IEEE Power Electronics Magazine - June 2023 - 42

The inductance, which links the magnetic flux and the
electric current (Figure 1c), can be expressed as
LN A
=⋅ +

2 µ0


l
µr


−1
lg 
.
The rms current through the coil imposes that large
wires and few turns be used to limit losses. On the
other hand, the peak current through the coil dictates
that a large number of turns (small due to the limited
window area) be used to avoid magnetic saturation
which would result in a drop in permeability-and
therefore inductance. As a matter of fact, unwanted
saturation may lead to over currents, increased losses,
and control issues. The device must be designed such
that, given the peak current
density B ˆ , given by
I ˆ , the peak magnetic flux
ˆ
ˆ
B =
IL
NA
,
remains below the saturation flux density Bsat of the material
of the core. In practice, this means selecting a core of
sufficient size.
When considering dynamically tunable magnetics, the
models must be updated to account for the energy stored
through the inductance-tuning mechanism, and that may
sometimes be recovered. In particular, the relation between
the voltage vL and the current iL reads
vt N()=⋅ =
L
dφ ()⋅ L =⋅ +⋅
dt
dL i
dt
L
di
dt
L
iL
dL
dt
,
where L is a function of time. However, most applications
currently operate the tunable magnetics in quasistatic
conditions, whereby the second term
can be neglected. When considering tunable
technics with low enough response time,
this effect could nonetheless be leveraged,
for instance for active filtering.
The following section highlights how these
variables can be tuned online to achieve
dynamic inductance control.
Physical Effects and Technologies for
Inductance Control
The basic model presented in the previous section
provides guidance on how the inductance
can be dynamically tuned by changing the
number of turns, the length of the air gap, the
permeability of the magnetic core, and the
overall shape of the inductor.
Tapped coils with tap changers
(mechanical or solid-state) constitute tunable
inductances that can take discrete
42 IEEE POWER ELECTRONICS MAGAZINE z June 2023
values. Practical use in power applications remains
limited to voltage regulating distribution transformers
due to the high cost and complexity of discrete implementations
[7]. On-chip integration of both the winding
and the tap-changer (MEMS- or solid-state-based) have
been demonstrated for low-value inductances, benefiting
from the high-density integration enabled by waferlevel
technology [8].
Magnetic cores are usually used in their linear region,
where they exhibit constant permeability. However, when
increasing the field and approaching saturation, the permeability
drops (Figure 2). This is exploited to create tunable
inductances, whereby a magnetic core is submitted
to a biasing field by means of an auxiliary winding, that
polarizes the material on its B-H curve around the target
operating point. This technology has, in particular, been
used to regulate power converters [9]. The main shortcoming
of this approach is the significantly increased
power loss in the dc-biased portion of the core. An application
is detailed in the section " use case " below [10].
Mechanical variability of the inductance is achieved
by various types of micromanipulators that vary the
length of the air gap (Figure 3) or move a magnetic shunt
into or out of it. Due to the large influence of the air gap
on the inductance, this allows efficient and wide tunability.
Other concepts rely on modifying the coil dimensions.
Yet, the use of mechanical means results in high
response times and is also considered a critical reliability
issue. This, added to the high cost and potentially high
power consumption, limits the practical applicability of
the concept.
If piezoelectric materials can be used as actuators
for voltage-controlled mechanical adjustment of the
air gap, they offer another way to tune the inductance.
In combination with magnetostrictive core materials,
FIG 2 First quadrant magnetization curve of a typical ferrite for power electronics
(N87): (BH) curve (blue) and incremental permeability µr,i , (red). External
biasing allows to move the operation point along the (BH) curve and
thereby tune the inductance, which is inversely proportional to µr,i.
IEEE Power Electronics Magazine - June 2023

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