# Design, Fabrication and Testing of a Multifrequency Microstrip RFID Tag Antenna on Si
## 2 Design Procedure and Parametric Study  
To provide some background information on the physical working mechanism of the antenna, a pair of equal-width microstrip lines (1, 2) is shown in Figure 1a, with \(s\) the spacing between them, \(\varepsilon_{r}\) the dielectric constant and \(h\) the thickness of the substrate material; a typical equivalent circuit is presented in Figure 1b. The characteristic impedance of each line is determined by the self-inductances \(L_{11},L_{22}\), and the self-capacitances \(C_{11},C_{22}\). The electromagnetic coupling between the lines, depending on spacing, is described by the mutual inductance \(L_{12}\) and capacitance \(C_{12}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Frequency (GHz)** & **Gain (dBi)** & **Material** \\ \hline
[6] & 20–25 & 1.29 & SiPh \\
[14] & 5–8.5 & 3.31 & FR-4 \\
[15] & 2.4–5.8 & 2–3.5 & FR-4 \\
[16] & 0.915–2.45 & 2.2 & FR-4 \\
[17] & 2.45 & 2.29 & FR-4 \\
[18] & 2.4–5.8 & 2.47–1.98 & Jeans \\
[19] & 2.4–5.8 & 3.6–3.3 & RT–Duroid 5880 \\
[20] & 5.8 & 3.62 & FR-4 \\
[21] & 5.8 & 2.84 & FR-4 \\
[22] & 2.4–5.8 & 1–3.7 & FR-4 \\ Proposed work & 2.4–5.8 & 3.47–4.18 1 & Si (High resistivity) \\ \hline \hline \end{tabular} \({}^{1}\)Simulated values.  
\end{table} 
Table 1: Comparison of present results with previous work. According to the stimulation (symmetric/antisymmetric), the line operates in different modes (even/odd), each with a different effective dielectric constant [23; 24]. A typical view of the corresponding electric and magnetic fields is depicted in Figure 2. In the odd mode, capacitances of adjacency [23; 24] appear due to the edge phenomenon (\(C_{ga},C_{gd}\)), and \(C_{f}\) is the fringe capacitance [24]. The mutual coupling between two lines is primarily attributed to the fields present at the air-dielectric interface and depends on their relative position, as in the case of slot or wire antennas; hence, the spacing (s) is of critical importance in this regard. An outline of the design process is shown in Figure 3. According to the rationale discussed in the Introduction, we based the design of the tag antenna on a Si substrate. A rectangular spiral shape was chosen for space saving and compactness. In short, we optimized the process by varying a specific parameter at a time and running the simulation; the procedure was repeated for all design parameters, arriving at a (partially) optimized value set. Then, the whole process was reiterated. At each iteration of the parametric search, the \(S_{11}\) and gain parameters, as well as the overall behavior of the radiation pattern in the 2.4 GHz and 5.8 GHz frequencies, were checked. We started from an initial design with \(b_{sub}=0.32\) mm, copper plate thickness \(b_{cop}=3\) um, \(b_{sp}=0.018\) mm and \(b_{dielectric}=3\) um. The total area of the antenna component was \(70\times 78.5\) mm\({}^{2}\). The spacing (s) and width (\(x\)) of the spiral lines were 0.9 mm and 2.7 mm, respectively. Figure 1: (a) Coupled microstrip lines and (b) equivalent circuit at odd mode and the arrows indicate the fringe capacitance. Figure 2: Electric and magnetic fields in coupled lines. A first search was performed by varying x for resonances at 2.4 GHz and 5.8 GHz. Figure 4 shows the results of a series of simulations for \(S_{11}\), where \(s=0.85\) mm and \(x\) is varied from 2.3 mm to 2.7 mm. The value of \(x=2.5\) mm yielded the best results. Further on, a search varying \(s\) from 0.8 mm to 0.95 mm was performed, as shown in Figure 5 for the simulated values of \(S_{11}\). The value of 0.85 was found to achieve good tuning close to 5.8 GHz. Figure 4: Example of parametric simulation for x.  
Figure 5: Example of parametric simulation for s.  
Figure 3: Design methodology flowchart. A subsequent search was carried out for \(x_{0},y_{0}\), which related to the overall dimension of the antenna, as shown in Figure 6, with \(x_{0}\) values ranging from 16 mm to 18 mm and \(y_{0}\) values ranging from 19 mm to 20 mm.