Lecture 44 - Transverse Electric Mode in Rectangular Waveguide Notes | EduRev

: Lecture 44 - Transverse Electric Mode in Rectangular Waveguide Notes | EduRev

 Page 1


 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
    Objective
   In this course you will learn the following
Cutt-off  Frequecy of TE and TM mode.
Page 2


 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
    Objective
   In this course you will learn the following
Cutt-off  Frequecy of TE and TM mode.
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
The analysis of TE mode in a rectangular waveguide can be carried out on the line similar to that of the the TM mode.
  
For TE mode we have
---------- (6.57 )
The wave equation is solved for  in this case.
 
In the case of TM mode the wave equation was solved for  which was tangential to all the four walls of the
waveguides. We therefore had boundary conditions on .
  
In the TE case however the independent component  is tangential two the walls of the waveguide which do not
impose any boundary conditions on . The tangential component of magnetic field is balanced by the appropriate
surface currents on the walls of the waveguides.
 
The analysis procedure for TE mode therefore is slightly different than that of the TM mode. We have seen in the case of
parallel plane waveguide that the tangential component of the magnetic field is maximum at the waveguide walls. Also in
Cartesian co-ordinate system the solution to the wave equation are sinosoidal in nature.
 
One can note that for  = , (vertical walls) and for , (horizontal walls) the tangential component of
magnetic field is maximum.
 
Substituting for  from
---------- (6.58)
 
and  = 0 in 6.31, 6.32, 6.33 and 6.34, we get the transverse field components as
 
 
---------- (6.59)
 
---------- (6.60)
 
---------- (6.61)
 
---------- (6.62)
 
 
In this case also we get
---------- (6.63)
Following observations can be made regarding the TE mode :
 
(1)
The fields for the TE modes have similar behaviour to the fields of the TM modes i.e they exist in the form of discrete
Page 3


 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
    Objective
   In this course you will learn the following
Cutt-off  Frequecy of TE and TM mode.
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
The analysis of TE mode in a rectangular waveguide can be carried out on the line similar to that of the the TM mode.
  
For TE mode we have
---------- (6.57 )
The wave equation is solved for  in this case.
 
In the case of TM mode the wave equation was solved for  which was tangential to all the four walls of the
waveguides. We therefore had boundary conditions on .
  
In the TE case however the independent component  is tangential two the walls of the waveguide which do not
impose any boundary conditions on . The tangential component of magnetic field is balanced by the appropriate
surface currents on the walls of the waveguides.
 
The analysis procedure for TE mode therefore is slightly different than that of the TM mode. We have seen in the case of
parallel plane waveguide that the tangential component of the magnetic field is maximum at the waveguide walls. Also in
Cartesian co-ordinate system the solution to the wave equation are sinosoidal in nature.
 
One can note that for  = , (vertical walls) and for , (horizontal walls) the tangential component of
magnetic field is maximum.
 
Substituting for  from
---------- (6.58)
 
and  = 0 in 6.31, 6.32, 6.33 and 6.34, we get the transverse field components as
 
 
---------- (6.59)
 
---------- (6.60)
 
---------- (6.61)
 
---------- (6.62)
 
 
In this case also we get
---------- (6.63)
Following observations can be made regarding the TE mode :
 
(1)
The fields for the TE modes have similar behaviour to the fields of the TM modes i.e they exist in the form of discrete
pattern, they have sinosoidal variations in  and  directions, indices  and  represent number of half cycles of the
field amplitudes in  and  direction respectively and so on.
 
(2)
Unlike TM mode both indices  and  need not be non-zero for the existence of the TE mode. However, of both the
indices zero makes the magnetic field independent of space and therefore cannot exist. In other words,  mode
cannot exist but  and  modes can exist.
 
(3)
The lowest order mode for the TE case therefore would be  and .
Page 4


 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
    Objective
   In this course you will learn the following
Cutt-off  Frequecy of TE and TM mode.
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
The analysis of TE mode in a rectangular waveguide can be carried out on the line similar to that of the the TM mode.
  
For TE mode we have
---------- (6.57 )
The wave equation is solved for  in this case.
 
In the case of TM mode the wave equation was solved for  which was tangential to all the four walls of the
waveguides. We therefore had boundary conditions on .
  
In the TE case however the independent component  is tangential two the walls of the waveguide which do not
impose any boundary conditions on . The tangential component of magnetic field is balanced by the appropriate
surface currents on the walls of the waveguides.
 
The analysis procedure for TE mode therefore is slightly different than that of the TM mode. We have seen in the case of
parallel plane waveguide that the tangential component of the magnetic field is maximum at the waveguide walls. Also in
Cartesian co-ordinate system the solution to the wave equation are sinosoidal in nature.
 
One can note that for  = , (vertical walls) and for , (horizontal walls) the tangential component of
magnetic field is maximum.
 
Substituting for  from
---------- (6.58)
 
and  = 0 in 6.31, 6.32, 6.33 and 6.34, we get the transverse field components as
 
 
---------- (6.59)
 
---------- (6.60)
 
---------- (6.61)
 
---------- (6.62)
 
 
In this case also we get
---------- (6.63)
Following observations can be made regarding the TE mode :
 
(1)
The fields for the TE modes have similar behaviour to the fields of the TM modes i.e they exist in the form of discrete
pattern, they have sinosoidal variations in  and  directions, indices  and  represent number of half cycles of the
field amplitudes in  and  direction respectively and so on.
 
