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Dielectric resonator methods constitute one of the most useful techniques for the measurement of electromagnetic material properties in the microwave frequency range. Several geometric configurations are used for this purpose and, in the present paper, we consider the case of a dielectric rod enclosed in a cylindrical metallic enclosure. To carry out dielectric measurements in this system it is necessary to know the highest permittivity constant value for which the resonance condition still can be attained into the cavity. Using an approach based on magnetic and electric Hertzian potentials we have derived the set of TE and TM modes for the relevant geometry described and, then we have calculated the valid dielectric permittivity constant range of measurements for low-loss materials in a cylindrical cavity using a simple resonance frequency condition. Finally, we present a simple application of this method in order to determine the dielectric permittivity constant of heavy oil with 11 API.

When dielectric measurements are performed in a shielded dielectric resonator, the shift of the resonant frequency caused by the presence of the sample (with a given dimension and dielectric permittivity constant), must be taken into account. The electromagnetic field must satisfy the boundary conditions required at the metal shielding (usually assumed as a perfect electric conductor) and at the material sample, thus obtaining a standing wave inside the cavity. The resonant frequency of this configuration decreases with respect to the resonant frequency of the empty cavity because the electric field increases due to the presence of a low loss dielectric material, so that forcing the wavelength to be smaller in order to achieve the border condition at the metal enclosure. For any specific combination of the diameter and the dielectric permittivity constant (see

In this paper we employ the formalism of the Hertzian potentials to find the electromagnetic solution of this configuration. Then we determine the limit of the real dielectric permittivity constant by numerically solving the nonlinear electromagnetic equations, thus obtaining the wave number in both materials inside the cavity (dielectric rod and air) and independently calculating the frequency inside those volumes.

Hertzian potentials have been used to solve different electromagnetic problems: in the study of the properties of aperture array systems [

In [

The electromagnetic field in the material sample (region 1) and in air (region 2) are determined by using the relevant Hertzian potentials in their general mathematical form. The proper boundary conditions are then applied to the derived equations which can then be solved numerically. In practice, the resonant frequency and the system quality factor are measured then to determine the electromagnetic properties of materials [

In this paper we first present a brief introduction to Hertz’s potentials, and then we apply this theory to the classical case of a cylindrical waveguide [

In a charge free region with constant permittivity, the divergence of E is zero, thus implying that this field can

be expressed as the curl of an auxiliary vector potential. This is the base for Hertz potential formulation [

Then from Maxwell’s equations,

where

Now, we can use

Using the following identity

Since

A second Helmholtz equation can be obtained from the fact that the divergence of H is always zero. This fact can be expressed in terms of the curl of another vector potential

Using Maxwell’s equations and combining with (6), we obtain,

where

In the case of TE modes, if we observe the magnetic-type Hertzian potential, the only way to generate a zero component in longitudinal axis of E field is that the Hertzian potential has zero components in radial and azimuthal direction. It means that magnetic-type Hertzian potential can be expressed like:

In the case of resonators the perfect conductor boundary conditions at

Due to this fact, when the cavity geometry coincides with the coordinate system, we can use the separation variable method in order to achieve the boundary conditions [

Substituting (11) in (5), we obtain the Bessel differential equation

A possible P dependence with

The differential equation shown above is known as the Bessel Differential Equation, whose general solution is,

where,

The Helmholtz wave Equation (5) is valid for both regions of the shielded resonator, with the requirement that in the enclosed section

It is important to note that

The boundary conditions of the tangential electric field at

Solving for B and C in (18), substituting it in (19) and (20), and then dividing side by side these two ex- pressions, we can obtain the following relation,

where

By making

Equations (21) and (23) give the relation between the electromagnetic parameters of the unknown medium and the resonant frequency. The cut off wave number can be determined if the dielectric permittivity constant is known or viceversa. This system is solved numerically using the Matlab function fsolve. The solution given in Equation (21) reduces to the simple form given in [

The treatment is similar for TM modes but using electric-type Hertzian potential as was presented on page 3. The general solution for this potential is in principle the same as in (15). E and H fields can be obtain using (6) and (7) and after applying the electromagnetic edge conditions, it gives,

where,

It is also worth pointing out that for this given circular symmetry of the set of E and H fields, which is easily appreciable in the corresponding Hertzian potentials (15), we can expect pure TE and TM as stated in [

Previously, we have mentioned that the frequency in both regions (air and dielectric) must be equal in order to represent a real resonant condition. However since we set

