# Documentation

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# AC Power Electromagnetics

Solve PDEs that model harmonic electrical fields in conductors

The AC power electromagnetics problems are found when studying motors, transformers, and conductors carrying alternating currents. The Helmholtz's equation

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

describes the propagation of plane electromagnetic waves in imperfect dielectrics and good conductors (σ » ωε). Here, ω is the angular frequency, ε is the coefficient of dielectricity, µ is the magnetic permeability, and σ is the conductivity. Complex permittivity εc is εc = ε-/ω. The conditions at material interfaces with abrupt changes of ε and µ are typical for the variational formulation and do not require special attention.

The boundary conditions associated with AC power electromagnetics are a Dirichlet boundary condition, specifying the value of the electric field Ec on the boundary, and a Neumann condition, specifying the normal derivative of Ec. This is equivalent to specifying the tangential component of the magnetic field ${H}_{t}=\frac{j}{\omega }n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(\frac{1}{\mu }\nabla {E}_{c}\right)$.

For details, see AC Power Electromagnetics.

## Apps

 PDE Solve partial differential equations in 2-D regions

## Topics

### Programmatic Workflow

Wave Equation

Solve the wave equation using the PDE app and the command line.

Helmholtz's Equation on a Unit Disk with a Square Hole

Solve a Helmholtz equation using the solvepde function.

### PDE App Workflow

AC Power Electromagnetics

Analyze the skin effect produced by AC current carried by a wire with circular cross section.

Wave Equation

Solve the wave equation using the PDE app and the command line.

Scattering Problem

Solve a simple scattering problem computing the waves reflected from an object illuminated by incident waves.

### Concepts

Hyperbolic Equations

Mathematical definition and discussion of the hyperbolic equation