where the Wagner absorption factor K is given by (T1+t2-τ)t (55.13) t1t2 where T, and T2 are the relaxation times of the two contiguous layers or strata of respective thicknesses d, and d2; T is the overall relaxation time of the two-layer combination and is defined by t=(ef d2+e2 di)(o,d2 +odi), where E, E2, On and o2 are the respective real permittivity and conductivity parameters of the two discrete layers. Note that since ef and e? are temperature- and frequency-dependent and o, and o, are, in addition, lso voltage-dependent, the values of T and e will in turn also be influenced by these three variables. Space charge processes involving electrons are more effectively analyzed, using dc measurement techniques. If retrap- ping of electrons in polymers is neglected, then the decay current as a function of time t, arising from detrapped electrons, assumes the form [ Watson, 1995] dr)=mE (55.14) where n(E)is the trap density and v is the attempt jump frequency of the electrons. The electron current displays the usual tI dependence and the plot of i(t)t versus kTIn(vt) yields the distribution of trap depths. g.(55.14)represents an approximation, which underestimates the current associated with the shallow traps and overcompensates for the current due to the deep traps. The mobility of the free charge carriers is determined by the depth of the traps, the field resulting from the trapped charges, and the temperature. As elevated mperatures and low space charge fields, the mobility is proportional to exp(-AH/kT] and at low temperatures to(T)4 [LeGressus and Blaise, 1992]. A high trapped charge density will create fields. which will in urn exert a controlling influence on the mobility and the charge distribution profile In polymers, shallow traps are of the order of 0.5 to 0.9 eV and deep traps are ca 1.0 to 1.5 ev, while the activation energies of dipole orientation and ionic conduction in solid and liquid dielectrics fall within the same range. It has been known that most charge trapping in the volume occurs in the vicinity of the electrodes; this can now be confirmed by measurement, using thermal and electrically stimulated acoustical pulse methods [Bernstein, 1992]. In the latter method this involves the application of a rapid voltage pulse across a dielectric specimen. The resulting stress ve propagates at the velocity of sound and is detected by a piezoelectric transducer. This wave is assumed ot to disturb the trapped charge; the received electrical signal is then correlated with the acoustical wave to determine the profile of the trapped charge. Errors in the measurement would appear to be principally caused by the electrode surface charge effects and the inability to distinguish between the polarization of polar dipoles and that of the trapped charges [wintle, Temperature influences the real value of the permittivity or dielectric constant e insofar as it affects the density of the dielectric material. As the density diminishes with temperature, e falls with temperature in accordance with the Clausius-Mossotti equation [P]= (e′-1)M (55.15) (E′+2)d where [P] represents the polarization per mole, M the molar mass, d, the density at a given temperature, and E'=E Equation(55. 14)is equally valid, if the substitution E=(n) is made; here nis the real value of the index of refraction. In fact, the latter provides a direct connection with the dielectric behavior at optical ith the complex permittivity, the index of refraction is also a complex quantity, and its imaginary value n"exhibits a loss peak at the absorption frequencies; in contrast with the evalue which an only fall with frequency, the real index of refraction n'exhibits an inflection-like behavior at the absorption frequency. This is illustrated schematically in Fig. 55.2, which depicts the knor n"and n-l values as a function of frequency over the optical frequency regime. The absorption in the infrared results from atomic c 2000 by CRC Press LLC
© 2000 by CRC Press LLC where the Wagner absorption factor K is given by (55.13) where t1 and t2 are the relaxation times of the two contiguous layers or strata of respective thicknesses d1 and d2; t is the overall relaxation time of the two-layer combination and is defined by t = (e¢ 1 d21e¢ 2 d1)/(s1d21s2d1), where e¢ 1, e¢ 2 , s1, and s2 are the respective real permittivity and conductivity parameters of the two discrete layers. Note that since e¢ 1 and e¢ 2 are temperature- and frequency-dependent and s1 and s2 are, in addition, also voltage-dependent, the values of t and e² will in turn also be influenced by these three variables. Space charge processes involving electrons are more effectively analyzed, using dc measurement techniques. If retrapping of electrons in polymers is neglected, then the decay current as a function of time t, arising from detrapped electrons, assumes the form [Watson, 1995] (55.14) where n(E) is the trap density and n is the attempt jump frequency of the electrons. The electron current displays the usual t–1 dependence and the plot of i(t)t versus kTln(nt) yields the distribution of trap depths. Eq. (55.14) represents an approximation, which underestimates the current associated with the shallow traps and overcompensates for the current due to the deep traps. The mobility of the free charge carriers is determined by the depth of the traps, the field resulting from the trapped charges, and the temperature. As elevated temperatures and low space charge fields, the mobility is proportional to exp[–DH/kT] and at low temperatures to (T)1/4 [LeGressus and Blaise, 1992]. A high trapped charge density will create intense fields, which will in turn exert a controlling influence on the mobility and the charge distribution profile. In polymers, shallow traps are of the order of 0.5 to 0.9 eV and deep traps are ca. 1.0 to 1.5 eV, while the activation energies of dipole orientation and ionic conduction in solid and liquid dielectrics fall within the same range. It has been known that most charge trapping in the volume occurs in the vicinity of the electrodes; this can now