Making Material Measurements with a VNA

Figure 1 Free-space measurement test setup. Source: Compass Technology Group.

Knowing a material’s RF properties, such as its complex permittivity and permeability, can be very important. The radome protecting an outdoor radar from the elements must have a known permittivity and thickness to pass radar frequencies with minimal attenuation. How will a thin coating of water on the radome effect performance? The protective cover over the radar embedded in an automobile bumper must be designed to pass the mmWave frequencies emitted and received. How will paint affect the RF transmission through the cover? Antennas are encased in our cellular phones. How do the plastic materials affect the antenna performance? Material measurements with a vector network analyzer (VNA) can measure material properties and answer these important questions.

HOW IS A VNA USED TO MEASURE MATERIAL PROPERTIES?

Microwave or mmWave signals are applied to a flat material sample with known thickness. Signals incident on the sample should be plane waves applied normal to the sample’s surface. This way, RF signals that transit the material pass through the known thickness and not some longer path at an angle. It is sufficient to measure the thickness and the transmission and reflection characteristics of a material to determine its complex permittivity. Permeability may also be determined, but the calculation is greatly simplified if it can be assumed to be unitary.

Figure 2 SwissTo12 MCK measurement system.

Figure 3 SPEAG DAK measurement system.

Figure 4 SPEAG DAK-TL2 measurement system.

The VNA is either attached to a pair of antennas for a free-space measurement or to a waveguide with a sample holder. In the free-space method, a pair of horns may be used to send and receive the signal and dielectric lenses are positioned to focus the beams onto the specimen. In Figure 1, the sample is held by a fixture in the center between the antennas of a system offered by Compass Technology Group and at a position where the focused beams are most planar. Calibration is performed using a shorting plate as the reflect and an empty sample holder as the through. Time domain gating is used around the sample position in bandpass mode to eliminate stray reflections and multipath. The sample is then inserted and S-parameter measurements are made.

A material might also be measured by inserting it in a waveguide path. The MCK measurement system from SwissTo12, shown in Figure 2, is configured in this manner. Two corrugated horn antennas operating over a given waveguide bandwidth are placed end to end and a material under test (MUT) is placed between them. These systems measure the S-parameters, reflection and transmission through the MUT and a mathematical inversion converts these measurements to a complex permittivity.

SPEAG’s Dielectric Assessment Kit (DAK) product line, based on the open coaxial probe method, provides high-precision dielectric parameter measurements, including permittivity, conductivity and loss tangent, over a wide frequency range from 4 MHz to 67 GHz. The advanced hardware technology and user-friendly software of DAK instruments are designed for accurate, precise and non-destructive measurements, making them ideal for use in telecommunications, material science, bioelectromagnetics, biomedical research and industries like automotive, electronics and food.

The DAK System (DAKS) is the first system capable of measuring thin-layer materials and small liquid volumes, as well as DAK single probes. DAKS is a low-cost, portable and easy-to-use dielectric assessment system kit that combines the DAK technology, shown in Figure 3, with the miniature portable R60 and R140B vector reflectometers from Copper Mountain Technologies. The direct and rigid connection of the probe to the reflectometer allows the probe to be moved to the MUT after calibration, greatly simplifying measurements in the lab. The DAK product line includes the DAK-TL2, shown in Figure 4.

HOW IS THE PERMITTIVITY INVERSION COMPUTATION DONE?

The computation is an inversion because the material’s complex permittivity determines the transmission and reflection of the RF waves. The inversion must solve for the unique permittivity that causes the measured reflections. Uniqueness is an important consideration. In any inversion problem, multiple parameter values could create the same outcome, so seeding the problem with a best-guess solution is usually necessary.

How is complex permittivity calculated from the S-parameters? First, the transmission and reflection properties at the interfaces and within the material are modeled when illuminated by a plane wave. As detailed by Dr. Schultz¹ and shown in Figure 5, the MUT can be divided into three zones: Zone A is an infinitely thin left-hand surface to model signal reflection, Zone B is a central section to model signal transmission and Zone C is another infinitely thin surface to model a second reflection with a 180-degree phase shift compared to the first reflection. Reflections occur whenever a wave passes from a medium with one dielectric constant to another, ε₀ to ε_m and ε_m back to ε₀, in this case.

Figure 5 Modeled material zones.

In this analysis, ε₀, the permittivity of air, is 8.854 pF/m or may be normalized to 1.0. The material permittivity, ε_m, will also be normalized by this value. For non-magnetic materials, the permeability, μ_m, can be set to 1.0.

S-parameter matrices can be built for the three zones, but transfer parameters are more useful since they can be multiplied to obtain the composite result for the entire slab of material.

Transfer parameters relate the a and b used in Figure 4 by Equation 1:

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 ε_m = 1 and μ_m = 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 ε_m and μ_m, 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 ε_m and μ_m, have been determined as measured from one surface of the MUT to the other. The Nicolson-Ross-Weir (NRW) algorithm^2,3 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, ε_m, 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 ε_m 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.