**4. Scattering parameters measurements and methods for permittivity and permeability extractions**

When an electromagnetic wave interacts with a material sample of finite thickness, reflected and transmitted signals can be registered. On the air-material sample interface, part of the signal reflects back, while the other part penetrates into the material. Inside the material the signal attenuates and on the second material sample-air interface part of the signal reflects back into the sample while the other part continues to go forward, ie. transmits. Scattering parameters (transmission and reflection) depend on permittivity (*ε*) and permeability (*μ*) of the material and by measuring them we can extract *ε* and *μ*.

### **4.1. Transmission/reflection waveguide measurements**

Coaxial line, rectangular and cylindrical waveguide measurements are widely used broadband measurement techniques because of their simplicity ([5]), ([3]). In these methods, the material sample is placed into the section of a waveguide or a coaxial line and scattering parameters are measured with network analyzer.

#### 6 Will-be-set-by-IN-TECH 78 Microwave Materials Characterization

A major problem in transmission line measurements is the possible existance of air gaps between the sample and walls of the waveguide. Samples must be very precisely machined. For example, in X band waveguide measurements (8 to 13GHz, waveguide dimensions 22.9 x 10.2mm), uncertainty in the sample cutting should be in the range of 20*μ*m ([5]).

Important problems with transmission line measurements are also half wavelength resonance and overmoding ([5]). Propagation of single mode resonates at integer multiplication of one half wavelength in the sample. Some techniques for permittivity and permeability extractions break down in the presence of half wavelength resonance. An additional problem is overmoding which means appearance and propagation of higher order modes in the closed waveguide structure. In waveguides and coaxial lines the asymmetry of the sample and machining precision promotes higher order mode propagation. Their appearance determines dips in the reflection coefficient caused by resonances of the excited higher order mode which cause failure of, for example, point by point technique for permittivity and permeability extraction.

### *4.1.1. Free space measurements*

Free space measurements give a noninvasive broadband technique for transmission and reflection parameters measurements. Scattering parameters are measured of the sample that is plane parallel. Measurement setup consists of two identical antennas that operate in certain frequency range and network analyzer. For measurements, corrugated horn antennas can be used. Antennas are aligned and one of them transmits signal while the other antenna works as receiver. Material sample is placed between the two antennas, the incident signal passes through material and is registered by the other antenna. On that way, free space transmission coefficient is measured. Depending on frequency and sample size, focusing lenses can be used.

For reflection measurements, one antenna is connected to the network analyzer via directional coupler. The antenna is sending signal and also measures reflection from the sample that is in front of the antenna.

### **4.2. Methods for permittivity and permeability extractions**

### *4.2.1. Analytical approach - Nicholson Ross Weir derivation*

The Nicholson Ross Weir (NRW) derivation is an analytical method that calculates permittivity and permeability from measured *S*<sup>11</sup> and *S*<sup>21</sup> parameters. Dependence of scattering parameters from material properties is derived considering multiple reflections of the wave incident upon the air-sample interfaces when the sample is in free space or inside of waveguide ([5]), ([3]). Equations (18) to (29) represent short version of the NRW derivation for scattering parameters measured in free space. If we consider a system of air/sample/air, then the incident wave travels and a first partial reflexion on the air-sample interface occurs. The remaining portion of the signal continues to travel through the sample and on the second air-sample interface part of the signal transmits and the other part reflects back and travel through the sample toward the first air-sample interface. After simplification of expressions that include all terms of multiple reflections and transmissions, the final expression for the total reflection parameter, *S*11, and the total transmission parameter, *S*21, are given with Eq. (18) and Eq. (19).

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A major problem in transmission line measurements is the possible existance of air gaps between the sample and walls of the waveguide. Samples must be very precisely machined. For example, in X band waveguide measurements (8 to 13GHz, waveguide dimensions 22.9 x

Important problems with transmission line measurements are also half wavelength resonance and overmoding ([5]). Propagation of single mode resonates at integer multiplication of one half wavelength in the sample. Some techniques for permittivity and permeability extractions break down in the presence of half wavelength resonance. An additional problem is overmoding which means appearance and propagation of higher order modes in the closed waveguide structure. In waveguides and coaxial lines the asymmetry of the sample and machining precision promotes higher order mode propagation. Their appearance determines dips in the reflection coefficient caused by resonances of the excited higher order mode which cause failure of, for example, point by point technique for permittivity and permeability

