**2.4. The rectangular DRA**

34 Dielectric Material

**Figure 8.** Input impedance (a) and S11 parameter (b) of the DRA

frequency equals 510 MHz with 3.6% of bandwidth at -10 dB.

radiation pattern of the cylindrical DRA using the simulator.

**Figure 9.** 3D radiation pattern

mode radiates like a short magnetic monopole.

the antenna.

It allows knowing the matching frequency as well as the impedance bandwidth, basically corresponding to the working frequencies of the antenna. It this case, the matching

Another important issue is the radiation pattern. It can be expressed in spherical coordinates by using equivalent magnetic surface currents [7]. This can only be done until the DRA is mounted on an infinite ground plane. Using the electromagnetic software is another way of accessing to the 3D radiation pattern. This method is more accurate because it is taking into account the realistic structure (i.e. the finite ground plane). The Figure 9 shows the 3D

The simulated 3D radiation pattern provides antenna information such as the radiation efficiency, which is defined as the ratio between the radiated to accepted (input) power of

The HE11δ mode of the cylindrical DRA radiates like a short horizontal magnetic dipole. Concerning other modes, the TM01δ radiates like a short electric monopole, while the TE01<sup>δ</sup>

The rectangular DRA has one degree of freedom more than the cylindrical DRA. Indeed, it is characterized by three independent lengths, i.e. its length *a*, its width *b* and its height *d*. Thus, there is great design flexibility, since the large choice of both dielectric materials and different lengths ratios.

Usually the dielectric waveguide model is used to analyze the rectangular DRA [8-10]. In this approach, the top surface and two sidewalls of the DRA are assumed to be perfect magnetic walls, whereas the two other sidewalls are imperfect magnetic walls. Since the considered case is a realistic one (Figure 10), the DRA is mounted on a ground plane, thus, an electric wall is assumed for the bottom surface.

**Figure 10.** Rectangular DRA

The modes in an isolated rectangular dielectric resonator can be divided in two categories: TE and TM modes, but in case of the DRA mounted on a ground plane, only TE modes are typically excited. The fundamental mode is the TE111. As the three dimensions of the DRA are independent, the TE modes can be along the three directions: x, y and z. By referring to the Cartesian coordinate system presented Figure 10, if the dimensions of DR are such as a>b>d, the modes in the order of increasing resonant frequency are TEz111, TEy111 and TEx111. The analysis of all the modes is similar. The example of the TEz111 mode is discussed in [8], the field components inside the resonator and resonant frequencies are analytically presented.

As for the cylindrical case, CST MS can be used to see both E and H fields, as presented Figure 11.

**Figure 11.** E field (a) and H field (b) of the TE111 mode

The resonant frequencies definition is reminded hereafter:

$$f\_0 = \frac{c}{2\pi\sqrt{\varepsilon\_r\mu\_r}}\sqrt{k\_x^2 + k\_y^2 + k\_z^2} \tag{3}$$

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 37

 **r =100 r =90 r =80 r =70 r =60 r =50 r =40 r =30 r =20 r =10 r =2**

(6)

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

acceptable voltage standing wave ratio (VSWR).

DRAs.

used to estimate the fractional bandwidth of an antenna using:

Figure 13 is thus the result of a large number of simulations.

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

**Q factor**

**0.5 <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> <sup>0</sup>**

**a/h**

Q factor is increasing with εr and reaching a maximum for a/h=1.05. This Q factor can be

0 *<sup>f</sup> <sup>s</sup>* <sup>1</sup> *BW f Q s*

Where Δf is the absolute bandwidth, f0 is the resonant frequency and s the maximum

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

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

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

It is found by solving the following transcendental equation:

$$k\_z \tan(\frac{k\_z a}{2}) = \sqrt{(\varepsilon\_r - 1)k\_0^2 - k\_z^2} \tag{4}$$

where 0 <sup>0</sup> <sup>2222</sup> 0 0 <sup>2</sup> , , and *r r x y xyz <sup>f</sup> kkk kkkk abvc* 

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 material properties having significant influences on antenna performances.
