PERMITTIVITY AND MEASUREMENTS 3693 00 0 PERMITTIVITY AND MEASUREMENTS tan de ¼ e =e . Mechanisms that contribute to the dielectric loss in heterogeneous mixtures include polar, electronic, atomic, and Maxwell–Wagner responses [7]. At RF and V. K OMAROV microwave frequencies of practical importance and cur- S. WANG rently used for applications in material processing (RF J. TANG Washington State University 1–50 MHz and microwave frequencies of 915 and 2450 MHz), ionic conduction and dipole rotation are domi- nant loss mechanisms [8]: 1. INTRODUCTION 00 00 00 00 s e ¼ ed þ es ¼ ed þ ð2Þ Propagation of electromagnetic (EM) waves in radio e0o frequency (RF) and microwave systems is described mathematically by Maxwell’s equations with corres- where subscripts ‘‘d’’ and ‘‘s’’ stand for contributions due to ponding boundary conditions. Dielectric properties of dipole rotation and ionic conduction, respectively; s is the lossless and lossy materials influence EM field distri- ionic conductivity in S/m of a material, o is the angular bution. For a better understanding of the physical frequency in rad/s, and e0 is the permittivity of free space processes associated with various RF and microwave or vacuum (8.854  10–12 F/m). Dielectric lossy materials devices, it is necessary to know the dielectric properties convert electric energy at RF and microwave frequencies of media that interact with EM waves. For telecommuni- into heat. The increase in temperature (DT) of a material cation and radar devices, variations of complex dielectric can be calculated from [9] permittivity (referring to the dielectric property) over a wide frequency range are important. For RF and micro- DT rC ¼ 5:563  10À11fE2e00 ð3Þ wave applicators intended for thermal treatments of dif- p Dt ferent materials at ISM (industrial, scientific, medical) frequencies, one needs to study temperature and moisture where Cp is the specific heat of the material in content dependencies of the permittivity of the treated JkgÀ 1 1 C À 1, r is the density of the material in kg/m3, materials. E is the electric field intensity in V/m, f is the frequency in Many techniques have been developed for the measure- Hz, Dt is the time duration in seconds, and DT is the tem- ment of material permittivity. Some general descriptions perature rise in the material in 1C. It is clear from Eq. (3) of those methods are provided in the literature [1–4]. The that the rise in temperature is proportional to the mate- measurement results for diverse dielectric media are com- rial’s dielectric loss factor, in addition to electric field in- piled in the literature [5–7]. tensity squared, frequency, and treatment time. The objective of this article is to provide a brief descrip- In dielectric materials, the electric field strength de- tion of basic physical theory for the measurement of per- creases with distance z from the surface and is written as mittivity associated with three popular experimental Àaz methods and to review the permittivities for different ma- E ¼ E0e ð4Þ terials. Most of these data are taken from the literature, but some of permittivity data were measured in Biological The attenuation factor a depends on the dielectric prop- Systems Engineering Department of Washington State erties of the material [5] and is given by University. The measurements were conducted by means 2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 of open-ended coaxial probe method with commercial 1=2 00 2 instrumentation. 2p 41 0@ e A5 a ¼ e 1 þ 0 À 1 ð5Þ l0 2 e 2. THEORY OF PERMITTIVITY where l0 is the free-space wavelength. Substituting Eq. (4) Permittivity is a quantity used to describe dielectric prop- into power equation Eq. (3), one obtains erties that influence reflection of electromagnetic waves at interfaces and the attenuation of wave energy within ma- À2az P ¼ P0e ð6Þ terials. In frequency domain, the complex relative permit- tivity eà of a material to that of free space can be expressed Penetration depth of microwave and RF power is de- in the following form: fined as the depth where the power is reduced to 1/e (e ¼ 2.718) of the power entering the surface (Fig. 1). The eà ¼ e0 À je00 ð1Þ penetration depth dp in meters of RF and microwave en- ergy in a lossy material can be calculated by [5] The real part e0 is referred to as the dielectric constant and represents stored energy when the material is ex- vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 00 dp ¼ u 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ð7Þ posed to an electric field, while the dielectric loss factor e , u which is the imaginary part, influences energy absorption u e00 2 pffiffiffiffiffiffiffi 2pf t2e04 1 þ À 15 and attenuation, and j ¼ À1. One more important para- e0 meter used in EM theory is a tangent of loss angle: Encyclopedia of RF and Microwave Engineering, Edited by Kai Chang ISBN 0-471-27053-9 r 2005 John Wiley & Sons, Inc. 3694 PERMITTIVITY AND MEASUREMENTS