A method of measuring and calculating complex dielectric at broad microwave bands
Hao Weixing
Inst. Of Remote Sensing Application, CAS
P.O. BOX 775, Beijing, 100101, P.R. China
Abstract
In this paper, an approach to getting high microwave dielectric constants with mathematical first approximation of integral function. The formula of Macuvitz is developed to fitfor the problems of non-continuous boundary and lossy materials which usually result in plurality mathematically. The math, tools of integration and root-found of complex function are applied to calculate the dielectric constants of varies types of materials with the experimental data. A good agreement is achieved between the inversional values and the ones theoretically predicated for some canonical materials, especially in liquid such as distill water, othanediol, formamide, methanol et alwhose dielectric constants and lossy are much high at the waveband from 2 GHz to 12 GHz. One of the characteristics of this paper is the combination of two advantages of other methods, one is broadband is peculiar to coaxial line, and the other is the capability of measurement of all types of materials easily by applying opened-end probe structure.
The optimum measuring condition is investigated also. It is found that the best precision is obtained when the lower dielectric materials are being measured. The argument of reflect is very sensitive to the susceptance of the effective loads.
Introduction
Microwave Remote Sensing is being developed from quantitative to qualitative, which demands computer to know ground data. Among them the dielectric property of sensed object is an important parameter. There are two characteristics of the data in Remote Sensing. One is that very high precision is not required. Another is versatility of the methods. The reason for the first is that sensed object is distributed in very large are which are composed by varies materials but not pure things. The reason for the second is that the materials existed in nature being sensed and being measured are in liquid, solid, and other different kinds and types. Up till now, many methods of measuring microwave complex dielectric contents have been studied by those in many fields who are interested n the study of materials’ electric property (1). Measured materials were placed in the fixed structures such as waveguide or coaxial line resonator to measure reflectant coefficient of the wave, transmitting parametersor resonate frequency, Q’s value to determine the material property(2). Free space wave method is also used by certain the shifting of the reflect wave. Some of others used open-ended waveguide or coaxial line to measure the reflect amplitude and phase of the wave and this method is appreciated much because this kind of
structure can provide convenience for measuring all types of materials such as gas, liquid and solid and non-destructive measurement, also because of the characteristic of its broad wave bands especially for coaxial line(3).
But the non-continuous border of the open-ended of probe, especially the free margin made it a problem to find rigous resolution. As a result, some approximation method has to be employed to replace the truth resolution. For example, linear equivalent circuit used to imitate the microwave circuit of this structure(4). It approximated well in the lower microwave frequency. But at a higher band, a parallel capacitance and a radiation conductance as a compensation circuit haven been added to the primary equivalent circuit in order to make up its non-linear characteristic under higher measuring wavebands and thus decreasing the error produced by the equivalent circuit method. But it is difficult to estimate the effective range in high frequency to this method. Numerical simulate method was also used to solve this problem(5). But it is not suitable for practicable measurement because of time consuming although exact resolution can be acquired by this method. The purpose of this paper is trying to find out an approach which could take account of both the broad microwave bands and versatile measurement easily and quickly.
Lewin(6) et al. has studied the problem of aperture radiation. This theory is developed to the radiation to free space at an open-ended coaxial line by Marcuvitz(7) and Gale is(8). On the base, we will be going on expending the free space to non-continuous medium.
Principle
Generally speaking, the formula of radiation with open-ended coaxial line to air to based on the condition of continuous medium by Mareuvitz+. And in fact the equivalent equation of the admittance have already been derived by Gale js(8) which means that filling materials in waveguide or coaxial transmitting line can be different from that in free space. The same equation can be derived from electromagnetic theory and same structure boundary conditions when radiation not into free space but into some kind of material assumed to be full of space. Here it should be noted that dielectric parameter e0 not to be of air but of measured materials. Since the wave number k is related to the material in the form of
where k
0 is wave number of free space and
em is the relative dielectric constant. If f
e is defined as effective frequency, it varies with change of the dielectric. Since appearing of f is always accompanied by
em is equivalent to he decrease of the free space wave number leading to the increase of the radiation loss through the probe. If the linear equivalent circuit is still used to calculated high dielectric constant, measured error will be increased evidently due to he non-linearity of admittance with frequency, and the radiation conductance can not be neglected yet under this condition. therefore true wave frequency is replaced by effective frequency f
e considering the effect of dielectric.
where Y is normalized equivalent admittance;
B is normalized equivalent susceptance;
G is normalized equivalent conductance;
J
0(x) zero order Bessel function of x;
Si(x) is sine integral of x;
a,b is conductor radius of inner and outer respectively;
em is the relative dielectric constant of medium filling outside of right semi-infinite space;
ei is the relative dielectric constant filing in the coaxial line;
k is the wave number in the medium to be measured;
Fig. 1 is a cutaway view of a open-ended coaxial line probe. At the right of the probe is measured material full of semi-infinite space. According to the test of adding two dielectric slices in different thickness in sequence and observing the change of the reflected signal on receiver the medium can assumed infinitive If the change is small to be neglected. In fig. 2 is the curves normalized load of coaxial line probe expressed in admittance to effective frequency calculated with equation (2) and (3). It is very clear that under the effective frequency of 20 GHz, equivalent admittance
B/
Ö>e is linearly varied with f
e, and equivalent conductance
G /
Öe is small to be neglected in calculation. So linear circuit model is valid in this range. But once the effective frequency is over 40 GHz, the curve of Y /
Ö e--f
e is non-linearity with frequency, strong radiation conductance strengthened as well. Big error must be generated if the linear model is still be used. Therefore this model is valid only for measuring small dielectric materials at lower fluency, that is
fe < 40 GHz
