**2.3. The cylindrical DRA**

30 Dielectric Material

height of the probe.

DRA

chapter.

issue.

*2.2.3. Aperture coupling* 

*2.2.2. Microstrip feeding line and coplanar waveguide* 

placing the line under the resonator as shown Figure 3.

Microstrip line

**Figure 3.** Microstrip feeding line and coplanar waveguide

presenting a rectangular DRA excited by a coplanar waveguide.

Substrate

When the excitation probe is inside the resonator, particular attention has to be paid to the air gap between the probe excitation and the dielectric material. Indeed, an air gap results in a lower effective dielectric constant, which entails both a decrease in the Q-factor and a shift of the resonance frequencies [23-24]. The probe location allows choosing the intended excited mode and the coupling of the mode can be optimized by adjusting both length and

The principle is similar to the probe excitation case. A microstrip line placed close to the DRA can couple the magnetic field of the DRA mode. However, this latter can affect the antenna polarisation and can thus increase the parasitic radiation. This could be reduced by

Another way is to replace the microstrip line by a coplanar waveguide, the Figure 3 is

In these both cases, the mode coupling can be optimized by changing the resonator position and/or its dielectric permittivity. For low dielectric permittivity materials (which allows obtaining a wide bandwidth), it is somewhat difficult to excite the mode. There are different solutions to obtain both miniaturization and good coupling, they will be explained in this

An important point is that these excitation methods are disturbing DRA modes by introducing electrical boundary conditions. This issue is all the more sensitive since the antenna is miniature. The last part of the chapter will show how to take advantage of this

A common method of exciting a DRA is acting through an aperture in the ground plane. The Figure 4 shows an example of the excitation of the TE111 mode of a rectangular DRA with a rectangular slot. To achieve relevant coupling, the aperture has to be placed in a DRA

DRA

Coplanarwaveguide

Metallization

It offers great design flexibility, where both resonant frequency and Q-factor are depending on the ratio of radius/height. Various modes can be excited within the DRA and a significant amount of literature is devoted to their field configurations, resonant frequencies and radiation properties [27-28]. This part will present a complete and concrete study of a cylindrical DRA.

Like most realistic cases, the cylindrical DRA presented Figure 5, is mounted on a finite ground plane. Because dielectric material properties will be studied in very great depth in the third part, the dielectric permittivity εr of the DRA is fixed and chosen equal to 30. It is also characterized by its height d and its radius a. To keep the chapter concise while remaining comprehensive, only relevant results and equations will be given.

**Figure 5.** Cylindrical DRA

#### 32 Dielectric Material

First of all, the study can begin with a modal analysis. This allows determining the most appropriate excitation method. Modes of cylindrical DRA can be divided into three types: TE, TM and hybrid modes, i.e. EH or HE [9-10,30]. The latter have a dependence on azimuth φ, while TE and TM modes have no dependence on azimuth. To identify fields' variations according φ (azimuth), r (radial) and z (axial) directions, subscripts respectively noticed n, p and m are following the mode notation. Because TE and TM modes have no azimuthal variation, n=0 for these modes. Finally, all cylindrical DRA modes can be defined such as: TE0pm+δ, TM0pm+δ, HEnpm+δ and EHnpm+δ. The δ value is ranging between 0 and 1, it approaches 1 for high εr. It should be noted that n, m and p are natural numbers. The modal analysis of a DRA can be deducted either with analytical calculations or thanks to electromagnetic simulators like CST Microwave Studio (CST MS).

#### *2.3.1. Modal analysis*

 Analytical equations derive from the analytical calculations of a cylindrical dielectric resonator by assuming perfect magnetic and/or electric walls on resonator faces. The perfect magnetic wall boundary condition was demonstrated to be valid for high ε<sup>r</sup> values [30], it remains accurate for lower values as well.

Fields equations inside the DRA, resonant frequencies and Q factor are detailed in [7,9] and [12]. Resonant frequencies of all modes are provided hereafter:

$$
\begin{pmatrix} f\_{\text{TMupun}} \\ f\_{\text{TEupun}} \end{pmatrix} = \frac{c}{2\pi a \sqrt{\varepsilon\_r \mu\_r}} \sqrt{\begin{pmatrix} X\_{np}' \\ X\_{np} \end{pmatrix}^2 + \left( \frac{(2m+1)\pi.a}{2d} \right)^2} \tag{1}
$$

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 33

**Figure 6.** E field (a) and H field (b) of the HE11δ mode

straight to the electromagnetic study of the DRA.

along the E field inside the DRA.

*2.3.2. Electromagnetic study* 

In light of this above, the best way to excite the HE11δ mode is to integrate a coaxial probe

Now that the modal analysis was explained and the excitation determined, we can go

Since input impedance and also S11 parameter cannot be calculated with the magnetic wall model, their study is solely possible with an electromagnetic simulator or of course can be experimentally done. Moreover, electromagnetic simulators facilitate accurate and efficient antenna analysis by providing the complete electric and magnetic fields inside and outside the antenna taking into account the finite ground plane. The impedance, the S11 parameter, the power radiated and the far field radiation pattern are determined everywhere and at any

frequency in a single analysis thanks to the Finite Integration Temporal method.

d

a

Using the CST MS software, the input impedance is presented Figure 8. It shows that the fundamental mode is excited at the resonant frequency fairly corresponding to the predicted value by the modal analysis of 504 MHz. The minor shift between resonant frequencies deducted with modal analysis and electromagnetic study is due to magnetic wall model

An important data for antenna designers is the S11 parameter (Figure 8). It is directly

The Figure 7 presents the cylindrical DRA excited by a coaxial probe.

Feedingprobe

used during the modal analysis, which is not absolutely accurate.

