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Transverse resonance method

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					    ECE 5317-6351
Microwave Engineering
         Fall 2011
          Fall 2011
     Prof. David R. Jackson
          Dept. of ECE


                   Notes 13
          Transverse Resonance
                Method



                                 1
   Transverse Resonance Method

This is a general method that can be used to help us calculate
various important quantities:



  § Wavenumbers for complicated waveguiding structures (dielectric-
    loaded waveguides, surface waves, etc.)
  § Resonance frequencies of resonant cavities




 We do this by deriving a “Transverse Resonance Equation (TRE).”




                                                                      2
Transverse Resonance Equation (TRE)
To illustrate the method, consider a lossless resonator formed by a
transmission line with reactive loads at the ends.
                                           R




                             x           x = x0         x=L

                   R = reference plane at arbitrary x = x0




  We wish to find the resonance frequency of this transmission-line resonator.



                                                                                 3
                         TRE (cont.)
                                           R




                              x            x = x0    x=L

Examine the voltages and currents at the reference plane:

                         Il       R            Ir

                         +                      +
                         Vl                     Vr
                         -                      -

                                  x = x0
                                                            4
                     TRE (cont.)
                           R         Ir
                     Il

                     +                +
                     Vl               Vr
                     -                -
                                                      x
                            x = x0


Define impedances:    Boundary conditions:   Hence:




                                                          5
TRE (cont.)
     R




     TRE

              Note about the reference
              plane: Although the
              location of the reference
or            plane is arbitrary, a
              “good” choice will keep
              the algebra to a minimum.

                                 6
                           Example
Derive a transcendental equation for the resonance frequency of this
transmission-line resonator.

                                  L




                          x



                We choose a reference plane at x = 0+.




                                                                       7
        Example (cont.)
                     L


             R



                 x


Apply TRE:




                          8
Example (cont.)




                  9
                 Example (cont.)

 After simplifying, we have




Special cases:




                                   10
              Rectangular Resonator
Derive a transcendental equation for the resonance frequency of a
rectangular resonator.
                                 z


                                                         PEC boundary
  Orient so that
   b < a <h                 h
                                                         y
                                              a
                        x
                                     b


The structure is thought of as supporting RWG modes bouncing back
and forth in the z direction.
                                                  The index p describes the
             We have TMmnp and TEmnp modes.       variation in the z direction.
                                                                               11
       Rectangular Resonator (cont.)

We use a Transverse Equivalent Network (TEN):

                    h




             z                                  We choose a reference
                                                plane at z = 0+.




                        Hence                                           12
Rectangular Resonator (cont.)
  Hence




                         h




                     z

                                13
         Rectangular Resonator (cont.)
Solving for the wavenumber we have




  Hence                               Note: The TMz and
                                      TEz modes have the
                                      same resonance
                                      frequency.
                                     TEmnp mode:

    or




The lowest mode is the TE101 mode.                   14
       Rectangular Resonator (cont.)

              TE101 mode:




Note: The sin is used to ensure the boundary condition on the PEC top and bottom plates:




    The other field components, Ey and Hx, can be found from Hz.
                                                                                           15
Rectangular Resonator (cont.)
                                  z
    Practical excitation
    by a coaxial probe
                                                  PEC boundary


                              h
                                                  y
                                          a
                              x
                                      b
      Lp (Probe inductance)




                  R    L C        Circuit model
 Tank (RLC) circuit

                                                        16
Rectangular Resonator (cont.)



             Q = quality factor of resonator




      Lp (Probe inductance)




                  R   L C             Circuit model
 Tank (RLC) circuit

                                                      17
Rectangular Resonator (cont.)




                                18
             Grounded Dielectric Slab
Derive a transcendental equation for wavenumber of the TMx surface
waves by using the TRE.


                               x



         h
                                                                     z


   Assumption: There is no variation of the fields in the y direction,
   and propagation is along the z direction.




                                                                         19
      Grounded Dielectric Slab

                 x



TMx    H
           E
                                 z




                                     20
       TMx Surface-Wave Solution
           h    R

                         The reference plane is
TEN:                     chosen at the interface.


           x




                                                21
TMx Surface-Wave Solution (cont.)
      TRE:




                                    22
TMx Surface-Wave Solution (cont.)
   Letting




    We have




     or




Note: This method was a lot simpler than doing the EM analysis and
applying the boundary conditions!
                                                                     23

				
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