# Numerical Modeling, Analysis, and Optimization of RFID Tags Functioning at Low Frequencies
## 5 Discussions  
Some of the structures presented above were created with the equipment from the laboratory, but due to their small dimensions, there were some limitations. This is why we considered creating some of the structures with s and w geometrical dimensions larger than 0.2 mm. The considered structures were the square planar coil ones due to the fact that they outperformed the other structures. Their inductance was calculated with the formulas, determined through numerical modeling and measured with an RLC meter. It can be seen that the values were in accordance (Table 5). The higher difference between the values obtained by the measurements was due to the presence of wires with a greater length at the terminals of the circuit. The values obtained for the capacitance necessary for the circuit to resonate at 13.56 MHz had higher values when obtained through numerical modeling, also due to the dimensions of the terminals of the circuit modeled in the Ansys program, but, as we know, the capacitors had specific values. Therefore, an approximation of this value was possible by analytical calculation or numerical modeling. The structures were created with the help of LPKF Plotter (LPKF Laser & Electronics SE, LPKF Laser & Electronics SE, Osteriede 7. D-30827 Garbsen), and they are presented in Figure 21. The capacitance and inductance values obtained experimentally, analytically, and through numerical modeling are also compared in the graphs in Figure 22. The measuring process of the inductance can be seen in Figure 23 for each of the structures analyzed. Figure 21: The square structures constructed with the help of LPKF Plotter. Figure 22: A comparison of the obtained values through measurement, numerical modeling, and analytical calculations of (**a**) inductivity and (**b**) capacitance. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(D_{\textbf{r}}\) (mm) & \(N\) & \(s\) (mm) & \(W\) (mm) & \begin{tabular}{c} **L Calculated** \\ **(uH)** \\ \end{tabular}  & \begin{tabular}{c} **C Calculated** \\ **(pF)** \\ \end{tabular}  & \begin{tabular}{c} **L Numerical** \\ **Modeling (uH)** \\ \end{tabular}  & \begin{tabular}{c} **C Numerical** \\ **Modeling (pF)** \\ \end{tabular}  & \ \begin{tabular}{c} **L Measured** \\ **(uH)** \\ \end{tabular}  & \begin{tabular}{c} **C Included in** \\ **Circuit (pF)** \\ \end{tabular}  \\ \hline
17 & 7 & 0.2 & 0.3 & 1.18 & 116.8 & 1.13 & 137 & 1.33 & 128 \\
19 & 6 & 0.2 & 0.3 & 1.17 & 117.56 & 1.13 & 133 & 1.275 & 128 \\
19 & 7 & 0.2 & 0.3 & 1.43 & 96.28 & 1.38 & 106 & 1.585 & 98 \\
19 & 7 & 0.2 & 0.4 & 1.24 & 111.50 & 1.21 & 120 & 1.345 &Figure 23: The measurements of the characteristic parameters of the square and circular planar inductors: (**a**) structures to be measured; (**b**) inductance of the square planar inductor; (**c**) inductance of the circular planar inductor; (**d**) S parameters for the square planar inductor with circuit elements to resonate at 13.56 MHz; (**e**) S parameters for the circular planar inductor with circuit elements to resonate at 13.56 MHz. In order to determine if the square structures were indeed better than the other shape planar structures, circular and square structures with an exterior diameter of 19 mm, nine turns, a spiral width of 0.3 mm, and a distance between the turns of 0.2 mm were constructed and compared. The structures are presented in Figure 23, along with the experimental measurements of the inductance values and the determination of the circuit elements placed on the planar structures in order for them to function at 13.56 MHz. The results analytically obtained for the structures are in accordance with those obtained through numerical modeling and experimentally. Therefore, any of these methods could lead to optimal structure determination. For the structures to function at 13.56 MHz due to the specific values of the capacitors on the market, two capacitors were placed in parallel in order to reach the needed value.