IEEE Power Electronics Magazine - September 2023 - 18

winding (per Ampere's law). Proportional constants in the
above mapping of dM dti / and Hz
to surrogate voltages Vi
and currents Iz can be chosen arbitrarily. Interestingly, by
choosing mapping proportions such that HI Hzzz z→= ∆
and dM dt Vx ydMdt
zz
//z
→= ()µ0∆∆
,
the current Iz
and voltage Vz in the equivalent circuit coincide with the
current and the voltage per turn of the physical inductor
(approximately for voltage given relative permeability
µr 1), allowing for direct integration of material modeling
into a circuit simulator to simulate a physical circuit.
The mapping of quantities in all three dimensions to the
surrogate circuit quantities is as follows:
Vy z Vx z
x
z
Vy x
∆∆
==
== ⋅
µµ
µ
00
∆∆
dM
dt
dM
dt
z /
z IH zzz ∆
,,
,.
x
y
∆∆
dM
dt
y
(2)
Substituting the circuit surrogates defined by (2) into the
three differential equations re-written from (1) and then
eliminating dM dt
using the z-projection equation, the
two remaining equations for x and y projections become:
V
=+
z∆
µγMz µ0∆∆
⋅
2
+
α
µ
0 =−
+
MM V
x
Vy
µγMz µ0∆∆
α∆ ⋅ x
2
µγ s∆∆ µµγ sz∆
2
xV
My z
z∆
+
22 + 22
0γMM µγzsz∆
xy x
+
∆ zx
xH Vdt
yzM
⋅∫
z
αMM Vxy y
⋅
22 +
Equivalent Circuit for LLG Equation
Equations (3) and (4) are actually KCL equations at ports x
and y of the three-port circuit when the excitation is applied
only to port z. The total currents at ports x and y each consist
of five components that add up to zero, expressed as
follows in terms of the equivalent circuit elements labeled in
Figure 4:
0=+
V
Z
x
g
0 =− +
V
Z
y
g
∫
Vdt
L
y
∫ x
Vdt
L
y 12
++ +
m
V
R
y
y
x 12
++ +
m
V
R
x
x
V
Z
y
R
x
x
The inductors Lx and Ly with coupling to the z port is thus
determined to be
Ly==
z
µµ
00 z
Lx
∆∆
∆
xzM
yH
z,, (7)
z
∆∆
∆
yzM
xH
along with additional elements: Rx1 and Ry1 are linear resistors,
while Rx2 and Ry2 are nonlinear resistors across port x
and y, respectively, given by
18 IEEE POWER ELECTRONICS MAGAZINE z September 2023
V
Z
x
R
y
y
.
V
(6)
V
(5)
MM 22z
Mx V
α
2∆ ⋅
(4)
µγMM yz
xx
sz
∆∆
.
∆ zy
yH Vdt
xzM
⋅∫
+
z
α
My V
MM xz
2∆ ⋅
yy
sz
∆∆
α∆ ⋅
µγ s∆∆
2
yV
Mx z
y
(3)
a gyrator with impedance Zg and a 1:1 transformer with all
equal self and mutual impedances Zm couple ports x and y:
ZM zZ
gz m
2
∆
==,.µγ
µγ
22
α
zz xyMxHzz , which, along with
==()()
Λ//µ0∆∆
∆
MM z
MM
sz
xy
∆
(9)
Notice that definitions of Vz and Iz in (2) directly result in
Ld dt
(7), indicate that the three coupled inductor share the same
flux φ that can be mapped to Mz.
Modeling Results
The equivalent circuit in Figure 4 can be constructed in any
circuit simulator that allows user-defined terminal functions.
The circuit requires one input at one port (z in the
example) representing the winding current, the dimension
of the single-domain core (i.e., Δx, Δy, and Δz), and two
material properties α and Ms, both of which can be experimentally
determined by measuring the frequency-dependent
magnetic susceptibility of the material. The outputs of
the model are the voltages and currents on each of the
three terminals (Vx, Vy, Vz, Ix, Iy, Iz), which correspond to
the magnetization and magnetic field along each direction
as suggested by (2). By varying the excitation signal
applied to the input side, the output can be measured in
voltage and examined dynamically in time-domain that
reflects what happens inside the microworld of the magnetic
material. This serves as a valuable tool to transparently
connect the material's properties to its function and
performance in a power electronics system in the language
of circuits.
Nonlinear Magnetization Process
As a preliminary test run of the model, a fictitious single
domain with a size of 200 μm × 50 μm × 38 mm, damping
constant of 5 × 10−4, and magnetization of 1750 Oe was
used. An external dc magnetic field was applied to the
domain in terms of a current source. The magnitude of the
dc field was increased from 0 to 100 Oe in the z direction,
and the domain was assumed to be aligned originally along
the x direction by defining the initial conditions of the voltages
on the three terminals. The magnetization process happens
when the domain starts to rotate and gets aligned with
the external field and eventually gets saturated, which can
all be observed vividly from the waveforms of Mx, My, and
Mz translated from the Vx, Vy, and Vz. As shown in Figure 5,
this process is plotted in 3D showing the trajectory of the
domain magnetization process from the initial state of alignment
with x (Mx = 1750 Oe) that then slowly spirals to the
final state of saturation along the z direction (Mz = 1750 Oe).
Rx1 ,,
α∆
µγ s∆∆
2
Rx2
µγ
22
2
α
sz
x
==
=
My z
x
∆x
Ry1
MM yz
M
∆∆
µγ s∆∆
α∆
2
Ry =
,;2
µγ
22
2
α
sz
y
MM xz
My
∆∆
∆
Mx z
y
(8)
IEEE Power Electronics Magazine - September 2023

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