2
z
Since e t  0
Or:
where

 ( k 2  k z )e t  0
2
k 2  kz
2
kc  k 2  k z  0
2
2
kc  k x  k y
2
2
2
For TEM mode
TEM Mode (7)
UCF
For +z direction propagating TEM mode:
E  e ( x , y ) e  jkz
H  h ( x , y ) e  jkz
t V  0
2
e   tV
From: H t  
1
j 
az (
E t
)
z
 H 
 

k




  jk
z
jk
j 
az E
 h
az e
wave impedance of the media
inside the TEM waveguide

UCF
From
TE, TM and Hybrid Modes (1)
   E   j  H
   H  j  E
  2 E  k 2 E  0


 2
2


E

0



H
k
H 0


   H  0
  2 E z  k 2 E z  0
For z component:
 2
2
  H z  k H z  0
  t2 E z  k c2 E z  0
or  2
  t H z  k c2 H z  0
2
2
2
2
2


k
,
k

k

k
since
z
c
z
2
z
UCF
TE, TM and Hybrid Modes (2)

For a mode propagating in +z direction:
  jk z
z
From   E   j  H
  H  j  E
 E z
  y  jk z E y  j  H x

E z

  j  H
   jk z E x 
x

 E y E x

  j  H z

y
 x
y
 H z
  y  jk z H y   j  E x

H z

 j  E y
  jk z H x 
x

 H y H x

 j  E z

y
 x
UCF
TE, TM and Hybrid Modes (3)
E
E
x
y
H
H
Or
 jk

k c2
t
E z
 j   H z

y
x
k c2
z
E z
j   H z

y
x
k c2
x

 jk
j   E z

y
k c2
k c2
y

 jk
 j   E z

x
k c2
k c2
Et 
H
 jk
k c2

z

 jk
k c2
z
tEz 
z
H z
x
z
H z
y
j 
a z  t H
2
kc
 jk
 j 
a
E


z
t
z
k c2
k c2
z
z
tH
z
UCF
Auxiliary Potential Approach
x
E
y
Actually:
Az
1 Fz
 

x

 y
k z Az
1 Fz
 

  y
 x
k z Fz
1 Az


x
 y

k z Fz
1 Az
 

 x
  y
k
In Balanis’s book: E
H
x
H
y
Ez   j
k c2

z
Az , H
z
  j
k c2

Fz
 t2 A z  k c2 A z  0 ,  t2 F z  k c2 F z  0
Balanis’s is my k .
TE Modes (1)
UCF
E
x
 
j   H z
x
k c2
E
y

 jk z  H z
x
k c2
H
x

x

 jk z  H z
y
k c2
H
y

y

Ex
Ey 
H
H
E
Ez  0
 H
2
t
Et 
t

k H

z
 0
H
z
j 
a z  t H
2
kc

2
t
z
 
H
1 Fz
 y
1 Fz
 x
k z Fz


x
k z Fz


y
0
 j   H z

y
k c2
2
c

a z H
kz
 jk
k c2
z
z
tH
t
z
z
  j
F
z
 k
k
2
c

2
c
F
z
A
z
 0
TE Modes (2)
UCF
For a mode propagating
in +z direction:
E  e ( x , y ) e  jk z z
H  h ( x, y )e
 jk z z
 t2 h z  k c2 h z  0 + Boundary Conditions
 j   h z
ex 
y
k c2
ey 
j   h z
k c2  x
hx 
 jk
k c2
hy 
 jk
k c2
ht 
 jk
k c2
et  
Z TE 

kz

kz
z
z
h z
x
z
h z
y
 thz
a z  h t   Z TE a z  h t
characteristic impedance
of TE modes
TM Modes (1)
UCF
Ex 
E
H
H
H
y
x
y
z
 jk
k c2
z
E z
x
E
z
E z
y
E
 jk

