Physical Principles of RFID Systems - RFID: Fundamentals and Applications in Contactless Smart Cards and Identification - page 88

C 1 R 1 i 1 i 2 R 2

M

u 0 u 1 L 1 L 2

u 2 Z 2 , Z TAG

Z 0

~ U Q2 = j w M × i 1

Figure 4.29 Simple equivalent circuit diagram of a transponder in the vicinity of a reader.
The impedance Z 2 of the transponder is made up of the load resistor R L (data carrier) and the
capacitor C 2

As it is generally impractical to work with the mutual inductance M , in a final step

we replace M with M = k √ L 1 · L 2 because the values k , L 1 and L 2 of a transponder
are generally known. We write:

ω 2 k 2

· L 1 · L 2
R 2 + jωL 2 + Z 2 · i 1

u 0 = R 1 · i 1 + ( 4 . 48 )

Dividing both sides of equation (4.48) by i 1 yields the total impedance Z 0 = u 0 /i 1
of the series resonant circuit in the reader as the sum of R 1 and the transformed
transponder impedance Z T . Thus Z T is found to be:

ω 2 k 2

· L 1 · L 2
R 2 + jωL 2 + Z 2

Z T = ( 4 . 49 )

Impedance Z 2 represents the parallel connection of C 2 and R L in the transponder.
We replace Z 2 with the full expression containing C 2 and R L and thus finally obtain
an expression for Z T that incorporates all components of the transponder and is thus
applicable in practice:

ω 2 k 2

· L 1 · L 2

Z T = ( 4 . 50 )

R L
1 + jωR L C 2

R 2 + jωL 2 +

4.1.10.2 Influencing variables of Z ' T

Let us now investigate the influence of individual parameters on the transformed
transponder impedance Z T . In addition to line diagrams, locus curves are also suitable
for this investigation: there is precisely one vector in the complex Z plane for every
parameter value x in the function Z T = f( x ) and thus exactly one point on the curve.