EE 41139 Microwave Techniques: TEM, TE and TM Waves in Transmission Lines and Waveguides (39

EE 41139 Microwave Technique 1
Lecture 2
 TEM, TE and TM Waves
 Coaxial Cable
 Grounded Dielectric Slab Waveguides
 Striplines and Microstrip Line
 Design Formulas of Microstrip Line

Lecture 2
 An Approximate Electrostatic
Solution for Microstrip Line
 The Transverse Resonance
Techniques
 Wave Velocities and Dispersion

TEM, TE and EM Waves
 transmission lines and waveguides are
primarily used to distribute microwave
wave power from one point to another
 each of these structures is characterized
by a propagation constant and a
characteristic impedance; if the line is
lossy, attenuation is also needed

 structures that have more than one conductor
may support TEM waves
 let us consider the a transmission line or a
waveguide with its cross section being
uniform along the z-direction
b
a


 the electric and magnetic fields can
be written as
 Where and are the transverse
components and and are the
longitudinal components
E x y z e x y ze x y e
H x y z h x y zh x y e
t z
j z
t z
j z
( , , )  ( , )  ( , )] ( )
( , , )  ( , )  ( , )] ( )
   
   


[e
[e
t
t


1
2
et ht
et ht

 in a source free region, Maxwell’s
equations can be written as
 Therefore,
      
E j H H j E
 
,


 









E
y
j E j H
j E
E
x
j H
E
x
E
y
j H
z
y x
x
z
y
y x
z
  
   
  
,( )
,( )
,( )
3
4
5


 









H
y
j H j E
j H
H
x
j E
H
x
H
y
j E
z
y x
x
z
y
y x
z
 
  
 
,( )
,( )
,( )
6
7
8

 each of the four transverse
components can be written in terms
of and , e.g., consider Eqs. (3) and
(7): 

 
E
y
j E j H
z
y x
  
  
j H
H
x
j E
x
z
y





    
  
        
 
j H
H
x
j j H
E
y
j
k H j
E
y
j
H
x
H
j
k
E
y
H
x
k k
x
z
x
z
x
z z
x
c
z z
c



 



 












( ) / ( )
( )
( ) ( )
( )
2 2
2
2 2 2
9

 Similarly, we have
 is called the cutoff wavenumber
H
j
k
E
x
H
y
E
j
k
E
x
H
y
E
j
k
E
y
H
x
y
c
z z
x
c
z z
y
c
z z


    


    
     
2
2
2
10
11
12
( ) ( )
( ) ( )
( ) ( )


















kc

 Transverse electromagnetic (TEM)
wave implies that both and
are zero (TM, transverse magnetic,
=0, 0 ; TE, transverse electric,
=0, 0)
the transverse components are also zero
unless is also zero, i.e.,
Ez Hz
Hz Ez 
Ez Hz 
kc
k2 2 2
 
  

 now let us consider the Helmholtz’s
equation
 note that and therefore, for TEM
wave, we have
( ) ,
     







 
2 2
2
2
2
2
2
2
2
0 0
k E
x y z
k E
x x






2
2
2
z









2
2
2
2
0
x y
Ex








 

 this is also true for , therefore,
the transverse components of the
electric field (so as the magnetic
field) satisfy the two-dimensional
Laplace’s equation
Ey
     
t t
e
2 0 13
( )

 Knowing that and
, and we have
 while the current flowing on a
conductor is given by
 
t t
e
2 0
    
D e
t t
 0
 
t x y
2 0
( , )
V E dl
12 1 2 1
2 14
        

  ( )
I H dl
C
 
    ( )
15

 this is also true for , therefore, the
transverse components of the
electric field (so as the magnetic
field) satisfy the two-dimensional
Laplace’s equation
Ey
     
t t
e
2 0 13
( )

 Knowing that and
 , we have
 the voltage between two conductors is
given by
 while the current flowing on a conductor is
given by
 
t t
e
2 0     
D e
t t
 0
 
t x y
2 0
( , )
V E dl
12 1 2 1
2 14
        

  ( )
I H dl
C
 
    ( )
15

 we can define the wave impedance for the
TEM mode:
 i.e., the ratio of the electric field to the
magnetic field, note that the components
must be chosen such that E x H is pointing
to the direction of propagation
Z
E
H
TEM
x
y
      




