Discussion Forum

Hi Jan,

in my understanding it is required to use mef instead of mf as soon as the electric field between conductors has a significant impact on the current flow. The power_inductor.mph model in the application library may be an example. As soon as the frequency is high enough to result in high reactance in the coil we have to take capacitive effects between the windings into account. Those would be ignored by mf. So in this example mf might still be useful in the DC case or at very low frequencies. However, an inductor like this is typically used at frequencies that imply significant reactance. Then mef is appropriate. To some degree mef maybe seen half way between mf and the RF interfaces. The system is still small compared to the wavelength, so RF and thus wave propagation is not yet needed, but capacitive effects are strong enough, so solving for the electric potential is required. To my knowledge, the mf interface is not checking if mef would be the better choice and I don't think any physics interface is checking for its suitability. I hope this can help and I hope I haven't missed a point by myself.

Cheers Edgar

This is an intriguing question indeed.

Let's limit the discussion to the use of magnetic vector potential and electric scalar potential (discarding the, mathematically symmetric, case of electric vector potential and magnetic scalar potential aka T-omega formulation).

In statics, the electric field is curl free (no induction) and thus the electric field has to be described using a scalar potential (V). From the scalar potential distribution and conductivity, we can use Ohm's law to compute the current density distribution which is the source term for the magnetic vector potential (A). The magnetic vector potential will only be unique if we specify its divergence. That is, we have to specify the gauge. However, in statics, the electric field is unaffected by the gauge .

Once we introduce time-varying fields, the electric field will get contributions from both V and the time derivative of A, where the size of the contributions depend on the chosen gauge. In particular, it is possible to choose a gauge in which the scalar electric potential V vanishes. This is described in the section on Gauge Transformations in the AC/DC Module User’s Guide (p 311 in version 5.4).

The gauge where V vanishes is the implicitly chosen one in the frequency domain version of the Magnetic Fields interface which is a full Maxwell formulation that includes wave propagation effects.

The equation of continuity of is inherent in the equation solved for. If you take the divergence of the entire equation, the curl parts evaluate to zero and you are left with the current conservation equation.

If you on the other hand keep both A and V, as in the Magnetic and Electric Fields interface, the gauge is not specified so the system of equations is singular (the divergence of the equation for A yields the equation for V). You can still find a solution. If you use an all-iterative solver it will converge to a solution that depends on mesh, solver used and initial conditions. However, the resulting electric and magnetic fields are still uniquely determined. COMSOL will set up a suitable iterative solver for you. You can also add an explicit gauge fixing node (at some computational expense) and use a direct solver. Our A-V formulation is also a full Maxwell formulation.

So what are the advantages of using A only or both A and V?

The A only formulation is the result of choosing a particular dynamic gauge. It breaks down when approaching the static limit as electric fields then cannot be represented by a magnetic vector potential. In practice this happens when the induced current density term becomes numerically insignificant and it typically happens first in domains with zero conductivity (induced displacement current density only). By adding some artificial conductivity to such domains one can push the limit a bit but the price is that capacitive effects are not properly captured. The advantage is that, given the current density is high enough, it is inherently gauged (non-singular) and numerically stable and direct solvers can be used. For many eddy current problems, capacitive effects can be neglected.

The A-V formulation does not break down at low frequency but separates gracefully into two one-way coupled equations. On the other hand the inherent gauge uncertainty has to be handled either by using a correctly configured all iterative solver, or by applying the explicit gauge fixing node.

As you pointed out, there are also some practical considerations when it comes to available features and excitations. Through the explicit current conservation equation, one can use the Terminal, Ground and Electric potential subfeatures to the Magnetic Insulation boundary condition to excite the model. It is also possible to impose Electric Insulation or Contact Impedance subfeatures to the Magnetic Continuity boundary condition to model thin dielectric layers embedded in a metallic domain. The latter is not doable in a pure A formulation.

Some concluding remarks:

Note that, as the equation of continuity is imposed by both formulations (and by Maxwell's equations), open current loops are not allowed and will either be closed by induced domain currents or by the explicit gauge fixing or the model may refuse to solve. When a Terminal or Electric Potential condition is applied on an external Magnetic Insulation boundary, induced surface currents on the Magnetic Insulation boundary will effectively close the current loop for you. One may in that case interpret the Magnetic Insulation condition as a symmety condition (or as an external ground plane if the Ground subnode is applied).

In the time domain, we neglect displacement currents in the Magnetic Fields interface (quasi-static approximation) so then it does not support wave propagation. The Magnetic and Electric Fields interface does currently not support time domain modeling at all.