Complex Permittivity and Permeability

Energy lost per cycle Energy stored per cycle 

Each of the above conversion technique has different advantages and limitations. The selection of the technique depends on several factors such as the measured S-parameters, sample length, desired output properties, speed of conversion and accuracies in the converted results. Among above-mentioned procedures, Nicolson-Ross-Weir (NRW) technique [1,88,89] is the most widely used regressive/iterative analysis as it provides direct calculation of both the permittivity (e ) and permeability from the input S-parameters. It is the most commonly used technique for performing such conversions where the measurement of reflection ( T) and transmission (T) coefficient requires all four (S, S, S 22) or a pair (Sjj, S j) of S-parameters of the material under test to be measured. The procedure proposed by NRW method is deduced from the following set of equations  

Conversion technique Input S-parameters Output parameters 

Once these S-parameters are extracted from the NA, simultaneous solving of equation set (9.70) gives the reflection coefficient as  

The condition [ r 1] is used for finding the correct root of the quadratic equation so that parameter X can be expressed as  

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Courtney, W.E. (1970) Analysis and evaluation of a method of measuring the complex permittivity and permeability of microwave insulators, IEEE Trans. 

W. E. Courtney, Analysis and evaluation of a method of measuring the complex permittivity and permeability of microwave insulators, IEEE Trans. Microwave Theory Tech. 18 (1970) 476-485. B.W. Hakki and P.D. Coleman, A Dielectric Resonator Method of Measuring Inductive Capacities in the Millimeter range, IEEE Trans. Microwave Theory Tech. 8 (1960) 402-410. 

I6W. E. Courtney, Analysis and Evaluation of a Method of Measuring the Complex Permittivity and Permeability of Microwave Insulators, IEEE Trans. Microwave Theory Tech., 18,476-85 (1970). 

These books remain the reference works in measuring dielectric and magnetic constants of homogeneous materials. Methods used at that time were limited in frequency band measurements, permitting only the determination of the complex permittivity and permeability at fixed frequencies and sometimes versus temperature variations. 

Despite the first presentation of the network analyser in 1965, the large-band measurement techniques only appeared in 1974. After ameliorations on the accuracy and the development of Von Hippel s methods, the first data treatments were proposed. Weir [III] and Nicholson [112] used the reflection and transmission coefficients (S parameters) resulting when a test sample was inserted into a waveguide or a TEM transmission line as shown in Figure 8.7 From measurements, complex permittivity and permeability values were derived in the range from 50 MHz to 18 GHz. 

Often, free-space methods are complementary to use of the waveguide, coaxial, cavity, one-horn interferometer or open-ended coaxial probe. Indeed, due to their heterogeneity, small composite samples arc not representative of the whole material and ffee-space methods are non-destnictive and contactless. They are also suitable for complex permittivity and permeability measurements under high temperature conditions. 

Moreover ffee-space measurements permit calculation of the complex permittivity and permeability from the measurements of the amplitude and phase variations of a plane wave transmitted through or reflected by the sample (the unknown sample being located between the transmitting and receiving antennas). 

M. Hotta, M. Hayashi, K. Nagata, High temperature measurement of complex permittivity and permeability of Fe304 powders in the frequency range of 0.2 to 13.5 GHz, ISIJ Int. 2010, 5/(3), 491-497. 

Cavity perturbation techniques are used for resonator-based sensing systems [7]. When the complex permittivity or permeability of a region within the resonator is changed, the resonant frequency is shifted and the shift. A/, is given by... 

Figure 14 Components of the complex relative permittivity and relative permeability of a maple wood disc...

The complex permeability (ja) and complex permittivity (e) are two important parameters ... 

The two vital EM attributes of shielding material are complex permittivity [e = (e + je")] and complex permeability = j/+ j ")]> which consists of real (polarization or storage) and imaginary (relaxation or loss) parts as shown in Figure 9.6 [1]. 

The interaction between microwaves and materials is governed by the complex permittivity, = s -f je,", and complex permeability, p. = p + jp", of the matter. The real part of the complex permittivity is also called the dielectric constant and represents the ability of the material to align its electric dipoles when an external electric field is applied to the medium. 

The imaginary part of the complex permittivity is a function of the conductivity of the medium and is given by e" = cr/m, where cr is the electrical conductivity of the material and co is the angular frequency of the wave. The permeability of the medium is similarly defined for magnetic fields and dipoles. In a microwave spectrum, materials are usually classified as insulators or conductors. If e 3> c , the material behaves as an insulator, and if e 3> e", the material behaves as a conductor. 

It was also mentioned earlier, that quantitative information regarding the microwave-material interaction can be deduced by measuring the dielectric properties of the material, in particular of the real and imaginary part of the relative complex permittivity, f = — j , where the term ff includes conduction losses, as well as dielectric losses. The relative permeability is not a constant and strictly depends on frequency and temperature. A different and more practical way to express the degree of interaction between microwaves and materials is given by two parameters the power penetration depth (Dp) and the power density dissipated in the material (P), as defined earlier in a simplified version as follows ... 

The scattering parameters S, and Sji, which are automatically error-corrected in the Automated Network Analyzer (ANA), are measured as a function of frequency. The complex permittivity e and the permeability p are calculated from the measured S-parameters, using a modification of the Nicolson-Ross method, as summarized below. 

Most of the physical properties of the polymer (heat capacity, expansion coefficient, storage modulus, gas permeability, refractive index, etc.) undergo a discontinuous variation at the glass transition. The most frequently used methods to determine Tg are differential scanning calorimetry (DSC), thermomechanical analysis (TMA), and dynamic mechanical thermal analysis (DMTA). But several other techniques may be also employed, such as the measurement of the complex dielectric permittivity as a function of temperature. The shape of variation of corresponding properties is shown in Fig. 4.1. 

The permittivity e (and its magnetic counterpart, the permeability p) are complex in general, with an imaginary component accounting for losses (e.g., the finite electric conductivity). The relative permittivity Zy may be expressed as... 

These tables use the recommended SI (Systeme International) units, with some indication of their relation to other units in the literature, including electromagnetic units (e.m.u.) and the less common electrostatic units (e.s.u.), both in the centimeter gram second system (CGS). The relationships are complex four systems of equations have been used, and each of these has been written in nonrationalized and now rationalized forms/ With rationalization, explicit values and dimensions are given to the permittivity Sq of a vacuum, which was taken as unity and dimensionless in the e.m.u. system, and to the permeability taken as unity and dimensionless in the e.s.u. system. (These two systems are mutually inconsistent on the Maxwell theory the product Sq/ o is equal to c , where c is the speed of light, approximately 3 10 ms ) The physical relationships of the different systems of electrical and magnetic units have been ably described. ... 

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