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{ WAVEGUIDE.PDE

 

 This problem solves for the Transverse-Electric modes of a T-septate

 rectangular waveguide.

 

 Assuming that Z is the propagation direction, we can write

   E(x,y,z) = E(x,y)*exp(i*(omega*t-Kz*z))

   H(x,y,z) = H(x,y)*exp(i*(omega*t-Kz*z))

 where omega is the angular frequency and kz denotes the propagation constant.

 

 In a Transverse-Electric waveguide, the electric field component in the propagation

 direction is zero, or Ez = 0.

 

 Substituting these equations into the source-free Maxwell's equations and rearranging,

 we can write

   Ey =  -(omega*mu/kz)*Hx

   Ex = (omega*mu/kz)*Hy

   Hx = -i*dx(Hz)*kz/kt

   Hy = i*dy(Hz)*kz/kt

   with kt = [omega^2*eps*mu - kz^2]

 

 It can also be shown that in this case Hz satisfies the homogeneous Helmholtz equation

   dxx(Hz) + dyy(Hz) + Kt^2*Hz = 0

 together with the homogeneous Neumann boundary condition on the conducting wall

   dn(Hz) = 0  

 

 In order to avoid clutter in this example script, we will supress the proportionality factors.

 (The leading "i" in the definition of Hx and Hy is merely a phase shift.)

        -----  From J. Jin,  "The Finite Element Method in Electromagnetics", p. 197

}  

 

title "TE Waveguide"  

 

select  

 modes = 4     { This is the number of Eigenvalues desired. }  

 Hz:  del2(Hz) + lambda*Hz = 0       { lambda = Kt^2 }  

 

constraints  

 integral(Hz) = 0 { since Hz has only natural boundary conditions,

                       we need an additional constraint to make  

                       the solution unique }  

 

boundaries  

region 1  

  start(0,0)  

  natural(Hz) = 0     ! this condition applies to all subsequent segments

    ! walk the box body

    line to (L,0) to (L,1) to (0,1) to (0,h+g)  

    ! walk the T-septum

              to (s-g,h+g) to (s-g,h+g+tang) fillet(g/2) to (s+g,h+g+tang) fillet(g/2)

              to (s+g,h-g-tang) fillet(g/2) to (s-g,h-g-tang) fillet(g/2) to (s-g,h-g) to (0,h-g)

    line to close

 

monitors

contour(Hz)

 

plots

contour(Hz) painted

vector(Hx,Hy) as "Transverse H" norm

 contour(magnitude(Hx,Hy))

 contour(magnitude(Hx,Hy)) zoom(0.5,0.5,0.2,0.2)

vector(Ex,Ey) as "Transverse E" norm

 contour(magnitude(Ex,Ey))

 contour(magnitude(Ex,Ey)) zoom(0.5,0.5,0.2,0.2)

 

end