Waveguide (Helmholtz Equation - Electromagnetics)

Mathematical description of waveguides

Mathematical description of waveguides is given by the Maxwell’s equations

(1)\nabla \times {\pmb{H}} &= {\pmb{J}} + \frac{\partial {\pmb{D}}}{\partial t},

(2)\nabla \times {\pmb{E}} &= - \frac{\partial {\pmb{B}}}{\partial t},

(3)\nabla \cdot {\pmb{D}} &= \rho,

(4)\nabla \cdot {\pmb{B}} &= 0

where

(5){\pmb{B}} = \mu {\pmb{H}}, \ {\pmb{J}} = \sigma {\pmb{E}}, \ {\pmb{D}} = \varepsilon {\pmb{E}}.

Here \varepsilon means permittivity, \mu permeability and \sigma stands for electric conductivity. For waveguides analysis, material properties are often considered constant and isotropic. After substituting material properties (5) into equations (1) and (2), we get

(6)\nabla \times \frac{1}{\mu} {\pmb{B}} &= \sigma {\pmb{E}} + \varepsilon \frac{\partial {\pmb{E}}}{\partial t},

(7)\nabla \times {\pmb{E}} &= - \frac{\partial {\pmb{B}}}{\partial t}.

If the vector operator \mathrm{curl} is applied on the equation (7), it is possible to substitute \nabla \times \pmb{E} from the equation (6) and get the wave equation for the electric field in the form

(8)\nabla \times \nabla \times \pmb{E} = - \mu \sigma \frac{\partial {\pmb{E}}}{\partial t} - \mu \varepsilon \frac{\partial^2 {\pmb{E}}}{\partial t^2}.

In a medium with zero charge density \rho it is useful to apply the vector identity

(9)\nabla \times \nabla \times \pmb{E} = \nabla \nabla \cdot \pmb{E} - \Delta \pmb{E}.

Since \nabla \cdot \pmb{E} = 0), the wave equation (8) can be simplified to

(10)\Delta \pmb{E} - \mu \sigma \frac{\partial {\pmb{E}}}{\partial t} - \mu \varepsilon \frac{\partial^2 {\pmb{E}}}{\partial t^2} = \mathbf{0}.

For many technical problems it is sufficient to know the solution in the frequency domain. After applying the Fourier transform, equation (10) becomes

(11)\Delta \overline{\pmb{E}} - \mathrm{j} \mu \sigma \omega \overline{\pmb{E}} + \omega^2 \mu \varepsilon \overline{{\pmb{E}}} = \mathbf{0},

which is the Helmholtz equation.

Parallel plate waveguide is the simplest type of guide that supports TM (transversal magnetic) and TE (transversal electric) modes. This kind of guide allows also TEM (transversal electric and magnetic) mode.

Mathematical model - TE modes

Suppose that the electromagnetic wave is propagating in the direction z, then the component of the vector \pmb{E} in the direction of the propagation is equal to zero

(12)\overline{E_z} = 0,

thus it is possible to solve the electric field in the parallel plate waveguide as a two-dimensional Helmholtz problem

(13)\Delta \overline{\pmb{E}} - \mathrm{j} \mu \sigma \omega \overline{\pmb{E}} + \omega^2 \mu \varepsilon \overline{{\pmb{E}}} = \mathbf{0}.

The conducting plates (boundary \Gamma_1, \Gamma_2) are usually supposed to be perfectly conductive, which can be modeled using the perfect conductor boundary condition

(14)\pmb{n} \times \overline{\pmb{E}} = 0.

For the geometry in the above figure the expression (14) is reduced to a zero Dirichlet boundary condition

(15)\overline{E_x} = 0.

For the boundaries \Gamma_3, \Gamma_4, the following types of boundary conditions can be used:

Electric field (Dirichlet boundary condition)

(16)\overline{\pmb{E}}(\Gamma) = \overline{E_0} = \mathrm{const}.

