In Part I of this article,1 a number of experimental results were presented, suggesting that interactions in arrays of tag antennas play an important role in determining when a UHF RFID tag can be read, even when no other scattering is present in the environment. How can these effects be understood and predicted? It is first helpful to verify that, given the measured antenna geometry and reasonable values for the impedance of the integrated circuit loads, the observed behavior can be numerically reproduced from expected electromagnetic interactions. Such a demonstration both validates the interpretation of the measured data and limits the scope of physical effects that need to be considered to those incorporated in the numerical model. Guided by the results of numerical modeling, an attempt was also made to formulate a quasi-analytic model that is less accurate but computationally much less demanding, and thus easier to implement and scale to large arrays and complex geometries.

Numerical Simulations

An array of I-tag-like antennas was simulated using the full 3D field-solver CST Microwave Studio (Transient Solver), which uses the finite integration technique. The simulated antenna metal was assumed to be a perfect electric conductor. In the center of the antenna, where the RFID tag IC normally would be attached, a lumped load network was connected with a 1 k? resistor in parallel with a 1.6 pF capacitor. The simulated antenna geometry is shown in Figure 1. Figure 2 shows the arrangement of antennas for the single- and three-column array cases.

Simplified Analytic Model for Tag Arrays

The scattered voltage from the Jth tag antenna (self-voltage) is simply due to the radiation resistance and the series equivalent resistance of the IC

where

I0 = current along the antenna, presumed constant

The scalar potential contribution from the next-neighbor antenna (J-1) is approximately due only to the charge localized at the ends of the element. If ?y is small compared to L

The scalar potential contribution from more distant antennas (J-2), etc. will generally be negligible. The magnetic contribution is obtained from the vector potential, in this case along the axis of the antenna (so that only the x component is considered). To a reasonable approximation

Though for small ?y, one should slightly modify the contribution from the nearest antenna

The electric field due to the vector potential is –i?A; this potential is approximately constant along the antenna, so the associated voltage is obtained by multiplying by L. Summing all the contributions

The total scattered voltage, obtained by adding all the contributions above, is set equal to minus the incident voltage

Since each contribution is proportional to the current on the antenna I0, the resulting current is inversely proportional to the total impedance due to each contribution to the voltage. The power delivered to the end of the array relative to that incident on an isolated antenna in the same location is then

Because of the local Floquet assumption, no iteration is necessary: this model may readily be implemented in a spreadsheet or other general mathematical environment, and the computational time is negligible, even for large arrays. In order to apply the model, the fact that in general, the effective (radiating) length of a tag antenna is not the same as the physical end-to-end length, must be accounted for. Examples are shown in Figure 7 for the I-tag and Squiggle tag; reasonable estimates of the radiating lengths of these antennas are approximately 10 and 6 cm, respectively.

In Figure 8, the relative intensity for varying array sizes is shown, for two different values of the antenna length L, roughly corresponding to the I-tag and Squiggle tag, respectively, predicted from the simple model for a tag array. In all cases, the relative power is measured at the most distant tag (the far end of the array from the transmitting antenna). The model results are shown for a single column with a spacing of 5 cm in the propagation direction. Also shown is the experimental data for this configuration from Figure 19 in Part I. This very simple model provides a semi-quantitative agreement with the measured data, and explains the difference between the two types of tags primarily in terms of the difference in their effective radiating length. The simple analytic model is not nearly as accurate as the full numerical model, but can be scaled to large or complex arrays more readily than the numerical model.

When arrays other than a simple linear end-fire configuration are considered, both the phase offset between the element currents, due to the local Flocquet assumption, and the phase delay due to propagation from each remote element J to the element of interest must be explicitly accounted for. Equations 3 to 6 become slightly more complex, as phase factors must be included, and a full complex number implementation is needed to perform model calculations, but otherwise the model is not significantly altered. An example of a result for a more complex geometry is shown in Figure 9. Here a pair of plane arrays was modeled. Each plane contains a three by nine array of tag antennas, making this problem challenging to implement using a self-consistent numerical approach. The intensity in the center of the rear plane is plotted for parameters representing Squiggle and I-tag antennas. The expected half-wave periodicity can be observed and qualitative agreement with the data in Figures 10 and 12 in Part I is obtained.

In order to use this analytic approach as a systematic and accurate tool for general-purpose modeling of large RFID tag arrays, it will be necessary to establish a detailed correspondence with numerical simulations of various tag designs or measurements, obtained for example from radar cross-section data, so that appropriate parameters can be employed for each given combination of tag design and frequency of operation.

Conclusion

In this article, reasonable estimates of the load presented to the tag antenna by the integrated circuit have been shown permitting to reproduce the measured behavior of a simple array of single-dipole tags and providing theoretical confirmation that tags create substantial attenuation of incident signals in geometries reasonably representative of realistic commercial applications. Array effects are very sensitive to assumptions about the central IC impedance, and thus would be expected to vary with frequency and tag design. It is also shown that a simplified analytic theory for this interaction shows promise for providing a practical modeling tool, which can be implemented readily using general-purpose software tools and can deal with arrays much larger than those that are convenient to model using 3D simulation software.

Acknowledgments

The authors would like to thank Jim Buckner, Kathy Radke, Titus Wandinger, Jose Hernandez, Don MacLean, Dan Kurtz and Walter Strifler for their assistance with this work.

References

1. D.M. Dobkin and S.M. Weigand, “UHF RFID and Tag Antenna Scattering, Part I: Experimental Results,” Microwave Journal®, Vol. 49, No. 5, May 2006, pp. 170–190.

3. B. Munk, R. Kouyoumjian and L. Peters, “Reflection Properties of Periodic Surfaces of Loaded Dipoles,” IEEE Transactions on Antennas and Propagation, Vol. 19, No. 5, September 1971, p. 612.

4. D. Hill and C. Cha, “Quasi-infinite Array Scattering Technique,” 1987 IEEE Antennas and Propagation Symposium Digest, paper AP06-2, p. 264.

5. J. Jin and J. Volakis, “Electromagnetic Scattering by a Perfectly Conducting Patch Array on a Dielectric Slab,” IEEE Transactions on Antennas and Propagation, Vol. 38, No. 4, 1990, p. 556.

6. B. Tomasic and A. Hessel, “Analysis of Finite Arrays – A New Approach,” IEEE Transactions on Antennas and Propagation, Vol. 47, No. 3, 1999, p. 555.