**Free-Space Transmission Method for the Characterization of Dielectric and Magnetic Materials at Microwave Frequencies**

Irena Zivkovic and Axel Murk

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51596

## **1. Introduction**

72 Microwave Materials Characterization

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> Materials that absorb microwave radiation are in use for different purposes: in anechoic chambers, for electromagnetic shielding, in antenna design, for calibration targets of radiometers, etc. It is very important to characterize them in terms of frequency dependent complex permittivity and permeability for a broad frequency range.

> Widely used absorbing materials are CR Eccosorb absorbers from Emerson&Cuming Company. Permittivity and permeability of these materials are characterized by manufacturer up to frequency of 18GHz, but it is important (in absorbing layer design purposes, for example) to know these values at much higher frequencies.

> In this work we will present new method, retrieved results and validation for complex and frequency dependent permittivity and permeability parameter extraction of two composite, homogeneous and isotropic magnetically loaded microwave absorbers (CR Eccosorb). Permittivity and permeability are extracted from free space transmission measurements for frequencies up to 140GHz. For the results validation, reflection measurements (samples with and without metal backing) are performed and are compared with simulations that use extracted models. The same method is applied in complex and frequency dependent permittivity model extraction of commercially available epoxies StycastW19 and Stycast 2850 FT.

> The proposed new method solves some shortcomings of the popular methods: extracts both permittivity and permeability only from transmission parameter measurements, gives good results even with noisy data, does not need initial guesses of unknown model parameters.

## **2. Definition of permittivity and permeability**

In the absence of dielectric or magnetic material, there are the following relations:

$$D = \varepsilon\_0 \cdot E \tag{1}$$

$$B = \mu\_0 \cdot H \tag{2}$$

©2012 Zivkovic and Murk, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Zivkovic and Murk, licensee InTech. This is a paper distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

where *D* is electric induction, *E* is electric field, *B* is magnetic induction and *H* is magnetic field. *ε*<sup>0</sup> and *μ*<sup>0</sup> are permittivity and permeability of free space, respectively. Values of *ε*<sup>0</sup> and *<sup>μ</sup>*<sup>0</sup> are: *<sup>ε</sup>*0=8.854 · <sup>10</sup>−<sup>12</sup> [ *<sup>F</sup> <sup>m</sup>* ] and *<sup>μ</sup>*0=4 · *<sup>π</sup>* · <sup>10</sup>−<sup>7</sup> [ *<sup>V</sup>*·*<sup>s</sup> <sup>A</sup>*·*<sup>m</sup>* ]. Dielectric permittivity and magnetic permeability of the free space are related to each other in the following way:

$$c^2 = \frac{1}{\mu\_0 \cdot \varepsilon\_0} \tag{3}$$

where *<sup>c</sup>* is the speed of light in a vacuum and its value is *<sup>c</sup>* <sup>≈</sup> <sup>3</sup> · <sup>10</sup><sup>8</sup> [ *<sup>m</sup> s* ].

If an electromagnetic field interacts with material that is dielectric or magnetic, equations (1) and (2) can be represented as follows:

$$D = \varepsilon\_0 \cdot \varepsilon \cdot E \tag{4}$$

$$B = \mu\_0 \cdot \mu \cdot H \tag{5}$$

where *ε* and *μ* are relative permittivity and permeability of the observed material and can be real or complex numbers (Eq. (6) and (7)).

$$
\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}' - \mathbf{j} \cdot \boldsymbol{\varepsilon}'' \tag{6}
$$

$$
\mu = \mu' - \mathbf{j} \cdot \boldsymbol{\mu}'' \tag{7}
$$

where *j* is imaginary unit and *j* 2= -1.

Permittivity is a quantity that is connected to the material's ability to transmit ('permit') an electric field. The real part, *ε* � , is related to the ability of material to store energy, while the imaginary part, *ε* �� , describes losses in material. Permeability is a parameter that shows the degree of magnetization that a material obtains in response to an applied magnetic field. Analogous to the real and imaginary permittivity, the real permeability, *μ* � , represents the energy storage and the imaginary part, *μ* �� , represents the energy loss term.

