# Constraints of Using Conductive Screen-Printing for Chipless RFID Tags with Enhanced RCS Response
## 2 Theoretical Background  
The theoretical analysis presented in this section is used to describe a methodology to maximize the quality factor \(Q\) for the simplest of the structures investigated below--the planar strip above the infinite ground plane; see Section 3.3. For even more complex structures, it would be very difficult to develop an analytical model, but the behavior here will have comparable tendencies in principle. Designing a structure with the largest possible \(Q\), and at the same time the largest possible resonant peaks/drops in the RCS response, is the way to enable the realization of such a structure using conductive screen-printing, since the use of screen-printing to realize structures introduces a significant degradation in the quality factor, as well as the level of the individual peaks/drops in the RCS response, as can be seen in the following sections. Quality Factor Based on Coupled-Mode Theory  
The phenomenon of a significant increase in the quality factor for this structure can be easily explained by the image principle. The original dipole (strip) and the image ground current have the same amplitude and opposite phases [27]. Consequently, the electromagnetic field distribution promotes energy storage and subsequently the required increase in the quality factor \(Q\)[28; 29; 30; 31; 32]. As derived in detail in [33], a single grounded scatterer, see Figure 1, which is excited by a plane electromagnetic wave, can be imagined as a resonant cavity that excites the TM\({}_{01}\) patch mode. The presence of this mode is related to conductivity and dielectric losses due to the dissipation inside the cavity. The mode also causes the radiation of energy in the form of a propagating wave. According to the theory presented in [27], such a structure can be represented by the reflection coefficient at the interface, at the reference plane of the patch/dipole surface (between the resonator structure and the free space) according to the following relation:  
\[\Gamma(f)=\frac{\frac{1}{Q_{r}}-\frac{1}{Q_{0}}-\frac{2i(f-f_{r})}{f_{r}}}{ \frac{1}{Q_{r}}+\frac{1}{Q_{0}}+\frac{2i(f-f_{r})}{f_{r}}}\;, \tag{1}\]where \(Q_{t}\) is the radiation (external) quality factor, \(Q_{0}\) is the unloaded quality factor of the patch cavity, and \(f_{t}\) is the resonant frequency. The CAD formula for the space wave radiation quality factor was derived by [PERSON] [32] as:  
\[Q_{r}=\frac{3}{16}\frac{\varepsilon_{r}}{pc_{1}}\frac{L_{eff}}{W_{eff}}\frac{ \lambda_{0}}{h}, \tag{2}\]  
where \(\varepsilon_{r}\) is the relative permittivity and \(p\) is the ratio of radiated power by the patch to the radiated power by the equivalent dipole with the same dipole moment as the patch. The constant \(c_{1}\) is expressed in (4), \(L_{\text{eff}}\) is the patch effective length, and \(W_{\text{eff}}\) is the patch width. \(\lambda_{0}\) represents the free space wavelength, and \(h\) is the dipole substrate thickness between the dipole and the metallic plate. \[ \begin{split} p=1+\frac{\varepsilon_{2}}{10}\Big{(}kW_{eff} \Big{)}^{2}+\frac{3}{560}\big{(}a_{2}^{2}+2a_{4}\big{)}\Big{(}kW_{eff}\Big{)} ^{4}+\frac{1}{5}c_{2}\Big{(}kL_{eff}\Big{)}^{2}\\ +\frac{1}{\gamma_{0}}\Big{(}kW_{eff}\Big{)}^{2}\Big{(}kL_{eff} \Big{)}^{2}\,\end{split}  \tag{3}\]  
where \(k\) is the free space wave constant, \(a_{2}=-0.16605\), \(a_{4}=0.00761\), \(c_{2}=-0.0914153\),  
\[c_{1}=\frac{1}{n^{2}}+\frac{2}{5n^{4}}\, \tag{4}\]  
and \(n=\sqrt{\mu_{r}\varepsilon_{r}}\) is the index of refraction of the substrate. The unloaded quality factor \(Q_{0}\) expresses both conductor and dielectric losses of the cavity and is given by [27]  
\[Q_{0}=\frac{Q_{c}Q_{d}}{Q_{c}+Q_{d}},Q_{c}=h\sqrt{\frac{\omega}{2}\mu\sigma}, \ \ \ Q_{d}=\frac{1}{tan\delta} \tag{5}\]  
where \(\omega\) is the angular frequency, \(\mu\) represents the permeability, and \(\sigma\) is the conductivity of the metal parts, \(\tan\delta\) is the loss tangent of the dielectric substrate. The total (i.e., loaded) quality factor \(Q_{t}\) of the cavity can be calculated according to (6).