IEEE Power Electronics Magazine - September 2023 - 20

Scaling up to Bulk Materials and Refining the Model
The above examples showcase the potential of the proposed
modeling framework, in simple cases where the core
can be treated as having a uniform M. In the discussion of
scaling up, we refer to a region of uniform M governed by
(1) in a core material as a " domain, " which does not necessarily
identify with a physical magnetic domain although
conceptually similar. We first point out that a single-domain
model can already simulate some special cases. A demagnetization
field Hd dependent on M can be added to simulate
the shape anisotropy of a long, thin core made of a soft magnetic
material, because the long-range interaction is embedded
in the M dependence in Hd. This can be implemented
by utilizing ports x and y in Figure 4. At the relatively low
frequencies of interest to power electronics (even for fastswitching
converters enabled by future magnetic materials),
the effect of initial M orientations is insignificant therefore
the single-domain model is expected to achieve adequate
accuracy. This model will work even better for a thin torus,
with the cross section area and axial perimeter substituting
∆∆
than physical magnetic domains to lower the computational
cost in most cases, as in a previous LLG-based model
[19], where the (0.1 mm)3 discretized region is already much
larger than typical physical domains while only a 16 × 16 × 8
array of such regions was modeled to represent a magnetic
core orders of magnitude larger in volume.
The on-going efforts of this work involve adding physical
xy and ∆z, respectively. Other core shapes can be simulated
by using the appropriate Hd versus M dependence in
accordance with the demagnetizing tensors. Furthermore, a
magnetocrystalline anisotropy field Hani can be added as
exemplified above (Figure 6), with an orientation determined
by the fabrication process. Worth pointing out is the
flexibility afforded by the LLG equation in accommodating
various physics through various effective fields. We mention
in passing that Hd, while acting similarly, is conceptually
not an effective field, but rather a part of H, which is determined
by the winding current per Ampere's law.
Multiple domains will be needed in general. For example,
the easy and hard axes may be oriented in different directions
with regard to the core geometry, requiring different
Hani in different regions. In a multi-domain model, each
domain n is represented by a three-port circuit simulating
the dynamics of its magnetization Mn, and the susceptibility
is extracted by χ=∑ ⋅()
[]ˆ
nn nz∆VV H
zM // , where
∆Vn and V are the volume of domain n and the total volume,
respectively. We expect the use of " domains " larger
mechanisms for the model to be adequate in more and more
application scenarios to predict core loss and nonlinear permeability
with a small number of physically meaningful and
measurable parameters, in contrast to the extensive curve
fitting as currently practiced. Down the road, neighboring
domain interactions will be incorporated as coupling
circuit elements to account for domain wall motion under
exchange interactions within a single-crystal material, a
crystal grain of polycrystalline materials, or an amorphous
material. Finally, the domain wall motion impeding effects
of grain boundaries, as visualized in cartoon illustration
(Figure 7c), as well as other defects, will be captured. This
framework is versatile to accommodate various physical
mechanisms through effective fields, thanks to the flexibility
of the single-domain circuit (Figure 4): while the excitation
is applied only to port z, ports x and y can be connected
to current sources representing effective fields. Separately
on the single-domain level, the stochastic process of thermal
relaxation may need to be incorporated. Overall, the
parameters of the circuit model will be physical properties
that can be extracted from material characterizations such
as saturation flux density, ferromagnetic resonance quality
factor, etc. The loss and permeability of the magnetic core
can be simulated and extracted as the real and imaginary
parts of the input impedance seen by the excitation source
in the circuit model. By fine-tuning the model parameters,
the loss and permeability should match bulk material and
component measurement results. This model calibration is
fundamentally different from empirical curve fitting, in that
our model parameters bear physical meanings. Therefore,
the accuracy gained by model calibration will be transferrable
to components based on the same material but of different
geometries and under different operation conditions,
FIG 7 Illustration of the scale-up process from the circuit model of (a) single magnetic domain to (b) single-crystal materials and
(c) polycrystal magnetic materials.
20 IEEE POWER ELECTRONICS MAGAZINE z September 2023

IEEE Power Electronics Magazine - September 2023

Table of Contents for the Digital Edition of IEEE Power Electronics Magazine - September 2023