Electromagnetic waves

Maxwell’s equations

$$\text{Gauss's law}$$$$\nabla \cdot \boldsymbol{D} = \rho$$$$\text{Gauss's law for magnetism}$$$$\nabla \cdot \boldsymbol{B} = 0$$$$\text{Faraday's law}$$$$\nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}$$$$\text{Ampère-Maxwell law}$$$$\nabla \times \boldsymbol{H} = \boldsymbol{J} + \frac{\partial \boldsymbol{D}}{\partial t}$$

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Electric field

Strong formulation

We start from Maxwell’s equations along with the material relations

$$\begin{align} \boldsymbol{B} &= \boldsymbol{\mu} \boldsymbol{H} + \boldsymbol{B_r} \\[5pt] \boldsymbol{D} &= \boldsymbol{\varepsilon} \boldsymbol{E} + \boldsymbol{D_r} \\[5pt] \boldsymbol{J} &= \boldsymbol{\sigma} \boldsymbol{E} + \boldsymbol{J_r}\ . \end{align}$$

Where $\boldsymbol{\varepsilon}$ is the electric permittivity tensor, $\boldsymbol{\mu}$ is the magnetic permeability tensor and $\boldsymbol{\sigma}$ is the electric conductivity tensor. Quantities $\boldsymbol{D_r}$, $\boldsymbol{B_r}$ and $\boldsymbol{J_r}$ are typically associated with remanent effects. We assume 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 this assumptions we can rewrite Faraday’s law as

$$\begin{align} \nabla \times \boldsymbol{E} &= -\boldsymbol{\mu} \frac{\partial \boldsymbol{H}}{\partial t} \\[10pt] \boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E}) &= -\frac{\partial \boldsymbol{H}}{\partial t}. \end{align}$$

We apply the curl operator to both sides and assume the curl and time derivative can be interchanged using the time derivative of a curl identity

$$\begin{align} \nabla \times (\boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E})) &= -\frac{\partial}{\partial t} (\nabla \times \boldsymbol{H}). \end{align}$$

Substituting in the Ampère-Maxwell, material relations (2) and (3) and by rearranging we obtain

$$\begin{align} \nabla \times (\boldsymbol{\mu}^{-1} (\nabla \times \boldsymbol{E})) + \boldsymbol{\sigma} \frac{\partial \boldsymbol{E}}{\partial t} + \boldsymbol{\varepsilon} \frac{\partial^2 \boldsymbol{E}}{\partial t^2} &= 0. \end{align}$$

If we assume isotropic material, all tensor quantities become scalars, the equation results in

$$\begin{align} \nabla \times (\frac{1}{\mu} (\nabla \times \boldsymbol{E})) + \sigma \frac{\partial \boldsymbol{E}}{\partial t} + \varepsilon \frac{\partial^2 \boldsymbol{E}}{\partial t^2} &= 0. \end{align}$$

Weak formulation

To obtain the weak formulation of (7) we multiply the equation by a test function $\boldsymbol{E}'$ and integrating over the electromagnetic waves physics domain $\Omega$:

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

We can rewrite the first term using the divergence of cross product identity

$$\begin{align} \int_{\Omega}\ \nabla \cdot ((\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \times \boldsymbol{E}^{\prime})\ d\Omega + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime}), \end{align}$$

where the divergence theorem can rewritten using the divergence theorem

$$\begin{align} \int_{\Gamma}\ ((\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \times \boldsymbol{E}^{\prime}) \cdot \boldsymbol{n}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega. \end{align}$$

Substituting in the Faraday’s law and using the material relation (1) we obtain

$$\begin{align} -\int_{\Gamma}\ (\frac{\partial \boldsymbol{H}}{\partial t} \times \boldsymbol{E}^{\prime}) \cdot \boldsymbol{n}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega. \end{align}$$

Finally, by using the scalar triple product identity we derive the weak formulation

$$\begin{align} -\int_{\Gamma}\ (\boldsymbol{n} \times \frac{\partial \boldsymbol{H}}{\partial t}) \cdot \boldsymbol{E}^{\prime}\ d\Gamma + \int_{\Omega}\ (\boldsymbol{\mu}^{-1} \nabla \times \boldsymbol{E}) \cdot \ (\nabla \times \boldsymbol{E}^{\prime})\ d\Omega \\[10pt] + \int_{\Omega}\ \boldsymbol{\sigma} \frac{\partial \boldsymbol{E}}{\partial t} \cdot \boldsymbol{E}^{\prime}\ d\Omega + \int_{\Omega}\ \boldsymbol{\epsilon} \frac{\partial^2 \boldsymbol{E}}{\partial t^2} \cdot \boldsymbol{E}^{\prime}\ d\Omega &= 0. \end{align}$$

Available Interactions

Constraint

Fixes the $\boldsymbol{E}$ field value at the defined region, enforcing idealized behavior such as perfect reflection.