Free space measurements give a noninvasive broadband technique for transmission and reflection parameters measurements. Scattering parameters are measured of the sample that is plane parallel. Measurement setup consists of two identical antennas that operate in certain frequency range and network analyzer. For measurements, corrugated horn antennas can be used. Antennas are aligned and one of them transmits signal while the other antenna works as receiver. Material sample is placed between the two antennas, the incident signal passes through material and is registered by the other antenna. On that way, free space transmission coefficient is measured. Depending on frequency and sample size, focusing lenses can be

For reflection measurements, one antenna is connected to the network analyzer via directional coupler. The antenna is sending signal and also measures reflection from the sample that is in

The Nicholson Ross Weir (NRW) derivation is an analytical method that calculates permittivity and permeability from measured *S*<sup>11</sup> and *S*<sup>21</sup> parameters. Dependence of scattering parameters from material properties is derived considering multiple reflections of the wave incident upon the air-sample interfaces when the sample is in free space or inside of waveguide ([5]), ([3]). Equations (18) to (29) represent short version of the NRW derivation for scattering parameters measured in free space. If we consider a system of air/sample/air, then the incident wave travels and a first partial reflexion on the air-sample interface occurs. The remaining portion of the signal continues to travel through the sample and on the second air-sample interface part of the signal transmits and the other part reflects back and travel through the sample toward the first air-sample interface. After simplification of expressions that include all terms of multiple reflections and transmissions, the final expression for the

**4.2. Methods for permittivity and permeability extractions**

*4.2.1. Analytical approach - Nicholson Ross Weir derivation*

10.2mm), uncertainty in the sample cutting should be in the range of 20*μ*m ([5]).

extraction.

used.

front of the antenna.

*4.1.1. Free space measurements*

$$S\_{11} = \frac{G1 \cdot (1 - z^2)}{(1 - G1^2 \cdot z^2)} \tag{18}$$

$$S\_{21} = \frac{z \cdot (1 - G1^2)}{(1 - G1^2 \cdot z^2)}\tag{19}$$

$$z^2 = e^{-2 \cdot j \cdot \gamma \cdot d} \tag{20}$$

$$G1 = \frac{(Z-1)}{(Z+1)}\tag{21}$$

$$Z = \sqrt{\frac{\mu}{\varepsilon}} \tag{22}$$

where *z* is unknown variable that depends on the propagation constant *γ*, *d* is the sample thickness, *G*1 is the first partial reflection on the air-sample interface, *Z* is characteristic impedance of the material and depends on *ε* and *μ* of the material as given with Eq. (22). Relation of the propagation constant *γ* with *ε* and *μ* of the material is given by Eq. (23).

$$\gamma = \frac{j \cdot 2 \cdot \pi \cdot \sqrt{\varepsilon \cdot \mu}}{\lambda} \tag{23}$$

$$
\lambda = \frac{c}{f} \tag{24}
$$

where *λ* is a free space wavelength, *c* is speed of light in vacuum and *f* is the frequency.

$$
\gamma = -\frac{1}{d} \cdot \log \frac{1}{z} \tag{25}
$$

From the Eq. (20), *γ* is expressed as a function of *z* and *d* (Eq. (25)). *d* is material slab thickness and *z* is calculated from Eq. (18) and Eq. (19).

*Nm* is material's refractive index and can be expressed in terms of permittivity and permeability as:

$$N\_m = \sqrt{\mu \cdot \varepsilon} \tag{26}$$

By combining Eq. (23) and Eq. (26) we obtain expression for material's refractive index (Eq. (27)). *G*1 is expressed as a funcion of *S*<sup>11</sup> and *S*<sup>21</sup> and *Z* is expressed as a function of *G*1. Finally, *ε* and *μ* are expressed as a functions of *Nm* and *Z* (Eqs. (28) and (29)).

$$N\_m = \frac{-j \cdot \lambda \cdot \gamma}{2 \cdot \pi} \tag{27}$$

$$
\varepsilon = \frac{N\_m}{Z} \tag{28}
$$

$$
\mu = \mathbf{N}\_{\text{m}} \cdot \mathbf{Z} \tag{29}
$$

The numerical shortcoming of the NRW derivation comes from Eq. (25) where the natural logarithm of a complex number has to be calculated and there is no unique solution. The