Lossy material factor (Fig. 2). For moist dielectric materials ionic conduc- tivity plays a major role at lower frequencies (e.g., o200 MHz), whereas both ionic conductivity and dipole Po Decay of EM power according to Lambert's law rotation of free water play a combined role at microwave frequencies. Maxwell–Wagner polarization arises from EM Po*1/e buildup of charges in the interface between components radiation in heterogeneous systems. The Maxwell–Wagner polariza- tion effect peaks at about 0.1 MHz [7], but in general, its dp Depth into the material z, m contribution is small compared to that of ionic conductiv- ity. For low-moisture media, bound water plays a major Figure 1. Typical penetration depth inside a large-sized material role in dielectric heating in the frequency range between (larger than the wavelength). 20 and 30 GHz [10,11] at room temperature (201C) [12]. For a pure liquid, the Debye model [13] describes the frequency-dependent dielectric properties in the rectan- where c is the speed of light in free space (3  108 m/s). gular form After obtaining the dielectric properties, the penetration depths of electromagnetic energy into selected materials e À e ðe À e Þot eà ¼ e þ s 1 À j s 1 ð8Þ can be calculated at the required frequency. Given fixed 1 1 þ o2t2 1 þ o2t2 dielectric properties, the penetration depth of a material is inversely proportional to frequency. It is, therefore, ex- where e1 is the infinite or high frequency relative permit- pected that in general deeper penetration corresponds to tivity, es is the static or zero-frequency relative permit- lower frequencies, and that higher frequencies result in tivity, and t is the relaxation time in seconds of the greater surface heating. It should be noted that the material. In general, the larger the molecules, the longer dielectric properties of lossy materials vary with frequen- the relaxation time. For a pure liquid, such as water, the cy but penetration depth does not vary exactly as 1/f. dielectric loss factor reaches the maximum at a critical Nevertheless, EM waves with short wavelength do not frequency related to the relaxation time ðfc ¼ 1=2ptÞ.Wa- penetrate deeply into most moist media [7], where the di- ter molecules are polar, and the most important constitu- electric constants and loss factors are relatively high. ent that contributes to the dielectric properties of moist dielectrics. Water molecules bound to the surface of polar materials in monolayers or multilayers have much longer 2.1. Frequency Effect relaxation times than do free water molecules. For exam- Dielectric properties of materials are affected by many ple, the relaxation time of bound water in different food factors, including frequency, temperature, and moisture materials at 201C is determined to be between 0.98 and content. Dielectric properties can vary significantly with 2.00 ns, corresponding to a peak in e00 at B100 MHz, frequency, which will be discussed in detail in this section. whereas the relaxation time t of free water in those foods The frequency-dependent trend of dielectric properties at 201C is between 0.0071 and 0.00148 ns, corresponding can provide important information of the material char- to a peak in e00 at around 16000 MHz [14]. Harvey and acteristics. In theory, electric conduction and various po- Hoekstra [10] found that e00 of monolayer bound water in larization mechanisms (including dipole, electronic, ionic, lysozyme at 251C peaked at 300 MHz and e00 for the second and Maxwell–Wagner) contribute to the dielectric loss layer bound water peaked at about 10000 MHz. The two Conductivity Effect of increasing temperature Log (ε") Free water Effect of Maxwell-Wagner increasing effect temperature Bound water Figure 2. Contributions of various mecha- nisms of the loss factor ðe00Þ of moisture mate- 0.1 100 20,000 MHz. rials as a function of frequency (f) [12]. Log (f) PERMITTIVITY AND MEASUREMENTS 3695 15 Frequency-dependent characteristics of complex biological materials, including moist foods, cannot be described with simple mathematical expressions. 10 ' 2.2. Temperature Effect 5 Temperature of a material has a significant effect on the dielectric properties. Generally, the loss factor increases 0 with increasing temperature at low frequencies due to ionic conductance and decreases with increasing temper- 5 ature at high frequencies due to free-water dispersion " (Fig. 2). In multidispersion materials, for example, the 0 transition is gradual because of the combined effects of 107 33108 109 331010 relaxation and the ionic conduction and there is a U-shape 00 Frequency (Hz) frequency response in e (Fig. 6). Debye related the relax- ation time for the spherical molecule to viscosity and tem- Figure 3. Dielectric constant ðe0Þ and loss factor ðe00Þ as a function perature as the result of randomizing agitation of the of frequency for packed lysozyme samples containing nearly two layers of bound water at 251C [10].