 ( )
16

 for TEM field, the E and H are related
by
h x y
Z
z e x y
TEM
( , )  ( , ) ( )
     
1
17

why is TEM mode desirable?
 cutoff frequency is zero
 no dispersion, signals of different
frequencies travel at the same
speed, no distortion of signals
 solution to Laplace’s equation is
relatively easy

 a closed conductor cannot support
TEM wave as the static potential is
either a constant or zero leading to
 if a waveguide has more than 1
dielectric, TEM mode cannot exists as
cannot be zero in all regions
et  0
k k
ci ri
 
( ) /
 
2 2 1 2

 sometime we deliberately want to
have a cutoff frequency so that a
microwave filter can be designed

TEM Mode in Coaxial Line
 a coaxial line is shown here:
 the inner conductor is at a potential
of Vo volts and the outer conductor
is at zero volts
b
a

V=0
V=Vo

 the electric field can be derived from
the scalar potential ; in cylindrical
coordinates, the Laplace’s equation
reads:
 the boundary conditions are:

1 1





 







      
2
2
2
0 18

( )
 
( , ) ( ), ( , ) ( )
a V b
o
 
         
19 0 20

 use the method of separation of variables, we
let
 substitute Eq. (21) to (18), we have
 note that the first term on the left only
depends on  while the second term only
depends on 
( , ) ( ) ( ) ( )
   
    
R P 21
)
22
(
0
2
P
2
1
R
P
R

























 if we change either  or , the RHS
should remain zero; therefore, each
term should be equal to a constant
)
25
(
0
2
k
2
k
),
24
(
2
k
2
P
2
1
)
23
(
2
k
R
P
R








































 now we can solve Eqs. (23) and (24) in
which only 1 variable is involved, the final
solution to Eq. (18) will be the product of
the solutions to Eqs. (23) and (24)
 the general solution to Eq. (24) is
P A k B k
( ) cos( ) sin( )
  
 
 

 boundary conditions (19) and (20)
dictates that the potential is
independent of , therefore must be
equal to zero and so as
 Eq. (23) is reduced to solving
k
k





R





  0

 the solution for R() now reads
R C D
A B
( ) ln
( , ) ln
 
  
 
 




( , ) ln
( , ) ln ln
/ ln( / )
( , )
ln( / )
(ln ln ) ( )
a V A a B
b A b B B A b
A V b a
V
b a
b
o
o
o


  
  
     

    
0
26

 the electric field now reads
 adding the propagation constant back,
we have
e
V
b a
t t
o
( , ) ( , )   
ln /
    








    





 

1
E e e
V e
b a
t
j z o
j z
( , ) ( , ) 
ln /
( )
    



    


27

 the magnetic field for the TEM mode
 the potential between the two
conductors are
H
V e
b a
o
j z
( , ) 
ln /
( )
  


   

28
)
29
(
e
V
d
)
,
(
E
V z
j
o
b
a
ab 







 



 the total current on the inner conductor
is
 the surface current density on the outer
conductor is
I H ad
b a
V e
a o
j z
     




  


0
2 2
30
( , )
ln( / )
( )
J H b
z
b b a
V e
s o
j z
   
 
 ( , )

ln( / )
 



 the total current on the outer conductor
is
 the characteristic impedance can be
calculated as
I J bd
b a
V e I
b sz o
j z
a
  

 






0
2 2
ln( / )
Z
V
I
b a
o
o
a
    


ln( / )
( )
2
31

 higher-order modes exist in coaxial line but is
usually suppressed
 the dimension of the coaxial line is controlled so
that these higher-order modes are cutoff
 the dominate higher-order mode is mode,
the cutoff wavenumber can only be obtained by
solving a transcendental equation, the
approximation is often used in
practice
TE11
k a b
c  
2/ ( )

Surface Waves on a
Grounded Dielectric Slab
 a grounded dielectric slab will generate
surface waves when excited
 this surface wave can propagate a long
distance along the air-dielectric interface
 it decays exponentially in the air region when
move away from the air-dielectric interface

