AC power electromagnetics problems are found when studying motors, transformers and conductors carrying alternating currents.

Let us start by considering a homogeneous dielectric, with coefficient
of dielectricity ε and magnetic permeability *µ*,
with no charges at any point. The fields must satisfy a special set
of the general Maxwell's equations:

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

For a more detailed discussion on Maxwell's equations, see Popovic,
Branko D., *Introductory Engineering Electromagnetics*,
Addison-Wesley, Reading, MA, 1971.

In the absence of current, we can eliminate **H** from
the first set and **E** from the second
set and see that both fields satisfy wave equations with wave speed $$\sqrt{\epsilon \mu}$$:

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

We move on to studying a charge-free homogeneous dielectric,
with coefficient of dielectrics *ε*, magnetic
permeability *µ*, and conductivity *σ*.
The current density then is

$$J=\sigma E$$

and the waves are damped by the Ohmic resistance,

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

and similarly for **H**.

The case of time harmonic fields is treated by using the complex
form, replacing **E** by

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

The plane case of this Partial Differential Equation Toolbox™ mode has $${E}_{c}=\left(0,0,{E}_{c}\right),\text{}J=\left(0,0,J{e}^{j\omega t}\right)$$, and the magnetic field

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

The scalar equation for *E _{c}* becomes

$$-\nabla \text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla {E}_{c}\right)+\left(j\omega \sigma -{\omega}^{2}\epsilon \right){E}_{c}=0.$$

This is the equation used by Partial Differential Equation Toolbox software
in the AC power electromagnetics application mode. It is a complex
Helmholtz's equation, describing the propagation of plane electromagnetic
waves in imperfect dielectrics and good conductors (*σ* » *ωε*).
A *complex permittivity* *ε _{c}* can
be defined as

The PDE parameters that have to be entered into the PDE Specification
dialog box are the *angular frequency* *ω*,
the magnetic permeability *µ*, the conductivity *σ*,
and the coefficient of dielectricity *ε*.

The boundary conditions associated with this mode are a Dirichlet
boundary condition, specifying the value of the electric field *E _{c}* on
the boundary, and a Neumann condition, specifying the normal derivative
of

$${H}_{t}=\frac{j}{\omega}n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla {E}_{c}\right).$$

Interesting properties that can be computed from the solution—the
electric field **E**—are the current
density **J** = *σ***E** and the magnetic flux density

$$B=\frac{j}{\omega}\nabla \times E.$$

The electric field **E**, the current
density **J**, the magnetic field **H** and the magnetic flux density **B** are available for plots. Additionally, the
resistive heating rate

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

is also available. The magnetic field and the magnetic flux density can be plotted as vector fields using arrows.

The example shows the *skin effect *when
AC current is carried by a wire with circular cross section. The conductivity
of copper is 57 · 10^{6},
and the permeability is 1, i.e., *µ* = 4*π*10^{–7}.
At the line frequency (50 Hz) the *ω*^{2}*ε*-term
is negligible.

Due to the induction, the current density in the interior of
the conductor is smaller than at the outer surface where it is set
to *J _{S}* = 1, a Dirichlet
condition for the electric field,

$$J={J}_{S}\frac{{J}_{0}\left(kr\right)}{{J}_{0}\left(kR\right)},$$

where

$$k=\sqrt{j\omega \mu \sigma}.$$

*R* is the radius of the wire, *r* is
the distance from the center line, and *J*_{0}(*x*)
is the first Bessel function of zeroth order.

Start the PDE app and set the application mode to **AC
Power Electromagnetics**. Draw a circle with radius 0.1 to
represent a cross section of the conductor, and proceed to the boundary
mode to define the boundary condition. Use the **Select
All** option to select all boundaries and enter `1/57E6`

into
the **r** edit field in the Boundary Condition dialog
box to define the Dirichlet boundary condition (*E* = *J*/*σ*).

Open the PDE Specification dialog box and enter the PDE parameters.
The angular frequency *ω* =
2*π* · 50.

Initialize the mesh and solve the equation. Due to the skin
effect, the current density at the surface of the conductor is much
higher than in the conductor's interior. This is clearly visualized
by plotting the current density *J* as a 3-D plot.
To improve the accuracy of the solution close to the surface, you
need to refine the mesh. Open the Solve Parameters dialog box and
select the **Adaptive mode** check box. Also, set
the maximum numbers of triangles to `Inf`

, the maximum
numbers of refinements to 1, and use the triangle selection method
that picks the worst triangles. Recompute the solution several times.
Each time the adaptive solver refines the area with the largest errors.
The number of triangles is printed on the command line. The following
mesh is the result of successive adaptations and contains 1548 triangles.

**The Adaptively Refined Mesh**

The solution of the AC power electromagnetics equation is complex.
The plots show the real part of the solution (a warning message is
issued), but the solution vector, which can be exported to the main
workspace, is the full complex solution. Also, you can plot various
properties of the complex solution by using the user entry. `imag(u)`

and `abs(u)`

are
two examples of valid user entries.

The skin effect is an AC phenomenon. Decreasing the frequency of the alternating current results in a decrease of the skin effect. Approaching DC conditions, the current density is close to uniform (experiment using different angular frequencies).

**The Current Density in an AC Wire**

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