# Numerical Modeling, Analysis, and Optimization of RFID Tags Functioning at Low Frequencies
## 2 Materials and Methods  
A study from University College Cork in Ireland [28] was taken as the starting point, and an NFC tag operating at a frequency of 13.56 MHz was designed, practically created, and analyzed. Similar values were obtained through numerical modeling in Ansys HFSS and measurements with domain-specific instruments, such as an RLC meter and a vector network analyzer. The analysis used two programs from the Ansys suite, namely Q3D Extractor and HFSS. The first of the programs was used to determine the inductance, and the second one was used to determine the frequency range over which the tag operates, and subsequently, to identify the circuit elements that should be added to make it resonate at the target frequency. Thus, a square coil with an outer diameter of 44 mm was modeled, with 3 turns. The thickness of the winding and the distance between the turns was 0.5 mm, as can be seen in Figure 1.  
The dielectric used was FR4 Epoxy with dimensions of 47 \(\times\) 47 \(\times\) 1.51 mm. Following the analysis with Ansys Q3D Extractor (HEADQUARTERS Southpointe 2600 Ansys Drive Canonsburg, PA15317, USA), it was determined that the inductance value obtained through numerical modeling was close to the value calculated with Formula (1), as shown in Figure 2[8; 29; 30]. Figure 1: _Cont._  
After adapting the model to achieve resonance at 13.56 MHz, a capacitance value of 136 pF and a resistance of 50 \(\Omega\) were used to be connected in parallel with the planar inductor. The capacitance value calculated analytically using Formula (2) was \(C=132.464054\) pF. Validation was also performed experimentally to determine the accuracy of the results. The structure was created using the LPFK S103 PCB production line, and its inductance was measured with an RLC Meter (Aim-TTi Glebe Road, Huntingdon, Cambridgeshire PE29 7 DR, United Kingdom), as shown in Figure 2.  
Figure 3 shows the testing setup, including the VNA and the measured S-parameters after a capacitor value of 130 pF and a resistance of 50 \(\Omega\) were added to the terminals of the structure. It can be seen that results are in accordance with the results obtained analytically; thus, the formulas are verified. Figure 1: The structures modeled and analyzed in Ansys: (**a**) the structure from Q3D Extractor; (**b**) the structure constructed in HFSS. Figure 2: The inductance values of the analyzed coil: (**a**) the values obtained from numerical modeling; (**b**) the values measured with the RLC meter. In Figure 3c, the resonant frequencies obtained after numerically modeling the structures with Ansys HFSS (HEADQUARTERS Southpointe 2600 Ansys Drive Canonsburg, PA 15317, USA) are shown. Figure 3: The values of the circuit elements added to the circuit to achieve resonance at 13.56 MHz: (**a**) the values obtained from numerical modeling; (**b**) the resonance frequency measured after adding the circuit elements at the tag terminals; (**c**) the S parameters that confirm the operation of the structure at a resonance frequency very close to the desired value. The formulas for the inductance and capacitance are presented as follows:  
\[L=\left(\frac{\mu\cdot N^{2}\cdot d_{m}\cdot c_{1}}{2}\right)\cdot\left(\log\left( \frac{c_{2}}{\rho}\right)+c_{3}\cdot\rho+c_{4}\cdot\rho^{2}\right) \tag{1}\]  
\[C=1/\left(4\cdot\pi^{2}\cdot f^{2}\cdot L\right) \tag{2}\]  
where \(L\) is the inductance, \(\mu\) is the permeability of the free space, \(N\) is the number of turns, and \(d_{m}\) represents the average diameter of the inductor. The latter is calculated as the mean of the inner and outer diameters, calculated with Formula (3), where \(D_{e}\) is the exterior diameter of the inductor, \(D_{i}\) is the interior diameter of the inductor, \(\rho\) is the fill factor, which is the ratio of the inner diameter to the outer diameter of the spiral inductor calculated with Formula (4), and \(c_{i}\) are the coefficients that depend on the inductor's shape.