Surface Waves on a
 while it does not support a TEM mode, it excites
at least 1 TM mode
 assume no variation in the y-direction which
implies that
 write equation for the field in each of the two
regions
 match tangential fields across the interface
x
z
r
d
 
/ y  0

Surface Waves on a
 for TM modes, from Helmholtz’s equation
we have
 which reduces to






2
2
2
2
2
2
2
0
x y z
k Ez
  







 


 



2
2
2 2
2
2
2 2
0 0
0
x
k E x d
x
k E d x
r o z
o z
 







   
 







    
,
,

Surface Waves on a
 Define
   
k k h k
c r o o
2 2 2 2 2 2
    
  
,




2
2
2
2
2
2
0 0 32
0 33
x
k E x d
x
h E d x
c z
z








      








       
, ( )
, ( )

Surface Waves on a
 the general solutions to Eqs. (32) and
(33) are
 the boundary conditions are
 tangential E are zero at x = 0 and x
 tangential E and H are continuous at x = d
e x y A k x B k x x d
e x y Ce De d x
z c c
z
hx hx
( , ) sin cos ,
( , ) ,
   
    

0
 

Surface Waves on a
 tangential E at x=0 implies B =0
 tangential E = 0 when x implies
C = 0
 continuity of tangential E implies
 tangential H can be obtained from Eq.
(10) with
 
A k d De
c
hd
sin ( )
   
 34
Hz  0

Surface Waves on a
 tangential E at x=0 implies B =0
 tangential E = 0 when x implies
C = 0
 continuity of tangential E implies
 
A k d De
c
hd
sin ( )
   
 34

Surface Waves on a
 continuity of tangential H implies
 taking the ratio of Eq. (34) to Eq. (35)
we have
r
c
c
hd
k
A k d
h
h
De
cos ( )



  

2
35
k k d h
c c r
tan ( )
   
 36

Surface Waves on a
 note that
 lead to
 Eqs. (36) and (37) must be satisfied
simultaneously, they can be solved for by
numerical method or by graphical method
   
k k h k
c r o o
2 2 2 2 2 2
    
  
,
k h k
c r o
2 2 2
1 37
     
( ) ( )


Surface Waves on a
 to use the graphical method, it is more
convenient to rewrite Eqs. (36) and (37)
into the following forms:
k d k d hd
c c r
tan ( )
   
 38
( ) ( ) ( )( ) ( )
k d hd k d r
c r o
2 2 2 2
1 39
      


Surface Waves on a
 Eq. (39) is an equation of a circle with a
radius of , each interception
point between these two curves yields a
solution
( )
r o
k d
 1
/2  kcd
hd
r
Eq.(38)
Eq. (39)

Surface Waves on a
 note that there is always one intersection
point, i.e., at least one TM mode
 the number of modes depends on the radius r
which in turn depends on the d and
 h has been chosen a positive real number,
we can also assume that is positive
 the next TM will not be excited unless
r o
k
,
kc
r k d
r o
  
( )
 
1

Surface Waves on a
 In general, mode is excited if
 the cutoff frequency is defined as
TMn
r k d n
r o
  
( )
 
1
( )( / )
  
r c
f c d n
 
1 2 
f
nc
d
n
c
r



2 1
0 1 2

, , , ,...--- (40)

Surface Waves on a
 once and h are found, the TM field
components can be written as for
kc
0  
x d
E A k xe
E
j
k
A k xe
H
j
k
A k xe
z c
j z
x
c
c
j z
y
o r
c
c
j z
    


   


   



sin ( )
cos ( )
cos ( )




 
41
42
43

Surface Waves on a
 For
 similar equations can be derived for TE
fields
d x
  
E A k de e
E
j
h
A k de e
H
j
h
A k de e
z c
h x d j z
x c
h x d j z
y
o
c
h x d j z
    


  


   
  
  
  
sin ( )
sin ( )
sin ( )
( )
( )
( )





