**4.3. Circular sector DRAs**

To clearly explain this miniaturization technique, we need to take up the cylindrical DRA example with the equation 1 of the resonant frequencies. In [32], DRA size and resonance frequencies significant reductions have been demonstrated by using cylindrical sector DRA which is shown in Figure 18.

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 43

Following these results, resonance frequencies depend on whether parts of some faces are coated with a metal or not. A good trade-off between antenna size and low resonant frequency is to choose the third line of the Table 4, when one face is coated by metal. Finally, this study has demonstrated that the metallization of some DRA faces allows creating new resonant modes, which have resonant frequencies lower than the fundamental modes inside a classical cylindrical DRA. It should be noted that these results have been done with cavity resonator model, i.e. the outer surfaces of the cavity are approximated by perfect magnetic walls. They thus show an approximate analysis of the fields inside the resonator. However,

this model provides reasonable accuracy for prediction of resonant frequencies.

HE21 2 1 3.054 1.27 GHz

HE11 1 1 1.841 872 MHz

HE1/2 1 1/2 1 1.166 687 MHz

HE1/3 1 1/3 1 0.910 629 MHz

HE1/4 1 1/4 1 0.769 602 MHz

**Table 4.** Excited modes with the corresponding DRA shapes

**5. New approaches for wireless applications** 

to integrate them inside mobile handheld.

Mode ν p X'p fνpm Shape

Antenna design for mobile communications is often problematic by the necessity to

Some of currently antennas integrated in portable wireless systems have a planar structure based on microstrip patches or PIFAs [33-35]. These kinds of antennas present a low efficiency, especially when small, because of metallic losses. DRAs do not suffer from such losses, which makes them a good alternative for these more conventional antennas. That is why, in recent years, much attention has been given to DRA miniaturization [16,36] in order

implement multiple and/or ultra wideband applications on the same small terminal.

**Figure 18.** Circular sector DRA

As shows the Figure 18, a cylindrical sector DRA shape consists of a cylindrical DRA of radius *a* and height *d* mounted on a metallic ground plane, with a sector of dielectric material removed. β is the angle between the face 1 and 2, which can be metalized or left open. Thus, a cylindrical sector DRA is formed when β < 2π. For such a DRA, considering the equation 1, the n subscript can be substitute by the ν subscript, which is a positive real number that depends on the boundary conditions on the sector faces as well as the sector angle β. In this case, first excited modes are writing such as HEpm+� and the corresponding resonant frequencies as defined by the following equation:

$$f\_{\nu\rho\underline{m}} = \frac{c}{2\pi\sqrt{\varepsilon\_r\mu\_r}}\sqrt{\left(\frac{X\_{\nu\rho}'}{a}\right)^2 + \left(\frac{(2m+1)\pi}{2d}\right)^2} = \frac{c}{2\pi a\sqrt{\varepsilon\_r\mu\_r}}\sqrt{\left(X\_{\nu\rho}'\right)^2 + \left(\frac{(2m+1)\pi.a}{2d}\right)^2} \tag{7}$$

It should be noted that for the β = 2π case, ν=n.

An important point has to be highlighted: For a given cylindrical sector DRA (radius, height, permittivity and permeability), the resonant frequency is only depending on the X'p value. The lower the X'p value will be, the lower the resonant frequency will be. X'p values are summarized in the Table 3.


**Table 3.** X'p values according to the ν and p values

By applying the good boundary conditions on each faces, fundamental modes of the different shapes presented in Table 4 can be determined. This table is reminding the X'<sup>p</sup> values with the corresponding resonant frequencies for a DRA such as a=40mm and d=45mm with a dielectric permittivity equals to 10.

Following these results, resonance frequencies depend on whether parts of some faces are coated with a metal or not. A good trade-off between antenna size and low resonant frequency is to choose the third line of the Table 4, when one face is coated by metal. Finally, this study has demonstrated that the metallization of some DRA faces allows creating new resonant modes, which have resonant frequencies lower than the fundamental modes inside a classical cylindrical DRA. It should be noted that these results have been done with cavity resonator model, i.e. the outer surfaces of the cavity are approximated by perfect magnetic walls. They thus show an approximate analysis of the fields inside the resonator. However, this model provides reasonable accuracy for prediction of resonant frequencies.


**Table 4.** Excited modes with the corresponding DRA shapes

42 Dielectric Material

**4.3. Circular sector DRAs** 

which is shown in Figure 18.

**Figure 18.** Circular sector DRA

> 

summarized in the Table 3.

a

Face 2

resonant frequencies as defined by the following equation:

*p*

It should be noted that for the β = 2π case, ν=n.

**Table 3.** X'p values according to the ν and p values

d=45mm with a dielectric permittivity equals to 10.

d

Face 1

To clearly explain this miniaturization technique, we need to take up the cylindrical DRA example with the equation 1 of the resonant frequencies. In [32], DRA size and resonance frequencies significant reductions have been demonstrated by using cylindrical sector DRA

β

 

(7)

<sup>2</sup> 2 2 <sup>2</sup> (2 1) (2 1) .

Face 1

Face 2

Ground plane

As shows the Figure 18, a cylindrical sector DRA shape consists of a cylindrical DRA of radius *a* and height *d* mounted on a metallic ground plane, with a sector of dielectric material removed. β is the angle between the face 1 and 2, which can be metalized or left open. Thus, a cylindrical sector DRA is formed when β < 2π. For such a DRA, considering the equation 1, the n subscript can be substitute by the ν subscript, which is a positive real number that depends on the boundary conditions on the sector faces as well as the sector angle β. In this case, first excited modes are writing such as HEpm+� and the corresponding

2 2 2 2

An important point has to be highlighted: For a given cylindrical sector DRA (radius, height, permittivity and permeability), the resonant frequency is only depending on the X'p value. The lower the X'p value will be, the lower the resonant frequency will be. X'p values are

> ν=0 ν=1/4 ν=1/3 ν=1/2 ν=2/3 ν=1 ν=2 p=1 3.832 0.769 0.910 1.166 1.401 1.841 3.054 p=2 7.016 4.225 4.353 4.604 4.851 5.331 6.706

By applying the good boundary conditions on each faces, fundamental modes of the different shapes presented in Table 4 can be determined. This table is reminding the X'<sup>p</sup> values with the corresponding resonant frequencies for a DRA such as a=40mm and

 

*<sup>c</sup> <sup>X</sup> m c m a f X a d <sup>a</sup> <sup>d</sup>*

*pm p r r r r*