The polarization response of the matter to an electromagnetic excitation cannot precede the cause, so Kramers-Kronig relations given by Eqs. (8) and (9) ([8]), for the real and imaginary parts of permittivity (permeability) have to be fulfilled.

$$
\epsilon'(\omega) = \epsilon\_{\infty} + \frac{2P}{\pi} \int\_0^{\infty} \frac{\omega' \epsilon'' \left(\omega'\right)}{\omega'^2 - \omega^2} d\omega' \tag{8}
$$

$$
\epsilon''(\omega) = -\frac{2\omega P}{\pi} \int\_0^\infty \frac{\epsilon'(\omega') - \epsilon\_\infty}{\omega'^2 - \omega^2} d\omega' \tag{9}
$$

where *ω* is angular frequency, *ε*<sup>∞</sup> is permittivity when *ω* → ∞ and *P* stands for the Cauchy principal value. The convention of the time variation is exp(*jωt*) and the time derivative is equal to multiplication by *jω*.

The Kramers-Kronig relations connect the real and imaginary parts of response functions. If we know the real/imaginary part of permittivity/permeability on the complete frequency range, the other unknown part can be calculated using the Kramers-Kronig relations. Because of the causality constraint, when we develop models for frequency dependent permittivity or permeability, they must satisfy the Kramers-Kronig relations.

### **3. Modeling of frequency dependent permittivity and permeability**

Dielectric and magnetic loss mechanisms can be represented in the frequency domain as relaxation or resonant type. In the microwave frequency range dielectric losses usually exhibit relaxation behavior ([1]), while magnetic losses exhibit resonant behavior ([6]), ([7]).

### **3.1. Debye relaxation model**

2 Will-be-set-by-IN-TECH

where *D* is electric induction, *E* is electric field, *B* is magnetic induction and *H* is magnetic field. *ε*<sup>0</sup> and *μ*<sup>0</sup> are permittivity and permeability of free space, respectively. Values of *ε*<sup>0</sup> and

> *<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> *μ*<sup>0</sup> · *ε*<sup>0</sup>

If an electromagnetic field interacts with material that is dielectric or magnetic, equations (1)

where *ε* and *μ* are relative permittivity and permeability of the observed material and can be

Permittivity is a quantity that is connected to the material's ability to transmit ('permit') an

degree of magnetization that a material obtains in response to an applied magnetic field.

The polarization response of the matter to an electromagnetic excitation cannot precede the cause, so Kramers-Kronig relations given by Eqs. (8) and (9) ([8]), for the real and imaginary

> 2*P π*

> > ∞ 0

where *ω* is angular frequency, *ε*<sup>∞</sup> is permittivity when *ω* → ∞ and *P* stands for the Cauchy principal value. The convention of the time variation is exp(*jωt*) and the time derivative is

The Kramers-Kronig relations connect the real and imaginary parts of response functions. If we know the real/imaginary part of permittivity/permeability on the complete frequency range, the other unknown part can be calculated using the Kramers-Kronig relations. Because

 ∞ 0

> *�* � (*ω*�

*ω*� *�* �� (*ω*� )

*ω*�

*ω*�

*ε* = *ε* � − *j* · *ε* ��

*μ* = *μ* � − *j* · *μ* ��

permeability of the free space are related to each other in the following way:

where *<sup>c</sup>* is the speed of light in a vacuum and its value is *<sup>c</sup>* <sup>≈</sup> <sup>3</sup> · <sup>10</sup><sup>8</sup> [ *<sup>m</sup>*

2= -1.

Analogous to the real and imaginary permittivity, the real permeability, *μ*

(*ω*) = *ε*<sup>∞</sup> +

(*ω*) = <sup>−</sup>2*ω<sup>P</sup>*

*π*

��

�

parts of permittivity (permeability) have to be fulfilled.

