Environmental Engineering Reference
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where
L
1
and
L
2
are the inductances of the primary and secondary coils, re-
spectively.
M
is the mutual inductance of the coils, derived from Biot-Savart's
law using simple approximations [172], which is expressed as
N
2
r
2
r
b
o
M
=
(6.8)
2
r
2
D
2
1
.
5
+
10
−
7
NA
−
2
is the permittivity of free space;
r
and
r
b
are the
primary and secondary coil radii, respectively.
N
is the number of turns in
the coil, and
D
is the distance between coils. Another consideration factor
is the intrinsic loss rate
×
where
o
=4
of the WPT system, which is discussed [163] with
the following equations, representing the ohmic or absorption loss
R
ohmic
and
radiation loss
R
radiative
. The intrinsic loss rate is defined by the following equa-
tion:
R
ohmic
+
R
radiative
2
L
=
(6.9)
For a coil with
N
turns, radius
r
, and height
h
and made of an electri-
cally conducting wire with conductivity of
, the ohmic resistance
R
ohmic
is
expressed as
o
2
o
2
l
rN
2
a
R
ohmic
=
=
(6.10)
4
a
∗
where the total length
l
of the wire with radius of
a
can be estimated as 2
N
.
Other than ohmic loss, there is power loss in the radiation resistance, which
is given by
r
2
o
4
N
2
12
r
c
2
h
c
R
radiative
=
+
(6.11)
3
3
o
According to Kurs et al. [163], the first term in
Equation 6.11
is a magnetic
dipole radiation term (assuming
r
, where
c
is the speed of light),
and the second term is due to the electric dipole of the coil. The second term
is much smaller than the first term for these WPT system parameters, so the
second term of
Equation 6.11
is ignored for simplicity. Hence, by substituting
for
2
c
/
and
c
, the radiative resistance is simplified to
4
4
r
2
N
2
r
4
5
N
2
3
o
=
=
R
radiative
15600
(6.12)
As mentioned, for a system to operate in the strongly coupled regime, the
term
2
must be greater than 1 to achieve an efficient WPT system. Refer-
ring to
Equations 6.7
and
6.9
,
the coupling-to-loss ratio
2
/
/
can be expressed
as follows:
=
ML
R
radiative
)
√
L
1
L
2
(6.13)
(
R
ohmic
+
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