Iterative Technique for Analysis and Design of Circular Leaky-Wave Antenna for the 2.45 GHz RFID Applications

Abstract

This chapter proposes another scaled-down receiving wire for Radio Frequency Identification uses of 2.45 GHz. Our design comprises of a roundabout microstrip fix radio wire which integrates two concentric annular openings imprinted on multi-facet substrates. The transmission capacity, one of the main qualities of radio wire, can be essentially improved by utilizing a multi-facet dielectric setup. We point by this review to show that the impact of emanating structure stacked by annular rings for the fixed size decrease as well as the benefit of this construction is to make a roundabout polarization toward maximal radiation design. The wave idea iterative method is utilized to examine this new radio wire. Utilizing the proposed technique, less figuring time and memory are expected to work out the electromagnetic boundaries of our plan. The approval of the consequences of our created model was checked with realized business programming called “CST Microwave Studio Software” trailed by a trial test. As per the arrived results, we can decide that our new plan radio wire is reasonable for RFID applications in the 2.45 GHz band.

1. Introduction

The exponential evolution of the need for microwave devices in modern communication systems has prompted manufacturers to invest more and more in this field in order to meet recent market trends. These systems are often multi-band in order to meet several communication standards. Their multiplicity on the same carrier means that more and more attention is being paid to reducing their size. The antenna is one of the essential components in wireless systems for both civil and military applications, space and terrestrial [1, 2, 3, 4].

The diversity of the fields of activity using these antennas has increased the need to develop antennas that are agile (in frequency, pattern, or polarization) and multistandard while keeping a compact appearance. The radiation of a circularly polarized wave is often of interest in order to overcome depolarization phenomena that can occur during propagation [5, 6, 7, 8].

In order to meet the various challenges, a multitude of avenues has been explored to adapt to the performance requirements of the applications. Indeed, various topologies and techniques have been investigated in order to meet the required specifications, namely: miniaturization, multi-band operation, and circular polarization radiation while maintaining optimal radio performances (gain, radiation efficiency, reflection coefficient). In addition, a promising way to improve the performance of an antenna is to integrate innovative materials. Many research works have explored this way. Our objective is, therefore, to design miniature microstrip antenna structures for RFID readers in the UHF band with the correct performance [9, 10, 11].

Circularly polarized antennas are the most commonly used antennas in RFID communications. In addition, for optimal operation, these antennas must have a particular radiation pattern. It must be omnidirectional in any azimuthal cutting plane, and maintain a high gain, although the choice of a circular antenna is justified by the fact that it has the advantage of being able to achieve circular polarization [12, 13, 14, 15, 16, 17, 18, 19, 20].

The rigorous analysis and design of microwave circuits require first of all a rigorous resolution of the equations governing the electromagnetic field, in order to be able to reconstruct the closest to real behavior of the fields in the devices. To satisfy this need, in this chapter, we focus on the study of the circular antenna by the wave concept iterative method (W.C.I.P). This method is based on the manipulation of the incident and reflected waves instead of the electromagnetic field. It is based on the back and forth between the spatial and spectral domains, using the Fourier transform FMT followed by the Hankel transform in the cylindrical coordinate system [21, 22, 23, 24, 25, 26].

2. WCIP presentation

2.1 Formulation

To begin, Figure 1 shows the starting structure.

Figure 1.

WCIP structure.

The incident wave and the scattered waves are calculated from the tangential magnetic and electric field at the surface:

$$\left[\begin{array}{c}{A}_i\\ B_i\end{array}\right]=\frac{1}{2\sqrt{Z_{0i}}}\left[\begin{array}{cc}1& \sqrt{Z_{0i}}\\ 1& -\sqrt{Z_{0i}}\end{array}\right]\left[\begin{array}{c}{E}_i\\ J_i\end{array}\right]$$E1

Where ${\mathrm{B}}_{\mathrm{i}}$ and ${\mathrm{A}}_{\mathrm{i}}$ are the incident and reflected waves associated with the discontinuity plane ${\mathrm{\Omega }}_1$. ${\mathrm{Z}}_{0\mathrm{i}}$ is the characteristic impedance of the medium i (i = 1, 2) given by ${\mathrm{Z}}_{0\mathrm{i}}=\sqrt{{\mathrm{\mu }}_0/{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{ri}}}$ (Figure 2).

