It is important to note that after calculation the Epsilon/Mu spectra should be analyzed and revised before further use. In some cases, the calculation algorithm may provide incorrect non-physical results, in which case the measurements should be repeated and recalculated. The general analysis of broadband dielectric spectroscopy predicts that usually, for non-resonant materials, at normal conditions, without external magnetic field, at room temperature there are 4 most common types of spectra listed below with typical examples of S-parameters spectra and pointed out some problem places for further Epsilon/Mu calculation.

## 1. Dielectrics

The typical spectrum (in linear scale) of dielectric materials is presented in Fig.1. It was obtained by measuring a 9.75 mm thick PTFE sample inside a coaxial measurement cell of length Lc=33.9mm with Lpos=15mm (see source file here).

Figure 1. - PTFE sample 9.75 mm thick inside a coaxial cell of length Lc=33.9mm with Lpos=15mm.

The expected value of complex dielectric permittivity of PTFE in the microwave range is about 2 - 0.02j [1]. However, not all calculation algorithms convert the spectrum shown in Fig. 1 to the expected value of permittivity over the entire frequency range of 0.01-18 GHz. Some calculation problems may arise in regions I and II.

In region I (for frequencies below 1 GHz) the wavelength of the initial signal is large and the phase of S11 parameter may be measured with large uncertainty. Other methods of measuring permittivity in the region of frequencies below 1 GHz are usually used.

In region II the frequency corresponds to half of the wavelength fit inside the sample and thus a numeric instability of the NRW algorithm arises. This instability is caused by large S11 phase uncertainties and is more likely to happen in low-loss materials. The NRW algorithm will fail to give trustworthy results in region II.

It is better to choose sample thickness Ls less than half a wavelength within the frequency range during experiments. It may be estimated by the simple formula:

Ls<c/(\sqrt(ε)2f_{max}),

where c=3*10^{8} is the speed of light, f_{max} - maximum frequency in your experiment, ε - the expected value of dielectric permittivity.

Only iterative algorithm with a nonmagnetic option (it uses only the complex S21 parameter data for calculations) gives reliable result (ε = 2 - 0.02j for spectra from Fig. 1).

General rules for dealing with **dielectrics** inside **coaxial** measurement cell:

- it is better to avoid measurements for frequencies below 1 GHz.
- select the thickness of the sample Ls less than half a wavelength within the measured frequency range.

Typically, Re(ε) and Im(ε) are almost constant or weakly dispersive for common types of plastics and usually Re(ε)>>Im(ε). An example of a good spectrum of a 4mm thick sample of PTFE inside the Lc=34mm cell with Lpos=20mm is provided here. All available methods convert this type of spectra to Epsilon/Mu without errors. An example of calculation for a PTFE sample using the iterative algorithm with a non-magnetic option is presented in Fig.2.

Figure 2. - Dielectric permittivity of PTFE in 1-18 GHz range

## 2. Lossy materials

Lossy materials are widely used for electromagnetic shielding and absorption in the range of 1-18 GHz. The typical examples of such materials are polymer composites with conductive fillers (carbon black, carbon fibers, SiC, etc.) [2] with concentrations below the percolation threshold (i.e. bulk material is not conductive in DC). Re(ε) is generally decreasing and Im(ε) usually is increasing with frequency for such materials.

The typical example of a spectrum of a lossy composite based on carbon fibers with concentration below the percolation threshold is provided here. The result of the calculation using the iterative algorithm for such materials with a non-magnetic option turned on is presented in Fig.3.

## 3. Lossy DC-conductive materials

If a studied sample is DC-conductive, then it's complex dielectric permittivity demonstrates pronounced dispersion and both Re(ε) and Im(ε) are typically decreasing with frequency. For composites just above the percolation threshold with relatively small conductivity (<0.01 S/m) typically Re(ε) > Im(ε). Otherwise, for highly conductive and, correspondingly, highly reflective materials Re(ε) < Im(ε). The typical example of a spectrum of a conductive MWCNT-based epoxy resin composite is provided here. The result of the calculation using the iterative algorithm with a non-magnetic option is shown in Fig.4.

## 4. Magnetic composites

Most materials have Mu=1 in the microwave frequency range. Nevertheless, some types of ferrites and carbonyl iron may demonstrate pronounced magnetic properties in the frequency range above 1 GHz.

According to [3-4], carbonyl iron composites in the dielectric matrix demonstrate large almost constant values of Re(ε), while Re(mu) and Im(mu) decrease with frequency. The typical example of a spectrum is shown here. The result of the calculation using the iterative algorithm with a magnetic material option turned on is presented in Fig.5.

Figure 5. - Complex dielectric permittivity and magnetic permeability of a carbonyl iron-based material in 1-18 GHz frequency range.

**References:**

[1] Riddle, Bill, James Baker-Jarvis, and Jerzy Krupka. "Complex permittivity measurements of common plastics over variable temperatures." *IEEE Transactions on Microwave theory and techniques* 51.3 (2003): 727-733.

[2] Bychanok, D., et al. "Characterizing epoxy composites filled with carbonaceous nanoparticles from dc to microwave." *Journal of Applied Physics* 113.12 (2013): 124103.

[3] Wang, Aimin, et al. "Facile preparation, formation mechanism and microwave absorption properties of porous carbonyl iron flakes." *Journal of Materials Chemistry C* 2.19 (2014): 3769-3776.

[4] Yang, Ruey-Bin, and Wen-Fan Liang. "Microwave properties of high-aspect-ratio carbonyl iron/epoxy absorbers." *Journal of Applied Physics* 109.7 (2011): 07A311.