3. Open resonator with a dielectric rod

#### 3.1. Resonator model

The method of the solution is based on the same physical principles as in the case of the resonator with the dielectric bead. Thus, we consider a resonance cavity with the boundary made of a spherical, a conical and a cylindrical perfectly conducting surface. There is a cylindrical rod extended all along the resonator axis (area 2 in Figure 7), which is assumed to be a homogeneous and isotropic medium, having the material parameters ε2, μ2. The rest part of the resonator is assumed to be empty ε<sup>1</sup> ¼ μ<sup>1</sup> ¼ 1 (area 1).

We confine ourselves to the analysis of axially-symmetric oscillations of TE-mode, which in cylindrical coordinates, where axis z coincides with the resonator symmetry axis, have Eφ-, Hrand Hz- components of the electromagnetic field. For this oscillation mode, the initial problem for the Maxwell equations can be reduced to finding wave numbers k, for which there exist the nontrivial solutions U<sup>1</sup> and U<sup>2</sup> of the two-dimensional Helmholtz equations.

Figure 7. Geometric model of OR with the dielectric rod.

$$\Delta\_{rz}l I\_1 + \left(k^2 \varepsilon\_1 \mu\_1 - \left(1/r^2\right)\right) l I\_1 = 0, \ 0 < z < l, \ \ a < r < b(z), \tag{16}$$

$$\Delta\_{rz}\mathcal{U}\_2 + \left(k^2 \varepsilon\_2 \mu\_2 - \left(1/r^2\right)\right)\mathcal{U}\_2 = 0, \ 0 < z < l, \ 0 < r < a,\tag{17}$$

that satisfy the boundary conditions:

$$\mathcal{U}I\_{1,2}(r,0) = 0, \quad \mathcal{U}\_{1,2}(r,l) = 0, \quad \mathcal{U}\_1(b(z),z) = 0, \quad \mathcal{U}\_2(0,z) = 0,\tag{18}$$

and the field matching conditions at r ¼ a:

$$\mathcal{U}\_1(a, z) = \mathcal{U}\_2(a, z), \quad \frac{1}{\mu\_1} \frac{\partial \mathcal{U}\_1(a, z)}{\partial z} = \frac{1}{\mu\_2} \frac{\partial \mathcal{U}\_2(a, z)}{\partial z}. \tag{19}$$

permittivity placed in the resonator weakly influences the distribution of the standing wave electric field components. The structure of the field at the placing of the rod, having diameter 2а ¼ 1 mm, into the resonator is shown in Figure 8a. Actually, it is identical to the structure of the field in the resonator at the rod diameter 2а ¼ 2 mm (Figure 8b). The difference of the ТЕ<sup>0115</sup> mode resonance frequencies at the increase of the rod diameter from 1 to 2 mm is insignificant. It decreases from f ¼ 67.07 to f ¼ 66.97 GHz. That says about the low correlation of the resonant oscillation with the dielectric rod and could be explained by the fact that the sample is located in the area with low electric field intensity. An increase of the sample permittivity can change the situation. In Figure 8c, the lines of equal amplitudes of E<sup>φ</sup> electromagnetic field component for the same mode ТЕ<sup>0115</sup> in the resonator with the same

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

http://dx.doi.org/10.5772/intechopen.73643

43

<sup>2</sup> = 3.824) at the diameter 2а ¼ 2 mm, are

dimensions for the rod, made of fused quartz (ε<sup>0</sup>

Figure 8. Field distribution in a cavity with a dielectric rod.

Here, а is the rod radius.

The numerical algorithm for the problem (Eq. (16))÷(Eq. (19)) utilizes the method of Bubnov-Galerkin, as described earlier, and the problem is reduced to the system of linear algebraic equalizations (Eq. (5)) [6, 21].

The developed algorithm, as in the case with the dielectric bead, was tested by the passing to the limit from the geometry of the considered resonator to the cavities of spherical and cylindrical shape. The correctness of the described approach is validated by the papers [9, 12, 22]. Moreover, the algorithm convergence rate was estimated numerically for the growing dimensional representation of the algebraic problem (Eq. (5)).