How to use:

You can use either the matrix editor or the expression editor.

With the matrix editor, you must provide a 3×1 matrix for a 3D problem (2×1 for 2D, etc.). Add electric field strength vector values for each dimension.

In the expression editor, you can write the direct expression for this 3D case:

Example: [1; 5; 0]

Both styles are equivalent and apply a 3D electric field strength constraint of $E_x = 1\:\frac{V}{m}$, $E_y = 5\:\frac{V}{m}$ and $E_z = 0\:\frac{V}{m}$ to specified node or a region. Same principles apply for 2D.

Unit: Electric field strength in Volts per meter (V/m)

Perfectly matched layer

Implements an absorbing boundary region to simulate infinite space and prevent reflections from truncated boundaries. Placed at the outer boundary of the air domain, it absorbs outgoing waves without reflection.

Key properties:

• Used to simulate open boundaries for radiating structures.

Two types available:

• AML type: Suitable for smooth boundaries.
• Box PML: Suitable for rectangular or cornered boundaries.
• The number of PML layers can be tuned in the Shared PML settings.

Rectangular port

A special case of the Eigenmode port, used when the port geometry and material are well-defined and homogeneous. Provides analytical solutions for the field distribution under certain geometric assumptions.

Assumptions:

• The port surface must be rectangular.
• The dielectric across the port must be uniform.
• All four port edges must be bounded by perfect conductors.

Key properties:

• Perfectly absorbing (impedance matched, no reflection).
• Provides electric field $\boldsymbol{E}$ and magnetic field$\boldsymbol{H}$ access, but not voltage $V$ or current $I$.

Perfect conductor

Imposes a Dirichlet boundary condition on the electric field. Sets $\boldsymbol{E} = 0$ on the target boundary or region.

Key properties:

• Models highly conductive materials as ideal conductors.
• Reduces the number of unknowns and computational cost.

Eigenmode port

Defines a port that perfectly absorbs the excited mode with matched impedance (no reflection). This port type is typically used when the cut-off frequency or mode shapes are unknown. It supports both low and high-frequency signals.

Key properties:

• No voltage $V$ or current $I$ output — only electric field $\boldsymbol{E}$ and magnetic field $\boldsymbol{H}$ are available.

Dielectric loss

As the EM wave propagates through a dielectric, some of the energy is absorbed by the dielectric and is dissipated as heat. This loss of EM energy is simulated using the Dielectric loss interaction.

The target region for this interaction is the dielectric region. In Allsolve, the dielectric loss is parameterized using the loss tangent ($\tan \delta$):

$\tan \delta = \varepsilon^{\prime\prime} / \varepsilon^{\prime}$

• $\varepsilon^{\prime}$ is the real part of the complex permittivity.
• $\varepsilon^{\prime\prime}$ is the negative of the imaginary part of the complex permittivity.
• The complex electric permittivity is given by $\varepsilon = \varepsilon^{\prime} - j\varepsilon^{\prime\prime}$

For script users:

• The predefinedemwave function in Allsolve takes as arguments, among many, the real part and imaginary part of the complex permittivity.
• The real part ($\varepsilon^{\prime}$) is given by the permittivity defined in the material properties of the dielectric.
• The imaginary part ($-\varepsilon^{\prime\prime}$) is defined as $-\tan \delta * \varepsilon^{\prime}$

Lump V/I

Defines a lumped voltage or current source for feeding compact structures where the port dimensions are much smaller than the wavelength. This type of port assumes that the electric potential $v$ is physically meaningful and that the local electric field is derived from $\boldsymbol{E} = –\nabla v$.

Key properties:

• Provides direct access to voltage $V$ and current $I$ at the port.
• Allows connecting external circuit elements (RLC components).
• Suitable for compact feed structures such as microstrip, CPW, or GCPW.

Periodicity

Imposes periodic boundary conditions on the electromagnetic $\boldsymbol{E}$ and $\boldsymbol{H}$ fields between two boundaries. Reduces the computational domain size for geometrically symmetric or antisymmetric problems, avoiding the need to model the full geometry.

Example:

Periodicity is similar in every physics section. This example is from $\varphi$-formulation, but workflows are identical:

The periodicity of an electric motor allows modeling only a fraction of the full geometry, such as one pole pair or one quarter, while still capturing the complete field behavior.

Periodicity in electric motor

Absorbing boundary

Boundary impedance

Use cases

• Electromagnetic waves: Patch antenna
• Electromagnetic waves: Eigenmode ports and S-parameters
• Electromagnetic waves: Lumped ports and RLC components