An application of a mode dielectric resonator is described for precise measurements of complex permittivity and the thermal effects on permittivity for isotropic dielectric materials. The Rayleigh-Ritz technique was employed to find a rigorous relationship between permittivity, resonant frequency, and the dimensions of the resonant structure, with relative computational accuracy of less than . The influence of conductor loss and its temperature dependence was taken into account in the dielectric loss tangent evaluation. Complex permittivities of several materials, including cross-linked polystyrene, polytetrafluoroethylene, and alumina, were measured in the temperature range of 300-400 K. Absolute uncertainties of relative permittivity measurements were estimated to be smaller than 0.2%, limited mainly by uncertainty in the sample dimensions. For properly chosen sample dimensions, materials with dielectric loss tangents in the range of to can be measured using the mode dielectric resonator.

https://iopscience.iop.org/article/10.1088/0957-0233/9/10/015/pdf

A dielectric resonator for measurements of complex permittivity of low loss dielectric materials as a function of temperature

This review covers approximately 15 years of development in the techniques used to measure dielectric properties of materials over the frequency range 1 MHz to 1500 GHz. An introductory section summarizes the broad development trends and is followed by short sections which deal with developments at a more detailed level. The approaches described include time- and frequency-domain methods; reflection, transmission, and resonant methods, guided and free-space methods; discrete-frequency and broad-band methods, especially Fourier Transform Spectroscopy. Measurements on magnetic materials are also briefly discussed.

The measurement of the properties of materials

Overview of frequency domain measurement techniques of the complex permittivity at microwave frequencies is presented. The methods are divided into two categories: resonant and non-resonant ones. In the first category several methods are discussed such as cavity resonator techniques, dielectric resonator techniques, open resonator techniques and resonators for non-destructive testing. The general theory of measurements of different materials in resonant structures is presented showing mathematical background, sources of uncertainties and theoretical and experimental limits. Methods of measurement of anisotropic materials are presented. In the second category, transmission–reflection techniques are overviewed including transmission line cells as well as free-space techniques.

https://iopscience.iop.org/article/10.1088/0957-0233/17/6/R01/pdf

Frequency domain complex permittivity measurements at microwave frequencies

Whispering-gallery modes are used for very accurate permittivity, dielectric loss, and temperature coefficient of permittivity measurements for both isotropic and uniaxially anisotropic dielectric materials. The relationship between resonant frequencies, dimensions of the resonant structure, and permittivity of the sample under test is calculated with a radial mode-matching technique. The relative accuracy of these computations is better then 10/sup -4/. The influence of conductor losses on dielectric loss tangent determination is treated for both whispering-gallery-mode and TE/sub 01/spl delta//-mode dielectric-resonator techniques. Two permittivity tensor components of sapphire and their temperature dependence were measured from 4.2 to 300 K. The total uncertainty in permittivity when use is made of whispering-gallery modes was estimated to be less than 0.05%. The uncertainty was limited principally by uncertainty in sample dimensions. Experimental and calculated resonant frequencies of several whispering-gallery modes differed by no more than 0.01%. The dielectric loss tangent of sapphire parallel and perpendicular to its anisotropy axis was calculated to be less than 10/sup -9/ at 4.2 K. The permittivity and dielectric loss tangent of a commercially available low-loss high-permittivity ceramic material has also been measured at S- and C-band frequencies using a large number of whispering-gallery modes.

Use of whispering-gallery modes for complex permittivity determinations of ultra-low-loss dielectric materials

It is no longer necessary to use extrapolated microwave dielectric data when designing millimeter-wave components, devices, and systems. Precision measurements can now be made to generate highly accurate millimeter-wave (5 to 1/2 mm) continuous spectra on complex refractive index, complex dielectric permittivity, and loss tangent for a variety of materials such as common ceramics, semiconductors, crystalline, and glassy materials. The continuous spectra reveal an increase in dielectric loss with increase in frequency in this wavelength range for most materials. Reliable measurements also reveal that the method of preparation of nominally identical specimens can change the dielectric losses by many factors. These broad-band measurements were carried out employing dispersive Fourier transform spectroscopy applied to a modular two-beam polarization interferometer. Data obtained with Fabry-Perot open resonator methods at wavelengths of 5 mm and longer will also be compared.

Dielectric Measurements of Millimeter-Wave Materials

A method for the measurement of permittivity at microwave frequencies using an open resonator is proposed. The theory of the method is developed, and an experimental procedure is evolved and justified. It is shown that the present method has advantages over other open resonator methods which have been proposed, and a thorough experimental study of the method has shown that experimental errors can be made very small indeed. Since the very beginnings of modern microwave techniques in the 1940s, many alternative methods of measuring the complex permittivity of gases, liquids and solids have been devised. Most of these have employed a cavity resonator or a length of waveguide as a means of defining the configuration of the electromagnetic field with the precision needed for accurate quantitative analysis of the experimental results. Although very satisfactory results have been obtained using such methods, they become increasingly difficult to apply as the wavelength decreases. For the solid dielectric materials with which the present paper is exclusively concerned, a major problem is the accurate machining of a specimen of the material to fit closely into the resonator or waveguide. Even very small air gaps between the dielectric material and the metal wall can cause large errors, and this problem is obviously more serious at shorter wavelengths since the fractional error corresponding to a given absolute error in the dimensions of the specimen is inversely proportional to the wavelength. Another difficulty is encountered at very high frequencies in the resonant cavity method of measurement of the loss tangent of a dielectric material when this has a small value. The accuracy of measurement begins to decrease when the loss tangent falls below the reciprocal of the Q factor of the empty resonant cavity. Since this Q factor varies as fo , where f0 is the resonant frequency it is clear that the method is unsatisfactory for low-loss materials at high frequencies. The pioneer work of Culshaw & Anderson (I962) on the measurement of permittivity and dielectric loss with a millimetre wave Fabry-Perot interferometer was the first successful attempt to use an open resonator for this purpose. However, the use of parallel-plane mirrors rather than spherical mirrors gives rise to a field distribution within the resonator which is not readily amenable to mathematical analysis. Moreover, the diffraction losses are much larger than is the case when spherical mirrors are used in a suitable configuration.

The accurate measurement of permittivity by means of an open resonator

The objective of this paper is to provide a reliable theoretical foundation for the open-resonator method of measuring permittivity and loss-tangent of dielectric materials. From an exact solution of Maxwell's equations obtained by means of the complex-source-point method, simplified formulae for the six Cartesian components of the electromagnetic field are obtained. These are then used in a variational formula to obtain an accurate formula for the resonant frequency of an open resonator with spherical mirrors (of equal radius of curvature) having a parallel-plane slab of dielectric material located centrally between the two mirrors. It is assumed that the mirrors are sufficiently large for diffraction effects to be negligible. In the following companion paper II, experimental results are presented which verify the accuracy, within the usual limits of practical measurement, of the simple correction formulae deduced here.

P. K. Yu

Papers

The accurate measurement of permittivity by means of an open resonator

Measurement of Permittivity by Jeans of an Open Resonator. I. Theoretical