#### 3.2. Numerical and experimental results

The computations have been carried out for a resonator having the same dimensions, as in the case of the resonator with the bead. In Figure 8, the lines of equal amplitudes E<sup>φ</sup> - components of the electromagnetic field of the mode ТЕ<sup>0115</sup> in the resonator with rods, made of Teflon and having permittivity ε<sup>0</sup> <sup>2</sup> = 2.07, are presented. Apparently, the rod with relatively low

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies http://dx.doi.org/10.5772/intechopen.73643 43

Figure 8. Field distribution in a cavity with a dielectric rod.

<sup>Δ</sup>rzU<sup>1</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

Figure 7. Geometric model of OR with the dielectric rod.

ΔrzU<sup>2</sup> þ k

that satisfy the boundary conditions:

Here, а is the rod radius.

having permittivity ε<sup>0</sup>

equalizations (Eq. (5)) [6, 21].

and the field matching conditions at r ¼ a:

3.2. Numerical and experimental results

ε1μ<sup>1</sup> � 1=r

42 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

<sup>U</sup>1ð Þ¼ <sup>a</sup>; <sup>z</sup> <sup>U</sup>2ð Þ <sup>a</sup>; <sup>z</sup> , <sup>1</sup>

dimensional representation of the algebraic problem (Eq. (5)).

μ1

The numerical algorithm for the problem (Eq. (16))÷(Eq. (19)) utilizes the method of Bubnov-Galerkin, as described earlier, and the problem is reduced to the system of linear algebraic

The developed algorithm, as in the case with the dielectric bead, was tested by the passing to the limit from the geometry of the considered resonator to the cavities of spherical and cylindrical shape. The correctness of the described approach is validated by the papers [9, 12, 22]. Moreover, the algorithm convergence rate was estimated numerically for the growing

The computations have been carried out for a resonator having the same dimensions, as in the case of the resonator with the bead. In Figure 8, the lines of equal amplitudes E<sup>φ</sup> - components of the electromagnetic field of the mode ТЕ<sup>0115</sup> in the resonator with rods, made of Teflon and

∂U1ð Þ a; z <sup>∂</sup><sup>z</sup> <sup>¼</sup> <sup>1</sup>

ε2μ<sup>2</sup> � 1=r

2

<sup>2</sup> <sup>U</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; l, a <sup>&</sup>lt; <sup>r</sup> <sup>&</sup>lt; b zð Þ, (16)

<sup>2</sup> <sup>U</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; l, <sup>0</sup> <sup>&</sup>lt; <sup>r</sup> <sup>&</sup>lt; a, (17)

μ2

<sup>2</sup> = 2.07, are presented. Apparently, the rod with relatively low

∂U2ð Þ a; z

<sup>∂</sup><sup>z</sup> : (19)

U1,2ð Þ¼ r; 0 0, U1, <sup>2</sup>ð Þ¼ r; l 0, U1ð Þ¼ b zð Þ; z 0, U2ð Þ¼ 0; z 0, (18)

permittivity placed in the resonator weakly influences the distribution of the standing wave electric field components. The structure of the field at the placing of the rod, having diameter 2а ¼ 1 mm, into the resonator is shown in Figure 8a. Actually, it is identical to the structure of the field in the resonator at the rod diameter 2а ¼ 2 mm (Figure 8b). The difference of the ТЕ<sup>0115</sup> mode resonance frequencies at the increase of the rod diameter from 1 to 2 mm is insignificant. It decreases from f ¼ 67.07 to f ¼ 66.97 GHz. That says about the low correlation of the resonant oscillation with the dielectric rod and could be explained by the fact that the sample is located in the area with low electric field intensity. An increase of the sample permittivity can change the situation. In Figure 8c, the lines of equal amplitudes of E<sup>φ</sup> electromagnetic field component for the same mode ТЕ<sup>0115</sup> in the resonator with the same dimensions for the rod, made of fused quartz (ε<sup>0</sup> <sup>2</sup> = 3.824) at the diameter 2а ¼ 2 mm, are presented. It could be seen that, in this case, the field is located in the rod, and resonance frequency decreases down to f ¼ 57.36 GHz. Particularly, it concerns the cylindrical part of the considered resonator. For the resonator without a rod, as shown earlier, frequency f ¼ 71.382 GHz.

considering the cylindrical resonator with a rod was solved using the method of variable