#### 8 Will-be-set-by-IN-TECH 80 Microwave Materials Characterization

choice of the correct root is essential in order to find the correct solution of the complex permittivity and permeability. Comparison of the calculated and measured time delays can help in resolving this problem ([5]). The second problem arises when *S*<sup>11</sup> and *S*<sup>21</sup> are noisy. The NRW derivation calculates *ε* and *μ* from *S*<sup>11</sup> and *S*<sup>21</sup> for each frequency point. Small measurement errors in the dataset result in significant errors on the calculated values of *ε* and *μ*. Figure 4 represents an example of permittivity and permeability calculation with NRW algorithm when *S*<sup>11</sup> and *S*<sup>21</sup> are noisy. Frequency dependent permittivity and permeability represented with Debye and Lorenzian models are included into the Fresnel's equations based algorithm. Fresnel's equations based algorithm then calculates *S*<sup>11</sup> and *S*<sup>21</sup> (Figure 3). Noise with relative amplitude of 0.05 is added to simulated *S*<sup>11</sup> and *S*21. Figure 4 represents original *ε* and *μ* as well as *ε* and *μ* calculated with NRW derivations.

**Figure 3.** Noisy *S*<sup>11</sup> and *S*<sup>21</sup> that are used for permittivity and permeability calculations with NRW derivation.

### *4.2.2. Numerical approach*

Taking various linear combinations of scattering parameters it is possible to calculate unknown permittivity and permeability ([5]). In order to obtain values for both parameters, we need to have two different measurements of scattering parameters: *S*<sup>11</sup> and *S*<sup>21</sup> measured on one material sample, or one port shorted circuit line measurement of material sample at two different positions in the line, or measurement of one scattering parameter but for the same material and two different thicknesses, etc. For the root determination of the equations (that are similar to the NRW equations) Newton's numerical method can be used ([5]). This iterative approach works well if good initial guesses are available.

One of the numerical methods for both permittivity and permeability determination is based on nonlinear least square optimization technique and is described in details in ([4]). Complex permittivities and permeabilities are represented as a sum of resonance and relaxation terms. Measured scattering parameters are fitted with simulations in the nonlinear least square sense and unknown free parameters from permittivity and permeability models are extracted. It is important to have a good initial guesses of the unknown parameters contained in the *ε* and *μ* models in order of optimization to converge to the correct solution. The initial guesses should 80 Microwave Materials Characterization Free-Space Transmission Method for the Characterization of Dielectric and Magnetic Materials at Microwave Frequencies <sup>9</sup> 81 Free-Space Transmission Method for the Characterization of Dielectric and Magnetic Materials at Microwave Frequencies

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choice of the correct root is essential in order to find the correct solution of the complex permittivity and permeability. Comparison of the calculated and measured time delays can help in resolving this problem ([5]). The second problem arises when *S*<sup>11</sup> and *S*<sup>21</sup> are noisy. The NRW derivation calculates *ε* and *μ* from *S*<sup>11</sup> and *S*<sup>21</sup> for each frequency point. Small measurement errors in the dataset result in significant errors on the calculated values of *ε* and *μ*. Figure 4 represents an example of permittivity and permeability calculation with NRW algorithm when *S*<sup>11</sup> and *S*<sup>21</sup> are noisy. Frequency dependent permittivity and permeability represented with Debye and Lorenzian models are included into the Fresnel's equations based algorithm. Fresnel's equations based algorithm then calculates *S*<sup>11</sup> and *S*<sup>21</sup> (Figure 3). Noise with relative amplitude of 0.05 is added to simulated *S*<sup>11</sup> and *S*21. Figure 4 represents original

*ε* and *μ* as well as *ε* and *μ* calculated with NRW derivations.

S11 no noisy S11 noisy

−20

−15

−10

S21 [dB]

**Figure 3.** Noisy *S*<sup>11</sup> and *S*<sup>21</sup> that are used for permittivity and permeability calculations with NRW

Taking various linear combinations of scattering parameters it is possible to calculate unknown permittivity and permeability ([5]). In order to obtain values for both parameters, we need to have two different measurements of scattering parameters: *S*<sup>11</sup> and *S*<sup>21</sup> measured on one material sample, or one port shorted circuit line measurement of material sample at two different positions in the line, or measurement of one scattering parameter but for the same material and two different thicknesses, etc. For the root determination of the equations (that are similar to the NRW equations) Newton's numerical method can be used ([5]). This

One of the numerical methods for both permittivity and permeability determination is based on nonlinear least square optimization technique and is described in details in ([4]). Complex permittivities and permeabilities are represented as a sum of resonance and relaxation terms. Measured scattering parameters are fitted with simulations in the nonlinear least square sense and unknown free parameters from permittivity and permeability models are extracted. It is important to have a good initial guesses of the unknown parameters contained in the *ε* and *μ* models in order of optimization to converge to the correct solution. The initial guesses should

−5

0

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> −25

(b)

Frequency [GHz]

S21 no noisy S21 noisy

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> −14

(a)

Frequency [GHz]

iterative approach works well if good initial guesses are available.

