Filling in the values for a1, a2, b1 and b2 from the forward and reverse case of Zone A in Figure 5 and noting that Γ- = -Γ+, Equation 2 can be used to solve for the four transfer parameters:

The tangent voltages of the plane wave must be the same on each side of the interface for both forward and reverse waves. Substituting Γ- = -Γ+ and equating the results in Equation 3:

Eliminating the transmission parameters t+ and t-, the transfer matrix for Zone A can be written in terms of Γ+ alone. Dropping the “+” and doing the substitution yields Equation 4:

The reflection at the interface is a known function of the normalized permittivity and permeability and it is given by Equation 5:

Note that if εm = 1 and μm = 1, then Γ = 0, or no reflection for an air-to-air interface.

From the forward and reverse wave cases for Zone B, the transfer matrix can be determined as in Equation 6:

The propagation velocity of a wave is a function of the permittivity and permeability. For free space, ,

which is the speed of light. Using normalized εm and μm, the wave speed in the MUT is given by Equation 7:

The “t” in Equation 6 may be expressed in terms of the permittivity and permeability as given by Equation 8:

Where: Material wave number, rad/m

and d is the thickness of the material in meters.

Finally, substituting -Γ for Γ into Equation 4 gives the transfer parameters for Zone C as shown in Equation 9:

Multiplying all three matrices gives the expression in Equation 10:

Finally, converting the transfer parameters to S-parameters using the standard conversion formula gives the expression in Equation 11:

With these equations, the S-parameters, in terms of εm and μm, have been determined as measured from one surface of the MUT to the other. The Nicolson-Ross-Weir (NRW) algorithm2,3 or an iterative solver can determine the permittivity and permeability that fits the measured data. In the free-space methods, the NRW method is not recommended and a four-parameter method is preferred since it eliminates the need to precisely position the sample under test.

For a non-magnetic sample, it is sufficient to measure S11 and S21, guess at the permittivity, εm, calculate Γ and t and then use an iterative method to improve the guess and minimize the errors in Equation 12 and Equation 13:

As a check, note that S21 = t if there is no reflection, Γ. Also, S11 = Γ if there is no transmission, t. After a potential solution is found, it is a good idea to plot the calculated versus measured values of S11 and S21 to assess the accuracy of the solution.

The primary issue with this method is that for materials with a low loss tangent, the imaginary part of εm is small. Changing the value of this quantity has only a tiny effect on the complex value of S21. Most optimizer strategies will struggle with this problem. Compass Technologies, SwissTo12 and SPEAG have all overcome this issue with their software.

The equations detailed in this article provide a helpful background for an engineer who needs to perform material measurements. Integration experts at Compass Technologies, SwissTo12 and SPEAG provide measurement systems and software to perform these measurements and calculate permittivity. For a nominal fee, they can also make batch measurements for those who do not wish to procure a dedicated system.

PRACTICAL CONSIDERATIONS

Sometimes troublesome resonances may occur at frequencies where the sample thickness is an integer multiple of half wavelengths and measurements may contain singularities. This occurs when using both reflection and transmission inversions on non-magnetic specimens. A video demonstrating the focused-beam measurement technique is available on the Copper Mountain Technologies website.4

Several practical considerations may prove helpful:

  • If TRL calibration is performed, it is helpful to normalize the S21 response while viewing it in the Smith Chart format. To do this, place a marker in the middle of the frequency band. Move one of the antennas until a 90-degree phase shift is attained for the “line” standard. For the “through” standard, move the antenna back until the phase is zero once again.
  • Time domain gating, a standard feature of all Copper Mountain Technologies VNAs except the “M” series, should be applied to the area occupied by the MUT to eliminate multipath reflections from other surfaces in the lab.
  • Different material measurements require different solutions. Lower frequency measurements might be performed with a focused-beam system from Compass Technologies. Liquid materials would best be measured with a system from SPEAG. mmWave measurements could be made with the MCK system from SwissTo12 or a table-top free-space measurement system from Compass.
  • The waveguide measurement fixture from SwissTo12 can measure plain, coated or multilayer solids, liquids and powders.

CONCLUSION

There are many ways to make material measurements. Copper Mountain Technologies has a wealth of experience with metrology-grade VNAs covering frequencies from 1.5 to 330 GHz. The best solution may also require fixturing and other areas of expertise in addition to the measurement techniques. To account for the impact that RF material properties, such as their complex permittivity and permeability, may have on a design and minimize their effects, it is often helpful to enlist a partner with expertise in these areas.

References

  1. J. W. Schultz, “Focused Beam Methods,“ 2012, First ed., pp. 44–48.
  2. W. B. Weir, “Automatic Measurement of Complex Dielectric Constant and Permeability at Microwave Frequencies,”  Proceedings of the IEEE, Vol. 62, No. 1, January 1974, pp. 33–36, doi: 10.1109/PROC.1974.9382.
  3. A. N. Vicente, G. M. Dip and C. Junqueira, “The step-by-step Development of the NRW Method,” SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference, 2011, Web: https://doi.org/10.1109/IMOC.2011.6169318.
  4. “Free Space Material Characterization at Millimeter Wave: Using Frequency Extenders with CTG’s Focused Beam System,” Copper Mountain Technologies, Web: coppermountaintech.com/webinar/free-space-material-characterization-at-millimeter-wave-using-frequency-extenders-with-ctgs-focused-beam-system.