Digital Signal Processing Reference
In-Depth Information
Figure 4.13) at frequencies well below the resonant frequencies of both circuits, but
that when the resonant frequency is reached, voltage
u
2
increases by more than a
power of ten in the parallel resonant circuit compared to the voltage
u
2
for the coil
alone. Above the resonant frequency, however, voltage
u
2
falls rapidly in the parallel
resonant circuit, even falling below the value for the coil alone.
For transponders in the frequency range below 135 kHz, the transponder coil
L
2
is
generally connected in parallel with a chip capacitor (
C
2
=
20 - 220 pF) to achieve the
desired resonant frequency. At the higher frequencies of 13.56 MHz and 27.125 MHz,
the required capacitance
C
2
is usually so low that it is provided by the input capacitance
of the data carrier together with the parasitic capacitance of the transponder coil.
Let us now investigate the influence of the circuit elements
R
2
,
R
L
and
L
2
on
voltage
u
2
. To gain a better understanding of the interactions between the individual
parameters we will now introduce the Q factor (the Q factor crops up again when we
investigate the connection of transmitter antennas in Section 11.4.1.3). We will refrain
from deriving formulas because the electric resonant circuit is dealt with in detail in
the background reading.
The
Qfactor
is a measure of the voltage and current step-up in the resonant circuit
at its resonant frequency. Its reciprocal 1/Q denotes the expressively named
circuit
damping d
. The Q factor is very simple to calculate for the equivalent circuit in
Figure 4.13. In this case
ω
is the angular frequency (
ω
=
2
πf
) of the transponder
resonant circuit:
1
1
Q
=
C
2
L
2
+
L
2
C
2
=
(
4
.
31
)
R
2
ωL
2
+
ωL
2
R
L
1
R
L
·
R
2
·
A glance at equation (4.31) shows that when
R
2
→∞
and
R
L
→
0, the Q factor
also tends towards zero. On the other hand, when the transponder coil has a very low
coil resistance
R
2
→
0 and there is a high load resistor
R
L
0 (corresponding with
very low transponder chip power consumption), very high Q factors can be achieved.
The voltage
u
2
is now proportional to the quality of the resonant circuit, which means
that the dependency of voltage
u
2
upon
R
2
and
R
L
is clearly defined.
Voltage
u
2
thus tends towards zero where
R
2
→∞
and
R
L
→
0. At a very low
transponder coil resistance
R
2
→
0 and a high value load resistor
R
L
0, on the other
hand, a very high voltage
u
2
can be achieved (compare equation (4.30)).
It is interesting to note the path taken by the graph of voltage
u
2
when the inductance
of the transponder coil
L
2
is changed, thus maintaining the resonance condition (i.e.
C
2
=
1
/ω
2
L
2
for all values of
L
2
). We see that for certain values of
L
2
, voltage
u
2
reaches a clear peak (Figure 4.15).
If we now consider the graph of the Q factor as a function of
L
2
(Figure 4.16), then
we observe a maximum at the same value of transponder inductance
L
2
. The maximum
voltage
u
2
=
f(L
2
)
is therefore derived from the maximum Q factor,
Q
=
f(L
2
)
,at
this point.
This indicates that for every pair of parameters (
R
2
,
R
L
), there is an inductance
value
L
2
at which the Q factor, and thus also the supply voltage
u
2
to the data carrier,
is at a maximum. This should always be taken into consideration when designing
a transponder, because this effect can be exploited to optimise the energy range of
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