Faraday’s Law

Any change to the magnetic fluxgenerates an electric field strengthEi. This characteristic of the magnetic field is described byFaraday’s law.

The effect of the electric field generated in this manner depends upon the material properties of the surrounding area. Figure 4.11 shows some of the possible effects (Paul, 1993):

• Vacuum: in this case, the field strength E gives rise to an electric rotational field. Periodic changes in magnetic flux (high-frequency current in an antenna coil) generate an electromagnetic field that propagates itself into the distance.

• Open conductor loop: an open-circuit voltage builds up across the ends of an almost closed conductor loop, which is normally calledinduced voltage. This voltage corresponds with the line integral (path integral) of the field strengthE that is generated along the path of the conductor loop in space.

• Metal surface: an electric field strengthE is also induced in the metal surface. This causes free charge carriers to flow in the direction of the electric field strength. Currents flowing in circles are created, so-callededdy currents. This works against the exciting magnetic flux (Lenz’s law), which may significantly damp the magnetic flux in the vicinity ofmetal surfaces. However, this effect is undesirable in inductively coupled RFID systems (installation of a transponder or reader antenna on a metal surface) and must therefore be prevented by suitable countermeasures (see Section 4.1.12.3).

In its general form Faraday’s law is written as follows:

ui=

Ei·ds= −d(t)

dt (4.21)

For a conductor loop configuration withN windings, we can also say thatui=N·d/dt. (The value of the contour integral∫Ei·ds can be increased N times if the closed integration path is carried outN times; Paul, 1993).

To improve our understanding of inductively coupled RFID systems we will now consider the effect of inductance on magnetically coupled conduction loops.

A time-variant current i1(t) in conduction loop L1 generates a time-variant magnetic flux d(i1)/dt. In accordance with the inductance law, a voltage is induced in the conductor loops L1andL2 through which some degree of magnetic flux is flowing. We can differentiate between two cases:

• Self-inductance: the flux change generated by the current change din/dt induces a voltage un

in the same conductor circuit.

• Mutual inductance: the flux change generated by the current change din/dt induces a voltage in the adjacent conductor circuitLm. Both circuits are coupled by mutual inductance.

U_i

Flux change dΦ/dt

Conductor (e.g. metal surface) Eddy current,

Current density S Open conductor loop

Nonconductor (vacuum), Induced field strength E_i

=> Electromagnetic wave

Figure 4.11 Induced electric field strengthE in different materials. From top to bottom: metal surface, conductor loop and vacuum

u₁

Figure 4.12 Left, magnetically coupled conductor loops; right, equivalent circuit diagram for magnetically coupled conductor loops

Figure 4.12 shows the equivalent circuit diagram for coupled conductor loops. In an inductively coupled RFID systemL1would be the transmitter antenna of the reader.L2represents the antenna of the transponder, whereR2is thecoil resistanceof the transponder antenna. The current consumption of the data memory is symbolised by the load resistorR_L.

A time varying flux in the conductor loopL1 induces voltageu2iin the conductor loopL2due to mutual inductance M. The flow of current creates an additional voltage drop across the coil resistanceR2, meaning that the voltageu2 can be measured at the terminals. The current through the load resistorRLis calculated from the expressionu2/RL. The current throughL2also generates an additional magnetic flux, which opposes the magnetic flux1(i1). The above is summed up in the following equation:

Because, in practice,i1andi2are sinusoidal (RF) alternating currents, we write Equation (4.22) in the more appropriate complex notation (whereω=2πf):

u2=j ωM·i1−j ωL2·i2−i2R2 (4.23)

The voltage u₂ induced in the transponder coil is used to provide the power supply to the data memory (microchip) of a passive transponder (see Section 4.1.8.1). In order to significantly improve the efficiency of the equivalent circuit illustrated in Figure 4.12, an additional capacitor C2 is connected in parallel with the transponder coilL2to form aparallel resonant circuitwith aresonant frequency that corresponds with the operating frequency of the RFID system in question.¹ The resonant frequency of the parallel resonant circuit can be calculated using the Thomson equation:

1However, in 13.56 MHz systems with anticollision procedures, the resonant frequency selected for the transponder is often 1– 5 MHz higher to minimise the effect of the interaction between transponders on overall performance.

f = 1 2π√

L2·C2

(4.25) In practice, C2 is made up of a parallel capacitorC₂ and a parasitic capacitanceCp from the real circuit.C2=(C₂+Cp). The required capacitance for the parallel capacitorC₂ is found using the Thomson equation, taking into account the parasitic capacitanceCp:

C₂ = 1

(2πf )²L2 −Cp (4.26) Figure 4.13 shows the equivalent circuit diagram of a real transponder.R2is the natural resistance of the transponder coilL2and the current consumption of the data carrier (chip) is represented by the load resistorRL.