44
45
46

Striplines and Microstrip Lines
 various planar transmission line
structures are shown here:
stripline slot line
microstrip coplanar
li
line line

 the strip line was developed from the
square coaxial
coaxial square coaxial
rectangular line flat stripline

 since the stripline has only 1 dielectric, it
supports TEM wave, however, it is difficult to
integrate with other discrete elements and
excitations
 microstrip line is one of the most popular
types of planar transmission line, it can be
fabricated by photolithographic techniques
and is easily integrated with other circuit
elements

 the following diagrams depicts the
evolution of microstrip transmission line
+
-
two-wire line
+
-
single-wire above
ground (with image)
+
-
microstrip in air
(with image)
microstrip with
grounded slab

 a microstrip line suspended in air can support
TEM wave
 a microstrip line printed on a grounded slab
does not support TEM wave
 the exact fields constitute a hybrid TM-TE
wave
 when the dielectric slab become very thin
(electrically), most of the electric fields are
trapped under the microstrip line and the
fields are essentially the same as those of the
static case, the fields are quasi-static

 one can define an effective dielectric constant
so that the phase velocity and the
propagation constant can be defined as
 the effective dielectric constant is bounded by
, it also depends on the slab
thickness d and conductor width, W
v
c
p
e
    

( )
47
 
   
ko e ( )
48
1  
 
e r

Design Formulas of Microstrip Lines
 design formulas have been derived for
microstrip lines
 these formulas yield approximate values
which are accurate enough for most
applications
 they are obtained from analytical expressions
for similar structures that are solvable exactly
and are modified accordingly

 or they are obtained by curve fitting
numerical data
 the effective dielectric constant of a
microstrip line is given by

 
r
r r
d W





   
1
2
1
2
1
1 12
49
/
( )

 the characteristic impedance is given by
 for W/d 1
 For W/d 1

Z
d
W
W
d
o
r
 





    
60 8
4
50

ln ( )

 
Z
W d W d
o
r

  
 
120
1 393 0 667 1 444
51

 / . . ln( / . )
( )

 for a given characteristic impedance
and dielectric constant , the W/d
ratio can be found as
for W/d<2
Zo
r
W d
e
e
A
A
/ ( )


   
8
2
52
2

 for W/d > 2
 Where
 And
W d B B
B
r
r
r
/ [ ln( )
{ln( ) .
.
}] ( )
    


      
2
1 2 1
1
2
1 0 39
0 61
53




A
Zo r r
r r






60
1
2
1
1
0 23
0 11
 
 
( .
.
)
B
Zo r

377
2



 for a homogeneous medium with a
complex dielectric constant, the
propagation constant is written as
 note that the loss tangent is usually very
small
   
     
   
  
d c
c o o r
j k
k j
2 2
2 2 1
( tan )
 
  
k k jk
c
2 2 2 tan

 Note that where x is small
 therefore, we have
( ) /
/
1 1 2
1 2
  
x x


  

  
k k
jk
k k
c
c
2 2
2
2 2
2
54
tan
( )

 Note that
 for small loss, the phase constant is
unchanged when compared to the
lossless case
 the attenuation constant due to
dielectric loss is therefore given by
 Np/m (TE or TM) (55)
j k k
c
  
2 2



d
k

2
2
tan

 For TEM wave , therefore
 Np/m (TEM) (56)
 for a microstrip line that has
inhomogeneous medium, we multiply
Eq. (56) with a filling factor
k  


d
k

tan
2
 
 
r e
e r
( )
( )


1
1

 = (57)
 the attenuation due to conductor loss is given by
 (58) Np/m where
 is called the surface resistance of the conductor

 
d
o e
k

tan
2
 
 
r e
e r
( )
( )


1
1
ko r e
e r
  
 
( ) tan
( )


1
2 1
c
s
o
R
Z W

Rs o
  
/ ( )
2
Rs

 note that for most microstrip substrate,
the dielectric loss is much more
significant than the conductor loss
 at very high frequency, conductor loss
becomes significant

An Approximate Electrostatic
Solution for Microstrip Lines
 two side walls are sufficiently far away that
the quasi-static field around the microstrip
would not be disturbed (a >> d)
y
x
a/2
d
-a/2
W
r

 we need to solve the Laplace’s equation
with boundary conditions
 two expressions are needed, one for
each region
     