k c2
x
 
y
 
j   E z

k c2  y
H
x
 j   E z

x
k c2
H
y
H
z
 0
 Ez  k Ez  0
2
t
2
c
 jk z
Et 
tEz
k c2
Ht 

 j 
a z  t E z
2
kc

kz
a z E t
E
z

2
t
k
Az
x
Az
y
z

k
z

1 Az
 y
1 Az
 
 x
 0

  j
A
z
k
2
c
A

 k
2
c
A
z
z
 0
TM Modes (2)
UCF
For a mode propagating
in +z direction:
E  e ( x , y ) e  jk z z
H  h ( x, y )e
 jk z z
 t2 e z  k c2 e z  0
ex 
 jk
k c2
ey 
 jk
k c2
hx 
j   e z
k c2  y
hy 
 j   e z
x
k c2
et 
ht 
z
e z
x
z
e z
y
+ Boundary Conditions
 jk z
 tez
2
kc

kz
a z E t 
Z TM 
kz

1
Z TM
a z E t
characteristic impedance
of TM modes
Hybrid Modes
UCF
For a mode propagating
in +z direction:
E  e ( x , y ) e  jk z z
H  h ( x , y ) e  jk z z
  t2 e z  k c2 e z  0
 2
2
h
k


c hz  0
 t z
+ Boundary Conditions
ex 
 jk z  e z  j   h z

2
x
y
k c2
kc
ey 
 jk z  e z
j   h z

k c2  y
k c2  x
hx 
j   e z  jk z  h z

2
k c y
k c2  x
hy 
 jk z  h z
 j   e z

x
k c2
k c2  y
et 
 jk z
j 

e

a z  t h z
t z
2
2
kc
kc
ht 
 jk
 j 


a
e
z
t z
k c2
k c2
z
 thz
UCF
Phase Velocity and Group Velocity (1)
For a single frequency:
e
 jk z z
cos(  t  k z z )
Phase velocity: the velocity of constant phase plane (t-kzz=const)
vp 

kz
UCF
Phase Velocity and Group Velocity (2)
Waveguide
f (t ) cos( 0 t )

 Re f (t )e j0t

Ae
 jk z z
s (t )
S ( )  F (   0 ) Ae  jk z z
F (   0 )
F  f (t )  F ( )
F ( )
 m
m

1

Re  S ( )e jt d 
 s (t ) 
 

2
 0  m
1
Re[ 

S ( )e jt d ]
0  m
2
 0  m
1

Re[ 
AF (   0 )e j (t  k z z ) d ]
0  m

2
UCF
Phase Velocity and Group Velocity (3)
 0  m
1
s (t ) 
Re[ 
AF (   0 )e j (t  k z z ) d ]
0  m
2
Expand
Let
dk z
k z ( )  k z ( 0 ) 
 0 (   0 )  
d
dk z
 k z ( 0 ) 
 0 (   0 )
d
k z 0  k z ( 0 )
dk z
k 'z0 
d
  0
 k z ( )  k z 0  k z' 0 (   0 )
UCF
Phase Velocity and Group Velocity (4)
 0  m
1
jt  j k z 0  k z' 0 (  0 ) z
s (t ) 
Re[ 
AF (   0 )e e
d ]



0
m
2
p    0 A
j ( 0 t  k z 0 z )  m
j t  k z' 0 z  p

Re[e
F ( p )e
dp ]



m
2
 A Re f (t  k z' 0 z )e j (0t  k z 0 z )


 Af (t  k z' 0 z ) cos( 0 t  k z 0 z )
Information with group velocity
k z 0  k z ( 0 )
1
1
vg 

k ' z 0 dk z
  0
d
Carrier with phase velocity
vp 
0
k z0
UCF
Phase Velocity and Group Velocity (5)
f (t  k z' 0 z ) cos( 0 t  k z 0 z )
t
z
Cutoff Frequency in
Metallic Waveguide
UCF
For
e
When
 jk z z
k  0,
2
z
the mode is an evanescent mode.
Find cutoff frequency
k z2  k 2  k c2  0
k c is cutoff wavenumber.
From k c  2f c 
cutoff frequency f c 
,
kc
2 
UCF
Degenerate Modes
If two or more modes have the same eigenvalue
(propagation constant kz) but different
eigenvectors (field patterns), they are called
degenerate modes.