Note that for TE modes (and for the geometry shown above), a natural boundary condition is described by the expression

(17)\overline{E}_x(y) = \overline{E_0} \cos\left(\frac{y \cdot n \pi}{h} \right),

where n stands for a mode.

Impedance matching (Newton boundary condition)

For harmonic TE mode waves the following relation holds:

(18)\overline{\pmb{E}} = Z_0 (\overline{H_y} \pmb{i} - \overline{H_x} \pmb{j}) = Z_0 \cdot \pmb{n} \times \overline{\pmb{H}},

where Z_0 is the wave impedance. At the same time the second Maxwell equation

(19)\nabla\times \overline{{\pmb{E}}} = -j \omega \mu \overline{\pmb{H}}

must be satisfied. From quations (18) and (19) it is possible to derive impedance matching boundary condition in the form

(20)\pmb{n} \times \nabla \times \overline{\pmb{E}} = \frac{j \omega \mu }{Z_0} \overline{\pmb{E}} = j \beta \overline{\pmb{E}}.

For a given geometry the equation (20) can be reduced to the Newton boundary condition in the form

(21)\frac{\partial \overline{E_x}}{\partial y} = j \beta \overline{E_x}.

Material parameters

const double epsr = 1.0;                    // Relative permittivity
const double eps0 = 8.85418782e-12;         // Permittivity of vacuum F/m
const double mur = 1.0;                     // Relative permeablity
const double mu0 = 4*M_PI*1e-7;             // Permeability of vacuum H/m
const double frequency = 3e9;               // Frequency MHz
const double omega = 2*M_PI * frequency;    // Angular velocity
const double sigma = 0;                     // Conductivity Ohm/m

Boundary conditions

There are three possible types of boundary conditions:

• Zero Dirichlet boundary condition on boundaries \Gamma_1, \Gamma_2:

bc_types.add_bc_dirichlet(Hermes::vector<int>(BDY_PERFECT, BDY_LEFT));
BCValues bc_values_r;
bc_values_r.add_const(BDY_PERFECT, 0.0);
BCValues bc_values_i;
bc_values_i.add_const(BDY_PERFECT, 0.0);

Note: There are two ways how to approach to complex problems. The first one is using the type complex from the STL library. The second approach is demonstrated - two matrices and two right side vectors (one for real parts and second for imagnary parts) are build.

• Nonzero Dirichlet boundary conditions:

The boundary condition on the boundary \Gamma_3 is specified according to the expression (17).:

scalar essential_bc_values(double x, double y)
{
  return cos(y*M_PI/0.1)*100;
}

int main()
{
  ...
  bc_values_r.add_function(BDY_LEFT, essential_bc_values);
  ...
}

• Newton boundary condition

bc_types.add_bc_newton(Hermes::vector<int>(BDY_IMPEDANCE));

Weak forms

• registration (in function main())

  WeakForm wf(2);
  wf.add_matrix_form(0, 0, callback(magnetic_matrix_form_real_real));
  wf.add_matrix_form(0, 1, callback(magnetic_matrix_form_real_imag));
  wf.add_matrix_form(1, 1, callback(magnetic_matrix_form_imag_imag));
  wf.add_matrix_form(1, 0, callback(magnetic_matrix_form_imag_real));
  wf.add_matrix_form_surf(0, 1, callback(magnetic_vector_form_surface_imag_real), BDY_IMPEDANCE);
  wf.add_matrix_form_surf(1, 0, callback(magnetic_vector_form_surface_real_imag), BDY_IMPEDANCE);

The function magnetic_matrix_form_real_real describes behaviour of the of the component of electric field \overline{E_x} and magnetic_matrix_form_imag_imag describes behaviour of the imaginary part of the component of electric field \overline{E_x} in this code. Functions magnetic_matrix_form_imag_real and magnetic_matrix_form_real_imag represent the conection between real and imaginary part. The functions magnetic_vector_form_surface_imag_real and magnetic_vector_form_surface_imag_real express the Newton boundary condition and also the conection between the real and the imaginary part of the component of the electric field \overline{E_x}.

Sample results