−12 −10 −8 −6 −4 −2 0

derivation.

*4.2.2. Numerical approach*

S11 [dB]

**Figure 4.** Real and imaginary permittivity and permeability calculated from noisy *S*<sup>11</sup> and *S*<sup>21</sup> by using NRW derivation. Black line represents original values of permittivity and permeability which were used to simulate the S-parameters.

be within 10 to 20% of the true values. The problem is when we examine material without any *a priori* knowledge about its properties, then we are not able to give good initial guesses of its unknown parameters.

## **5. Developed method for permittivity and permeability extraction**

There are uncertainties in free space scattering parameters measurements and they have some frequency dependence with higher frequencies having larger uncertainties. More sensitive to measurement uncertainties is the *S*<sup>11</sup> parameter ([5]), ([3]), ([4]). For that reason we use free space transmission *S*<sup>21</sup> parameter for permittivity and permeability model retrival, and free space reflection (*S*<sup>11</sup> and *S*11*m*, reflections from the samples with and without metal backing) parameter for extracted permittivity and permeability models validation.

Measurements and parameters extraction are concerned Eccosorb CR110, CR114 and CRS117 samples. The CR materials are a two composite-mixture of epoxy and magnetic inclusions and are not flexible materials. A higher product number indicates a higher filling factor of the magnetic loading and therefore the higher absorption. The CRS materials are flexible silicon based materials. They should have the same electrical properties as hard epoxy based materials.

In our work we use the assumption that all material samples are two composite, homogeneous and isotropic. Dimensions and densities of the samples that we examined are summarized in the Table 1. The samples are 2.00mm thick with plane parallel circular surfaces.

#### 10 Will-be-set-by-IN-TECH 82 Microwave Materials Characterization


**Table 1.** CR samples dimensions and densities

**Figure 5.** Microscope view of CR110 (left) and CR114 (right) samples. Photographs are taken with the same magnification for both samples.

Figure 5 represents microscopic view of CR110 and CR114 materials. Different densities of materials can be noticed. For these images, both samples had been polished to approximately 20*μ*m thick slabs.

### **5.1. Scattering parameters measurements**

#### *5.1.1. Transmission measurements*

Free space transmission measurements are performed with a setup explained in section 4.1.1. Measurements are performed at frequencies from 22 to 140GHz. Low frequency measurements are limited by the sample size.

Instead of placing material sample between two aligned antennas, we do transmission measurements by placing sample on the aperture of one of the antennas. Corrugated horn antennas are used because they have less near field effects at the aperture than rectangular horns. Calibration for *S*<sup>21</sup> measurements is done with 'through' measurement Eq. (30). 'Through' signal is measured when antennas are separated only by air, without sample.

$$\mathbf{S\_{21cal}} = \frac{\mathbf{S\_{21meas}}}{\mathbf{S\_{21through}}} \tag{30}$$

where *S*21*cal* is the calibrated signal, *S*21*meas* is the transmission parameter measured through material sample and *S*21*through* is the through measurement, with the signal received by the other antenna when no sample is between the two antennas.

### *5.1.2. Reflection measurements*

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*Eccosorb material Diameter* [*mm*] *Density* [ *<sup>g</sup>*

**Table 1.** CR samples dimensions and densities

same magnification for both samples.

*5.1.1. Transmission measurements*

**5.1. Scattering parameters measurements**

measurements are limited by the sample size.

other antenna when no sample is between the two antennas.

20*μ*m thick slabs.

*CR*110 91.70 1.60 *CR*114 92.80 2.88 *CRS*117 75.60 4.16

(a) (b)

**Figure 5.** Microscope view of CR110 (left) and CR114 (right) samples. Photographs are taken with the

Figure 5 represents microscopic view of CR110 and CR114 materials. Different densities of materials can be noticed. For these images, both samples had been polished to approximately

Free space transmission measurements are performed with a setup explained in section 4.1.1. Measurements are performed at frequencies from 22 to 140GHz. Low frequency

Instead of placing material sample between two aligned antennas, we do transmission measurements by placing sample on the aperture of one of the antennas. Corrugated horn antennas are used because they have less near field effects at the aperture than rectangular horns. Calibration for *S*<sup>21</sup> measurements is done with 'through' measurement Eq. (30). 'Through' signal is measured when antennas are separated only by air, without sample.