*ε* �

*ε* ��

*<sup>m</sup>* ] and *<sup>μ</sup>*0=4 · *<sup>π</sup>* · <sup>10</sup>−<sup>7</sup> [ *<sup>V</sup>*·*<sup>s</sup> <sup>A</sup>*·*<sup>m</sup>* ]. Dielectric permittivity and magnetic

*s* ].

*D* = *ε*<sup>0</sup> · *ε* · *E* (4)

*B* = *μ*<sup>0</sup> · *μ* · *H* (5)

, is related to the ability of material to store energy, while the

�

, represents the

, describes losses in material. Permeability is a parameter that shows the

, represents the energy loss term.

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>d</sup>ω*�

) − *ε*<sup>∞</sup>

<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>d</sup>ω*�

(3)

(6)

(7)

(8)

(9)

*<sup>μ</sup>*<sup>0</sup> are: *<sup>ε</sup>*0=8.854 · <sup>10</sup>−<sup>12</sup> [ *<sup>F</sup>*

and (2) can be represented as follows:

real or complex numbers (Eq. (6) and (7)).

where *j* is imaginary unit and *j*

electric field. The real part, *ε*

equal to multiplication by *jω*.

��

energy storage and the imaginary part, *μ*

imaginary part, *ε*

The simplest one pole Debye relaxation model is represented with the Eq. (10).

$$\varepsilon(f) = \varepsilon\_{\infty} + \frac{\varepsilon\_{s} - \varepsilon\_{\infty}}{1 + j\frac{f}{f\_{r}}} \tag{10}$$

where *ε<sup>s</sup>* is a static dielectric permittivity, *ε*<sup>∞</sup> is permittivity at infinite frequency (optical permittivity), *fr* is relaxation frequency. Figure 1 gives an example of Eq. (10) with *ε*<sup>∞</sup> = 7, *ε<sup>s</sup>* = 17 and *fr* = 9 (in GHz). Imaginary part of permittivity in Figure 1 is represented as a positive number.

**Figure 1.** Real and imaginary permittivity represented with Debye model.

There are some modified Debye relaxation models that include asymmetrical and damping factors ([2]). These models are Cole-Cole, Cole-Davidson and Havriliak-Negami. They are

#### 4 Will-be-set-by-IN-TECH 76 Microwave Materials Characterization

presented with equations (11), (12) and (13), respectively.

$$\varepsilon(f) = \varepsilon\_{\infty} + \frac{\varepsilon\_{s} - \varepsilon\_{\infty}}{1 + j\left(\frac{f}{f\_{r}}\right)^{1-a}} \tag{11}$$

$$\varepsilon(f) = \varepsilon\_{\infty} + \frac{\varepsilon\_{s} - \varepsilon\_{\infty}}{\left(1 + j\left(\frac{f}{f\_{\prime}}\right)\right)^{\beta}} \tag{12}$$

$$\varepsilon(f) = \varepsilon\_{\infty} + \frac{\varepsilon\_{s} - \varepsilon\_{\infty}}{\left(1 + j\left(\frac{f}{f\_{r}}\right)^{1-\alpha}\right)^{\beta}} \tag{13}$$

The terms *α* and *β* are empirical parameters and their values are between 0 and 1. *α* is a damping factor and describes the degree of flatness of the relaxation region. *β* is an asymmetric factor and describes relaxation properties asymmetric around relaxation frequency.

In our work, we will model dielectric permittivities of the samples with Debye relaxation model given with Eq. (10).

#### **3.2. Lorenzian resonance model**

Lorenzian resonant model is represented with Eq. (14) and Eq. (15). Graphical representation of Eq. (14) is in Figure 2 which is an example with *μ<sup>s</sup>* = 9 and *fr* = 25 GHz. Imaginary part of permeabilty in Figure 2 is represented as positive number.

$$\mu(f) = 1 + \frac{\mu\_s - 1}{\left(1 + j\frac{f}{f\_r}\right)^2} \tag{14}$$

where *μ<sup>s</sup>* is static permeability and *fr* is resonant frequency.

