Documentation |
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 ε_{c} = ε-jσ/ω. The conditions at material interfaces with abrupt changes of ε and µ are the natural ones for the variational formulation and need no special attention.
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 E_{c}. This is equivalent to specifying the tangential component of the magnetic field H:
$${H}_{t}=\frac{j}{\omega}n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(\frac{1}{\mu}\nabla {E}_{c}\right).$$
Interesting properties that can be computed from the solution—the electric field E—are the current density J = σE 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 = 4π10^{–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, E_{c} = 1/σ. For this case an analytical solution is available,
$$J={J}_{S}\frac{{J}_{0}\left(kr\right)}{{J}_{0}\left(kR\right)},$$
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 = 2π · 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