EM wave simulator

Introduction and Use cases

Patricio Villar

Motivation

• Site surveys require planning and resource allocation ahead of time, even for the simplest propagation footprint.
• Avoiding a site survey might end up in coverage holes (bad source distribution) or hidden interference sources (rogue EM sources), which lead to bad user experience.
• Site surveys determine required hardware at the design stage, so it's a mandatory task before procurement.

How do we simulate a EM standing wave field?

Take Maxwell's equations in the presence of nonmagnetic matter, and the constitutive relations as follows:

\begin{eqnarray} \nabla \cdot \mathbf{D} &=& 0 \\ \nabla \cdot \mathbf{B} &=& 0 \\ \nabla \times \mathbf{E} &=& -\frac{\partial\mathbf{B}}{\partial t} \label{eqn:faraday} \\ \nabla \times \mathbf{H} &=& \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t} \\ \mathbf{D} &=& \epsilon_0\epsilon_r\mathbf{D} \\ \mathbf{H} &=& \frac{1}{\mu_0}\mathbf{B} \end{eqnarray}

How do we simulate a EM standing wave field? Cont'ed

Substituting everything in for Eq.:

\nabla \times \mathbf{E} &=& -\frac{\partial\mathbf{B}}{\partial t} \label{eqn:faraday}

and assuming that:

\partial/\partial t \rightarrow i\omega

How do we simulate a EM standing wave field? Cont'ed

We get the following:

\begin{equation} \left(\nabla^2 + n^2k0^2\right)\mathbf{E} + \nabla\left(\frac{1}{n^2}\left(\mathbf{E}\cdot\nabla n^2 \right) \right) = \mu_0i\omega_0\mathbf{J} \end{equation}

where n(x,y) = \sqrt{\epsilon_r} is the refractive index and (\omega_0, k_0) are the frequency and vacuum wavevector of the radiation, related by \omega_0 = ck_0. In the presence of matter, the radiation field keeps the same frequency but the wavevector changes to k(x,y) = nk_0, causing refraction, reflection etc.

Application details

• A floor map is required to start the simulation.
• The floor map should be color-coded, in order to differentiate materials (and therefore their respective refractive index) - air, concrete, wooden walls are valid examples.
• Algorithm complexity is m x n, so ~ to wave resolution.

Application output: Qualitative

• This plot can be used as a rough aproximation for AP placement:

Application output: Quantitative

• This contour plot can be used to quantify coverage holes size and impact.

Future Enhancements:

• Multiple AP capability. (algorithm complexity increases to N x m x n)
• Multi-Threading and Grid support.
• Support for directional and quad antenas.
• GUI interface with Tk.

Biblio:

• Computational Electromagnetics - Rylender, Bondenson, Ingelstrom.
• Computational Electrodynamics: The FDTD method - A. Taflove
• Computational Electromagnetics for RF and Microwave Engineering - D. Davidson.

QA