$$\mu(f) = 1 + \frac{\mu\_s - 1}{1 + j\frac{f}{f\_1} - \left(\frac{f}{f\_2}\right)^2} \tag{15}$$

Eq. (15) comes when we develop Eq. (14). If 2 *fr*1= *fr*2, Eq. (14) is equivalent to the Eq. (15). Damping and asymmetric factors are introduced in the following equations ([2]).

$$\mu(f) = 1 + \frac{\mu\_s - 1}{1 + j\gamma \frac{f}{f\_r} - \left(\frac{f}{f\_r}\right)^2} \tag{16}$$

$$\mu(f) = 1 + \frac{\mu\_s - 1}{\left(1 + j\gamma \frac{f}{f\_r} - \left(\frac{f}{f\_r}\right)^2\right)^k} \tag{17}$$

**Figure 2.** Real and imaginary permeability represented with Lorenzian model.

4 Will-be-set-by-IN-TECH

1 + *j f fr*

 1 + *j f fr*

 1 + *j f fr* <sup>1</sup>−*<sup>α</sup>*

The terms *α* and *β* are empirical parameters and their values are between 0 and 1. *α* is a damping factor and describes the degree of flatness of the relaxation region. *β* is an asymmetric factor and describes relaxation properties asymmetric around relaxation

In our work, we will model dielectric permittivities of the samples with Debye relaxation

Lorenzian resonant model is represented with Eq. (14) and Eq. (15). Graphical representation of Eq. (14) is in Figure 2 which is an example with *μ<sup>s</sup>* = 9 and *fr* = 25 GHz. Imaginary part of

> *<sup>μ</sup>*(*f*) = <sup>1</sup> <sup>+</sup> *<sup>μ</sup><sup>s</sup>* <sup>−</sup> <sup>1</sup> 1 + *j <sup>f</sup> fr*

*<sup>μ</sup>*(*f*) = <sup>1</sup> <sup>+</sup> *<sup>μ</sup><sup>s</sup>* <sup>−</sup> <sup>1</sup> 1 + *j <sup>f</sup> fr*<sup>1</sup> − *f fr*2

Eq. (15) comes when we develop Eq. (14). If 2 *fr*1= *fr*2, Eq. (14) is equivalent to the Eq. (15).

*<sup>μ</sup>*(*f*) = <sup>1</sup> <sup>+</sup> *<sup>μ</sup><sup>s</sup>* <sup>−</sup> <sup>1</sup>

*<sup>μ</sup>*(*f*) = <sup>1</sup> <sup>+</sup> *<sup>μ</sup><sup>s</sup>* <sup>−</sup> <sup>1</sup> 

1 + *jγ <sup>f</sup>*

1 + *jγ <sup>f</sup>*

*fr* − *f fr*

*fr* − *f fr* 2

Damping and asymmetric factors are introduced in the following equations ([2]).

*ε<sup>s</sup>* − *ε*<sup>∞</sup>

*ε<sup>s</sup>* − *ε*<sup>∞</sup>

*ε<sup>s</sup>* − *ε*<sup>∞</sup>

<sup>1</sup>−*<sup>α</sup>* (11)

*<sup>β</sup>* (12)

*<sup>β</sup>* (13)

<sup>2</sup> (14)

<sup>2</sup> (15)

<sup>2</sup> (16)

*<sup>k</sup>* (17)

*ε*(*f*) = *ε*<sup>∞</sup> +

*ε*(*f*) = *ε*<sup>∞</sup> +

*ε*(*f*) = *ε*<sup>∞</sup> +

presented with equations (11), (12) and (13), respectively.

frequency.

model given with Eq. (10).

**3.2. Lorenzian resonance model**

permeabilty in Figure 2 is represented as positive number.

where *μ<sup>s</sup>* is static permeability and *fr* is resonant frequency.

The *γ* factor is an empirical constant and represents damping factor of a resonance type. The term *k* is also an emirical constant which values are between 0 and 1. It is asymmetrical factor of a resonance type.

In our work we will model complex permeability with Lorenzian resonant model and we will 'tune' models by involving empirical factors *k* and *γ*.
