A method of measuring and calculating complex dielectric at broad microwave bands
In the microwave bands, the maximum fe may be reached to 90 GHz. So a better model is needed in this case. No any other better model is found except for the calculating equation (1) directly, including numerical integration and complex root extraction, and combined the measured data to inversion the parameters required. Steps as follows:
I – D = 0 (4)
Where D is the data taken from experiment and i is the results of integration of equation (1). I can be expressed as:
I = I1 + jI2 (5)
I
1 is the real part of I and I
2 is the Imagery of I. If equation (1) is departed into two parts:
Y = Y1 + jY2 (6)
where Y
1 is the apparent real part of I
1 Y
2 is apparent imagery of I. There will be:
Y1 = I1 (7)
Y2 = I2 (8)
when
e” = 0. This is for the lossless material.
Generally speaking, there always are:
e = e’ - Je”
e” ¹ 0
therefore:
G = G
1 + jG
2 (9)
B = B
1 + jB
2 (10)
I
1 = G
2 + B
2 (11)
I
2 = B
1 + G
2 (12)
It can be seen that because of the complex dielectric, loss is increased. G2 is not real lossy in conductance, and B2 is lossy in susceptance. The and suseptance should be given to I1 and I2 respectively but to G and B which are only in form.
Due to the complicated mathematical calculation, a simple method is hard to be found out that is related to integration of complex Bessel function and using inversion method to extract root, formed a complicated calculation net. In this paper Chebyshev integration and Guass numerical integration are used considering the feature of the integrands. Then Muller method is applied to extract root to calculate complex dielectric constant, these methods are very effective in practice.
3 Measuring System
Measuring system is composed of HW network analyzer made in China, PC computer controlling signature, HP8673B signature source and coaxial line probe. The PC computer was used for sign control, including sending out and receiving of the sign, set intermittence, sampling and data collecting. It can be operated automatically. But deficiency is that it was unstable sometimes. For data comparing, HP8510A network analyzer was also utilized. Test materials are Teflon, Plexiglas rocks which are in slid and distill water, methanol, ethanol, formamide and ethanediol which are in liquid. The constants of these materials are spread in small to very large, lossless to strong loss and also these data can be found in references for comparing (9)
4. Experiments and Analysis
Results of distill water, methanol, ethanol, formamide measurements and theoretical calculation are shown in fig. 3. It can be seen that the two parts of the results are closely fitted. Experimental results are smoothed by mathematical method. In the very broad range of 2-12 GHz,
high constants of the distill water and methanol can made their effective frequency fe very high up to 60 GHz. So that large error must be made by normal, linear model. The experiments demonstrated that the developed equation of Marcuvitz is suitable for the high dielectric materials at high frequency. It will still be noted that zero order approximation was used when use Marcuvitz equation to find equivalent loads of open-ended coaxial line and neglect the effects of high order loss modes which are cut-off modes in coaxial line and contribute to radiation energy in the boundary mutation. Primary characteristic of high cut-oof modes are energy storage and equivalent to lossless loads. These modes are contributable to the conductance and admittance of the equivalent loads which increased with the frequency increasing. Therefore those effects should not be neglected at higher frequency.
The average error of theoretical with experimental are small enough to 3% except for the imagery of ethanediol. It must results in different in some parameter(s) or different measurement conditions, I believe.
5. Conclusion
In this paper, we introduced a new method to study dielectric property of materials in any forms or types by open-ended coaxial line probe. The characteristic of this method compared with others with the same structures is that it can expend measurement frequency from 2 GHz to 12 GHz in sweeping frequency mode easily and quickly. And it is also valid for the high dielectric materials with high loss. Measurement results showed that this method can give out satisfied precision for Remote Sensing below the frequency of 12GHz with relative dielectric constants between 1 to 100. The problem of measurement error is solved in some extent due to high frequency radiation. It is convenient to determine the moisture of soil, rocks, vegetable and et al (10). We also utilized Muller root-extracting, Chebyshev integration and other mathematical tools to solve plural problems successfully.
Reference
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