Resonant Systems for Measurement of Electromagnetic Properties of Substances at V-Band Frequencies

The figure demonstrates an obvious similarity of the curves for the both resonator types. It proves the fact that the nature of the physical processes occurring in the resonators is similar, both in the resonator under consideration and in the cylindrical cavity. It is an indirect evidence of the adequacy of our theoretical considerations. One more point to emphasize is that small changes of samples' permittivity, which are critical for oil quality control and food stuff, should be measured at the areas with a stronger disperse dependence, which is solely up to the

The block diagram of the experimental unit used in the research is given in Figure 4. Only, in this case, the cylindrical sample was used. A sample was inserted into the cavity through a hole in the middle of the waveguide plunger, with a guide for the precise alignment of the

The measured shift of the resonance frequencies and the calibration curves in Figure 9 were used to determine the dielectric permittivities of two cylindrical samples made of fused quartz and silicate glass. The value of resonance frequencies obtained in the experimental measurements for ТЕ0116- and ТЕ0115-modes for the case when the cylindrical samples of 2 mm and 1.5 mm in diameter were located along the resonator axis are marked with squares at the calculated curve. The results of measuring the dielectric permittivity of cylindrical samples of

<sup>2</sup> Reference value of ε<sup>0</sup>

http://dx.doi.org/10.5772/intechopen.73643

45

<sup>2</sup> Difference (%)

4. Open resonator with a segment of rectangular waveguide

Fused quartz 2 ТЕ<sup>0116</sup> 3.67 (а) 3.8 [7] 3.4 Fused quartz 2 ТЕ<sup>0115</sup> 3.72 (c) 3.8 [7] 2.1 Soda-lime glass 1.5 ТЕ<sup>0116</sup> 6.61 (b) 6.7 [23] 1.3 Soda-lime glass 1.5 ТЕ<sup>0115</sup> 6.56 (d) 6.7 [23] 2.0

control the quality of food stuff in millimeter and in sub-millimeter ranges.

A new electrodynamic system appears when inserting the segment of the short-circuited rectangular waveguide in the center of one of the ОR mirrors [24]. Cross-section sizes of the waveguide a � b are chosen by the condition of the peak efficiency of the ТЕ<sup>10</sup> mode excitation by the fundamental mode ТЕМ00q. One can consider such ОR as a resonant cell for measurement of composite materials and biological liquids electromagnetic specifications as well as to

<sup>2</sup> of the dielectric rods from Teflon and Plexiglas.

We consider the hemispherical ОR with a rectangular waveguide located in the center of the flat mirror. Reflection from the waveguide horn is neglected. We consider the resonator

separation, that is, its resonance frequency was described by rigorous formulas.

diameter of the cylindrical sample (Figure 9).

sample along the resonator axis.

various diameters are listed in Table 3.

Table 3. The measured values of the ε<sup>0</sup>

Material Sample diameter (mm) Mode Measured value of ε<sup>0</sup>

The carried out research (Figure 8c) demonstrated the strong relation of the sample with the electromagnetic field and high filling coefficient of the resonator on the electromagnetic field [7]. In such a manner, for qualitative control of liquid samples, the composition of which includes water, it is necessary to use pipes, made of the material having low permittivity, as compared with the substance under studies.

On the basis of the analysis carried out, one can say that sensitivity of the resonant cell is defined by the diameter of the cylindrical sample and by the value of its permittivity. From Figure 8, one can see that electric field intensity near the conical metal surface (dotted lines) is low. It indicates that the considered cavity is equivalent to the ОR, in which high- Q oscillations, having low diffraction losses, are excited.

The experimental measurements of the permittivity of materials can be performed with the aid of calibration curves, that is, the dependencies of the resonator frequency shift on the permittivity of the cylindrical samples of various diameters, introduced into the resonance cavity. These characteristics are shown in Figure 9.

The upper part of the figure presents the series of curves plotted for the ТЕ<sup>0116</sup> mode by the rod diameter of 2 mm (curve 1) and 1.5 mm (curve 2). The bottom part presents the ТЕ<sup>0115</sup> mode by the same diameters of the samples, that is, 2 mm (curve 3) and 1.5 mm (curve 4). The dotted lines in the same figure, being almost parallel to those described earlier, illustrate similar dependencies for ТЕ<sup>0116</sup> and ТЕ<sup>0115</sup> modes in a cylindrical resonance cavity, with the cylindrical test pieces of the said size arranged along the resonator axis.