(2)
Unlike TM mode both indices  and  need not be non-zero for the existence of the TE mode. However, of both the
indices zero makes the magnetic field independent of space and therefore cannot exist. In other words,  mode
cannot exist but  and  modes can exist.
 
(3)
The lowest order mode for the TE case therefore would be  and .
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
Cutt-off  Frequecy of TE and TM mode
For both  and  modes the phase constant is given by
   
 
                                                             
--------- (6.64 )
 
For the mode to be travelling   has to be a real quantity. If   becomes imaginary then the fields no more remain
travelling but become exponentially decaying
 
The frequency at which  changes from real to imaginary is called the cut-off frequency of the mode. At cut-off
frequency therefore   giving
  
--------- (6.65 )
 
 
--------- (6.66 )
 
 
The cut-off frequencies for lowest TM and TE modes i.e   can be obtained from eqn. 6.70 as
  
--------- (6.67 )
 
--------- (6.68 )
 
 
--------- (6.69)
 
 
Since by definition we have   we get the frequencies as
--------- (6.70)
 
We can make an important observation that if at all the electro magnetic energy travels on a rectangular waveguide its
frequency has to be more than the lowest cut-off frequency i.e .
 
As the order of the mode increases the cut-off frequency also increases i.e with increasing frequency there is possibilty
of existence of higher order mode.
  
The very first mode that propagates on the rectangular waveguide is  mode and therefore this mode is called the
dominant mode of the rectangular waveguide. The cut-off frequency for the dominant mode is
                                                                                               
--------- (6.71)
Page 5


 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
    Objective
   In this course you will learn the following
Cutt-off  Frequecy of TE and TM mode.
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
The analysis of TE mode in a rectangular waveguide can be carried out on the line similar to that of the the TM mode.
  
For TE mode we have
---------- (6.57 )
The wave equation is solved for  in this case.
 
In the case of TM mode the wave equation was solved for  which was tangential to all the four walls of the
waveguides. We therefore had boundary conditions on .
  
In the TE case however the independent component  is tangential two the walls of the waveguide which do not
impose any boundary conditions on . The tangential component of magnetic field is balanced by the appropriate
surface currents on the walls of the waveguides.
 
The analysis procedure for TE mode therefore is slightly different than that of the TM mode. We have seen in the case of
parallel plane waveguide that the tangential component of the magnetic field is maximum at the waveguide walls. Also in
Cartesian co-ordinate system the solution to the wave equation are sinosoidal in nature.
 
One can note that for  = , (vertical walls) and for , (horizontal walls) the tangential component of
magnetic field is maximum.
 
Substituting for  from
---------- (6.58)
 
and  = 0 in 6.31, 6.32, 6.33 and 6.34, we get the transverse field components as
 
 
---------- (6.59)
 
---------- (6.60)
 
---------- (6.61)
 
---------- (6.62)
 
 
In this case also we get
---------- (6.63)
Following observations can be made regarding the TE mode :
 
(1)
The fields for the TE modes have similar behaviour to the fields of the TM modes i.e they exist in the form of discrete
pattern, they have sinosoidal variations in  and  directions, indices  and  represent number of half cycles of the
field amplitudes in  and  direction respectively and so on.
 
(2)
Unlike TM mode both indices  and  need not be non-zero for the existence of the TE mode. However, of both the
indices zero makes the magnetic field independent of space and therefore cannot exist. In other words,  mode
cannot exist but  and  modes can exist.
 
(3)
The lowest order mode for the TE case therefore would be  and .
 Module 6 : Wave Guides
Lecture 44 : Transverse Electric Mode in Rectangular Waveguide
 
Cutt-off  Frequecy of TE and TM mode
For both  and  modes the phase constant is given by
   
 
                                                             
--------- (6.64 )
 
For the mode to be travelling   has to be a real quantity. If   becomes imaginary then the fields no more remain
travelling but become exponentially decaying
 
The frequency at which  changes from real to imaginary is called the cut-off frequency of the mode. At cut-off
frequency therefore   giving
  
--------- (6.65 )
 
 
--------- (6.66 )
 
 
The cut-off frequencies for lowest TM and TE modes i.e   can be obtained from eqn. 6.70 as
  
--------- (6.67 )
 
--------- (6.68 )
 
 
--------- (6.69)
 
 
Since by definition we have   we get the frequencies as
--------- (6.70)
 
We can make an important observation that if at all the electro magnetic energy travels on a rectangular waveguide its
frequency has to be more than the lowest cut-off frequency i.e .
 
As the order of the mode increases the cut-off frequency also increases i.e with increasing frequency there is possibilty
of existence of higher order mode.
  
The very first mode that propagates on the rectangular waveguide is  mode and therefore this mode is called the
dominant mode of the rectangular waveguide. The cut-off frequency for the dominant mode is
                                                                                               
--------- (6.71)
The equation suggest that for propagation of an electro magnetic wave inside a rectangular waveguide the width of a
 
waveguide should be greater than half the wave length of the wave.
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