*<sup>S</sup>*21*cal* <sup>=</sup> *<sup>S</sup>*21*meas*

where *S*21*cal* is the calibrated signal, *S*21*meas* is the transmission parameter measured through material sample and *S*21*through* is the through measurement, with the signal received by the

*S*21*through*

(30)

*cm*<sup>3</sup> ]

Reflection measurements are performed by placing the material sample on the aperture of the antenna. We did two types of reflection measurements: with sample and metal backing (a metalic reflector placed behind the sample) placed on the aperture and only with the sample placed on the antenna's aperture.

The antenna is connected to the vector network analyzer through directional coupler. For calibration purposes we measure the reference signal (metal plate is on the aperture of the antenna, 100% reflection) and the signal when low reflectivity pyramidal foam absorber (*S*<sup>11</sup> *f oam* < -50dB) is in front of the antenna's aperture (to calibrate directivity). Calibration of the measured reflection parameter is given by Eq. (31).

$$S\_{11cal} = \frac{S\_{11meas} - S\_{11foam}}{S\_{11alu} - S\_{11foam}} \tag{31}$$

where *S*11*cal* is the calibrated signal, *S*11*meas* is the measured reflection parameter of the sample, *S*<sup>11</sup> *f oam* is the measured reflection when foam absorber is on the top of aperture and *S*11*alu* is reference reflection measurement, when metal is on the aperture of the antenna.

### **5.2. Procedure for parameters extraction**

We do not have any information about epoxy and magnetic inclusions properties. Our assumptions on the examined material samples are described as follows:


Epoxy is low loss dielectric material, while magnetic inclusions have magnetic and also dielectric properties. With previous assumptions and according to ([8]), if magnetic dispersion of inclusions is of resonance Lorenzian type than dispersion law for the composite will be Lorenzian as well. Analogous to that, dielectric dispersion of the composite can be modeled with relaxation Debye model.

At high frequencies (> 60-70GHz) there are no magnetic losses because magnetization is not possible since applied field is very fast and magnetic domains cannot follow the field. Permeability is equal to 1. It means that material samples exhibit only dielectric losses. We model dielectric losses with simple Debye model (Eq. (11)). Next step is to fit measured transmission parameter, with simulated *S*21, both amplitude and phase. Fitting is based on minimization of the differences (in both amplitude and phase) between simulated and measured transmission data. Model of free space propagation is required to relate the material properties (permittivity and permeability) to the transmission (reflection) parameters. For that purpose we use routine based on Fresnel's equations. There are three unknowns in Debye model for permittivity calculation, *εs*, *ε*<sup>∞</sup> and *fr*. Static permittivity, *εs*, is calculated for all samples by measuring capacitance of the sample at very low frequency. We

#### 12 Will-be-set-by-IN-TECH 84 Microwave Materials Characterization

measure the sample's capacitance in a calibrated capacity bridge operating between 10Hz and 20kHz. Capacitance measurement works good if wavelength is much longer then the sample thickness. It is satisfied in our case because capacitance measurements are performed at 1kHz frequency and samples thicknesses are 2mm. One problem in capacitance measurements is given by the fringing fields. To eliminate them, we measure capacitance of the sample *C* and then capacitance of the capacitor with air instead of material sample *Cair*. Static permittivity *ε<sup>s</sup>* is expressed by Eq. (32).

$$
\varepsilon\_{\rm s} = \frac{\mathcal{C}}{\mathcal{C}\_{air}} \tag{32}
$$

where *C* is the measured capacitance of the material sample and *Cair* is the capacitance between parallel capacitor plates which are separated for a distance equal to the thickness of the material sample, but instead of material there is air.

We include measured *ε<sup>s</sup>* into permittivity Debye model. The next step is to do measured and simulated data fitting at high frequencies (*μ*=1), both amplitude and phase, and to extract two other unknown parameters of Debye model, *fr* and *ε*∞. Once we have full Debye dielectric permittivity model, we can extract permeability.

Starting guess is that permeability of the CR110 sample satisfies Debye relaxation model. As a matter of fact, CR110 is the sample with the smallest amount of magnetic inclusions so permeability behavior should change from resonance to relaxation. Permeability models of CR114 and CRS117 samples are presumed to be of Lorenzian type. With these guesses, we do fitting of measurements and simulations at frequencies where permeability is different than 1, while for permittivity we use a model extracted in the previous step (from fitting with high frequency data). From fitting, we obtain free parameters of presumed permeability models.

