## AC Power Electromagnetics Equations

Consider a homogeneous dielectric with the coefficient of dielectricity ε, the magnetic permeability µ, and no charge at any point. The fields must satisfy a special set of general Maxwell's equations:

`$\begin{array}{c}\nabla ×E=-\mu \frac{\partial H}{\partial t}\\ \nabla ×H=\epsilon \frac{\partial E}{\partial t}+J\end{array}$`

Here, E is the electric field, H is the magnetic field, and J is the current density. In the absence of a current, you can eliminate H from the first set and E from the second set and see that both fields satisfy wave equations with the wave speed $\sqrt{\epsilon \mu }$:

`$\begin{array}{c}\Delta E-\epsilon \mu \frac{{\partial }^{2}E}{\partial {t}^{2}}=0\\ \Delta H-\epsilon \mu \frac{{\partial }^{2}H}{\partial {t}^{2}}=0\end{array}$`

Consider the case of a charge-free homogeneous dielectric with the coefficient of dielectricity ε, magnetic permeability µ, and conductivity σ. The current density is

`$J=\sigma E$`

and the waves are damped by the Ohmic resistance,

`$\Delta E-\mu \sigma \frac{\partial E}{\partial t}-\epsilon \mu \frac{{\partial }^{2}E}{\partial {t}^{2}}=0$`

Equations for H are similar.

For time-harmonic fields, use the complex form of the equations, replacing E with

`${E}_{c}{e}^{j\omega t}$`

For a plane, the electric field is , and the magnetic field is

`$H=\left({H}_{x},{H}_{y},0\right)=\frac{-1}{j\mu \sigma }\nabla ×{E}_{c}$`

The scalar equation for Ec becomes

`$-\nabla \text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(\frac{1}{\mu }\nabla {E}_{c}\right)+\left(j\omega \sigma -{\omega }^{2}\epsilon \right){E}_{c}=0$`

The PDE Modeler app uses this equation when it is in the AC Power Electromagnetics application mode . The equation is a complex Helmholtz equation that describes the propagation of plane electromagnetic waves in imperfect dielectrics and good conductors (σ » ωε). A complex permittivity εc can be defined as εc = ε/ω. The conditions at material interfaces with abrupt changes of ε and µ are the natural conditions for the variational formulation and need no special attention.

The boundary conditions associated with this mode are:

• The Dirichlet boundary condition specifying the value of the electric field Ec on the boundary

• The Neumann boundary condition specifying the normal derivative of Ec, which is equivalent to specifying the tangential component of the magnetic field H:

`${H}_{t}=\frac{j}{\omega }n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(\frac{1}{\mu }\nabla {E}_{c}\right)$`

The solution is the electric field E. Using the solution, you can compute the current density J = σE and the magnetic flux density

`$B=\frac{j}{\omega }\nabla ×E$`

Using the PDE Modeler app, you can plot the electric field E, the current density J, the magnetic field H, and the magnetic flux density B. You also can plot the resistive heating rate

`$Q={E}_{c}^{2}/\sigma$`

You can plot the magnetic field and the magnetic flux density as vector fields by using arrows. 