Coaxial air line is the most widely used trans- mission line in the characterization of permittivity and permeability of materials. In a coaxial air-line method, the toroidal sample is inserted between the inner and outer conductors of the coaxial line.
Coaxial air-line method has obvious advantages in that it, theoretically, can work down to zero fre- quency and can cover a wide frequency range. In the choice of a coaxial air line, we should con- sider the characteristic impedance and the working frequency range.
4.2.1 Coaxial air lines with different diameters As discussed in Chapter 2, the characteristic impedance Zc of a coaxial line is given by
Zc= 60
√εr
ln b
a
(4.47) where bandaare the outer and inner radii of the coaxial line and εr is the relative dielectric per- mittivity of the insulator filled between the inner and outer conductors. As in most measurement systems, the characteristic impedance of the mea- surement circuit is 50 and the impedance of coaxial line is often chosen as 50. To decrease the insertion loss of a coaxial line, low-loss dielec- tric, usually Teflon, is used as the insulator between the inner and outer conductors.
For a coaxial air line often used in materials property characterization, the insulation dielectric between the inner and outer conductors is air. From Eq. (4.47), we can get the relationship between the inner and outer diameters of a 50coaxial line:
b=2.3ãa (4.48) Usually, a coaxial line is named according to its outer diameter, for example, 3.5-mm coaxial line, 7-mm coaxial line, and 14-mm coaxial line.
Besides the fundamental TEM mode, TE and TM modes can also propagate in a coaxial cable. To ensure the measurement accuracy and sensitivity, the coaxial line for materials property characterization should work in a pure TEM mode.
The working frequency ranges for coaxial air lines with different dimensions are listed in Table 4.1.
The coaxial air line with smaller outer diameter has wider working frequency range, but it has more strict requirements on sample fabrication than the coaxial air line with larger outer diameter.
Table 4.1 The working frequency ranges of 50 coaxial air lines with different outer diameters
Outer diameter (mm)
Working frequency range (GHz)
3.5 0–34.5
7.0 0–18.2
14.0 0– 8.6
For composite samples with large size inclusions, coaxial air lines with larger outer diameters are more suitable.
4.2.2 Measurement uncertainties
As transmission/reflection methods have closed- form solutions for the calculation of complex per- mittivity and permeability, its uncertainty analy- sis can be conducted systematically. The uncer- tainty sources of transmission/reflection method mainly include algorithm uncertainty, air gap, uncertainty of sample position, and uncertainties ofS-parameter measurement.
4.2.2.1 Differential analysis of algorithm
Differential analysis on the algorithms for transmis- sion/reflection method has been conducted by many researchers, including Baker-Jarvis (1990) Baker-Jarviset al. (1993), Smith (1995) and Youngs (1996). As an example, we discuss Nicolson–Ross–
Weir algorithm. In Eqs. (4.18)–(4.23), five parame- ters are used to deduce the complex permittivity and complex permeability: amplitude and phase of com- plex reflection (S11andφ11), amplitude and phase of complex transmission (S21andφ21), and the thick- ness of sampleD. In the uncertainty analysis, we should consider the contributions from each of the five parameters. The uncertainties of real and imag- inary parts of permittivity and permeability can be generally expressed as
Uε′r=
n
n=1
Re ∂εr
∂ei ãUei 2
(4.49)
Uε′′r =
n
n=1
Im ∂εr
∂ei ãUei
2
(4.50)
Uà′r=
n
n=1
Re ∂àr
∂ei ãUei 2
(4.51)
Uà′′r =
n
n=1
Im ∂àr
∂ei ãUei
2
(4.52) whereei areS11, φ11, S21, φ21andD;Ua indicates the uncertainty of parameter a; and parameter
a refers to the real and imaginary parts of permittivity and permeability, or parameters ei. 4.2.2.2 Effect of air gaps
For a coaxial air-line method, the uncertainties caused by the air gaps are serious. As shown in Figure 4.6, in a coaxial measurement cell, there may exist air gaps between the sample and the inner conductor, and between the sample and the outer conductor.
The effects of air gaps, shown in Figure 4.6, to the measurement results of permittivity and permeability can be analyzed using the layered capacitor model (Baker-Jarvis et al. 1993). In this model, the air gaps are assumed to be uniform, with circular symmetry, and the segment of coaxial air-line filled with sample can be taken as capacitors in series. The relationships between the measurement results (εmandàm)and the corrected results (εcandàc)are given by (Youngs 1996)
ε′c=ε′mã
L2 L3−εm′L1
(4.53) εc′′=
ε′cεm′′
εm′
ã L3
L3−L1εm′[1+(εm′′/ε′m)2] (4.54) à′c=à′mãL3−L1
L2
(4.55) à′′c=à′′mãL3
L2 (4.56)
d1 d2 d3 d4
Figure 4.6 Air gaps in a coaxial-line sample holder.
