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RF 000 - Electromagnetic waves equations

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

Maxwell’s equations

 \,
∇⋅D=ρ\nabla \cdot \boldsymbol{D} = \rho

∇⋅B=0\nabla \cdot \boldsymbol{B} = 0

∇×E=−∂B∂t\nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{B}}{\partial t}

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

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

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

μ−1 ∇×E=−∂H∂t\boldsymbol{\mu}^{-1} \, \nabla \times \boldsymbol{E} = -\frac{\partial \boldsymbol{H}}{\partial t}

∇×H=σE+Jr+ϵ∂E∂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:

∇×(μ−1 ∇×E)=−∂ (∇×H)∂t\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:

∇×(μ−1 ∇×E)+σ∂ E∂t+ϵ∂2 E∂t2=0\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 E′\boldsymbol{E}' and integrating over the electromagnetic waves physics domain Ω\Omega:

∫Ω∇×(μ−1 ∇×E)⋅E′ dΩ+∫Ωσ∂ E∂t⋅E′ dΩ+∫Ωϵ∂2 E∂t2⋅E′ dΩ=0\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:

∫Ω(μ−1 ∇×E)⋅∇×E′ dΩ+∫Ωσ∂ E∂t⋅E′ dΩ+∫Ωϵ∂2 E∂t2⋅E′ dΩ −∫∂Ωn×∂H∂t⋅E′ dΩ=0\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 n\boldsymbol{n} is the unit vector normal to the boundary ∂Ω\partial \Omega of Ω\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.