At the scale that we are interested in modeling, only 2 of the 5 fields, along with the free charge density, are unique. The non–unique fields are related through constitutive equations. Because the constitutive equations are intended to capture relationships observable at the macroscopic level, they are based on some material–dependent modeling assumptions. The proper modeling assumptions are highly dependent on the effects that are required to be captured accurately, and also on the time scale on which those effects occur.
In general, it will be assumed that the electric flux density and electric current density are linearly and isotropically related to the electric field intensity:
D = ǫ0E + P, (2.7)
J = σE, (2.8)
where ǫ0 is the vacuum permittivity, P is the polarization of the material, and σ is the
electrical conductivity of the material. The polarization is generally some function of E, the details of which turn out to be relatively unimportant for the applications we are interested in. This will be evident after the discussion of the magnetoquasistatic approximation in Section 2.3.
The modeled value of the conductivity σ can range from 0 for nonconducting materials to values on the order of 107 for high quality conductors such as copper. Since material
conductivity is strongly dependent on temperature, some assumption on the temperature distribution must be made in order to fix this value for each material. We will assume that each material region is at a constant uniform temperature, although discrete jumps may occur whenever material discontinuities are encountered. This is a fair assumption, as materials with large electrical conductivities tend to have large thermal conductivities, resulting in a
feedback mechanism that prevents the generation of appreciable temperature gradients over the spatial scales of interest when Joule heating is the dominant loss mechanism.
Several possible constitutive relationships exist between H and B. In nonmagnetic ma- terials, the relationship is simply
H = νoB, (2.9)
where ν0 = 10
−7
4π is the vacuum reluctivity. Often magnetic materials are modeled using the
linear relationship
H = ν · B (2.10)
where ν is the reluctivity tensor. A further simplification can be made if the material is assumed to be isotropic. Then it is common to write
H = νB (2.11)
where ν is now a scalar. The reluctivity values can be normalized to the vacuum reluctivity by introducing the relative reluctivity νr such that
ν= ν0νr (2.12)
It is also possible to write the linear B–H relationships using permeability properties, usually denoted µ, that is inversely related to the reluctivity:
µ= ν−1 (2.13)
This is perhaps more common, but for our purposes the permittivity model turns out to be more convenient.
No magnetic material is truly linear. While linear relationships are useful for analytical calculations, an essential detail that must be captured when operating at high field levels
is the presence of magnetic saturation. A standard way of doing this is to assume ν is a nonlinear function of B (or possibly H). Instead, the approach taken here is to start with a cursory consideration of another important magnetic phenomenon, hysteresis, and simplify it somewhat to obtain a logically equivalent model that somewhat better represents the physical underpinnings of magnetism.
In ferromagnetic materials, the most general relationship between B and H is governed by the evolution of the magnetic domains intrinsic to the material occurring on the mi- crometer level [104, 105]. The details of the magnetic domains are mesoscopic effects, which depend on defects in the crystal lattice structure and even local stresses and strains. This is more appropriately the subject of micromagnetics, and is too detailed a consideration for our purposes. If instead we consider the ensemble average effect of the individual magnetic domains, a quite general macroscopic constitutive relationship can be posited:
B (t) = µ0[H (t) + M (H (τ ) ; τ ≤ t)] , (2.14)
where M is the (macroscopic) magnetization. The arguments of the magnetization indicate that M may depend on the entire history of H.
The evolution of the magnetization is governed by the minimization of complex, non- convex energy potentials that are determined by the coupling of various phenomena occurring at disparate spatial scales. At the macroscopic level, this effect is apparent through the observation of hysteresis loops in plots of |M| or |B| versus |H|. Magnetic hysteresis is a significant loss mechanism in magnetic devices, and accurately capturing this effect is essential to optimizing certain designs.
Hysteresis modeling is fraught with difficulties, due to a dearth of empirical data and the lack of a clear theory for passing from first–principles micromagnetic models to the macro- scopic scale. In order to simplify our modeling, we shall assume the following (instantaneous)
constitutive relationship holds:
H = ν0B − M (B) . (2.15)
Notice that M is now considered as a function of B, which will prove useful later on. Equation (2.15) represents a general non–isotropic relationship where M and B do not necessarily point in the same direction. Such a relationship is necessary for modeling materials that demonstrate a strong “preferred” magnetization direction.
From (2.15), we make two more material dependent assumptions. First, for permanent magnet materials, we assume an affine relationship holds:
H = ν0B − Mr, (2.16)
where Mr is the remanent magnetization. This relationship for the permanent magnets is
only valid below the Curie temperature and when the applied field opposing Mr is not too
large. For non–grain–oriented steels, we make the further simplifying assumption that
M = χB, (2.17)
where χ is the magnetic susceptibility; comparing (2.17) to (2.15), evidently
χ= (1 − νr) = |M|
|B|. (2.18)
Note that this definition of χ is somewhat different than the standard definition of magnetic susceptibility as M = χH. Equation (2.18) demonstrates an equivalency between magne- tization and reluctivity modeling under these assumptions. We will assume all nonlinear magnetic materials are non–grain–oriented.
If we accept the fact that the effects of hysteresis are neglected, the preceding assumptions are reasonable when simulating electric machines. Most electric machines use non–grain– oriented steel as there is not a single preferred direction for the magnetic flux. Similarly, if
the field levels are high enough to appreciably demagnetize any permanent magnets, it is nec- essary to include hysteretic effects in order to accurately capture this phenomenon. Further, the demagnetization characteristics of hard magnetic materials are temperature dependent, which adds additional complications beyond the inclusion of temperature–independent hys- teresis effects.
Substituting these relationships into Equations (2.1)–(2.6) yields the following modified Maxwell’s equations: ǫ0∇ · E + ∇ · P = ρ, (2.19) ∇ · B = 0, (2.20) ∇ × E = −∂B∂t , (2.21) ν0∇ × B − ∇ × M = σE + ǫ0 ∂E ∂t + ∂P ∂t , (2.22) ∇ · (σE) = −∂ρ∂t. (2.23)
Notice that, because σ may contain finite discontinuous jumps at material interfaces, it is not possible to use the linearity of the divergence to put (2.23) into a simpler form. The magnetic nonlinearity is assumed to be included in M in the right hand side of (2.22) along with the remanent magnetization of any permanent magnets.