# Numerical Modeling, Analysis, and Optimization of RFID Tags Functioning at Low Frequencies
## 3 Software Program Used for Determining Possible Structures  
Given the standard dimensions of coils used as tags in RFID and considering their various shapes, as found in the specialized literature, calculation software was designed to generate all possible geometric configurations to model such coils [29]. The logical scheme on which this software is based can be observed in Figure 5.  
Thus, the exterior diameter (\(D_{e}\)) of the tag was considered to be a maximum of 20 mm for this study because of the usual dimension of such a tag, but the user can choose any other outer dimension. The parameters present in the logical scheme are as follows: N represents the number of turns of the coil, Di is the interior diameter of the coil, s is the distance between the coils, w is the width of the coil, dm is the medium diameter of the coil, and ro is the coil fill factor. The inductance of the coils (\(L\)) has to exceed 1 \(\mu\)H, and the inner diameter (\(D_{i}\)) has to be at least 1 cm to later accommodate the necessary capacitor and resistor to achieve a resonance frequency of 13.56 MHz. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shape** & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{c_{3}}\) & \(\mathbf{c_{4}}\) \\ \hline Square & 1.27 & 2.07 & 0.18 & 0.13 \\ Hexagon & 1.09 & 2.23 & 0.00 & 0.17 \\ Octagon & 1.07 & 2.29 & 0.00 & 0.19 \\ Circle & 1.00 & 2.46 & 0.00 & 0.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The \(c_{i}\) coefficients for the capacitance calculations based on the shape of the inductor. Figure 4: Shapes considered for the planar inductor: (**a**) circular; (**b**) rectangular; (**c**) hexagonal; (**d**) octagonal. In this case, circular, square, hexagonal, and octagonal planar coils were considered. The formulas for calculating the inductance are given as (1), where the \(c_{i}\) coefficients must be considered in accordance with the inductor's shape. The number of turns was limited to 20 in this study, but this can be reconfigured. The values of w and s were between 0.1 mm and 1 mm, considering a step of 0.1 mm. For the imposed conditions, the most possible structures were obtained for the square configurations, as can be seen in Figure 6, for structures operating at 13.56 MHz. Figure 5: The logical scheme underlying the program that determines the parameters of structures to meet the design conditions. Figure 6: The number of structures of different shapes that meet the design criteria for various values of the outer diameter. Based on this algorithm, an interface was created using MATLAB R2023b (Corporate Headquarters: Natick, Apple Hill Campus 1 Apple Hill Drive Natick, MA 01760-2098 UNITED STATES). A window will open for the user, allowing them to select the desired structure type, as can be seen in Figure 7a. Next, they can enter the outer diameter for the structure in the interface that opens, as presented in Figure 7b. Upon pressing the \"Calculate inductance and capacitance required to achieve the resonant frequency\" button, all relevant variables that meet the requirements outlined in the logical scheme in Figure 5 will be calculated and displayed in a table. The table will contain the exterior diameter, the number of turns, the distance between turns, the thickness of the winding, the inductivity of the spiral coil, and the capacitance value needed for the coil to reach the resonance frequency. To work more easily with the values, the user can press the \"Export to Excel\" button, and an Excel file with the relevant data will be automatically generated. The inductance as a function of the number of turns is represented when pressing the \"Graphical Representation L [uH]\" button. The information about the capacitance needed for the coil to resonate at 13.56 MHz is obtained in the second graph, which represents the capacitance of the structures as a function of the number of turns when the \"Graphical Representation C [PF]\" button is pressed.