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 Figure 1) of the central material there is a cut-off frequency which below that it is possible to have a resonant condition. Above this frequency the cavity will exhibit a high impedance at the excitation port so that any electromagnetic mode practically disappears. Thereby, it is necessary to know the value of the highest dielectric permittivity constant which could be measured in the cavity for a given radius of the central rod.

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 [1] , non-linear waveguides [2] , Green’s functions for multilayered media [3] and electromagnetic wave interaction with nanodevices [4] . They have also been applied for the determination of TE and TM modes of circular cylindrical cavities using magnetic-type and electric-type Hertzian potentials res- pectively [5] . Figure 1 shows the case of a cylindrical dielectric resonator enclosed by a metal shield, where b is the radius of the outer cylinder, a is the inner resonator radius and d is the height (length of the structure). The configuration can be regarded as a cylindrical waveguide enclosing a central sample of radius a, and terminated by perfectly conducting planes. The general solution for the axial E field in TM modes was discussed in [6] , proposing a general solution for the axial E field for TM modes.

In [7] the solution of this system was obtained by solving directly Maxwell’s equations and substituting the corresponding zero axial component of magnetic or electric field for TE or TM mode respectively. A similar approach was presented in reference [8] . The aforementioned methods have the disadvantage that the cut-off wave number appears in the denominator of the total power flow equation implying a contradiction since it means that the power will vanish when the radius is very small. This contradiction is solved by including a factor in the numerator which depends on the cut off wave number, as proposed by [9] for a similar system.

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 [6] [7] [10] - [12] .

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 [5] . We then discuss the waveguide which includes the central cylindrical sample. Finally we discuss the usable range of this experimental arrangement in the determination of dielectric and magnetic properties.

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 [13] . For a homogeneous, isotropic, and charge-free region, E can be determined from the curl of an auxiliary vector potential defined as magnetic-type Hertzian potential. Thus,

Then from Maxwell’s equations,

where is an auxiliary scalar potential, and.

Now, we can use, and substituting (1), we obtain:

Using the following identity and combining (2) and (3), we obtain:

Since is an auxiliary scalar potential we can select, obtaining the following Helmholtz equation,

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 called electric-type Hertzian potential. As in the previous case, the same analysis applies for the TM modes where the electric-type Hertzian potential should be used [13] . The magnetic field can be expressed as,

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

where satisfies the Helmholtz wave equation,

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 in Equations (16) and (17), this condition should be verified after obtaining the parameters from the Equations (21) and (23). The previous argument is the final condition to probe that the numerical solution satisfies the resonance condition inside the cavity, otherwise, it represents a non-real solution and it is not possible to support a resonant mode inside the cavity.

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 Table 1 it is considered the case of two empty cavities designed for the TE₀₁₁ and TM₀₁₁ modes respectively with a resonant frequency of 5 GHz, being twice the radius a of the length d. See the Figure 1.

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 and correspond to different frequency values according to Equations (21) and (23). When this happens we face a non-physical solution. This implies that the existent mode inside the cavity could correspond to higher order modes. This is the usable range of a particular resonant mode in a specific cavity. In the limit, the resonant frequency begins to be complex which really implies that it is an attenuate mode. The higher order modes for all design are given in the Table 2.

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, and respectively. The respective values for TM modes, Cavity 2, are

, and, which are in general lower than the previous case of TE modes.

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 TE₀₁₁ 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. Figure 3 shows the new usable range using a dielectric rod of radius 2 cm for both Cavity 1 and Cavity 2.

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 [15] .

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 [19] and [20] . However the method shown in this article to determine the usable range of a given cavity can be used to avoid those regions of higher associated uncertainty, simply changing the diameter of the dielectric rod and hence the position in the curve of the expected permittivity under measurement. This result may be of the interest for metrology in dielectric permittivity.

In this section we explored the previous results by using a double-cylindrical dielectric resonator, diameter is and height resonating at 5 GHz in the mode TE₀₁₁ to characterize the permittivity constant of heavy oil with 11 API. The cavity was made of Aluminium C330 whose maximum dielectric conductivity is. To determine the effective permittivity constant we used the method of the critical points [22] with the help of a Vector Network Analyzer Anritsu, model 37269D, 40 MHz - 40 GHz. Posteriorly, the theory of mixing of concentric cylinders was used to determine the real permittivity constant of heavy oil. As the holder for petroleum sample we selected quartz due to its low dielectric losses and a very well known real permittivity constant of 3.850 [25] . The selection of its diameter was done taking into consideration the analysis presented in this article. The resulted resonant frequency was 4.881 GHz due to imperfections in constructions of the cavity.

Based on references [23] -[25] we expected a value to be lower than 4. It can be noticed in the Figure 5 that an adequate radius for this purpose is 3 cm. It is not recommended a large radius due to the simple fact that the

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.