Figure 6 represents measured and fitted amplitude and phase of examined Eccosorb samples. A phase offset of one *π* (Figure 6 (b)) has been applied between phases of different samples for clarity.

**Figure 6.** Measured and fitted transmission parameter amplitudes (a) and phases (b) of CR110, CR114 and CRS117 samples.

### **5.3. Extracted permittivity and permeability of CR Eccosorb absorbers and results validation**

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measure the sample's capacitance in a calibrated capacity bridge operating between 10Hz and 20kHz. Capacitance measurement works good if wavelength is much longer then the sample thickness. It is satisfied in our case because capacitance measurements are performed at 1kHz frequency and samples thicknesses are 2mm. One problem in capacitance measurements is given by the fringing fields. To eliminate them, we measure capacitance of the sample *C* and then capacitance of the capacitor with air instead of material sample *Cair*. Static permittivity

> *<sup>ε</sup><sup>s</sup>* <sup>=</sup> *<sup>C</sup> Cair*

where *C* is the measured capacitance of the material sample and *Cair* is the capacitance between parallel capacitor plates which are separated for a distance equal to the thickness

We include measured *ε<sup>s</sup>* into permittivity Debye model. The next step is to do measured and simulated data fitting at high frequencies (*μ*=1), both amplitude and phase, and to extract two other unknown parameters of Debye model, *fr* and *ε*∞. Once we have full Debye dielectric

Starting guess is that permeability of the CR110 sample satisfies Debye relaxation model. As a matter of fact, CR110 is the sample with the smallest amount of magnetic inclusions so permeability behavior should change from resonance to relaxation. Permeability models of CR114 and CRS117 samples are presumed to be of Lorenzian type. With these guesses, we do fitting of measurements and simulations at frequencies where permeability is different than 1, while for permittivity we use a model extracted in the previous step (from fitting with high frequency data). From fitting, we obtain free parameters of presumed permeability models. Figure 6 represents measured and fitted amplitude and phase of examined Eccosorb samples. A phase offset of one *π* (Figure 6 (b)) has been applied between phases of different samples

> −5 −4 −3 −2 −1 0

**Figure 6.** Measured and fitted transmission parameter amplitudes (a) and phases (b) of CR110, CR114

CR110 CR114 CRS117

Phase [π]

<sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> −6

Frequency [GHz]

(b)

(32)

*ε<sup>s</sup>* is expressed by Eq. (32).

for clarity.

Amplitude[dB]

−35 −30 −25 −20 −15 −10 −5 0 5

and CRS117 samples.

CR110 CR114 CRS117

of the material sample, but instead of material there is air.

permittivity model, we can extract permeability.

<sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> −40

(a)

Frequency [GHz]

To validate extracted models for both permittivity and permeability, we compare simulated reflection parameters (samples with and without metal backing) with measurements. Comparisons are presented in Figures 7 and 8 and good agreement between measurements and simulations is achieved. Reflection measurements are performed in Ka, U and W band.

**Figure 7.** Measured and simulated reflection coefficient of CR110 absorber, with and without metal backing. Simulated reflections are represented with black solid line.

**Figure 8.** Measured and simulated reflection coefficient of CR114 (a) and CRS117 (b) absorbers, with and without metal backing. Simulated reflections are represented with black solid line.

There are some inconsistencies in measurements in different frequency bands. That could come from the fact that we used different corrugated horn antennas for different frequency bands. Another possible source for the inconsistency in measurements can be caused by the

#### 14 Will-be-set-by-IN-TECH 86 Microwave Materials Characterization

presence of the air gaps between samples and metal backing. Small air gaps exist because some of the examined samples are not completely flat, but are slightly bended.

By looking in amplitude behavior of transmission measurements (Figure 6 (a)) we can say that if permittivities and permeabilities are not frequency dependent, transmission coefficient decreases with increasing frequency. The fact that measured transmission coefficient decreases in some frequency range and increases in the other, says about frequency dependent material parameters.

Figures 9 and 10 represent extracted frequency dependent real and imaginary parts of permittivity and permeability of examined samples. As we mentioned, CR110 sample is low loss material with very small amount of magnetic particles that can produce losses. Because of the small concentration of magnetic particles, magnetic loss mechanism is transformed from resonance to relaxation (which is seen in Figure 10 (b), imaginary part of permeability). Both CR114 and CRS117 materials show Debye relaxation model for permittivity and Lorenzian model for permeability. The difference is that CRS117 material contains more magnetic inclusions compared to CR114 and thus showing the highest value of imaginary part of permeability.