Figure 2.

WCIP procedure.

The steps of the iterative method are:

1. Define ($\overrightarrow{{\mathrm{B}}_0}$).

2. Calculates $\overrightarrow{{\mathrm{B}}_{\mathrm{\rho },\mathrm{\phi }}}={\mathrm{\Gamma }}_{{\mathrm{\Omega }}_1}\overrightarrow{{\mathrm{A}}_{\mathrm{\rho },\mathrm{\phi }}}+\overrightarrow{{\mathrm{B}}_0}$

3. Apply HT and FFT: ${\mathrm{HT}}_{/\mathrm{\rho }}\left({\mathrm{FFT}}_{/\mathrm{\phi }}\left(\overrightarrow{{\mathrm{B}}_{\mathrm{\rho },\mathrm{\phi }}}\right)\right)=\overrightarrow{{\mathrm{B}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}}$

4. Calculates $\overrightarrow{{\mathrm{A}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}}={\mathrm{\Gamma }}_{\mathrm{i}}\overrightarrow{{\mathrm{B}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}}$

5. Apply IHT and IFFT: ${{\mathrm{HT}}_{/\mathrm{\rho }}}_{‐1}^{}\left({{\mathrm{FFT}}_{/\mathrm{\phi }}}_{‐1}^{}\left(\overrightarrow{{\mathrm{A}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}}\right)\right)=\left(\overrightarrow{{\mathrm{A}}_{\mathrm{\rho },\mathrm{\phi }}}\right)$

6. Repeat steps 2 to 5.

$$\left\{\begin{array}{c}{E}_{/\phi ,\rho }^k=\sqrt{Z_{0i}}\left(A_i^k+B_i^k\right)\\ J_{/\phi ,\rho }^k=\left(A_i^k+B_i^k\right)/\sqrt{Z_{0i}}\end{array}\right.$$E2

The expression of return loss at the upper and bottom side of box in the spectrum domain is given by:

$${\widehat{\mathrm{\Gamma }}}_i^{\mathit{TE},\mathit{TM}}=\frac{1-Z_{0i}{\widehat{Y}}_{m,n}^{\mathit{TE},\mathit{TM}}}{1+Z_{0i}{\widehat{Y}}_{m,n}^{\mathit{TE},\mathit{TM}}}$$E3

Where the admittance ${\widehat{\mathrm{Y}}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}$ is done by Table 1:

	Mode TE	Mode TM
${\widehat{\mathrm{Y}}}_{\mathrm{m},\mathrm{n}}^{\mathrm{TE},\mathrm{TM}}$	$‐{\mathrm{jY}}_{\mathrm{r}}\frac{\mathrm{\gamma }}{{\mathrm{K}}_{\mathrm{r}}}coth\left(\mathrm{\gamma h}\right)$	${\mathrm{jY}}_{\mathrm{r}}\frac{{\mathrm{K}}_{\mathrm{r}}}{\mathrm{\gamma }}coth\left(\mathrm{\gamma h}\right)$

Table 1.

Admittance expressions.

$k_r=\sqrt{{\omega }^2{\epsilon }_0{\epsilon }_r{\mu }_0},\gamma =k_{\rho }^2-k_r^2\mathit{\text{and}}{Y}_r=\sqrt{\frac{{\epsilon }_0{\epsilon }_r}{{\mu }_0}}$ are the admittance of each domain.

The equivalent circuit of the model is shown in the figure below:

see (Figure 3).

Figure 3.

Equivalent circuit of multilayered structure.

The coupling between two layers is characterized by the equivalent admittance $\mathrm{Y}$. The modal admittance seen at the interface ${\mathrm{\Omega }}_{\mathrm{i}}$ between layers i-2 and i-1 can be calculated by:

$$Y_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-1\right)={\widehat{Y}}_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-1\right)\left(\frac{Y_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-2\right)+{\widehat{Y}}_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-1\right)tanh\left({\gamma }_{m,n}^{i-1}{h}_{i-1}\right)}{{\widehat{Y}}_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-1\right)+Y_{m,n}^{\mathit{TE},\mathit{TM}}\left(\mathrm{i}-2\right)tanh\left({\gamma }_{m,n}^{i-1}{h}_{i-1}\right)}\right),\mathrm{i}=2,3$$E4

At the printed surface of the discontinuity, the boundary conditions of fields are expressed in terms of waves that consist of three conditions as:

The boundary conditions are given by (5):

$$\left\{\begin{array}{l}{E}_1=E_2=0\left(\mathit{\text{Metal}}\right)\\ J_1+J_2=0\left(\mathit{\text{Dielectric}}\right)\\ E=E_0-z_0\left(J_1+J_2\right)\left(\mathit{\text{Source}}\right)\end{array}\right.$$E5

The relationship between the incident and reflected waves in the spatial domain is given by:

$$B_i={\widehat{\Gamma }}_{\mathit{\Omega }}{A}_i+B_0$$E6

With ${\mathrm{B}}_0$ is the source excitation, the diffraction operator ${\mathrm{\Gamma }}_{\mathrm{\Omega }}$ is defined by:

$${\mathrm{\Gamma }}_{\mathrm{\Omega }}=\left(\begin{array}{cc}{\mathrm{\Gamma }}_{\mathrm{\Omega }11}& {\mathrm{\Gamma }}_{\mathrm{\Omega }12}\\ {\mathrm{\Gamma }}_{\mathrm{\Omega }21}& {\mathrm{\Gamma }}_{\mathrm{\Omega }22}\end{array}\right)$$E7

Where:

$$\left\{\begin{array}{l}{\mathrm{\Gamma }}_{\mathrm{\Omega }11}=-H_m-\frac{-1+n_1+n_2}{1+n_1+n_2}{H}_s+\frac{1-n^2}{1+n^2}{H}_d\\ {\mathrm{\Gamma }}_{\mathrm{\Omega }12}=\frac{2n}{1+n^2}{H}_d+\frac{2n}{1+n_1+n_2}{H}_s\\ {\mathrm{\Gamma }}_{\mathrm{\Omega }21}=\frac{2n}{1+n^2}{H}_d+\frac{2n}{1+n_1+n_2}{H}_s\\ {\mathrm{\Gamma }}_{\mathrm{\Omega }22}=-H_m-\frac{-1-n_1+n_2}{1+n_1+n_2}{H}_s+\frac{1-n^2}{1+n^2}{H}_d\end{array}\right.$$

Where H is Heaviside function (m: metal; d: dielectric; S: source)

$$\text{And}\mathrm{n}=\frac{\mathrm{Z}0}{\sqrt{{\mathrm{Z}}_{01}{\mathrm{Z}}_{02}}},{\mathrm{n}}_1=\frac{{\mathrm{Z}}_0}{{\mathrm{Z}}_{01}},{\mathrm{n}}_2=\frac{{\mathrm{Z}}_0}{{\mathrm{Z}}_{02}};$$E8

The passage between the spectral and spatial domain is given by the following Table 2:

Mode TE	Mode TM
${\mathrm{E}}_{\mathrm{\rho }}=\frac{{\mathrm{e}}^{\mathrm{j\phi }}}{\sqrt{\mathrm{\pi }}}\frac{1}{\mathrm{\Lambda \rho }}{\mathrm{J}}_1\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\right)$	${\mathrm{E}}_{\mathrm{\rho }}=‐\mathrm{j}\frac{{\mathrm{e}}^{\mathrm{j\phi }}}{\sqrt{\mathrm{\pi }}}\frac{{\mathrm{\kappa }}_{\mathrm{\rho m}}}{{\mathrm{\Lambda }}^{\text{'}}}{\mathrm{J}}_1^{\text{'}}\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\mathrm{\rho }\right)$
${\mathrm{E}}_{\mathrm{\phi }}=\mathrm{j}\frac{{\mathrm{e}}^{\mathrm{j\phi }}}{\sqrt{\mathrm{\pi }}}\frac{{\mathrm{\kappa }}_{\mathrm{\rho m}}}{\mathrm{\Lambda }}{\mathrm{J}}_1^{\text{'}}\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\mathrm{\rho }\right)$	${\mathrm{E}}_{\mathrm{\phi }}=\frac{{\mathrm{e}}^{\mathrm{j\phi }}}{\sqrt{\mathrm{\pi }}}\frac{1}{{\mathrm{\Lambda }}^{\text{'}}\mathrm{\rho }}{\mathrm{J}}_1\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\right)$

Table 2.

Standards mode.