If a voltageuQ2=u_iis induced in the coilL₂, the following voltageu₂can be measured at the data carrier load resistorRLin the equivalent circuit diagram shown in Figure 4.13:

u2= uQ2

1+(j ωL2+R2)· 1

R_L+j ωC2

(4.27)

We now replace the induced voltageuQ2=uiby the factor responsible for its generation,uQ2= u_i=j ωM·i₁=ω·k·√

L₁·L₂·i₁, thus obtaining the relationship between voltage u₂ and the magnetic coupling of transmitter coil and transponder coil:

u2= j ωM·i1

Figure 4.13 Equivalent circuit diagram for magnetically coupled conductor loops. Transponder coilL2and parallel capacitorC2form a parallel resonant circuit to improve the efficiency of voltage transfer. The transpon-der’s data carrier is represented by the grey box

This is because the overall resonant frequency of two transponders directly adjacent to one another is always lower than the resonant frequency of a single transponder.

u2= ω·k·√

Figure 4.14 shows the simulated graph ofu₂with and without resonance over a large frequency range for a possible transponder system. The current i1 in the transmitter antenna (and thus also (i1)), inductanceL2, mutual inductanceM, R2andRLare held constant over the entire frequency range.

We see that the graph of voltageu2for the circuit with the coil alone (circuit from Figure 4.12) is almost identical to that of theparallel resonant circuit(circuit from Figure 4.13) at frequencies well below the resonant frequencies of both circuits, but that when the resonant frequency is reached, voltageu2 increases by more than a power of ten in the parallel resonant circuit compared with the voltageu2for the coil alone. Above the resonant frequency, however, voltageu2falls 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 L2 is generally connected in parallel with a chip capacitor (C₂ =20 –220 pF) to achieve the desired resonant frequency. At the higher frequencies of 13.56 and 27.125 MHz, the required capacitance C2 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 elementsR2, RLandL2on voltageu2. To gain a better understanding of the interactions between the individual parameters we will now introduce theQfactor (theQfactor 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.

0.1

Figure 4.14 Plot of voltage at a transponder coil in the frequency range 1– 100 MHz, given a constant magnetic field strengthHor constant currenti1. A transponder coil with a parallel capacitor shows a clear voltage step-up when excited at its resonant frequency (fRES=13.56 MHz)

TheQ factor is a measure of the voltage and current step-up in the resonant circuit at its resonant frequency. Its reciprocal 1/Qdenotes the expressively namedcircuit damping d. TheQfactor is very simple to calculate for the equivalent circuit in Figure 4.13. In this case ω is the angular frequency (ω=2πf) of the transponder resonant circuit:

Q= 1 towards zero. On the other hand, when the transponder coil has a very low coil resistanceR2→0 and there is a high load resistor RL0 (corresponding with very low transponder chip power consumption), very high Qfactors can be achieved. The voltage u2 is now proportional to the quality of the resonant circuit, which means that the dependency of voltageu2uponR2 andRLis clearly defined.

Voltageu2thus tends towards zero whereR2→ ∞andRL→0. At a very low transponder coil resistanceR2→0 and a high value load resistorRL0, on the other hand, a very high voltage u2can be achieved (compare Equation 4.30).

It is interesting to note the path taken by the graph of voltage u2 when the inductance of the transponder coilL2is changed, thus maintaining the resonance condition (i.e.C2=1/ω²L2for all values ofL2). We see that for certain values ofL2, voltageu2reaches a clear peak (Figure 4.15).

If we now consider the graph of theQfactor as a function ofL2(Figure 4.16), then we observe a maximum at the same value of transponder inductanceL2. The maximum voltageu2=f (L2)is therefore derived from the maximum Q factor,Q=f (L2), at this point.

0

Figure 4.15 Plot of voltageu2for different values of transponder inductanceL2. The resonant frequency of the transponder is equal to the transmission frequency of the reader for all values ofL2(i1=0.5 A,f=13.56 MHz,R2=1)

0 5 10 15 20

L (H)

1× 10⁻⁷ 1× 10⁻⁶ 1× 10⁻⁵

Q factor

Q=f(L₂)

Figure 4.16 Graph of theQfactor as a function of transponder inductanceL2, where the resonant frequency of the transponder is constant (f=13.56 MHz,R2=1)

This indicates that for every pair of parameters (R2, RL), there is an inductance value L2 at which theQfactor, and thus also the supply voltageu2 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 an inductively coupled RFID system. However, we must also bear in mind that the influence of component tolerances in the system also reaches a maximum in theQmax range. This is particularly important in systems designed for mass production. Such systems should be designed so that reliable operation is still possible in the rangeQQmax at the maximum distance between transponder and reader.

R_L should be set at the same value as the input resistance of the data carrier after setting the

‘power on’ reset, i.e. before the activation of the voltage regulator, as is the case for the maximum energy range of the system.