**Figure 7.** Excitation of the HE11δ mode with a coaxial probe

deducted from the input impedance parameter.

Where *Xnp* and *Xnp* are Bessel's solutions, <sup>3</sup> *nmp* , , , *a* and *d* are the radius and the height of the dielectric resonator.

In the case presented Figure 5, the fundamental excited mode is the HE11δ and its resonant frequency equals:

$$f\_{110} = \frac{3.10^8}{2\pi\sqrt{30}} \sqrt{\left(\frac{X\_{11}'}{0.04}\right)^2 + \left(\frac{\pi}{2 \times 0.045}\right)^2} = 503.6 MHz \tag{2}$$

This method requires the resolution of both Maxwell and propagation equations. It was therefore reserved for simple-shaped DRA.

 The "Eigenmode solver" of CST MS allows viewing 3D fields of each mode and having their resonant frequencies. When considering the studied example and defining perfect magnetic boundary conditions on all DRA walls, except for the DRA bottom where perfect electric boundary condition is considered (due to the ground plane), the software gives 504 MHz for the resonant frequency value of the HE11δ mode. It is also possible to see both H and E fields of this mode, they are presented Figure 6. This valuable information allows designer to choose the most appropriate excitation of the considered mode.

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 33

**Figure 6.** E field (a) and H field (b) of the HE11δ mode

In light of this above, the best way to excite the HE11δ mode is to integrate a coaxial probe along the E field inside the DRA.

Now that the modal analysis was explained and the excitation determined, we can go straight to the electromagnetic study of the DRA.

### *2.3.2. Electromagnetic study*

32 Dielectric Material

*2.3.1. Modal analysis* 

Where *Xnp* and *Xnp*

frequency equals:

of the dielectric resonator.

considered mode.

110

therefore reserved for simple-shaped DRA.

simulators like CST Microwave Studio (CST MS).

values [30], it remains accurate for lower values as well.

[12]. Resonant frequencies of all modes are provided hereafter:

*TMnpm np TEnpm r r np*

 

First of all, the study can begin with a modal analysis. This allows determining the most appropriate excitation method. Modes of cylindrical DRA can be divided into three types: TE, TM and hybrid modes, i.e. EH or HE [9-10,30]. The latter have a dependence on azimuth φ, while TE and TM modes have no dependence on azimuth. To identify fields' variations according φ (azimuth), r (radial) and z (axial) directions, subscripts respectively noticed n, p and m are following the mode notation. Because TE and TM modes have no azimuthal variation, n=0 for these modes. Finally, all cylindrical DRA modes can be defined such as: TE0pm+δ, TM0pm+δ, HEnpm+δ and EHnpm+δ. The δ value is ranging between 0 and 1, it approaches 1 for high εr. It should be noted that n, m and p are natural numbers. The modal analysis of a DRA can be deducted either with analytical calculations or thanks to electromagnetic

 Analytical equations derive from the analytical calculations of a cylindrical dielectric resonator by assuming perfect magnetic and/or electric walls on resonator faces. The perfect magnetic wall boundary condition was demonstrated to be valid for high ε<sup>r</sup>

Fields equations inside the DRA, resonant frequencies and Q factor are detailed in [7,9] and

*f X c m a f X a d*

 

2 2

In the case presented Figure 5, the fundamental excited mode is the HE11δ and its resonant

2 2 8

3.10 503.6

*<sup>X</sup> <sup>f</sup> MHz*

This method requires the resolution of both Maxwell and propagation equations. It was

 The "Eigenmode solver" of CST MS allows viewing 3D fields of each mode and having their resonant frequencies. When considering the studied example and defining perfect magnetic boundary conditions on all DRA walls, except for the DRA bottom where perfect electric boundary condition is considered (due to the ground plane), the software gives 504 MHz for the resonant frequency value of the HE11δ mode. It is also possible to see both H and E fields of this mode, they are presented Figure 6. This valuable information allows designer to choose the most appropriate excitation of the

11

2 30 0.04 2 0.045

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

(2)

are Bessel's solutions, <sup>3</sup> *nmp* , , , *a* and *d* are the radius and the height

(1)

Since input impedance and also S11 parameter cannot be calculated with the magnetic wall model, their study is solely possible with an electromagnetic simulator or of course can be experimentally done. Moreover, electromagnetic simulators facilitate accurate and efficient antenna analysis by providing the complete electric and magnetic fields inside and outside the antenna taking into account the finite ground plane. The impedance, the S11 parameter, the power radiated and the far field radiation pattern are determined everywhere and at any frequency in a single analysis thanks to the Finite Integration Temporal method.

The Figure 7 presents the cylindrical DRA excited by a coaxial probe.

**Figure 7.** Excitation of the HE11δ mode with a coaxial probe

Using the CST MS software, the input impedance is presented Figure 8. It shows that the fundamental mode is excited at the resonant frequency fairly corresponding to the predicted value by the modal analysis of 504 MHz. The minor shift between resonant frequencies deducted with modal analysis and electromagnetic study is due to magnetic wall model used during the modal analysis, which is not absolutely accurate.

An important data for antenna designers is the S11 parameter (Figure 8). It is directly deducted from the input impedance parameter.

Dielectric Materials for Compact Dielectric Resonator Antenna Applications 35

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

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,

<sup>a</sup> <sup>b</sup>

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

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

(a) (b)

components inside the resonator and resonant frequencies are analytically presented.

d

**2.4. The rectangular DRA** 

different lengths ratios.

**Figure 10.** Rectangular DRA

Figure 11.

an electric wall is assumed for the bottom surface.

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

The resonant frequencies definition is reminded hereafter:

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

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

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 radiation pattern of the cylindrical DRA using the simulator.

**Figure 9.** 3D radiation pattern

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 antenna.

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> mode radiates like a short magnetic monopole.
