**3. Analysis on the dielectric material choice**

To supply satisfactory answers about the effects of dielectric material properties, this section will present a careful and extensive investigation into relevant cases. Indeed, properties of the dielectric material have an influence on antenna characteristics, i.e. impedance bandwidth, Q factor, resonant frequency and radiation efficiency. Thus, this part will allow the reader to correctly select a dielectric material for a targeted application.

The cylindrical DRA example presented Figure 5 with a radius a=40mm and a height h=45mm will be pursue here. However, all results included in this part are generally applicable to most shapes of DRAs. A first sub-section will detail the influence of the dielectric permittivity and a second one will be interested in the impact of the dielectric loss tangent on the DRA performances.

Dielectric material properties having an impact on antenna characteristics are dielectric permittivity values and loss tangents.

#### **3.1. Influence of the dielectric permittivity**

To show the real impact of dielectric permittivity values, this sub-section will deal with a loss less dielectric material. Analytical studies show that the Q factor of the HE11δ mode is defined as

$$Q = 0.01007. \varepsilon\_r \cdot \frac{a}{h} \cdot \left( 1 + 100. \exp(-2.05(\frac{a}{2h} - \frac{1}{80} \left(\frac{a}{h}\right)^2) \right) \tag{5}$$

It is plotted as a function of a/h for different values of εr in the Figure 12.

**Figure 12.** Q factor according the a/h values

36 Dielectric Material

222

2 2

(3)

*k k* (4)

<sup>0</sup> 2 *xyz r r <sup>c</sup> <sup>f</sup> kkk*

<sup>0</sup> tan( ) ( 1) <sup>2</sup>

*z rz k a <sup>k</sup>* 

0 0

Values of resonant frequencies predicted by using this model are close to the measured ones for moderate to high value of εr. A frequency shift appears for low εr but it remains a good approximation method. If more accuracy is required, the electromagnetic study with CST MS (for example) presented in the cylindrical DRA case will have to be undertaken. Moreover, it allows taking into account feeding mechanism and ground plane dimensions.

Now DRA research method has been initiated, presenting resonant frequencies, fields configuration and feeding mechanisms, the next part will focus on the relevant dielectric

To supply satisfactory answers about the effects of dielectric material properties, this section will present a careful and extensive investigation into relevant cases. Indeed, properties of the dielectric material have an influence on antenna characteristics, i.e. impedance bandwidth, Q factor, resonant frequency and radiation efficiency. Thus, this part will allow

The cylindrical DRA example presented Figure 5 with a radius a=40mm and a height h=45mm will be pursue here. However, all results included in this part are generally applicable to most shapes of DRAs. A first sub-section will detail the influence of the dielectric permittivity and a second one will be interested in the impact of the dielectric loss

Dielectric material properties having an impact on antenna characteristics are dielectric

To show the real impact of dielectric permittivity values, this sub-section will deal with a loss less dielectric material. Analytical studies show that the Q factor of the HE11δ mode is

> <sup>2</sup> <sup>1</sup> 0.01007. . . 1 100.exp( 2.05(2 80 *<sup>r</sup> a a <sup>a</sup> <sup>Q</sup> h h <sup>h</sup>*

(5)

 

*z*

It is found by solving the following transcendental equation:

where 0 <sup>0</sup> <sup>2222</sup>

 

material properties having significant influences on antenna performances.

the reader to correctly select a dielectric material for a targeted application.

<sup>2</sup> , , and *r r x y xyz <sup>f</sup> kkk kkkk*

*abvc*

**3. Analysis on the dielectric material choice** 

tangent on the DRA performances.

permittivity values and loss tangents.

defined as

**3.1. Influence of the dielectric permittivity** 

 

> Q factor is increasing with εr and reaching a maximum for a/h=1.05. This Q factor can be used to estimate the fractional bandwidth of an antenna using:

$$BW = \frac{\Delta f}{f\_0} = \frac{s - 1}{Q\sqrt{s}}\tag{6}$$

Where Δf is the absolute bandwidth, f0 is the resonant frequency and s the maximum acceptable voltage standing wave ratio (VSWR).

The Q factor equation is deriving from the cylindrical dielectric resonator model approach by assuming perfect magnetic and/or electric walls on resonator faces. These equations are not absolutely accurate but they offer a good starting point for the design of cylindrical DRAs.

Let's consider the electromagnetic study presented in the first section with the cylindrical DRA example (Figure 7). Previously, the dielectric permittivity was fixed and equaled 30, it is now a variable. The Figure 13 plots both resonant frequencies and impedance bandwidths according to the dielectric permittivity εr. Because the coupling of the mode is depending on both length and height of the probe, this latter has been optimized for each εr value. The Figure 13 is thus the result of a large number of simulations.

As expected (see equation 3), the resonant frequency decreases when the dielectric permittivity increases. Moreover, this Figure shows that the bandwidth is the widest for εr=10. Fields are less confined for a low dielectric permittivity DRA, it is thus more difficult to couple the mode inside the resonator. Indeed, for higher dielectric values (εr>10), strong coupling is achieved, however, the maximum amount of coupling is significantly reduced if the dielectric permittivity of the DRA is lowered. That is why the bandwidth is low for εr

under 10. For a dielectric permittivity over 10, the Q factor is increasing and therefore the impedance bandwidth is decreasing (see equation 5).

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 39

**Figure 14.** Impedance bandwidth according to the dielectric permittivity and loss tangent

**Figure 15.** Radiation efficiency according to the dielectric permittivity and loss tangent

Antenna radiation efficiency is all the more affected by the losses as the dielectric

For low dielectric permittivity, even in case of a high losses material, the radiation

To conclude this part, a DRA designer has to choose the dielectric material according to the application for which he is aiming. If he targets a wide bandwidth application, he could choose an alumina ceramic (εr ~10). Depending on the radiation efficiency he aims, the

Now, if he targets an ultra-miniature DRA, it will be in his interest to choose a dielectric material with a higher dielectric permittivity. In this case, the impedance bandwidth will be

Other information can be deduced from this new graph:

permittivity increase.

efficiency remains higher than 50%.

chosen ceramic would have more or less losses.

affected, even more if the losses are high.

Impedance bandwidth (%)

Radiation efficiency (%)

**Figure 13.** Resonant frequency according the dielectric permittivity values εr

Now that the influence of the dielectric permittivity has been shown, we can consider in the next sub-section more realistic cases by studying the impact of losses on the antenna characteristics.
