## Understanding the Magnetic Insulation Boundary Condition

When using the AC/DC Module, the most common interface used for solving electromagnetic fields is the *Magnetic Fields* interface. This interface solves for the magnetic vector potential field, and from that computes the electric field. This guide addresses the default boundary condition within the *Magnetic Fields* interface, the *Magnetic Insulation* (MI) boundary condition.

### The Mathematical Form of the Equation and Its Implications

Within the *Magnetic Fields* interface, the *Magnetic Insulation* boundary condition enforces the equation:

This equation means that the cross product of the normal vector to the surface, , and the magnetic vector potential, , has to equal zero on that surface. It implies that that magnetic vector potential (the -field) has to be perpendicular to the boundary. This magnetic vector potential is a field that is useful for the purposes of solving Maxwell’s equations — it isn’t directly measurable nor does it directly appear in any constitutive relations. The -field is excited via other boundary conditions, or domain conditions, that excite currents flowing through the model or along surfaces. The software solves for the -field that satisfies the domain and boundary conditions.

There are three quantities that are defined from the -field and are measurable: the magnetic field, the current, and the electric field. First, the magnetic field, , is the curl of the -field:

Since the -field is strictly perpendicular to the surface, the -field must be everywhere tangential to the surface, without any component in the normal direction. Next, the current on the boundary is defined as

where the -field is defined from the -field via the constitutive relation used in the adjacent domain, such as . Since the -field must be purely tangential to the surface, its curl is also tangential to the surface. So, there can be currents that flow along the surface, and these currents are perpendicular to the -field. The electric field, , is the time derivative of the -field:

This means that, for a steady-state model, the -field does not contain any information about, or contribute to, the -field. For time-dependent or frequency-domain models, the -field will be perpendicular to the face and parallel to the -field, but with 90° phase lag. The -, -, and -fields are all orthogonal to each other. For a more detailed introduction to the equations and derivations, see: "What is Gauge Fixing? A Theoretical Introduction".

It is useful to visualize these vector fields on a surface, as shown below. The fields are excited via a sinusoidally time-varying excitation from within the model, and this visualization considers only a small patch of the boundary. The -field is everywhere perpendicular to the surface and oscillates in time.

*An animation of the magnetic vector potential (gray arrows) over a surface (transparent blue) with the* Magnetic Insulation *boundary condition surrounding a sinusoidally time-varying source. The colored surface is a visualization of the potential field on the surface.*

*An animat ion of the magnetic vector potential (gray arrow) as well as the magnetic field (red arrow), the surface current (green arrow), and the electric field (blue arrow).*

There might appear to be a contradiction in the above equations, as they mean that current will flow tangentially to the surface, even though the electric field has no component tangent to the surface. This appears to be in contradiction with the constitutive relation:

where is the electric conductivity. However, if , then this constitutive relation does not have meaning, and we arrive at an interpretation of this being the surface of a perfectly electrically conductive material with infinite conductivity, or zero resistivity. This leads to several other useful interpretations.

### The Magnetic Insulation Boundary Condition as a Type of Symmetry Condition

A common example of a material with very high conductivity is the layer of metal on a mirror, a plane that produces a symmetric image. If the *Magnetic Insulation* boundary condition is applied at a planar boundary of the modeling domain, and if no portion of the modeling domain extends past that boundary, then it can be interpreted as a symmetry condition. All geometry of the modeled structure is mirrored about this plane, but the polarity of sources is switched.

As an example, consider a loop of conductive wire connected to a DC source (a battery) leading to an electric field and current flow through the material, and a magnetic field in the surrounding space. A *Magnetic Insulation* boundary condition on one side of this loop implies that all of the structure exists on the other side of the plane, but with the terminals of the battery switched. This leads to currents flowing along the mirrored path but in the opposite direction.

*A battery connected to a wire and placed next to a* Magnetic Insulation *boundary condition. The current (green arrows) leads to a magnetic field (red arrows) The current flows through the battery and wires. Note the polarity of the battery.*

*The implication of* Magnetic Insulation *as a symmetry boundary condition is that a second, mirrored, structure exists on the other side with the polarity of sources reversed, leading to currents flowing in the opposite direction along the mirrored paths. The current flowing through the battery is not pictured.*

This symmetry condition also holds if the conductors are going into the boundary. The figures below show how the fields are reflected when the wire intersects the boundary. Note how the polarity of the source is reversed. Since only half of the source (the battery) exists on one side of the boundary, the magnitude of the potential due to that source (the battery voltage) is halved but excites the same magnitude current through the system.

*The* Magnetic Insulation *boundary condition can cut through a source and imposes symmetry such that the polarity of the source is reversed.*

It is possible to combine multiple *Magnetic Insulation* boundary conditions on different sets of planar surfaces and interpret them as a symmetry condition as long as the effect of the reflection is to fill the entirety of space. Both Cartesian symmetry and symmetry about an axis are allowed, as illustrated in the two figures below. For the case of symmetry about an axis, the angle of the modeling space must be 180°/*N*, where *N* is an integer.

