# High-Density 3D Printable Chipless RFID Tag with Structure of Passive Slot Rings
## The Principle of the Design
### Characteristic of the Rectangular Slot Ring Resonator  
The relationship between the resonant frequency and the side length of the rectangular slot ring is defined as [33]:  
\[f=\frac{C}{4a}\sqrt{\frac{2}{\varepsilon_{r}+1}} \tag{1}\]  
where \(C\) is the speed of light in free space, \(\varepsilon_{r}\) is the relative dielectric constant of the substrate and \(a\) is the side length of the rectangular slot ring. The equation does not take into account the effects of slot width and dielectric thickness on the resonant frequency.  
First of all, we need to verify the resonant characteristic of the designed slot. For a slot with side length 30 mm, the resonance frequency is simulated to be 1.94 GHz, as shown in Figure 2b, which is simulated by High-Frequency Structure Simulator (HFSSTM). The nanoparticle silver paste with the resistivity 3 \(\upmu\Omega\)-cm [34] and the thickness 100 \(\upmu\)m is adopted to print the designed RFID tag on F4 BM substrate. It is observed that the 2 th, 3 th and 4 th harmonics do not appear in the output spectrum but the 5 th does, as shown in Figure 2b. Only the fundamental resonant and the fifth harmonic can be generated and observed. Due to the symmetrical and continuous structure of the rectangular slot ring resonator, the current flow paths for the second, the third and the fourth harmonics cannot be generated. This is the main reason that they cannot be observed in the RCS spectra. Only the fundamental frequency resonant path and the fifth harmonic path can be generated and observed. Therefore, the unique property shows one of the design rules on the upper limit of the resonant frequency when multiple slots are considered. Based on Equation (1), the calculated resonant frequency is 1.975 GHz when the side length is 30 mm. A slight difference between the theoretical calculation and simulation do exist but will not influence the data coding. The slight discrepancy mainly comes from the ignorance of the width of slot and the thickness of the dielectric substrate during calculation.  
When a notch is shown in the RCS spectrum of the corresponding rectangular slot, the logic state of the code is \"1\"; while there is not a notch, the logic state of the code is \"0.\" The different sizes of rectangular slot rings correspond to the different resonant frequency. So, a series of notches are shown at different positions of the spectra. By adding or removing the slot ring resonators of different sizes, the notches of the spectrum are shown in a coding manner.  
Figure 3a shows the simulation results of the relationship between the resonant frequency and the side length. The resonant frequency is changed from 3.97 GHz to 3.24 GHz with the side length increasing from 18 mm to 22 mm.  
The influence of the slot width on the resonant frequency is also investigated, as shown in Figure 3b, in which the side length is fixed to be 30 mm. The slot width is changed from 0.2 mm to 0.8 mm. When the slot width is 0.5 mm, the resonant frequency is 1.94 GHz. When the slot width is 0.6 mm, the resonant frequency is changed to be 1.66 GHz. While the slot width is 0.4 mm, the resonant frequency is increased to be 2.19 GHz accordingly. It can be concluded that the resonant frequency drops with the increment of the slot width. Although increasing the slot width can shrink down the operation band, a suitable width should be determined based on the requirement on the tag area and the consequent cost.  
Figure 3: (**a**) Relationship of side length and resonant frequency with \(W_{slot}\) = 0.5 mm. (**b**) Relationship between \(W_{slot}\) and resonant frequency with a = 30 mm.  
Figure 2: (**a**) Schematic of slot ring embedded on insulated substrate. The substrate is insulator, without any conductor on back side. (**b**) Spectrum of RCS of the single slot. The side length of rectangle a = 30 mm. The width of slot \(W_{slot}\) = 0.5 mm. Its resonant frequency is 1.94 GHz.