RF 000 - Equations for electromagnetic waves

What you’ll learn

• How Allsolve derives the wave equation from Maxwell’s equations, including the role of permittivity, permeability, and conductivity tensors.
• The weak form formulation used by the solver, and how the Neumann trace term enables waveguide port excitation.

Prerequisites: this is a theory reference — no simulation setup is needed. Familiarity with Maxwell’s equations and vector calculus is helpful.

The electromagnetic waves physics solves for the wave propagation in Maxwell’s equations.

Maxwell’s equations

$\,$
$\nabla \cdot \boldsymbol{D} = \rho$

$\nabla \cdot \boldsymbol{B} = 0$

$\nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}$

$\nabla \times \boldsymbol{H} = \boldsymbol{J} + \frac{\partial \boldsymbol{D}}{\partial t}$
$\,$
where $\boldsymbol{E}$ is the electric field [V/m], $\boldsymbol{D}$ is the electric displacement field [C/m²], $\boldsymbol{H}$ is the magnetic field [A/m], $\boldsymbol{B}$ is the magnetic flux density [T], $\rho$ is the electric charge density [C/m³] and $\boldsymbol{J}$ is the conduction current density [A/m²].

Without loss of generality, the materials can be modeled as $\boldsymbol{D} = \boldsymbol{\epsilon} \boldsymbol{E} + \boldsymbol{D_r}$, $\boldsymbol{B} = \boldsymbol{\mu} \boldsymbol{H} + \boldsymbol{B_r}$ and $\boldsymbol{J} = \boldsymbol{\sigma} \boldsymbol{E} + \boldsymbol{J_r}$, where $\boldsymbol{\epsilon}$ is the electric permittivity tensor [F/m], $\boldsymbol{\mu}$ is the magnetic permeability tensor [H/m] and $\boldsymbol{\sigma}$ is the electric conductivity tensor [S/m]. Quantities $\boldsymbol{D_r}$, $\boldsymbol{B_r}$ and $\boldsymbol{J_r}$ are typically associated with remanent effects.

Assumption of slow material properties changes

It is assumed that quantities $\boldsymbol{\epsilon}$, $\boldsymbol{\mu}$, $\boldsymbol{\sigma}$, $\boldsymbol{D_r}$, $\boldsymbol{B_r}$ and $\boldsymbol{J_r}$ change slowly over time compared to the electromagnetic waves frequency. This allows to neglect their time derivatives.

Using the above assumption, the last two equations can be rewritten as:

$\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{H}}{\partial t}$

$\nabla \times \boldsymbol{H} = \boldsymbol{\sigma} \boldsymbol{E} + \boldsymbol{J_r} + \boldsymbol{\epsilon} \frac{\partial \boldsymbol{E}}{\partial t}$

Taking the curl of the first equation gives:

$\nabla \times (\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E}) = -\frac{\partial \, (\nabla \times \boldsymbol{H})}{\partial t}$

Using the second equation, one gets the strong form of the wave equation used in Allsolve:

$\nabla \times (\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E}) + \boldsymbol{\sigma} \frac{\partial \, \boldsymbol{E}}{\partial t} + \boldsymbol{\epsilon} \frac{\partial^2 \, \boldsymbol{E}}{\partial t^2} = \boldsymbol{0}$

The solver in Allsolve requires the weak form of the above equation. This is obtained by multiplying by a test function $\boldsymbol{E}'$ and integrating over the electromagnetic waves physics domain $\Omega$:

$\displaystyle \int_\Omega \nabla \times (\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E}) \cdot \boldsymbol{E}' \, d\Omega + \displaystyle \int_\Omega \boldsymbol{\sigma} \frac{\partial \, \boldsymbol{E}}{\partial t} \cdot \boldsymbol{E}' \, d\Omega + \displaystyle \int_\Omega \boldsymbol{\epsilon} \frac{\partial^2 \, \boldsymbol{E}}{\partial t^2} \cdot \boldsymbol{E}' \, d\Omega = 0$

which can be rewritten as:

$\displaystyle \int_\Omega (\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E}) \cdot \nabla \times \boldsymbol{E}' \, d\Omega + \displaystyle \int_\Omega \boldsymbol{\sigma} \frac{\partial \, \boldsymbol{E}}{\partial t} \cdot \boldsymbol{E}' \, d\Omega + \displaystyle \int_\Omega \boldsymbol{\epsilon} \frac{\partial^2 \, \boldsymbol{E}}{\partial t^2} \cdot \boldsymbol{E}' \, d\Omega \\\,\\- \displaystyle \int_{\partial\Omega} \boldsymbol{n} \times \frac{\partial \boldsymbol{H}}{\partial t} \cdot \boldsymbol{E}' \, d\Omega = 0$

where $\boldsymbol{n}$ is the unit vector normal to the boundary $\partial \Omega$ of $\Omega$.

The above formulation is what is actually solved in the electromagnetic waves physics. The last term is the Neumann trace, which allows for example to feed modes into a waveguide port.

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