  
  
t x y x a y
x y x a
x y y
2
0 2 0
0 2
0 0



( , ) ,| | / ,
( , ) , /
( , ) , ,

 using the separation of variables and
appropriate boundary conditions, we
write


( , ) cos sinh ( ),
( , ) cos ( ),
,
,
/
x y A
n x
a
n y
a
y d
x y B
n x
a
e d y
n
n odd
n
n odd
n y a
      
       



 
1
1
59 0
60
 
 

 the potential must be continuous at y=d
so that
 note that this expression must be true
for any value of n
A
n d
a
B e
n n
n d a
sinh /
 
 

 due to fact that
 if m is not equal to n
cos cos
/
/ m x
a
n x
a
dx
a
a  

 
2
2 0


( , ) cos sinh ( ),
( , ) cos sinh ( ),
,
,
( )/
x y A
n x
a
n y
a
y d
x y A
n x
a
n d
a
e
d y
n
n odd
n
n odd
n y d a
      
  
  



  
1
1
61 0
62
 
  

 the normal component of the electric field is
discontinuous due to the presence of surface
charge on the microstrip, E y
y   
/
E A
n
a
n x
a
n y
a
y d
x y A
n
a
n x
a
n d
a
e
d y
y n
n odd
n
n odd
n y d a
    
 
  



  
1
1
0
,
,
( )/
cos cosh ,
( , ) cos sinh ,
  
   


 the surface charge at y=d is given by
 assuming that the charge distribution is
given by on the conductor and
zero elsewhere
   
s o y o r y
E x y d E x y
  
 
( , ) ( , )
 
  


s o n
n odd
r
A
n
a
n x
a
n d
a
n d
a
    


1
63
,
cos (sinh cosh ) ( )
s  1

 multiply Eq. (63) by cos mx/a and
integrate from -a/2 to a/2, we have

 


  



 

  

 


s
a
a
W
W
o
a
a
n
n odd
r
o n r
a
a
o n r
dx
m x
a
dx
m W a
m a
A
n
a
n x
a
n d
a
n y
a
dx
A
n
a
n d
a
n d
a
m x
a
n x
a
dx
A
n
a
n d
a
n d
a
a
m






   
   
   

/
/
/
/
/
/
,
/
/
cos
sin( / )
/
cos (sinh cosh )
(sinh cosh ) cos cos
(sinh cosh ) ,
2
2
2
2
2
2
1
2
2
2 2
2
 n
A
a m W a
n n d a n d a
n
o r


4 2
2
sin( / )
( ) [sinh( / ) cosh( / )

    

 the voltage of the microstrip wrt the
ground plane is
 the total charge on the strip is
V E x y dy A n d a
y n
n odd
d
    



( , ) sinh /
,
0
1
0 
dx W
W
W

 
/
/
2
2

 the static capacitance per unit length is
 this is the expression for
C
Q
V
W
a m W a n d a
n n d a n d a
o r
n odd
 



 4 2
2
1
sin( / )sinh( / )
( ) [sinh( / ) cosh( / )]
,
 
    
(64)
r  1

 the effective dielectric is defined as
 , where is obtained from
Eq. (64) with
 the characteristic impedance is given by
e
C
Co
 Co
r  1
Z
v C cC
o
p
e
 
1 

The Transverse Resonance
Techniques
 the transverse resonance technique
employs a transmission line model of
the transverse cross section of the
guide
 right at cutoff, the propagation constant
is equal to zero, therefore, wave cannot
propagate in the z direction

Techniques
 it forms standing waves in the
transverse plane of the guide
 the sum of the input impedance at any
point looking to either side of the
transmission line model in the
transverse plane must be equal to zero
at resonance

Techniques
 consider a grounded slab and its
equivalent transmission line model
x
z
r
d
to infinity
Za, kxa
Zd,kxd

Techniques
 the characteristic impedance in each of the
air and dielectric regions is given by
 and
 since the transmission line above the
dielectric is of infinite extent, the input
impedance looking upward at x=d is simply
given by
Z
k
k
a
xa o
o


Z
k
k
k
k
d
xd d
d
xd o
r o
 
 

