This example shows the *skin effect* when a wire with a circular cross
section carries AC current. In a solid conductor, such as the wire, AC current travels near
the surface of a wire and avoids the area close to the center of the wire. This effect is
called the skin effect. The example uses the PDE Modeler app.

The Helmholtz equation

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

describes the propagation of plane electromagnetic waves in imperfect dielectrics and good
conductors (*σ* » *ωε*). The coefficient of dielectricity
is *ε* = 8.8*10^{-12} F/m. The conductivity of copper is *σ* = 57 * 10^{6} S/m. The magnetic permeability of copper is close to the magnetic permeability
of a vacuum, *µ* = 4*π**10^{–7}
H/m. The
*ω*^{2}*ε*-term is negligible at
the line frequency (50 Hz).

Due to 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. The Dirichlet condition for the electric field is

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

Here,

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

*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.

To solve this problem in the PDE Modeler app, follow these steps:

Draw a circle with a radius of 0.1. The circle represents a cross section of the conductor.

pdecirc(0,0.05,0.1)

Set the

*x*-axis limit to`[-0.2 0.2]`

and the*y*-axis limit to`[-0.1 0.2]`

. To do this, select**Options**>**Axes Limits**and set the corresponding ranges. Then select**Options**>**Axes Equal**.Set the application mode to

**AC Power Electromagnetics**.Specify the Dirichlet boundary condition

*E*=*J*/_{S}*σ*= 1/*σ*for the boundary of the circle. To do this:Switch to the boundary mode by selecting

**Boundary**>**Boundary Mode**.Select all boundaries by using

**Edit**>**Select All**.Select

**Boundary**>**Specify Boundary Conditions**.Specify

`h = 1`

and`r = 1/57E6`

.

Specify the PDE coefficients. To do this, switch to the PDE mode by selecting

**PDE**>**PDE Mode**. Then select**PDE**>**PDE Specification**or click the**PDE**button on the toolbar. Specify the following values:Angular frequency

`omega = 2*pi*50`

Magnetic permeability

`mu = 4*pi*1E-7`

Conductivity

`sigma = 57E6`

Coefficient of dielectricity

`epsilon = 8.8E-12`

Initialize the mesh by selecting

**Mesh**>**Initialize Mesh**.Solve the PDE by selecting

**Solve**>**Solve PDE**or clicking the**=**button on the toolbar.The solution of the AC power electromagnetics equation is complex. When plotting the solution, you get a warning message.

Plot the current density as a 3-D plot. To do this:

Select

**Plot**>**Parameters**.Select the

**Color**and**Height(3-D plot)**options.Select

`current density`

from the**Property**drop-down menu for both the**Color**and**Height(3-D plot)**options.Select

**Show Mesh**to observe the mesh.

Due to the skin effect, the current density at the surface of the conductor is much higher than in the conductor's interior.

Improve the accuracy of the solution close to the surface by using adaptive mesh refinement. To do this:

Select

**Solve**>**Parameters**.In the resulting dialog box, select

**Adaptive mode**.Set the maximum numbers of triangles to

`Inf`

.Set the maximum numbers of refinements to

`1`

.Select the

`Worst triangles`

selection method.

Recompute the solution five times. Each time, the adaptive solver refines the area with the largest errors. The number of triangles is printed at the command line.

Plot the current density as a 3-D plot.

These plots show the real part of the solution, but the solution vector is the full complex solution. Plot the imaginary part of the solution. To do this:

Select

**Plot**>**Parameters**.Select the

**Color**and**Height(3-D plot)**options.Select

`user entry`

from the**Property**drop-down menu for both**Color**and**Height(3-D plot)**options.Type

`imag(u)`

in the corresponding**User entry**fields.Select

**Show Mesh**to observe the mesh.

Observe that the skin effect depends on the frequency of the alternating current. When you increase or decrease the frequency, the skin "depth" increases or decreases, respectively. At high frequencies, only a thin layer on the surface of the wire conducts the current. At very low frequencies (approaching DC conditions), almost the entire cross section area of the wire conducts the current.

Find the solution for the angular frequencies

`omega = 2*pi*1000`

,`omega = 2*pi*50`

, and`omega = 1E-6`

. Plot the real parts of the solutions in 2-D.**Current density for omega = 2*pi*1000****Current density for omega = 2*pi*50****Current density for omega = 1E-6**