Source: Baker-Jarvis, J. Janezic, M. D. Grosvenor, J. H.
Jr. and Geyer, R. G. (1993).Transmission/Reflection and Short-circuit Line Methods for Measuring Permittivity and Permeability, NIST Technical Note 1355 (revised), National Institute of Standards and Technology, Boul- der, CO
where
L1 =ln d2
d1
+ln d4
d3
(4.57) L2 =ln
d4 d1
(4.58) L3 =ln
d3
d2
(4.59) The correction factor is taken as the ratio of the corrected component to the measured uncor- rected component. Figure 4.7 shows the relation- ship between the correction component of dielec- tric constant and the air gap between the inner conductor and the sample. It shows that when the gap between the inner conductor and the sample increases, the correction component increases; and when the dielectric constant value increases, the correction factor increases. Similar conclusions can also be obtained for the correction components of other components of materials’ intrinsic properties.
It should be noted that the air gap between the inner conductor and the sample could cause larger uncertainties than the one between the sample and the outer conductor. As the electric and magnetic field are concentrated near the inner conductor, even a small air gap between the inner
1.56 1.50 1.44 1.38 1.31 1.25 1.19 1.13 1.06
0 0.01 0.02
(d1−d2)/d1
0.03 0.4045 1
e′m= 2 e′m= 5 ec′ /em′
e′m= 8
Figure 4.7 The gap correction calculated for various values of dielectric constants, where d1 and d2 are defined in Figure 4.6 (Baker-Jarviset al. 1993). Source:
Baker-Jarvis, J. Domich, M. D. and Geyer, R. G.
(1993). Transmission/Reflection and Short-circuit Line Methods for Measuring Permittivity and Permeability, NIST Technical Note 1355 (revised), National Institute of Standards and Technology, Boulder, CO
(a) (b)
Figure 4.8 Air gaps in a coaxial air line filled with a sample. (a) A sample with an air gap between the inner conductor and the sample, and an air gap between the outer conductor and the sample and (b) a sample without an air gap between the inner conductor and the sample, but with an air gap between the outer conductor and the sample
conductor and the sample may result in large errors. Furthermore, as shown in Figure 4.8(a), because of the existence of the air gap between the inner conductor and the sample, the sample may be not symmetrical in the coaxial air line, so the above corrections could not be applied. Therefore, we should try to eliminate the air gap between the inner conductor and the sample. Besides, in experiments, the outer diameter of the sample is often made a little smaller than the inner diameter of the outer conductor of the coaxial air-line, as shown in Figure 4.8(b), so that the sample could be easily inserted into the measurement fixture, and the effect of the air gap between the sample and outer conductor can be corrected using Eqs. (4.53)–(4.59).
4.2.2.3 Effect of sample placement
In actual measurements, there may be some dis- tances between the sample ends and the calibration planes. A phase correction is required when any length of transmission line is added beyond the calibration plane. For the general case shown in Figure 4.9, the phase corrections for S11 and S21
are given by
φ11=2aã2πf
c (4.60)
φ21=(a+b)ã 2πf
c (4.61)
S11 S11 MS11 US11 MS11 US11
(a) (b)
Figure 4.10 Effects of measurement uncertainties ofS11. (a)|S11|>|US11|and (b)|S11|<|US11|
Calibration planes Sample ends
a b Port 2
Port 1
Figure 4.9 Sample placement in the sample holder where f is the measurement frequency and c is speed of light in vacuum.
In most of the algorithms for transmission/reflec- tion methods,S11andS21are used in the calculation of the properties of materials. So the uncertainties of the sample position in the transmission line may cause uncertainties to the measurement results.
Meanwhile, it should be noted that only the phases ofS11andS21varies with the position of the sample within the transmission line, provided that the overall length of the transmission line is accurately known.
4.2.2.4 S -parameter measurement uncertainties The measurement uncertainties of S-parameters directly cause uncertainties of permittivity and per- meability values. The uncertainties ofS-parameters can be minimized by calibration, but cannot be elim- inated because of the limited dynamic range in an actual measurement instrument.
