Physical Principles of RFID Systems - RFID: Fundamentals and Applications in Contactless Smart Cards and Identification - page 90

cancel each other out. Z T acts as an ohmic (real) resistor — the locus curve thus
intersects the real x axis of the complex Z plane at this point.
In the frequency range above the transponder resonant frequency (f TX >f RES ) ,the
locus curve finally passes through quadrant IV of the complex Z plane — Z T has a
capacitive effect in this range.
The impedance locus curve of the complex transformed transponder impedance Z T
corresponds with the impedance locus curve of a damped parallel resonant circuit
with a parallel resonant frequency equal to the resonant frequency of the transponder.
Figure 4.31 shows an equivalent circuit diagram for this. The complex current i 2 in
the coil L 2 of the transponder resonant circuit is transformed by the magnetic mutual
inductance M in the antenna coil L 1 of the reader and acts there as a parallel resonant
circuit with the (frequency dependent) impedance Z T . The value of the real resistor
R in the equivalent circuit diagram corresponds with the point of intersection of the
locus curve Z T with the real axis in the Z plane.

Coupling coefficient k Given constant geometry of the transponder and reader
antenna, the coupling coefficient is defined by the distance and angle of the two coils
in relation to each other (see Section 4.1.5). The influence of metals in the vicinity
of the transmitter or transponder coil on the coupling coefficient should not be disre-
garded (e.g. shielding effect caused by eddy current losses). In practice, therefore, the
coupling coefficient is the parameter that varies the most. Figure 4.32 shows the locus
curve of the complex transformed transponder impedance for the range 0 ≤ k ≤ 1. We
differentiate between three ranges:

• k = 0: If the transponder coil L 2 is removed from the field of the reader antenna

L 1 entirely, then no mutual inductance occurs. For this limit case, the transformed
transponder impedance is no longer effective, that is Z T (k = 0 ) = 0.

• 0 <k< 1: If the transponder coil L 2 is slowly moved towards the reader antenna

L 1 , then the coupling coefficient, and thus also the mutual inductance M between
the two coils, increases continuously. The value of complex transformed transpon-
der impedance increases proportionately, whereby Z T ∼ k 2 .When f TX exactly

Z ′

L ′ R ′ C ′

Figure 4.31 The equivalent circuit diagram of complex

transformed transponder impedance Z T is a damped parallel

resonant circuit