One very important fact is that permeability of magnetic materials (in the range from 0 up to GHz frequencies) can contain one or multiple dispersion areas ([9]), ([10]). Below 20GHz we did not perform scattering measurements, so we have no data to be used in the fitting procedure. Furthemore, we do not have any information about materials that we examine so we cannot be completely sure of the correctness of the reconstructed model for permeability behavior at frequencies below 20GHz. For that reason, retrieved permittivity and permeability data should be used in the frequency range from 20 to 140GHz, ie. that adopted in the fitting procedure.

**Figure 9.** Real and imaginary parts of retrived permittivity of Eccosorb samples.

### **5.4. Extracted permittivity of epoxies Stycast W19 and Stycast 2850 FT**

Commercially available epoxies Stycast W19 ans Stycast 2850FT can be used as matrices in the synthesis of microwave absorbing materials. Carbonyl iron or steel particles can be used as

**Figure 10.** Real and imaginary parts of retrived permeability of Eccosorb samples.

14 Will-be-set-by-IN-TECH

presence of the air gaps between samples and metal backing. Small air gaps exist because

By looking in amplitude behavior of transmission measurements (Figure 6 (a)) we can say that if permittivities and permeabilities are not frequency dependent, transmission coefficient decreases with increasing frequency. The fact that measured transmission coefficient decreases in some frequency range and increases in the other, says about frequency dependent material

Figures 9 and 10 represent extracted frequency dependent real and imaginary parts of permittivity and permeability of examined samples. As we mentioned, CR110 sample is low loss material with very small amount of magnetic particles that can produce losses. Because of the small concentration of magnetic particles, magnetic loss mechanism is transformed from resonance to relaxation (which is seen in Figure 10 (b), imaginary part of permeability). Both CR114 and CRS117 materials show Debye relaxation model for permittivity and Lorenzian model for permeability. The difference is that CRS117 material contains more magnetic inclusions compared to CR114 and thus showing the highest value of imaginary part of

One very important fact is that permeability of magnetic materials (in the range from 0 up to GHz frequencies) can contain one or multiple dispersion areas ([9]), ([10]). Below 20GHz we did not perform scattering measurements, so we have no data to be used in the fitting procedure. Furthemore, we do not have any information about materials that we examine so we cannot be completely sure of the correctness of the reconstructed model for permeability behavior at frequencies below 20GHz. For that reason, retrieved permittivity and permeability data should be used in the frequency range from 20 to 140GHz, ie. that adopted in the fitting

some of the examined samples are not completely flat, but are slightly bended.

parameters.

permeability.

procedure.

Real permittivity

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>0</sup>

CR110 CR114 CRS117

**5.4. Extracted permittivity of epoxies Stycast W19 and Stycast 2850 FT**

Commercially available epoxies Stycast W19 ans Stycast 2850FT can be used as matrices in the synthesis of microwave absorbing materials. Carbonyl iron or steel particles can be used as

**Figure 9.** Real and imaginary parts of retrived permittivity of Eccosorb samples.

0.5 1 1.5 2 2.5 3 3.5 4

CR110 CR114 CRS117

Imaginary permittivity

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>0</sup>

Frequency [GHz]

(b)

Frequency [GHz]

(a)

fillers. The differences between two epoxies are in viscosity, dielectric and thermal properties. Stycast 2850FT epoxy is loaded with alumina for the higher thermal conductivity. Alumina loading makes Stycast 2850FT epoxy more viscose than Stycast W19, thus implying lower filling fraction of possible absorbing particles. Stycast W19 exhibits lower viscosity and lower thermal conductivity than Stycast 2850FT. In terms of dielectric properties, Stycast 2850FT exhibits higher real permittivity and losses than Stycast W19.

Mentioned epoxies are dielectrics and have one relaxation frequency at microwave frequencies. In order to extract frequency dependent permittivity, we model it with simple Debye model (Eq. (10)). With the same procedure described in section 5.2, we extract high frequency permittivity model of both Stycast W19 and Stycast 2850FT.

Figure 11 (a) represents measured and fitted *S*<sup>21</sup> of Stycast 2850FT in W band. Figure 11 (b) represents measured *S*<sup>21</sup> of Stycast 2850FT in Ka band together with simulations performed with extracted permittivity model from W band. Good agreement is achieved. Figure 12

**Figure 11.** Measured and fitted *S*<sup>21</sup> of Stycast 2850FT epoxy in W band (a) and measured and simulated *S*<sup>21</sup> in Ka band (b).

