3.2 COAXIAL-LINE REFLECTION METHOD
3.2.3 Coaxial-line-excited monopole probes
3.2.3.1 Theoretical basis
In a nonmagnetic medium with à=à0, the antenna-modeling theorem can be expressed as (Burdetteet al. 1980)
Z(ω, ε0εr)
η = Z(nω, ε0)
η0 (3.58) where Z(ω, ε) is the terminal impedance of the antenna. The complex intrinsic impedance of the dielectric medium is given by
η=
à0/(ε0εr) (3.59)
and the intrinsic impedance of free space is given by
η0 =
à0/ε0 (3.60) The complex index of refraction of the dielectric medium is given by
n=√
εr. (3.61)
Equation (3.58) is based on the assumption that the medium surrounding the antenna is infinite in extent, or conversely, the theorem is valid as long as the probe’s radiation field is contained completely within the medium. This theorem is applicable for any probe provided that an analytical expression for the terminal impedance of the antenna is known both in free space and in dielectric medium.
When the length of the probe antenna is approx- imately one-tenth wavelength or larger, a radiation field exists. In cases where the penetration depth in the medium under study is greater than the sample thickness, errors are introduced in the measure- ment of the complex impedance of the medium because the field is not totally within the sample.
Usually, short monopole antennas with length less than one-tenth wavelength are used in materials characterization, as shown in Figure 3.17.
The terminal impedance of a short antenna in free space can be expressed as
Z(ω, ε0)=Aω2+ 1
jCω (3.62) where A and C are constants determined by the physical dimensions of the antenna. The antenna constantsAandC can be determined analytically
(a) (b)
Figure 3.17 Short antennas used for materials pro- perty characterization. (a) Short coaxial monopole antenna and (b) very short coaxial monopole antenna
and experimentally. From the knowledge of the antenna constants and the complex impedance Z(ω, ε0εr) of the antenna in a lossy medium, the complex permittivity of the medium εr can be obtained from Eq. (3.58).
From Eqs. (3.58) and (3.62), we can get the antenna impedance in the medium:
Z(ω, ε0εr)=Aω2√
εr+ 1
jCωεr (3.63) In terms of dielectric constant and loss tangent, Eq. (3.63) can be rewritten as
Z(ω, ε0εr)=Aω2
ε′r(1−j tanδ)
+ 1
jCωε′r(1−j tanδ) (3.64) The antenna impedance Z(ω, ε0εr) can be deter- mined through the measurements of the input reflection coefficient S11. The complex Eq. (3.64) can be written in the form Z=R+jX, which results in two real equations
R= sin 2δ 2ε′rωC +A
εr′ω2
secδ+1
2 (3.65)
X= cos2δ ε′rωC +A
ε′rω2
secδ−1
2 (3.66)
In Eqs. (3.65) and (3.66), the parametersRandX are the real and imaginary parts of the measured impedance, A and C are the physical constants of the probe, and all the other parameters are known exceptεr′andδ. Because the inverse pair of equations corresponding to Eqs. (3.65) and (3.66) cannot be easily obtained, the solutions for ε′rand δ are often obtained using iterative method.
When the probe length decreases, the configura- tion approaches to the one shown in Figure 3.17(b), which is quite similar to an open-ended coaxial probe, and the method for analyzing the open- ended coaxial line can also be used.
3.2.3.2 Typical monopole antennas used in materials property characterization
To achieve higher accuracy and sensitivity and to satisfy various measurement requirements, many
(a) (b)
Figure 3.18 Two coaxial antennas that can be used for materials characterization. (a) Coaxial antenna with step transition and (b) hemispherical antenna. Modified from Wang, Y. and Fan, D. (1994). “Accurate global solutions of EM boundary-value problems for coaxial radiators”, IEEE Transactions Antennas and Propagation, 42 (5), 767–770.2003 IEEE
Teflon bead Teflon
bead
(a) (b)
Figure 3.19 Two types of coaxial antennas used in materials characterization. (a) Conical monopole antenna.
Modified from Smith, G. S. and Nordgard, J. D. (1985). “Measurement of the electrical constitutive parameters of materials using antennas”, IEEE Transactions Antennas and Propagation, 33 (7), 783–792. 2003 IEEE.
(b) Spheroidal antenna. Modified from Stuchly, M. A. and Stuchly, S. S. (1980). “Coaxial line reflection methods for measuring dielectric properties of biological substances at radio and microwave frequencies – A review”,IEEE Transactions on Instrumentation and Measurement,29(3), 176–183.2003 IEEE
types of antennas have been used in materials property characterization. Figure 3.18 shows two examples of coaxial antennas that can be used in materials property characterization (Wang and Fan 1994). A coaxial antenna mainly consists of three regions: feedline, transition region, and radiation region. The main purpose of the transition is to improve the radiation properties of the
antenna so that higher accuracy and sensitivity can be achieved.
Figure 3.19 shows two other types of coaxial antennas: conical monopole antenna and spheroidal antenna (Smith and Nordgard 1985; Bucci and Franceschetti 1974). Such antennas have the advantage that they can get more accurate values of effective permittivity of composite materials.
Aperture
plane Sample
Figure 3.20 Cross-sectional view of a conical-tip coaxial-line probe
However, the insertion of the probe to the mate- rial, unless it is a liquid, creates serious difficulty, especially in assuring that the gap is filled with the material under test.
There are many other types of monopole antennas used in materials property characteri- zation. Figure 3.20 shows a conical-tip coaxial- line probe (Keam and Holdem 1997). Compared to conventional monopole antennas, a conical-tip coaxial-line probe has obvious advantages. Since the conical-tip probe has a cone formed out of the coaxial-line dielectric, it tends to push aside mate- rial in the insertion procedure, so the influence of moisture due to cell damage is minimized. It does not require a flat sample surface, and it is suitable for measurements on deformable materials. Fur- thermore, the length of the probe may be optimized for a specific permittivity and frequency range.