*Three orthogonal* Magnetic Insulation *boundary conditions and their symmetry implication.*

*Symmetry about an axis is imposed via the* Magnetic Insulation *boundary conditions when the angle between the faces divides the unit circle into an even number of sectors.*

If the object is placed at the center of a cubical domain with all boundaries set to *Magnetic Insulation*, this implies that the entirety of an infinite space is filled with a 3D, periodically repeating pattern of the object, with spacing equal to the cube size.

*An alternative type of symmetry can be imposed via the* Perfect Magnetic Conductor *boundary condition. It implies that a second, mirrored, structure exists on the other side with the same polarity of sources, leading to currents flowing in the same direction along the mirrored paths.*

There are two additional boundary conditions that can be used to model other types of symmetry. First, the *Perfect Magnetic Conductor* boundary condition similarly imposes mirror symmetry on the structure but does not imply a switch of the polarity of sources. Second, the *Periodic Condition* boundary condition imposes that the fields on two boundaries of identical shape have identical fields. For additional details on these conditions, see: "Exploiting Symmetry to Simplify Magnetic Field Modeling".

### The Magnetic Insulation Boundary Condition as an Approximation of Infinite Free Space

The previous section showed that a model of a system sitting at the center of a cubical domain with *Magnetic Insulation* boundaries is equivalent to an infinite 3D periodic pattern of that same system. There is a second interpretation of the *Magnetic Insulation* boundary conditions that entirely surround the system of interest, and that is as an approximation of infinite free space. Under this interpretation it is not necessary to restrict the *Magnetic Insulation* boundaries to be planar, but they do have to define a convex domain entirely enclosing the system. A set of spherical surfaces enclosing the system represents a good way to understand this interpretation.

*Enclosing the system within a convex set of* Magnetic Insulation *boundary conditions represents an approximation of free space. Current flows both through the battery and wire, and the resultant magnetic field intensity drops off with distance.*

Assuming that the system sits within an infinite space, empty of any other structure, the magnetic fields due to the current flow will extend to infinity but drop off exponentially with increasing radial distance. Consider two points, one sitting just within the sphere and the other just outside. We place ourselves at the point outside of the sphere and, if we temporarily ignore the *Magnetic Insulation* boundary conditions, know that there must be some nonzero magnetic field due to the currents flowing through the system.

To get a zero magnetic field at this point, one needs to introduce a canceling current. One could introduce a canceling current that is exactly opposite the current flow already in the device, but this would simply lead to a system with no current flow and no magnetic fields. It is also possible to introduce currents on the surface of the sphere. An appropriate current distribution over this surface will lead to zero fields outside of the sphere. The *Magnetic Insulation* boundary condition, which allows currents to flow along the surface, can be thought of as solving for this canceling surface current.

*The surface currents on the* Magnetic Insulation *boundary condition are the currents that will lead to zero magnetic fields outside of the domain.*

As the radius of the domain is increased, the surface currents will decrease in magnitude and have less and less of an effect on the fields just within the boundaries. However, increasing the radius of the surrounding sphere will increase the domain size and overall computational cost, so it is desirable to keep this domain as small as possible. The quantities of interest, such as the system inductance and impedance, can be studied with increasing domain size; these will converge monotonically. The *Perfect Magnetic Conductor* boundary condition can alternatively be used to truncate the modeling domain. With increasing domain size, this approach will converge toward the identical solution. For additional details on this approach, as well as using infinite elements, and the tradeoffs involved, see: "How to Choose Between Boundary Conditions for Coil Modeling".

### The Magnetic Insulation Boundary Condition as an Electrically Unconnected Perfect Electric Conductor

The previous section showed that there are currents flowing along the surface of a *Magnetic Insulation* boundary condition, and that these exist even in the case of a DC analysis, implying no variation with respect to time. This leads to results that require some careful interpretation. Consider the system shown below, of a sphere placed *within* the center of a loop of the wire connected to a DC source, with free space between the wire and sphere. The currents on the surface of this sphere are plotted, along with the currents through the wire. This might appear counterintuitive — how can a DC current flowing through a wire induce current to flow in an electrically unconnected object?

*A spherical set of* Magnetic Insulation *boundary condition placed inside of a wire with DC current. There are surface currents induced on these boundaries.*

An appropriate way to think about this is to view a stationary model as a frequency-domain model being excited at a very low, albeit nonzero, frequency and to introduce the concept of skin depth. This is the distance into a conductor over which the majority of the induced current will flow. A simple expression for the skin depth within a good conductor is:

The *Magnetic Insulation* boundary condition means that , so it also implies that the skin depth is zero. That is, no matter how low the frequency, eddy currents will always be induced and always flow at the surface. So, for the case of a stationary DC model, the *Magnetic Insulation* boundary condition is an idealization that should typically not be used to approximate the boundaries of a highly conductive domain that is electrically unconnected.

On the other hand, when solving a time-domain problem, it can be valid to interpret these currents as an approximation of the currents flowing near the surface of a good conductor and the resultant shielding. This does depend on how the system is excited and the objectives of the model. Verifying the validity of the approximation will require comparing against a model that solves for the fields within the volume of the good conductor.

