IEEE Power Electronics Magazine - March 2020 - 40

+
-

FIG 5 A bizarre magnetic structure, which is difficult to analyze without a many-parameter physical model. While mathematically such a model must reduce to the three independent
parameters of the inductance matrix (equivalently, a necessary-and-sufficient circuit model), such a reduction is not
always straightforward.

L12

1:1
L13

v 1 Lm
-

1 : n2 i2
L23

+
v2

1 : n3

-
+
v3

Modeling With Three Windings

-
FIG 6 The extended cantilever model, a necessary-and-sufficient model, for a three-winding structure. This model extends
naturally to M ports and M (M + 1)/2 parameters. Moreover,
each parameter can be measured without performing numerically sensitive calculations, though some of the measurements
require current sensing which can be challenging at elevated
frequencies.

Table 1. Recommended measurements to identify
parameters in the extended cantilever model
with three windings. These recommendations
require short circuit current measurements
but no subtractions.
Apply Condition to Winding
1

Open circuit

v 2 /v 1

= n 2

Open circuit

v 3 /v 1

= n 3

Short circuit

v 1 /i 2

= j~l 12 # n 12

Short circuit

v 1 /i 3

= j~l 13 # n 13

Short circuit

v 2 /i 3

= j~l 23 # n 23

IEEE POWER ELECTRONICS MAGAZINE

structure of Figure 5 with many relevant flux paths. With
physical modeling, the analysis is straightforward, even
considering the flux that links only some turns of each
winding (for example, by modeling each turn as a separate winding, electrically connected in series). (Flux
linking only some turns in a winding can be important
to model, for instance, to understand the direction and
magnitude of B fields within the structure, to capture
uneven distribution of voltage on a winding, and so on.)
However, the 12 reluctances in the model of Figure 5 are
more than the number of parameters in a two-port or even
a four-port inductance matrix (three and 10, respectively).
Therefore, given a prototype of this structure, it is impossible to determine the values of all 12 equivalent circuit
inductances by experimentally testing the electrical port
behavior, though such tests can determine the values of
an inductance matrix or a necessary-and-sufficient circuit
model. Conversely, if the physical model had too few reluctances/inductors in it to capture the actual flux paths in a
practical device, then the resulting circuit model (and any
inductance matrix derived from it) would not well represent the behavior of the practical device.
This case shows that the translation between physical
and necessary-and-sufficient models can be difficult, and
the advantages of each approach become more apparent.
On paper, physical modeling is the easiest way to understand a structure. Given a physical prototype, experimentally characterizing the component in terms of inductance
matrix parameters or necessary-and-sufficient circuit
models is much more straightforward.

Measure
= j~L m

z March 2020

Moving to three windings changes the modeling landscape substantially. While the mathematical and physical
representations extend naturally to M windings, most
necessary-and-sufficient circuit representations do not
extend so cleanly [18]. The primary exception and perhaps
the most used of such models is the extended cantilever
model [19] (Figure 6). In this model, an M- port magnetic
structure is represented by a set of M nodes with an
inductance connecting each of them to each other, yielding M (M - 1)/2 parameters. Each winding couples to one
of the M nodes, with the primary having a shunt inductance and a 1:1 turns ratio and every other turns ratio
being an independent variable. This yields M more
parameters, for a correct total of M (M + 1)/2. This model
extends naturally to M windings, and any apparent complexity in the model results from the fact that port relationships of magnetic structures simply become quadratically more complicated as a function of M.
Identifying the M (M + 1)/2 parameters in any necessary-and-sufficient representation is typically done
experimentally, either through experiments on a practical
device or through numerical simulations of a model of the
device. The recommended measurements for the extended
cantilever model are shown in Table 1, with open-circuit

IEEE Power Electronics Magazine - March 2020

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