We now exemplify our statement with an example. Consider a resonant frequency calculation for TE and TM modes in a circular cylindrical cavity resonator with a central dielectric rod. In the _{011} and TM_{011} modes respectively with a resonant frequency of 5 GHz, being twice the radius a of the length d. See the

For a given dielectric permittivity constant/radius of the central cylinder, the resonant frequency is calculated as solution of the corresponding TE or TM equations set. Thus, for a selected working mode order n and a fixed radius of the dielectric rod, the resonant frequency decreases when we increase the dielectric permittivity con- stant. Then we reach a value of permittivity where the wave numbers

In the following we have determined the usable range for the Cavity 1 and Cavity 2. This is shown in Figures 2-4 for the set of modes 011, 021 and 031 respectively, where we have determined the dependence of the resonance frequency with respect to the dielectric permittivity constante of the central sample under mea- surement for a fixed radius. In the case of TE modes, Cavity 1, the cutoff relative dielectric permittivity constant are

Cavity | Frequency (MHz) | Radius (cm) | Length (cm) |
---|---|---|---|

Cavity 1 | 5000 | 7.023 | 3.512 |

Cavity 2 | 5000 | 6.420 | 3.210 |

Mode | Cavity 1 Frequency (MHz) | Cavity 2 Frequency (MHz) |
---|---|---|

TE/TM_{011} | 5000.0 | 5000.0 |

TE/TM_{021} | 6398.7 | 6215.7 |

TE/TM_{031} | 8124.0 | 7947.9 |

Notice in general how the dielectric range increases for higher resonant modes, however compromising the measurement of the tangent loss due to lower quality factors because cavity losses.

For instance, observe that it is not possible to use the TE_{011} mode in the cavity 1 to characterize materials with permittivity values higher than 1.6 using a dielectric rod of 3 cm. However this situation can be solved if the dielectric rod diameter is reduced and, as consequence, the permittivity constant range increases.

This method is valid for each configuration cavity-resonant mode-dielectric rod, however due to the non- linearity of Equations (21) and (23), or equivalently (24) and (25) for TM modes, it results difficult to predict the percentage of increment in the valid usable range for dielectric measurements due to a reductions in the dielectric rod’s diameter.

Here it is easily observable how the usable range is increased, allowing to carry out measurements of higher permittivity values. This method is in general valid for any mode and resonant frequency.

Some authors do not use TM modes because any air gap between the top surface of the central cylindrical sample and the top surface of the external cavity can distort deeply the resonant frequency [

One important aspect worth mentioning is that there are some flattened sections, portion with horizontal trends, of the curves shown in the Figures 2-4. Those portions represent regions of less sensitivity of the permittivity regarding the frequency measurement. This region is not a recommendable region to measure properties due to higher expected uncertainty associated to the measurement according results given in [

In this section we explored the previous results by using a double-cylindrical dielectric resonator, diameter is _{011} to characterize the permittivity constant of heavy oil with 11 API. The cavity was made of Aluminium C330 whose maximum dielectric conductivity is

Based on references [

resonant frequency of the cavity is lowered implying an electromagnetic characterization at another frequency than the required.

The real effective dielectric determined using the method of the critical points is 2.688 and the corresponding for heavy oil is 2.356 in total agreement with the tend of the previous results known for lower frequencies.

In this paper we presented a general methodology which may be useful to determine the usable dielectric range for material characterization. Due to the presence of the central cylinder, a displacement of the resonant fre- quency occurs proportional to the permittivity constant of the material until the resonant pattern of the mode under study can no longer exist. It means that for a given cavity and fixed rod’s radius to be located at the center of the cavity there exists a maximum value of measurable permittivity constant. No resonant conditions can be satisfied for higher permittivity values. The best alternative in order to increase the range of permittivity constant that can be measured reducing the radius of the central dielectric rod or choosing a higher resonant mode with the disadvantage that cavity loses can affect the results by changing the resonant frequency and the quality factor. We have shown that the behavior of the resonant frequency versus the dielectric constant should be taken into account when selecting the best appropriate region with high sensitivity in order to ensure the repeatability of measurement and to reduce the uncertainty associated to the measurement. Finally we have shown with a simple example the electromagnetic characterization of heavy oil with 11 API selecting a proper radious for a petroleum holder made of quartz of purity 99%.

This work was supported by FONACIT under the project No. 2011001317. We wish to thank to Roque Rodríguez for his contribution in this work.