Za

Techniques
 the impedance looking downward is the
impedance of a short circuit at x=0 transfers
to x=d
 Subtituting , we have
 Therefore,
Z Z
Z jZ l
Z jZ l
in o
L o
o L



tan
tan


Z Z Z k l d
L o d xd
   
0, , ,

Z jZ d
in d
 tan
k
k
j
k
k
k d
xa o
o
xd o
r o
xd
 

 
tan 0

Techniques
 Note that , therefore, we have
 From phase matching,
 which leads to
 Eqs. (65) and (66) are identical to that
of Eq. (38) and (39)
k jh
xa  
r xd xd
h k k d
   
tan ( )
65
k k
yo yd

r o xd o xa o
k k k k k h
2 2 2 2 2 2 66
        ( )

Wave Velocities and Dispersion
 a plane wave propagates in a medium at the
speed of light
 Phase velocity, , is the speed at
which a constant phase point travels
 for a TEM wave, the phase velocity equals to
the speed of light
 if the phase velocity and the attenuation of a
transmission line are independent of frequency,
a signal propagates down the line will not be
distorted
1/ 
vp   
/

 if the signal contains a band of
frequencies, each frequency will travel
at a different phase velocity in a non-
TEM line, the signal will be distorted
 this effect is called the dispersion effect

 if the dispersion is not too severe, a
group velocity describing the speed of
the signal can be defined
 let us consider a transmission with a
transfer function of
Z Ae Z e
j z j
( ) | ( )|
 
 
 
 

 if we denote the Fourier transform of a time-
domain signal f(t) by F(), the output signal
at the other end of the line is given by
 if A is a constant and  = a, the output will
be
f t F Z e d
o
j t
( ) ( )| ( )| ( )
 



1
2
  
 
f t A F e d Af t a
o
j t a
( ) ( ) ( )
( )
   



1
2
 


 this expression states that the output
signal is A times the input signal with a
delay of a
 now consider an amplitude modulated
carrier wave of frequency o
s t f t t f t e
o
j t
o
( ) ( )cos Re{ ( ) }
 
 

 the Fourier transform of
is given by
 note that the Fourier transform of s(t) is
equal to
f t ej t
o
( ) 
S F o
( ) ( )
  
 
1
2
{ ( ) ( )}
F F
o o
   
  

 The output signal , is given by
 for a dispersive transmission line, the
propagation constant  depends on
frequency, here A is assume to be
constant (weakly depend on 
s t
o( )
s t AF e d
o o
j t z
( ) Re ( ) ( )
  



1
2
  
 

 if the maximum frequency component of the
signal is much less than the carrier
frequencies,  can be linearized using a
Taylor series expansion
 note that the higher terms are ignored as the
higher order derivatives goes to zero faster
than the growth of the higher power of
   


 
 
( ) ( ) | ( ) ...
   

o o
d
d o
( )
 
 o

 with the approximation of
          
( ) ( ) '( )( ) ' 
    
o o o o o
s t A F e d
o
j t z z
o o
( ) Re{ () }
( '  )
 


 
1
2
 
   
s t A e F e d
s t A f t z e
o
j t z j t z
o o
j t z
o o o
o o
( ) Re{ () }
( ) Re{ ( ' ) }
( ) ( ' )
( )
 
 





1
2
 

   
 
s t Af t z t z
o o o o
( ) ( ' )cos( ) ( )
     
   67

 Eq. (67) states that the output signal is
the time-shift of the input signal
envelope
 the group velocity is therefore defined
as
v
d
d
g
o
o
  
1



 
'
|

 consider a grounded slab and its
equivalent transmission line model

EE 41139 Microwave Techniques: TEM, TE and TM Waves in Transmission Lines and Waveguides (39

Recommended

Recommended

More Related Content

Similar to EE 41139 Microwave Techniques: TEM, TE and TM Waves in Transmission Lines and Waveguides (39

Similar to EE 41139 Microwave Techniques: TEM, TE and TM Waves in Transmission Lines and Waveguides (39 (20)

Recently uploaded

Recently uploaded (20)

EE 41139 Microwave Techniques: TEM, TE and TM Waves in Transmission Lines and Waveguides (39