When solving a frequency-domain model, the *Magnetic Insulation* boundary condition has a more clear interpretation if modeling a real conductor where the skin depth is much smaller than the part dimensions. In this case, it is a reasonable approximation, as long as the losses on the conductive material are not of interest. Alternatively, the *Impedance Boundary Condition* models the boundary of a domain with finite conductivity.

*A set of* Magnetic Insulation *boundary conditions on interior boundaries. The magnetic vector potential is solved for on both sides, so fields and surface currents will be different on either side. For an AC excitation, this will reasonably approximate the shielding due to a thin conductor.*

It can also be reasonable to apply a *Magnetic Insulation* boundary condition to an interior boundary, where the magnetic vector potential is being solved for on both sides, and there will be a different magnetic field, electric field, and surface current on either side of the boundary. This can be appropriate for modeling a relatively thin-walled structure. In the frequency domain, this implies that the skin depth is much smaller than the wall thickness and thus represents a perfect shielding condition. Alternatively, the *Transition Boundary Condition* models interior boundaries of thin-walled lossy materials.

### The Magnetic Insulation Boundary Condition as an Electrical Ground Condition

Here, an electrical ground is meant to be any domain that is relatively large compared to the system in question, or has a relatively high electric conductivity, or both, that can act as an infinite source and sink of current. This includes both *earth ground* (such as bedrock, soil, or a body of water) as well as *chassis ground* (such as an airplane fuselage or car frame). Any part of the electrical circuit that is nearby or touching this boundary will lead to induced currents flowing along the surfaces.

A *Magnetic Insulation* boundary condition of any shape can be interpreted as this ground condition. It is important to keep in mind that the currents flowing on the surface are induced due to the structure of the nearby electrical system; they do not follow the shortest path. For time-domain and frequency-domain analyses, this is reasonable. On the other hand, for stationary analysis, the currents through the medium of the domain should be modeled if accuracy of the nearby fields is of interest.

*A* Magnetic Insulation *boundary condition of any shape can be interpreted as a ground condition. Currents are induced on the surface due to the time variation of currents flowing within the modeling domain.*

A structure within the modeling domain with the *Magnetic Insulation* boundary condition applied to the boundaries can be interpreted as a chassis ground. Connecting the electrical circuit to this structure will also lead to currents flowing along the surface, and these should similarly be interpreted as induced currents and hence will not necessarily take the shortest return path through the structure.

When modeling earth or chassis ground in frequency-domain models, the *Impedance Boundary Condition* and the *Transition* boundary conditions are alternatives that will additionally compute losses. In time-domain models, the entire volume of the material might need to be modeled. In DC models, it is often sufficient to only consider current flow through part of the domain.

*The* Magnetic Insulation *boundary condition on the boundaries of a structure sitting within the modeling domain can be interpreted as a chassis ground when electrically connected to a source. If this was instead a DC model, only the part of the frame between the wires would need to be modeled.*

### The Magnetic Insulation Boundary Condition as the Boundaries of a Source

There are many ways to excite a system. Currents can be imposed to flow along edges, boundaries, or through volumes. These currents, at any one instant in time, are flowing into one terminal of the source and out of the other terminal, thus maintaining conservation of the current. Although the details of the current flow through the interior of source are assumed to not be relevant, there must still always be a way for the current to flow in a solenoidal path. The *Magnetic Insulation* boundary condition can be applied at the boundaries of the source, and in this context it represents a current closure that shields the interior details of the source. This interpretation is valid in DC, AC, and transient modeling regimes. The geometry of the source can be simplified, es pec ially if the source is far away from the fields of interest.

*Current flowing around the loop of the wire are driven by a source. The currents flowing on the surface of the* Magnetic Insulation *boundary condition represent the interior of the source.*

### The Magnetic Insulation Boundary Condition as a Waveguiding Condition

When solving in the time or frequency domain, there will be a nonzero electric field, and as a consequence there will be a nonzero Poynting vector, or power flow, which is:

At a *Magnetic Insulation* boundary condition, this vector has to be parallel to the surface current and does not vary in time. The implication of this is that power can be flowing through the adjacent modeled medium and will be flowing parallel to the *Magnetic Insulation* boundary condition. This interpretation is more applicable at higher frequencies, when the size of the model is comparable to the wavelength within the material. When solving in the frequency domain, the *Impedance Boundary Condition* and the *Transition Boundary Condition* can also be used to model a waveguiding structure.

*A model of the interior of a coaxial cable. The power flow (magenta arrows) within the modeling domain is parallel to the current flow on the surfaces.*

### Summary

The *Magnetic Insulation* boundary condition has several different interpretations. You are free to choose whichever interpretation of this boundary condition that you want for the modeling purposes at hand. Currents are always free to flow along any set of adjacent *Magnetic Insulation* boundaries. These induced surface currents will often require different interpretations when evaluating the model, so it is important to be familiar with all of the implications of using this boundary condition.

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