Digital Signal Processing Reference
In-Depth Information
I
m
U
ZT
U
L1
U
C1
U
R1
R
e
U
0
Figure 4.28
The vector diagram for voltages in the series resonance circuit of the reader
antenna at resonant frequency. The figures for individual voltages
u
L1
and
u
C1
can reach much
higher levels than the total voltage
u
0
Due to the constant current
i
1
in the series circuit, the source voltage
u
0
can be
represented as the sum of the products of the individual impedances and the current
i
1
. The transformed impedance
Z
T
is expressed by the product
jωM
·
i
2
:
1
jωC
1
·
i
1
+
jωL
1
·
i
1
+
R
1
·
i
1
−
jωM
·
i
2
u
0
=
(
4
.
45
)
Since the series resonant circuit is operated at its
resonant frequency
, the individual
impedances
(j ωC
1
)
−
1
and
jωL
1
cancel each other out. The voltage
u
0
is therefore
only divided between the resistance
R
1
and the transformed transponder impedance
Z
T
, as we can see from the vector diagram (Figure 4.28). Equation 4.45 can therefore
be further simplified to:
u
0
=
R
1
·
i
1
−
jωM
·
i
2
(
4
.
46
)
We now require an expression for the current
i
2
in the coil of the transponder, so
that we can calculate the value of the transformed transponder impedance. Figure 4.29
gives an overview of the currents and voltages in the transponder in the form of an
equivalent circuit diagram:
The source voltage
u
Q2
is induced in the transponder coil
L
2
by mutual inductance
M
. The current
i
2
in the transponder is calculated from the quotient of the voltage
u
2
divided by the sum of the individual impedances
jωL
2
,
R
2
and
Z
2
(here
Z
2
represents
the total input impedance of the data carrier and the parallel capacitor
C
2
). In the
next step, we replace the voltage
u
Q2
by the voltage responsible for its generation
u
Q2
=
jωM
·
i
1
, yielding the following expression for
u
0
:
u
Q2
R
2
+
jωL
2
+
Z
2
=
R
1
·
i
1
−
jωM
·
jωM
·
i
1
R
2
+
jωL
2
+
Z
2
u
0
=
R
1
·
i
1
−
jωM
·
(
4
.
47
)
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