IEEE Power Electronics Magazine - March 2020 - 42

measurements used to find the magnetizing inductance and
nonphysical turns ratios and short circuit current measurements used to find the "leakage" inductances. This approach
is particularly good since subtractions of measurements
are avoided, which otherwise could cause small-differenceof-large-numbers problems which are particularly acute in
magnetic systems with very strong or very weak couplings
(see "Example-Characterizing a Flyback Transformer").
As a useful necessary-and-sufficient model, perhaps the
only substantive complaint that can be levied against the

+

i1

1 : n 2 i2

1:1
L2

L1

v 1 Lm

+
v2

-

-
L3
1 : n3 i 3

+
v3
-

FIG 7 Modification to the extended cantilever model with
the graphical dual of the inner network. For three windings,
the designer can characterize each parameter using only
one-port impedance and two-port voltage ratio measurements, without any subtractions. It has the same advantages
as the extended cantilever model with extended utility at
higher frequencies.

Table 2. Well-behaved measurements available for
the three-winding model in Figure 7.
Apply Condition to
Winding
1

2

3

Measure

m1

v1

Open
circuit

Open
circuit

Z 1 /j~

	= L m

m2

v1

Open
circuit

Open
circuit

v 2 /v 1

	= n 2

m3

v1

Open
circuit

Open
circuit

v 3 /v 1

	= n 3

m4

v1

Short
circuit

Open
circuit

v 3 /v 1

	= L 2 /(L 2 + L 1) # n 3

m5

v1

Open
circuit

Short
circuit

v 2 /v 1

	= L 3 /(L 3 + L 1) # n 2

m6

Short
circuit

v2

Open
circuit

Z 2 /j~

	= (L 2 + L 1) # n 22
2
3

m7

Short
circuit

Open
circuit

v3

Z 3 /j~

	= (L 3 + L 1) # n

m8

Short
circuit

v2

Open
circuit

v 3 /v 2

	= L 1 /(L 1 + L 2) # n 3 /n 2

Short
circuit

Open
circuit

v3

v 2 /v 3

m9

42	

IEEE POWER ELECTRONICS MAGAZINE	

	= L 1 /(L 1 + L 3) # n 2 /n 3

z	March 2020

extended cantilever model is the recommended process for
determining model parameters, which includes the need
to measure currents at short circuited ports [20]. Current
measurements typically require bulky sensors that impose
inductive and/or resistive impedances on the circuit, both
of which become more restrictive concerns as frequency
increases and size decreases. Given the complexity of the
model, it is difficult to predict a priori if external impedances or artificial inductances from measurement loops will
be negligible compared to what one is trying to measure.
Therefore, it would be advantageous to have a necessary-and-sufficient representation that only requires
voltage measurements, which are easier to make with
high confidence at even 10s of megahertz. In particular,
we would like to use only one-port impedance measurements or two-port voltage ratios that can be obtained
on an impedance analyzer. It is also preferable, where
possible, to have measurements that require opencircuit terminations rather than short circuit terminations, as these are easier to realize at high frequency.
(At sufficiently high frequencies it becomes difficult to
impose sufficiently good short or open circuits at ports;
it is partially for this reason that S-parameter measurements become the dominant approach at radio frequencies.) We would like any required calculations to avoid
small-differences-of-large-numbers problems as well.
One way to approach this problem is to observe that
the offending measurements in the extended cantilever
model arise from the connection of the internal nodes
(a delta connection in the three-winding case). We might
hypothesize, perhaps without perfect rigor, that creating a model with the graphical dual of the internal section (a Y connection) might allow for voltage measurements in place of currents. Such a model is shown in
Figure 7, with available well-behaved measurements
in Table 2. Combining these measurements tur ns
out to be fruitful in obtaining every parameter using
only one-port impedance and two-port voltage ratio
measurements and without any subtractions (Table 3).
For completeness, the mappings between this model
and the inductance matrix are provided in Table 4; this
complements the mappings provided for the cantilever
model in [19].

Table 3. Measurements from Table 2 can be
combined to yield the model parameters in
Figure 7 without resorting to subtractions and the
ensuing numerical dangers.
Lm

m1

n2

m2

n3

m3

L1

m 8 m 6 /(m 2 m 3) or m 9 m 7 /(m 2 m 3)

L2

m 4 m 6 /(m 3 m 22)

L3

m 5 m 7 /(m 2 m 23)
IEEE Power Electronics Magazine - March 2020

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