Digital Signal Processing Reference
In-Depth Information
Inductance is one of the characteristic variables of conductor loops (coils). The
inductance of a conductor loop (coil) depends totally upon the material properties
(permeability) of the space that the flux flows through and the geometry of the layout.
4.1.3.1 Inductanceof a conductor loop
If we assume that the diameter
d
of the wire used is very small compared to the
diameter
D
of the conductor coil (
d/D <
0
.
0001) a very simple approximation can
be used:
µ
0
R
·
ln
2
R
d
L
=
N
2
(
4
.
12
)
where
R
is the radius of the conductor loop and
d
is the diameter of the wire used.
4.1.4 Mutual inductance M
If a second conductor loop 2 (area
A
2
) is located in the vicinity of conductor loop 1
(area
A
1
), through which a current is flowing, then this will be subject to a proportion of
the total magnetic flux
flowing through
A
1
. The two circuits are connected together
by this partial flux or coupling flux. The magnitude of the coupling flux
ψ
21
depends
upon the geometric dimensions of both conductor loops, the position of the conductor
loops in relation to one another, and the magnetic properties of the medium (e.g.
permeability) in the layout.
Similarly to the definition of the (self) inductance
L
of a conductor loop, the
mutual
inductance M
21
of conductor loop 2 in relation to conductor loop 1 is defined as the
ratio of the partial flux
ψ
21
enclosed by conductor loop 2, to the current
I
1
in conductor
loop 1 (Paul, 1993):
21
(I
1
)
I
1
B
2
(I
1
)
I
1
M
21
=
=
·
d A
2
(
4
.
13
)
A2
Similarly, there is also a mutual inductance
M
12
. Here, current
I
2
flows through the
conductor loop 2, thereby determining the coupling flux
ψ
12
in loop 1. The following
relationship applies:
M
=
M
12
=
M
21
(
4
.
14
)
Mutual inductance describes the coupling of two circuits via the medium of a mag-
netic field (Figure 4.8). Mutual inductance is always present between two electric
circuits. Its dimension and unit are the same as for inductance.
The coupling of two electric circuits via the magnetic field is the physical prin-
ciple upon which inductively coupled RFID systems are based. Figure 4.9 shows a
calculation of the mutual inductance between a transponder antenna and three dif-
ferent reader antennas, which differ only in diameter. The calculation is based upon
the following values:
M
1
:
R
=
55 cm,
M
2
:
R
=
7
.
5cm,
M
3
:
R
=
1 cm, transponder:
R
=
3
.
5cm.
N
=
1 for all reader antennas.
The graph of
mutual inductance
shows a strong similarity to the graph of magnetic
field strength
H
along the
x
axis. Assuming a homogeneous magnetic field, the mutual
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