The length and the diameter of the cylindrical resonator were chosen to be equal to the length of the considered resonator and to the diameter of its cylindrical part (Figure 7). The problem

Figure 9. Dependencies of the resonance frequency on the permittivity of cylindrical bodies of various diameter for ТЕ<sup>0116</sup> and ТЕ<sup>0115</sup> modes.

considering the cylindrical resonator with a rod was solved using the method of variable separation, that is, its resonance frequency was described by rigorous formulas.

presented. It could be seen that, in this case, the field is located in the rod, and resonance frequency decreases down to f ¼ 57.36 GHz. Particularly, it concerns the cylindrical part of the considered resonator. For the resonator without a rod, as shown earlier, frequency f ¼

44 Emerging Microwave Technologies in Industrial, Agricultural, Medical and Food Processing

The carried out research (Figure 8c) demonstrated the strong relation of the sample with the electromagnetic field and high filling coefficient of the resonator on the electromagnetic field [7]. In such a manner, for qualitative control of liquid samples, the composition of which includes water, it is necessary to use pipes, made of the material having low permittivity, as

On the basis of the analysis carried out, one can say that sensitivity of the resonant cell is defined by the diameter of the cylindrical sample and by the value of its permittivity. From Figure 8, one can see that electric field intensity near the conical metal surface (dotted lines) is low. It indicates that the considered cavity is equivalent to the ОR, in which high- Q oscilla-

The experimental measurements of the permittivity of materials can be performed with the aid of calibration curves, that is, the dependencies of the resonator frequency shift on the permittivity of the cylindrical samples of various diameters, introduced into the resonance cavity.

The upper part of the figure presents the series of curves plotted for the ТЕ<sup>0116</sup> mode by the rod diameter of 2 mm (curve 1) and 1.5 mm (curve 2). The bottom part presents the ТЕ<sup>0115</sup> mode by the same diameters of the samples, that is, 2 mm (curve 3) and 1.5 mm (curve 4). The dotted lines in the same figure, being almost parallel to those described earlier, illustrate similar dependencies for ТЕ<sup>0116</sup> and ТЕ<sup>0115</sup> modes in a cylindrical resonance cavity, with the cylindri-

The length and the diameter of the cylindrical resonator were chosen to be equal to the length of the considered resonator and to the diameter of its cylindrical part (Figure 7). The problem

Figure 9. Dependencies of the resonance frequency on the permittivity of cylindrical bodies of various diameter for ТЕ<sup>0116</sup>

71.382 GHz.

and ТЕ<sup>0115</sup> modes.

compared with the substance under studies.

tions, having low diffraction losses, are excited.

These characteristics are shown in Figure 9.

cal test pieces of the said size arranged along the resonator axis.

The figure demonstrates an obvious similarity of the curves for the both resonator types. It proves the fact that the nature of the physical processes occurring in the resonators is similar, both in the resonator under consideration and in the cylindrical cavity. It is an indirect evidence of the adequacy of our theoretical considerations. One more point to emphasize is that small changes of samples' permittivity, which are critical for oil quality control and food stuff, should be measured at the areas with a stronger disperse dependence, which is solely up to the diameter of the cylindrical sample (Figure 9).

The block diagram of the experimental unit used in the research is given in Figure 4. Only, in this case, the cylindrical sample was used. A sample was inserted into the cavity through a hole in the middle of the waveguide plunger, with a guide for the precise alignment of the sample along the resonator axis.

The measured shift of the resonance frequencies and the calibration curves in Figure 9 were used to determine the dielectric permittivities of two cylindrical samples made of fused quartz and silicate glass. The value of resonance frequencies obtained in the experimental measurements for ТЕ0116- and ТЕ0115-modes for the case when the cylindrical samples of 2 mm and 1.5 mm in diameter were located along the resonator axis are marked with squares at the calculated curve. The results of measuring the dielectric permittivity of cylindrical samples of various diameters are listed in Table 3.


Table 3. The measured values of the ε<sup>0</sup> <sup>2</sup> of the dielectric rods from Teflon and Plexiglas.