The radial cut-off constants and normalisation constants for the modes and are given in Table 3:

	TE and electric wall; Or; TM and magnetic wall	TM and electric wall; Or; TE and magnetic wall
${\mathrm{\kappa }}_{\mathrm{\rho m}}$	${\mathrm{J}}_1\left({\mathrm{\kappa }}_{\mathrm{\rho }}\mathrm{a}\right)=0$	$J_1^{\text{'}}\left({\kappa }_{\rho }a\right)=0$
$\mathrm{\Lambda },{\mathrm{\Lambda }}^{\text{'}}$	${\mathrm{\Lambda }}^2=\frac{1}{2}{\mathrm{\kappa }}_{\mathrm{\rho }}^2{\mathrm{a}}^2{{\mathrm{J}}_1^{\text{'}}}^2\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\mathrm{a}\right)$	${\mathrm{\Lambda }}^{\text{'}2}=\frac{1}{2}\left({\mathrm{\kappa }}_{\mathrm{\rho m}}^2{\mathrm{a}}^2‐1\right){\mathrm{J}}_1^2\left({\mathrm{\kappa }}_{\mathrm{\rho m}}\mathrm{a}\right)$

Table 3.

Cut-off and modes normalisation.

The passage into the spectral domain then requires the transformation of Hankel’s incident waves [27]:

The passage into the spectral domain then requires the Hankel transform of the incident waves [7]:

$$\left(\begin{array}{c}{B}^{\mathit{TE}}\\ B^{\mathit{TM}}\end{array}\right)=\frac{e^{\mathit{j\phi }}}{\sqrt{2\pi }}\left(\begin{array}{cc}\frac{1}{\mathrm{\Lambda }\rho }{\mathit{TH}}_1& j\frac{{\kappa }_{\rho }}{\mathrm{\Lambda }}{\mathit{TH}}_1^{\text{'}}\\ -j\frac{{\kappa }_{\rho }^{\text{'}}}{{\mathrm{\Lambda }}^{\text{'}}}{\mathit{TH}}_1^{\text{'}}& \frac{1}{{\mathrm{\Lambda }}^{\text{'}}\rho }{\mathit{TH}}_1\end{array}\right)\left(\begin{array}{c}{B}_{\rho }\\ B_{\phi }\end{array}\right)$$E9

With $\mathrm{\kappa }$ and ${\mathrm{\kappa }}^{\text{'}}$ are the zeros of the Bessel functions and its first-order derivative, and are the normalisation constants of the TE and TM modes respectively and are given by:

$$\left\{\begin{array}{l}{\mathit{TH}}_1\left(f\left(\rho \right)\right)=\underset{0}{\overset{a}{\int }}f\left(\rho \right)J_1\left(k_{\rho }\rho \right)\mathit{\rho d\rho }\\ {{\mathit{TH}}_1}_{\text{'}}^{}\left(f\left(\rho \right)\right)=\underset{0}{\overset{a}{\int }}f\left(\rho \right){J_1}_{\text{'}}^{}\left(k_{\rho }\rho \right)\mathit{\rho d\rho }\end{array}\right.$$E10

The transition from the spectral domain to the space domain is mandatory. This passage is done by the inverse Hankel transform [7]. We note and respectively follows:

$$\left\{\begin{array}{l}{{\mathit{TH}}_1}_{-1}^{}\left(f\left({\kappa }_{\rho }\right)\right)=\underset{0}{\overset{a}{\int }}f\left({\kappa }_{\rho }\right)J_1\left(k_{\rho }\rho \right){\kappa }_{\rho }d{\kappa }_{\rho }\\ {{{\mathit{TH}}^{\text{'}}}_1}_{-1}^{}\left(f\left({\kappa }_{\rho }\right)\right)=\underset{0}{\overset{a}{\int }}f\left({\kappa }_{\rho }\right){J_1}_{\text{'}}^{}\left(k_{\rho }\rho \right){\kappa }_{\rho }d{\kappa }_{\rho }\end{array}\right.$$E11

