On the Study of Dielectrics at Microwave Frequencies by the Resonance Method
Abstract Generated abstract
This paper examines resonance methods for measuring dielectric properties at microwave frequencies, with emphasis on barium titanate based ceramics intended for waveguide accelerator applications. Using rectangular resonators with rod and plate specimens, including small perturbation measurements and calculations based on the electromagnetic field solution, the authors determine permittivity and loss tangent as functions of frequency, temperature, and vacuum pressure. The measurements indicate that the studied ceramic shows weak dispersion from 2200 to 3200 MHz, changes in permittivity and losses of no more than about 10 percent over the investigated temperature range, and no detectable dependence on pressure down to 10 to the minus 5 mm Hg. The work also presents formulas and comparative measurements for thin dielectric plates, supporting the applicability of the setup to studies under accelerator relevant conditions.
Full Text
On the Study of Dielectrics at Microwave Frequencies by the Resonance Method
P. I. Goskov, G. A. Uralov
At the present time, alongside their broad application in technology—and especially in various microwave devices and instruments—dielectrics appear to be highly promising for use in accelerating systems of waveguide synchrotrons, both as slowing structures and as multilayer dielectric walls shielding the controlling rapidly varying magnetic fields [1].
In the latter case, it is of interest to determine the possibility of experimentally investigating the properties of dielectrics under conditions close to operating ones, i.e., in the presence of vacuum, elevated temperature, powerful rapidly varying fields, and intense radiation. In the present work, the properties of ceramics with high permittivity and low losses were investigated as functions of frequency, temperature, and vacuum.
In measuring the properties of dielectrics at ultrahigh frequencies, resonance methods are widely used, since they make it possible to obtain accurate results comparatively simply; dielectric specimens are most often used either in the form of rods or in the form of plates [2]. Resonance methods for investigating dielectrics at ultrahigh frequencies are based on two methods: the small-perturbation method and a method using the rigorous solution of the equations of the electromagnetic field in a resonator in the presence of a specimen. A significant drawback of the latter method is the ambiguity of the results obtained.
In view of the fact that the dielectric in waveguide accelerating systems will have a rectangular form and relatively large geometrical dimensions, in the present work a rectangular resonator with an \(H_{10p}\)-wave was used for the measurements, and dielectric specimens were used in the form of plates precisely fitted to the internal cross section of the resonator. The plates were placed on the bottom of the resonator. To eliminate ambiguity in the results of measuring the properties of the barium-titanate-based ceramic of interest to us, a preliminary investigation was carried out on specimens in the form of a rectangular rod.
Various arrangements of specimens in a rectangular resonator, determined by the magnitude of the losses of the dielectrics under investigation, are considered in [3]. In the present work, rectangular rods were placed in the antinode of the electric field. Expressions for pro-
permittivity and loss tangent, obtained by the method of small perturbations, have in this case the following form:
\[ \varepsilon = 1+\frac{abL}{2a_1b_1L_1}\,\frac{\Delta f_p}{f_p}, \]
\[ \operatorname{tg}\delta = \frac{abL}{4\varepsilon a_1b_1L_1} \left(\frac{1}{Q_1}-\frac{1}{Q_0}\right), \tag{1} \]
where
\[ \begin{aligned} a,\ b,\ L &\text{ are the dimensions of the resonator;}\\ a_1,\ b_1,\ L_1 &\text{ are the dimensions of the rod;}\\ \Delta f_p &\text{ is the shift of the resonant frequency of the resonator } f_p \text{ when the sample under study is placed in it;}\\ Q_0 \text{ and } Q_1 &\text{ are the quality factors of the resonator without and with the sample.} \end{aligned} \]
At a frequency of 2250 MHz, for a ceramic rod with dimensions \(1.4\times0.65\times15\) mm, the following results were obtained: \(\varepsilon=66\); \(\operatorname{tg}\delta=1.83\times10^{-3}\), while for a rod of expanded polystyrene with dimensions \(8\times8\times34\) mm, respectively, \(\varepsilon=1.1\) and \(\operatorname{tg}\delta=5\times10^{-4}\), which is in good agreement with the literature data [4].
Subsequently, the second method was used for measurements. Plates made of the same ceramic, with a cross section of \(72\times34\) mm and a thickness of several mm, were used. As analysis of the second method for studying dielectrics showed, the smallest error in measuring the properties of such ceramics will be observed in the case of quarter-wave plates, i.e., for a plate thickness of about 3.5 mm.
Fig. 1. Dependence of permittivity and loss tangent on temperature and frequency
In an experimental investigation of the temperature and frequency dependences of the properties of this ceramic, the following was established (Fig. 1): in the temperature interval studied, the change in permittivity and losses does not exceed 10%. In the frequency interval \(2200\div3200\) MHz, this ceramic has weak dispersion. A certain maximum of the dependence \(\varepsilon(f)\) is observed at a frequency of about 2550 MHz.
Comparison of the measurement results in the pressure interval from atmospheric pressure down to \(10^{-5}\) mm Hg shows that the properties of the ceramic under study do not depend on pressure.
When thin plates are used, the properties of dielectrics (including ceramics based on barium titanate) can also be studied by the method of small perturbations. Then, when positioned
for a thin dielectric plate at the bottom of the resonator, the calculated expressions for the permittivity and losses have the form:
\[ \varepsilon = 1 + \frac{4}{n^2\pi^2}\left(\frac{L}{L_1}\right)^3 \frac{\Delta f_p}{f_p}, \]
\[ \operatorname{tg}\delta = \frac{2}{2\varepsilon n^2\pi^2} \left(\frac{L}{L_1}\right)^3 \left(\frac{1}{Q_1}-\frac{1}{Q_0}\right), \tag{2} \]
where
\(L\) is the length of the resonator, \(L_1\) is the thickness of the plate.
The results of measuring, by this method, the permittivity of getinax and organic glass in the range \(2000 \div 4000\) MHz are shown in Fig. 2.
All measurements in the present work were carried out by the shock-excitation method [5, 6].
Thus, the results of the experimental investigations carried out showed that the method used and the measuring setup [7] make it possible to investigate the properties of barium-titanate-based ceramics as a function of radiation and rapidly varying magnetic fields.
Fig. 2. Dependence of permittivity on frequency.
○ — organic glass, × — getinax
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