**Figure 12.** Measured and simulated reflection coefficient of Stycast 2850FT epoxy samples without (a) and with (b) metal backing. Good agreement between measurements and simulations is obtained which validate extracted permittivity model.

represents measured and simulated reflection of Stycast 2850FT with and without metal backing in Ka band. There is a good agreement between measured and simulated data which is proof for correctness of the extracted model. Figure 13 represents extracted real and imaginary frequency dependent permittivity of examined epoxies. Conclusion is that in the case of dielectric materials whose permittivity model exhibits one relaxation frequency in microwave frequencies, permittivity model extracted from the fitting at high frequencies is also valid at low frequencies.

**Figure 13.** Real and imaginary permittivity extracted from the fitting for both Stycast W19 and Stycast 2850FT.

### **6. Conclusions**

In this Chapter, we described a method for the extraction of frequency dependent permittivity and permeability parameters of magnetically loaded absorbing materials from free space transmission measurements. Our approach can be applied to noisy data and do not need any parameter to be known in advance. The starting assumption is based on the consideration of a two composite material (dielectric matrix and magnetic particles). According to ([8]) about models that represent composite materials, dielectric property of our samples was modeled with simple Debye relaxation model, while complex permeability was modeled with Lorenzian resonant model. Important thing was that we restored first permittivity models of the samples by fitting at high frequencies where permeability is constant and equal 1. After that step we did fitting at low frequencies to extract permeability model. The proposed method is also suitable for permittivity extraction of dielectric materials in those situations where no *a priori* information is known about material, except that material is two composite, homogeneous and isotropic. Also, for the first time we presented extracted complex and frequency dependent values of permittivities and permeabilities of Eccosorb absorbing materials (CR110, CR114 and CRS117) in the frequency range from 22 to 140 GHz. Since below 22 GHz we did not perform scattering parameters measurements and fitting, we can not say if the extracted models are also valid in that region. Future work will include investigation of permittivity and permeability frequency dependence at low frequencies (from 0 up to 22 GHz) and from 150 to 650 GHz.

## **Author details**

16 Will-be-set-by-IN-TECH

−9 −8 −7 −6 −5 −4 −3 −2 −1 0

S11m [dB]

**Figure 12.** Measured and simulated reflection coefficient of Stycast 2850FT epoxy samples without (a) and with (b) metal backing. Good agreement between measurements and simulations is obtained which

represents measured and simulated reflection of Stycast 2850FT with and without metal backing in Ka band. There is a good agreement between measured and simulated data which is proof for correctness of the extracted model. Figure 13 represents extracted real and imaginary frequency dependent permittivity of examined epoxies. Conclusion is that in the case of dielectric materials whose permittivity model exhibits one relaxation frequency in microwave frequencies, permittivity model extracted from the fitting at high frequencies is

> 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

**Figure 13.** Real and imaginary permittivity extracted from the fitting for both Stycast W19 and Stycast

In this Chapter, we described a method for the extraction of frequency dependent permittivity and permeability parameters of magnetically loaded absorbing materials from free space transmission measurements. Our approach can be applied to noisy data and do not need any

Imaginary permittivity

<sup>26</sup> <sup>28</sup> <sup>30</sup> <sup>32</sup> <sup>34</sup> <sup>36</sup> <sup>38</sup> <sup>40</sup> −10

(b)

Frequency [GHz]

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>110</sup> <sup>0</sup>

Imaginary permittivity s2850 Imaginary permittivity W19

Frequency [GHz]

(b)

model meas

model meas

<sup>26</sup> <sup>28</sup> <sup>30</sup> <sup>32</sup> <sup>34</sup> <sup>36</sup> <sup>38</sup> <sup>40</sup> −30

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>110</sup> 2.5

(a)

Frequency [GHz]

Real permittivity s2850 Real permittivity W19

(a)

validate extracted permittivity model.

also valid at low frequencies.

3 3.5 4 4.5 5 5.5

**6. Conclusions**

Real permittivity

2850FT.

Frequency [GHz]

−25 −20 −15 −10 −5 0

S11 [dB]

Irena Zivkovic and Axel Murk *Institute of Applied Physics, University of Bern, Switzerland*

### **7. References**


	- [10] Zhuravlev, V. A. &Suslyaev V. I. (2006b). Analysis of the microwave magnetic permeability spectra of ferrites with hexagonal structure, *Russian Physics Journal* Vol. 49: No. 9.

<sup>90</sup> Microwave Materials Characterization **Chapter 6** 