The transition to the spatial domain takes the following matrix form:

$$\left(\begin{array}{c}{B}_{\rho }\\ B_{\phi }\end{array}\right)=\frac{e^{\mathit{j\phi }}}{\sqrt{2\pi }}\left(\begin{array}{cc}{{\mathit{TH}}_1}^{-1}\left(\frac{1}{\mathrm{\Lambda }\rho }\right.& {{\mathit{TH}}_1^{\text{'}}}^1\left(j\frac{{\kappa }_{\rho }}{\mathrm{\Lambda }}\right.\\ {{\mathit{TH}}_1^{\text{'}}}^{-1}\left(-j\frac{{\kappa }_{\rho }^{\text{'}}}{{\mathrm{\Lambda }}^{\text{'}}}\right.& {{\mathit{TH}}_1}^{-1}\left(\frac{1}{{\mathrm{\Lambda }}^{\text{'}}\rho }\right.\end{array}\right)\left(\begin{array}{c}{B}^{\mathit{TE}}\\ B^{\mathit{TM}}\end{array}\right)$$E12

The numerical implementation of the developed method is given by the flowchart below (Figure 4):

Figure 4.

Flowchart of the WCIP simulation.

2.2 Validation

Figure 5 shows the structure of a patch antenna to validate the method which is already developed. This antenna is engraved on a dielectric with relative permittivity ${\mathrm{\epsilon }}_{\mathrm{r}1}=4,25$. The second dielectric is of relative permittivity ${\mathrm{\epsilon }}_{\mathrm{r}2}=3,3$. The dimensions of the patch are given in Table 4. We use a discretisation of 100*40 pixels (100 pixels for the radial direction and 40 for the ortho-radial direction).

Figure 5.

Circular patch excited by a variable source.

Parameter	Value (mm)
Lc	1
Lp	17
Ls	5.4
G	0.8
Hs1	2
Hs2	1

Table 4.

Dimensions of the structure.

In order to validate the iterative method, we started first with the comparison of the operation’s number between WCIP and MOM which the most used in scientific research [28, 29, 30].

The operation number of WCIP is given by:

$$N_{\mathit{OP}}=N\left(4P+12{log}_{2}P\right)$$E13

The operation number of MOM is given by:

$$N_{\mathit{OP}}={\left(\mathit{KP}\right)}^3/3$$E14

Where:

• P is the number of meshes

• N is the number of iterations

• K is the proportion of metallic surface in the circuit structure.

Figure 6 shows the number of operations required for WCIP and MOM (Figure 7).

Figure 6.

Operation’s number between WCIP and MOM.

Figure 7.

Convergence versus iterations number at 2.35 GHz.

The convergence of WCIP is completed at 22 iterations as shown in the figure above.

A comparative study is made between the developed method and the MOM method as well as between these two methods and two software CST and HFSS.

Based on Figure 8, it can be deduced that the iterative method is the most significant in terms of accuracy.

Figure 8.

Simulation return loss using WCIP, MOM, CST and HFSS.

Table 5 shows the simulation time and memory consumption of each method.

Methods	Response time (s)	Consumption memory (MB)
WCIP	480	450
CST	2100	620
MOM	1756	905
HFSS	629	479

Table 5.

Comparison between these methods.

3. Antenna design

Figure 9 depicts the geometry of the circular leaky-wave antenna. It is characterized by a circular patch of radius $\mathrm{Lp}=20\mathrm{mm}$ which incorporates two concentric annular slots of widths $\mathrm{G}1=\mathrm{G}2=1.058\mathrm{mm}$ and of successive inner radius $\mathrm{R}1=7.058\mathrm{mm}$ and $\mathrm{R}2=12.882\mathrm{mm}$ is printed on multilayer substrates. The first substrate layer is characterized by a height ${\mathrm{H}}_{\mathrm{s}1}=1\mathrm{mm}$ and a permittivity ${\mathrm{\epsilon }}_{\mathrm{r}1}=4.25$. The parameters $\left({\mathrm{H}}_{\mathrm{s}2},{\mathrm{\epsilon }}_{\mathrm{r}2}\right)$ of the second layer will be optimized later. The antenna is supplied by an annular planar source of inner radius $\mathrm{Ls}=1.941\mathrm{mm}$ and of width $\mathrm{G}=1.058\mathrm{mm}$.

Figure 9.

Antenna design.

To fix the dimensions of the antenna, a parametric study is necessary. Figures 10–13 show the variation of the antenna performance as a function of the height of the antenna substrate 2 and its permittivity.

Figure 10.

Simulation frequency and bandwidth for different height of substrate 2 using WCIP and CST.

Figure 11.

Simulation gain and efficiency for different height of substrate 2 using WCIP and CST.

