The inductance can again be calculated by integration and averaging of the vector
potential
A
over the conductor. A more convenient alternative however is the inte-
gration of the magnetic energy density
w
m
=
B
·
H
2
over the complete space. Here,
this integral is simplified to 16 times the integral over the modelled volume
V
mod
:
LI
2
2
=
16
N
2
V
mod
w
m
d
V
(2.70)

36
2 Magnetic Induction
with
N
the number of turns of one coil. The factor
N
2
arises, as always, from the
fact that
I
is divided by
N
to have an equal global current distribution and that the
induced emf is perceived over
N
turns in series. For the Helmholtz coil in Fig.
2.14
,
a total inductance
L
=
2
.
3
N
2
μH is calculated when both coils are connected in
series and
L
=
575
N
2
nH when they are connected in parallel.
2.5.4 Mutual Inductance
Thus far the developed FE models have been used to calculate self-inductance and
where appropriate the equivalent series resistance. The same FE models can be used
to calculate the mutual inductance of the modelled inductor with an arbitrary sec-
ondary coil. Like for the self-inductance, usually a DC model suffices to calculate
mutual inductances.
Once the magnetic field induced by a certain source current is known, the mutual
inductance can be calculated by simply integrating the magnetic flux density over
the surface of interest (Eq. (
2.28
)). This may involve exporting the FE solution to
more advanced numerical computation packages, like M
AT
L
AB
[149]. Integration
over surfaces that do not conform to the symmetry of the original model, can also
be conducted this way. Alternatively the vector potential can be integrated over a
certain contour (Eq. (
2.22
)). For a secondary coil with a certain winding thickness,
the calculated flux should be averaged over its winding cross-section, as explained
in Sect.
2.5.1.1
on page
27
.
2.6 Conclusions
The electric field
E
inside of a conductor can be split up into two parts: the conserv-
ative field
−∇
V
corresponding to a charge distribution
ρ
and the non-conservative
field
−
jω
A
that is magnetically induced at non-zero frequencies. This latter field
can be self-induced, i.e. the current through the conductor itself is responsible for
it, or it can be induced by an external current flow. In the former case, the self-
inductance
L
relates the magnetically induced emf across a piece of conductor with
the current through it. In the latter case, it is the mutual inductance
M
that relates
the generated emf to the external current distribution.
The current density
σ
E
in general is heterogeneously distributed over a conduc-
tor’s cross-section as
−
jω
A
is heterogeneous. The heterogeneity of
σ
E
rises with
the frequency and as a result, since the current is pushed through smaller effective
areas, the losses increase as well. This reflects in a higher equivalent series resis-
tance (ESR) of the inductive element. In principle, also the inductance values are
affected by this current redistribution. For typical wire-wound coils used in induc-
tive powering applications however, the effect on the inductance is negligible.

#### You've reached the end of your free preview.

Want to read all 26 pages?

- Spring '19
- Andrews
- Magnetic Field, Sc