Coaxial lines are used to measure constitutive parameters of materials, permeability (μ) and dielectric constant (ε), with broad operating bandwidth and good mode control. Square lines are a special class that allows measurement of anisotropic materials, such as honeycomb, and periodic structures, such as pyramidal radiation absorbing material (RAM). These applications are presented (illustrated) based upon Damaskos Inc.’s experience as a provider of materials measurement solutions.

General Description of a Square Coax Setup

Figure 1 50 Ω, 6.0", two-port square coaxial line with square donut test sample.

Figure 1 shows an example of a square coaxial line. Except for the shape of the inner and outer conductors, square lines are fundamentally similar to circular lines. Propagation in the TEM mode is desired, as the S-parameters of a material under test (MUT) are recorded in one- and two-port configurations. Both lines offer control of the impedance of the line via the ratio of conductor radii or edge dimensions; both are calibrated with similar standards; both require consideration of the upper useful frequency above which a change of mode may occur; both typically employ swept frequency CW measurements and subsequent processing in hardware or software to generate error corrected S-parameters; and both manipulate the S-parameters in the solution of the inverse problem for the μ and ε of the MUT by means of the same set of equations. However, the planar boundaries of the conductors of square lines can provide undistorted images of the MUT, and this can be a distinct advantage in the measurement of some materials that are spatially periodical.

With a, the outer conductor dimension and b, the inner conductor dimension, the characteristic impedance of circular and square lines is:

Zocirc (Ω) = 60 ln (b/a)
Zosq (Ω) = 47.086 (1-b/a) / (0.279+0.721b/a), within 1 percent for b/a>0.25.

Table 1 shows some typical square coaxial lines and their primary functions.

For 50 Ω circular lines, the cut-off frequency in GHz of the first interfering mode is 5.178/a where the diameter a is given in inches. For square coax lines, commonly used to measure material parameters, the first interfering mode is given in Table 1. Experience shows that, with carefully prepared isotropic materials samples and carefully constructed lines, the onset of moding is delayed well above these frequencies. See Figure 2, for example, where the material parameters are obtained where the loss of the TEM mode is delayed well above the first possible interfering mode. The glitch at 1100 MHz may occur at a half-wavelength sample thickness for low loss materials.

Some Specific Uses

Some uses of square coaxial lines include the measurement of isotropic materials μ and ε approximate measurement of anisotropic materials μ and ε; and measurement of transmission and reflection properties of periodic materials.

Figure 2 µ (a) and ε (b) measurements in a 50 Ω, 10.0" square coaxial line.

Figure 3 Geometries of square coaxial lines and square donut samples.

Isotropic Materials: For this class, sample preparation requires four well-machined rectangular blocks that make good contact to the conductors and also good contact to each other (see Figure 3). Block to block gaps affect μ more so than ε. Any line impedance can be used.

Anisotropic Materials: In this instance four trapezoidal samples are prepared and again require good contact to conductors and to each other. Low impedance lines offer a better approximation to preserving the local anisotropy at the conductor corners. Honeycomb core is measured in this manner, where the modeling assumes three orthogonal directionally dependent components of ε. A low impedance waveguide is commonly employed to measure rectangular solid samples, which are simpler to prepare and more true to the modeling assumption. They become impractical to use below approximately a few hundred MHz. Square lines range lower in frequency with much smaller sample size.

Periodic Materials: The 3:1 edge ratio, 60 Ω lines, have eight equal area regions. When the material in each region is identical and has the proper symmetry, it is repeated infinitely in the orthogonal directions to the line. Measurement of the S-parameters of some frequency selective impedance sheets is illustrated in Figure 4. The line is a two-port, 5.4" × 1.8", square coaxial line. The pyramidal RAM tester of Figure 5 is a one-port, 6' × 6' × 60' long, shorted square coaxial line. For these applications, measurements down to long wavelengths are made possible using small quantities of material.

Figure 4 Frequency selective surface samples for a 60 Ω, 5.4" × 1.8" square coaxial line.

Figure 5 60 Ω, 60' long, 6' square one-port square coaxial line for pyramidal RAM return loss measurements.

Conclusion

Several methods have been shown, in which square coaxial lines are employed to make materials properties measurements and made, for reference, comparisons to the use of the more familiar circular coax lines. Square lines permit good measurements of the properties of anisotropic materials and classes of periodic structures. They offer as high quality measurements of isotropic measurements as circular lines.

References

1. J. Uher, J. Bornemann and U. Rosenberg, Waveguide Components for Antenna Feed Systems, Artech House Inc., Boston, MA, 1993.
2. S. Ramo, J. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, John Wiley & Sons, New York, NY, 1967.

Nickander J. Damaskos is President of Damaskos Inc. and Program Manager of R&D projects in microwave measurements of materials design.

Benuel J. Kelsall is Vice President of Damaskos Inc. and Manager of the microwave laboratory. He develops materials measuring setups and software.

James E. Powell, Jr. has a master’s degree in mechanical engineering and has been with Damaskos Inc. for 25 years. He has been the Principal Designer of the various fabricated setups.