# Constraints of Using Conductive Screen-Printing for Chipless RFID Tags with Enhanced RCS Response
## 2 Theoretical Background
\(Q_{t}\) includes all loss phenomena except the effect of surface waves, the magnitude of which is negligible. \[\frac{1}{Q_{t}}=\frac{1}{Q_{r}}+\frac{1}{Q_{c}}+\frac{1}{Q_{d}} \tag{6}\]  
The reflection coefficient at the resonant frequency is derived from (1) as  
\[\Gamma(f_{t})=\frac{\frac{1}{Q_{t}}-\frac{1}{Q_{0}}}{\frac{1}{Q_{t}}+\frac{1} {Q_{0}}} \tag{7}\]  
Figure 1: Cross-section (**a**) and top view (**b**) of the dipole placed above an infinite metallic plate. Based on this analysis, three different conditions for the coupling of dissipation and leakage modes can be expressed:  
1. \(Q_{r}>Q_{0}\), under-coupled,
2. \(Q_{r}<Q_{0}\), over-coupled,
3. \(Q_{r}=Q_{0}\), critically-coupled. For the case under study, the third condition is of interest, i.e., the critical coupling at which the maximum energy absorption is attained, which consequently leads to the maximum achievable \(Q\) for a given geometry. We consider a low-loss foam dielectric substrate with a relative permittivity \(\varepsilon_{\mathrm{r}}\) = 1.3, loss tangent tan \(\delta\)\(\sim\) 10\({}^{-3}\), and two cases of metal conductivity--copper \(\sigma\) = 5.8 \(\times\) 10\({}^{7}\) S/m and screen-printing silver paste \(\sigma\) = 1.6 \(\times\) 10\({}^{6}\) S/m. A dipole of length \(L_{\mathrm{dip}}\) = 46 mm is situated on a foam substrate of variable thickness \(h\) over an infinitely large conductive plane. Figure 2 shows the quality factors of a cavity consisting of 0.2 mm and 1 mm wide dipoles placed above the conducting plane depending on the substrate height \(h\). Surface plots in Figure 3 depict the dependence of the reflection coefficient on frequency along with the distance of the dipole from the conducting plane. Figure 2: Quality factors vs. height of **(a)** a narrow dipole 46 \(\times\) 0.2 mm\({}^{2}\) and **(b)** a wide dipole 46 \(\times\) 1 mm\({}^{2}\) situated on a foam substrate above an infinite metallic plane for conductivity of copper (5.8 \(\times\) 10\({}^{7}\) S/m) and silver paste (1.6 \(\times\) 10\({}^{6}\) S/m). Intersection of the \(Q_{0}\) and \(Q_{r}\) curves correspond to critical coupling and deep resonant minima on the RCS curve. From Figure 2, it can be said that there is an optimal dipole height for achieving critical coupling, and hence maximizing the total quality factor \(Q_{\rm r}\). This height ranges from 0.5 mm to 1.2 mm for both dipole widths investigated with copper metallization. Through the use of the screen-printing silver paste metallization, the optimal height is noticeably shifted, and it ranges from 1.2 mm to 2.0 mm for both dipole widths. In addition, the narrowing of the dipole width applies the reflection coefficient degradation, as can be seen in Figure 3. Furthermore, it is worth noting that the increase in permittivity and losses in the substrate lead to the requirement of a larger distance of the dipole from the conducting plane in order to achieve critical coupling (\(Q_{\rm r}=Q_{0}\)). A detailed study about this effect can be found in [33]. Due to the reduction of the conductivity of the silver paste compared to the copper, the value of the reflection coefficient degrades; see the comparison of Figure 3a,b with Figure 3c,d. A similar effect can be observed when the dipole width is narrowed; see comparison of Figure 3a,c with Figure 3b,d. Based on these results, a height of \(h=1\) mm, which is the closest available foam substrate height, was chosen for implementing the geometry from Section 3.3 for both the standard \"copper on substrate\" as well as screen-printing samples to the desired value from Figures 2 and 3. Figure 3: Reflection coefficient vs. frequency and dipole height of a dipole situated on a foam substrate above an infinite metallic plane. (**a**) Narrow dipole 46 \(\times\) 0.2 mm\({}^{2}\), copper; (**b**) wide dipole 46 \(\times\) 1 mm\({}^{2}\), copper; (**c**) narrow dipole 46 \(\times\) 0.2 mm\({}^{2}\), silver paste; (**d**) wide dipole 46 \(\times\) 1 mm\({}^{2}\), silver paste.