# Design, Fabrication and Testing of a Multifrequency Microstrip RFID Tag Antenna on Si
## 2 Design Procedure and Parametric Study
For \(x_{0}=17.5\) mm, \(y_{0}=20\) mm, good behavior is achieved at 2.4 GHz and 5.8 GHz, resulting in the reduced total dimensions of the antenna (an additional benefit). In summary, the most important parameters appear to be \(x\) (width) and \(s\) (spacing). As mentioned above, these values are responsible for the electromagnetic coupling between the lines and have a special role in the design. The \(x_{0},y_{0}\) values are also important, significantly affecting the behavior of the antenna. More specifically, the modification of \(x_{0},y_{0}\), i.e., the reduction in the overall dimension, results in better tuning at 5.8 GHz, as well as an improved radiation pattern in both bands at 2.4 GHz and 5.8 GHz. Reducing the value of the x-dimension also improves, to some extent, the tuning at 2.4 GHz. Figure 7 shows the initial (a) and final (b) designs after optimization. Table 2 shows the final parameter values in detail. Figure 6: Example of parametric simulation for \(x_{0},y_{0}\). Figure 7: Evolution of RFID tag antenna: (**a**) initial design and (**b**) proposed design. Purple denotes Polymide and yellow the copper. The substrate material for the design is Si (high resistivity) and Polyimide PI2525 is used for the bridge dielectric. The dielectric constant \(\varepsilon_{r}\) for high-resistivity Si is 11.68 and that for PI2525 is 2.68. In Figure 8, \(b_{sub}\) is 0.320 mm, \(b_{cop}\) is 3 \(\upmu\)m, \(b_{sp}\) is 0.018 mm, \(b_{dielectric}\) is 3 \(\upmu\)m, and \(k\) (the correction factor, see below) is 3 mm. The total area of the antenna component is 35 \(\times\) 47.6 mm\({}^{2}\). The spacing (s) and width (x) of the spiral lines, as shown in Figure 9, are 0.85 mm and 2.5 mm, respectively, and h and j are 17.6 mm and 7.5 mm, respectively. The antenna operation is based on electromagnetic coupling, as described previously. A decrease in the gap between subsequent lines corresponds to a stronger electromagnetic coupling. For the parameterized design, the following formulas have been used:  
Figure 2.0  
Figure 2.1  
where \(N\) is the number of turns and  
\(N(i+1)=N+i\), where \(i\) = 0:3 with step 1,  
\(M(i-1)=M-i\), where \(i\) = 1:3 with step 1,  
\(N(j+1)=N+j\), where \(j\) = 0:3 with step 1,  
\(M(j-1)=M-j\), where \(j\) = 1:3 with step 1,  
for \(x_{i}\), \(i\) = 0:3 with step 1 and for \(y_{j}\), \(j\) = 0:3 with step 1. Equations (1) and (2) are the basic conditions for creating the spiral antenna shown in Figure 2.2 \(N\) corresponds to the number of turns, and \(M\) corresponds to the number of spaces between the metal paths (In Figure 9, \(M\) = 3). We note that Equation (1) refers to the \(x\) dimension and (2) to the \(y\) dimension, with \(x_{0}=X/2\), \(y_{0}=Y/2\), respectively, where \(X\) and \(Y\) are the maximum dimensions. We use (1) and (2) for each metal path. For the first inner metal path, we apply a correction parameter \(k\) (\(k\geq 3\)) when \(x_{0}y_{0}\leq 19\). Figure 2.3  
Equations (1)-(3) determine the design and lead to the final design of a parameterized antenna through a series of simulations. The substrate thickness \(b_{sub}\) along the \(z\) axis is also taken into account in the numerical simulations. The inductance is calculated by the following equations (see, e.g., [24] pp. 95-96):  
Figure 2.4  
Figure 2.5  
Figure 2.6  
Figure 2.7  
We use these equations, especially (5a) and (5b), for each one of the 19 metal paths. As already mentioned, the values of \(x\) (width) and \(s\) (space) for the lines, as shown in Figure 9, have been specifically selected for optimization of the \(S_{11}\) parameter at the frequencies of interest.