Now it's time to look closer at the loosely coupled transformer hypothesis. The minimum bandwidth requirements are valid only if the coupling factor between the tag and the base station antenna is kept low. As an example, consider some simulation results using the schematic depicted in Figure 3.

On the left, is a model of a base station antenna. On the right, is a model of a tag. For the sake of simplicity, both devices are identical, and the base station antenna is, for the time being, driven by a perfect current source that will not change its intrinsic properties. Because we are dealing with a magnetic coupling problem and know that the magnetic field induced by a coil is proportional to the current flowing through it, we shall visualize this current. Both the antenna and tag are tuned to 13.56 MHz. In the middle, the linear coupling factor k = 0.1 is introduced. The Q factor of both devices is equal to 9.

If the linear coupling factor were kind, then the result would be that depicted in Figure 4a. Unfortunately, “k” is not kind and the actual result is depicted in Figure 4b. Instead of having a single peak in the frequency response, there are two bumps. One corresponds to the tag tuned circuit, and the other to the base station antenna. It looks as if there were two different resonant frequencies, well separated. In fact, this annoying effect is what makes possible the design of a class of RF filters, made with cavities or helical resonators, slightly mistuned and carefully coupled to one another until the desired frequency response is obtained.

In this case, it's possible to imagine that the effect of the coupling factor will have serious consequences on the design of a usable RFID system. To begin with, let's look at a multiple run simulation, where the coupling factor is increased from 1% to 20% in a logarithmic fashion (Figure 5).

It is obvious that, for a coupling factor higher than 10% (corresponding to the second largest frequency spreading illustrated in Figure 5), the tag and the base station will have difficulties communicating with one another. The paradox is that such communication problems will arise in a situation where intuition dictates the opposite, e.g., when the tag and the base station are close to each other. As a result, this situation must be avoided. The good news is that through careful system design it is possible to avoid this scenario. Remember also that if the coupling factor is too low, no energy transfer will be possible, and the system will not work. One can now understand why it is not possible to design an RFID system by considering only one side of the problem, either the tag or the base station. Because of the coupling factor, both sides must be considered at once.

Since the coupling factor depends only on geometrical parameters, the inductance values and the number of turns of the coils, for example, are not involved in calculating its value. Consider the diagram in Figure 6.

On the left, the base station antenna coil has a diameter d1 = 2 × r1. On the right, the tag antenna coil has a diameter d2 = 2 × r2, tilted along the axis by an angle α. The coupling factor can be expressed as:

Of course, this equation is valid only for circular coils, but the design guidelines that will be inferred from it are valid whatever the shape. From the equation, it is obvious that the coupling factor will be maximum when r1 = r2.

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