Figure 4.10 schematically shows the effects of measurement uncertainties ofS11. In the figure, the actual S11 value is denoted as S11, the measure- ment uncertainty of S11 is denoted as US11, and the measured value ofS11is denoted asMS11. The vectorMS11is the vector sum ofS11andUS11. The amplitude and phase difference between the mea- suredS11and actualS11is due to the measurement uncertainty of S11. When|S11|>|US11| as shown in Figure 4.10(a), the amplitude and phase differ- ences between the measuredS11and the actualS11 are not large. Whereas, if |S11|<|US11|as shown in Figure 4.10(b), the amplitude and phase differ- ences between the measured S11 and actual S11
are large, and such differences cause great uncer- tainties in the calculation of permittivity and per- meability. In transmission/reflection method, when the sample length is an integral times of the half wavelength of the microwave in the sample, the value of S11 is very small, so the uncertainties of S11cause great uncertainties in the results of per- mittivity and permeability.
4.2.3 Enlarged coaxial line
For samples with coarse grains, such as soil, concrete, and rock, large samples are needed to average out the fluctuations in the dielectric properties of the heterogeneous materials. So, the coaxial lines need to be enlarged to host large samples. Two kinds of enlarged coaxial lines are often used: enlarged circular coaxial line and enlarged square coaxial lines.
Sample
(a) (b)
Figure 4.11 Schematic drawings of an enlarged circular coaxial line. (a) Longitudinal view and (b) cross- section view
4.2.3.1 Enlarged circular coaxial line
Chewet al. proposed an enlarged circular coaxial- line structure (Chew et al. 1991). As shown in Figure 4.11, the measurement cell mainly consists of three parts: transition from normal coaxial line to large coaxial line, large coaxial sample holder, and transition from large coaxial line to normal coaxial line.
In the design of an enlarged measurement cell, the outer conductor radius b is made as large as possible, while the inner conductor radius a is adjusted for a characteristic impedance of 50. The restriction for this is that the value of (b−a) should be no larger than the half-sample wavelength at the highest operating frequency to prevent the existence of higher-order propagating modes. Meanwhile, the desired sample length is as long as possible for low-frequency-phase accuracy, but short enough to avoid resonance and to ensure adequate transmission at high frequencies. For the measurement cell developed by Chew et al., the diameter of the inner conductor is 1.535 cm, and the diameter of the outer conductor is 4.992 cm.
Similar to a conventional coaxial measure- ment cell, an enlarged coaxial measurement cell requires calibration. As calibration standards for large coaxial lines are not commercially avail- able, special calibration method should be devel- oped in the design of large coaxial measurement cell. As a coaxial line can cover a wide fre- quency range, in calibration and measurement pro- cedures, we can divide the whole frequency range into several frequency subranges. Chewet al. pro- posed three methods for three frequency sub- ranges respectively: low frequency (1–30 MHz),
medium frequency (30–800 MHz), and high fre- quency (800 MHz–3 GHz).
Standard “short” is often used in different kinds of calibration methods. A standard “short” should provide good electrical contact to both inner and outer conductors, as there are electrical currents flowing between the inner and outer conductors through the “short”, and a small gap between the standard “short” and the inner or outer conductor may result in large errors. In the design of the structure of a large coaxial line, the calibration procedures and techniques should be taken into consideration.
4.2.3.2 Enlarged square coaxial line
Enlarged square coaxial line has also been devel- oped for the characterization of electromagnetic properties of materials. As shown in Figure 4.12, the structure of an enlarged square coaxial line is similar to that of an enlarged circular coaxial line. The measurement cell also consists mainly of three parts: transition from normal coaxial line to large square coaxial line, large square coaxial sample holder, and transition from large square coaxial line to normal coaxial line. In the design of a square coaxial line, the side lengthbof the outer conductor and the side length a of the inner con- ductor should be chosen to ensure suitable charac- teristic impedance and to avoid higher-order prop- agation modes. The impedance of square coaxial line is usually chosen as 50 or 60. Similar to the enlarged circular coaxial line method, in the design of a square coaxial line, we should consider the calibration techniques used in the enlarged square coaxial line method.
Sample
(a) (b)
Figure 4.12 Schematic drawings of an enlarged rectangular coaxial line. (a) Longitudinal view and (b) cross view The samples for square coaxial lines may con-
sist of several pieces of rectangle-shaped materials.
The fabrication of samples for square coaxial lines is usually easier than the fabrication of toroid- shaped samples for circular coaxial lines. Square coaxial lines are also ideal for the characterization of periodic structures such as pyramidal absorbers, honeycombs, circuit-analog (CA) sheets, and fre- quency selective surfaces (FSS). For these applica- tions, square coaxial lines with 60-characteristic impedance have obvious advantage. For a 60- square coaxial line, the side lengthbof the outer conductor is about three times of the side length a of the inner conductor. To simulate a sample with infinite array, we can arrange eight pieces of square samples with side length bin the space between the outer and inner conductors as shown in Figure 4.12(b).