Figure 12.

Simulation frequency and bandwidth for different permittivity of substrate 2 using WCIP and CST.

Figure 13.

Simulation gain and efficiency for different permittivity of substrate 2 using WCIP and CST.

The parametric study shows that the best compromise between gain, efficiency and bandwidth at 2.45 GHz is obtained when ${\mathrm{H}}_{\mathrm{s}2}=0.3\mathrm{mm}$ and ${\mathrm{\epsilon }}_{\mathrm{r}2}=2.2$.

4. Results and discussions

The results presented in Figures 14 and 15 show that the antenna has a resonance peak at the frequency 2.45GHz which corresponds to -38.32 dB with a bandwidth of 310MHz (ranging from 2.29 to 2.6GHz) as well as the measured resonance peak is 2.45GHz with –28.57 dB reflection coefficient, which cover the bandwidth of 350MHz (from 2.28 to 2.63GHz). So, the antenna is applicable for RFID applications at 2.45 GHz. In this proposed design, we notice a good agreement between the simulated and measured results.

Figure 14.

Simulation and measurement return loss using WCIP.

Figure 15.

Simulation and measurement radiation pattern at 2.45GHz using WCIP.

The measured and simulated radiation patterns at the resonance frequency are given by Figure 16. Following the analysis of the results the phi-planes show mostly unidirectional radiation patterns over the specified frequencies.

Figure 16.

Simulation and measurement gain using WCIP.

Figure 16 shows the measured and simulated gain versus the frequency of the proposed antenna. From the simulated results the maximum value of gain is 9.02dB at 2.45GHz as well as the measured value is 8.96dB at 2.45GHz.

The efficiency of the antenna is depicted in Figure 17. We obtain a satisfactory value of 88% over a wide frequency bandwidth.

Figure 17.

Simulation and measurement efficiency using WCIP.

In free-space, the power received by a tag antenna ${\mathrm{P}}_{\mathrm{a}}$ can be calculated using the Friis free-space [31] equation, where:

$${\mathrm{P}}_{\mathrm{a}}=\frac{{\mathrm{P}}_{\mathrm{r}}{\mathrm{G}}_{\mathrm{r}}{\mathrm{G}}_{\mathrm{a}}{\mathrm{\lambda }}^2}{{\left(4\mathrm{\pi }\right)}^2{\mathrm{d}}^2}$$E15

Where (Table 6):

Parameter	Designation
$\mathrm{\lambda }$	wavelength
${\mathrm{P}}_{\mathrm{r}}$	power transmitted
${\mathrm{G}}_{\mathrm{r}}$	reader antenna gain
${\mathrm{G}}_a$	tag antenna gain
d	distance between the tag and reader

Table 6.

Designation.

The Eq. (16) represents the read range ®:

$$\mathrm{r}=\frac{\mathrm{\lambda }}{4\mathrm{\pi }}\sqrt{\frac{{\mathrm{P}}_{\mathrm{r}}{\mathrm{G}}_{\mathrm{r}}{\mathrm{G}}_{\mathrm{a}}\mathrm{\tau }}{{\mathrm{P}}_{\mathrm{th}}}}$$E16

According to the results of our structure at 2.45 GHz, the numerical values are given in the Table 7:

Parameter	Values
$\mathrm{\lambda }$	0.12 m
${\mathrm{P}}_{\mathrm{r}}$	1w
$\tau$	0.92
${\mathrm{G}}_a$	8.96 dB
${\mathrm{P}}_{\mathit{th}}$	0.04 w
${\mathrm{G}}_r$	8 dB
r	9.58 m

Table 7.

The numerical values.

5. Conclusion

This chapter focuses on the design and production of antennas for radio frequency identification. This technology is experiencing a huge growth and requires the reduction of the cost of an electronic tag to meet the development of its market. Indeed, the reduction of the cost of an RFID tag is a crucial element for the unitary identification of goods. This cost must represent a negligible part compared to the product it identifies. An RFID tag is made of an electronic chip, a support and an antenna. To date, the manufacturing of RFID tags adopts the classic techniques of electronics. A solution for the reduction of the cost of an RFID tag could come from the use of printing technologies. The design and realization of RFID tags, especially of their antennas, by using this innovative technology has been demonstrated. In this chapter, we succeeded in designing a miniaturized antenna that meets the requirements of RFID technology.