Eddy Current Solver

To generate the resistance matrix associated with a lossy transmission line, the software uses the eddy current field simulator to compute the energy stored in the magnetic fields and energy losses in the line (the electric scalar potential), f, at all points in the problem region. From f it first derives the magnetic vector potential, Az, and then derives the B-field, H-field, and the current density J from Az.

Note:

Use the AC conduction solver to compute conduction currents resulting from time-varying electric fields in lossy dielectrics.
Eddy solver uses magnetic loss tangent and ignores the dielectric loss tangent.
In Q3D and 2D Extractor, displacement current is assumed to be negligible during Eddy current computations, and is therefore neglected in the RL solution.

The following two equations are solved by the eddy current field simulator using the finite element method:

and

where:

A is the magnetic vector potential.
Φ is the electric scalar potential.
μ is the magnetic permeability.
ω is the angular frequency.
σ is the conductivity.
I_T is the total current flowing in conductors.
Ω is the area of the conductor cross-section.

These are derived from Maxwell 's equations, and use the total current you specify in conductors connected to an external source. Phasor notation is used to represent complex quantities.

The eddy current solver makes the following assumptions about the field quantities for which it solves:

All quantities must have the same value of , but can have different phase angles and amplitudes (Fm). If a current is not a pure sinusoid, it is decomposed into sinusoidal harmonics, and solved separately at each frequency.
The time-varying electromagnetic quantities B and D are assumed to have the periodic waveform F(t) = Fm cos(t + q) .
All currents (source and eddy) are assumed to flow in the z-direction. Therefore, the magnetic fields associated with these currents lie within the xy-plane. As a result, the magnetic vector potential, A, has a z-component only.
Because no currents flow in the xy-plane, the electric field, E, has a z-component only. It follows that f is constant over the cross-section of each conductor in the problem.

Magnetic and Electric Fields (First Equation)

To solve for the magnetic vector potential A_z, the eddy current solver uses the definition:

Substituting this into the first of Maxwell 's equations, and neglecting the displacement current term,

The result is:

Substituting it into the second of Maxwell 's equations:

And using the definition of E, we find that

So:

Current and Current Density (Second Equation)

Notice that the first equation is given in the form of conductivity multiplied by the complex value of E:

Hence, the total complex current density (J_total) is defined as:

J_total + J_source +J_eddy

where J_source =

and J_eddy =

The jω term in J_eddy indicates that eddy current becomes increasingly significant as frequency increases.

The integral of this expression over the cross-section of a conductor is must equal the total current that you specify in that conductor when you set up a problem.

The total current, I_total, is defined as:

I_total = I_source +I_eddy

Where:

I_source is the source current, and J_source is the source current density, due to the potential difference –σ∇Φ generated by the external source. It is the current that the source would supply if you reduced the potential difference by the back EMF produced by eddy current in the conductor.
I_eddy is the eddy current –∫jωσAdΩ, and J_eddy is the eddy current density –jωσA, induced in the conductor by time-varying magnetic fields.

Since A only has a z-component in the eddy current solver, Φ is constant for each cross-section of a conductor. Therefore, the field simulator does not have to solve for Φ at every node.