be confirmed by measurement, using thermal and electrically stimulated acoustical pulse methods [Bernstein, 1992]. In the latter method this involves the application of a rapid voltage pulse across a dielectric specimen. The resulting stress wave propagates at the velocity of sound and is detected by a piezoelectric transducer. This wave is assumed not to disturb the trapped charge; the received electrical signal is then correlated with the acoustical wave to determine the profile of the trapped charge. Errors in the measurement would appear to be principally caused by the electrode surface charge effects and the inability to distinguish between the polarization of polar dipoles and that of the trapped charges [Wintle, 1990]. Temperature influences the real value of the permittivity or dielectric constant e¢ insofar as it affects the density of the dielectric material. As the density diminishes with temperature, e¢ falls with temperature in accordance with the Clausius-Mossotti equation (55.15) where [P] represents the polarization per mole, M the molar mass, do the density at a given temperature, and e¢ = es. Equation (55.14) is equally valid, if the substitution e¢ = (n¢)2 is made; here n¢ is the real value of the index of refraction. In fact, the latter provides a direct connection with the dielectric behavior at optical frequencies. In analogy with the complex permittivity, the index of refraction is also a complex quantity, and its imaginary value n² exhibits a loss peak at the absorption frequencies; in contrast with the e¢ value which can only fall with frequency, the real index of refraction n¢ exhibits an inflection-like behavior at the absorption frequency. This is illustrated schematically in Fig. 55.2, which depicts the kn¢ or n² and n¢ 21 values as a function of frequency over the optical frequency regime. The absorption in the infrared results from atomic K = ( ) t t tt tt +- - t t 1 2 12 1 2 i t kT vt ( ) = n E( ) [ ] ( ) ( ) P M do = ¢ - ¢ + e e 1 2
Time domain methods (ow and high frequency approaches) stormer ratio arm bridge Q-meter method Quasi Broad band t Optical methods and interferometers Log(f) 止的 12 ITHz 300 FIGURE 55.3 Frequency range of various dielectric test methods [Bartnikas, 198 resonance that arises from a displacement and vibration of atoms relative to each other, while an electronic resonance absorption effect occurs over the ultraviolet frequencies as a consequence of the electrons being orced to execute vibrations at the frequency of the external field. The characterization of dielectric materials must be carried out in order to determine their properties for various applications over different parts of the electromagnetic frequency spectrum. There are many techniques and methods available for this purpose that are too numerous and detailed to attempt to present here even in a cursory manner. However, Fig. 55. 3 portrays schematically the different test methods that are commonly used to carry out the characterization over the different frequencies up to and including the optical regime. a direct relationship exists between the time and frequency domain test methods via the Laplace transforms The frequency response of dielectrics at the more elevated frequencies is primarily of interest in the electrical communications field. In contradistinction for electrical power generation, transmission, and distribution, it is the low-frequency spectrum that constitutes the area of application. Also, the use of higher voltages in the electrical power area necessarily requires detailed knowledge of how the electrical losses vary as a function of the electrical field. Since most electrical power apparatus operates at a fixed frequency of 50 or 60 Hz, the main variable apart from the temperature is the applied or operating voltage At power frequencies the dipole losses are generally very small and invariant with voltage up to the saturation fields which exceed substantially the operating fields, being in the order of 107 kV cm-l or more. However, both the space charge polarization and onic losses are highly field-dependent. As the electrical field is increased, ions of opposite sign are increasingly <s gregated; this hinders their recombination and, in effect, enhances the ion charge carrier concentration. As dissociation rate of the ionic impurities is further augmented by temperature increases, combined rises in temperature and field may lead to appreciable dielectric loss. Thus, for example, for a thin liquid film bounded by two solids, tan8 increases with voltage until at some upper voltage value the physical boundaries finally limit the amplitude of the ion excursions, at which point tan& commences a downward trend wit (Boning-Garton effect). The interfacial or space charge polarization losses may evince a rather intricate field dependence, depending upon the manner in which the discrete conductivities of the contiguous media change c 2000 by CRC Press LLC
© 2000 by CRC Press LLC resonance that arises from a displacement and vibration of atoms relative to each other, while an electronic resonance absorption effect occurs over the ultraviolet frequencies as a consequence of the electrons being forced to execute vibrations at the frequency of the external field. The characterization of dielectric materials must be carried out in order to determine their properties for various applications over different parts of the electromagnetic frequency spectrum. There are many techniques and methods available for this purpose that are too numerous and detailed to attempt to present here even in a cursory manner. However, Fig. 55.3 portrays schematically the different test methods that are commonly used to carry out the characterization over the different frequencies up to and including the optical regime. A direct relationship exists between the time and frequency domain test methods via the Laplace transforms. The frequency response of dielectrics at the more elevated frequencies is primarily of interest in the electrical communications field. In contradistinction for electrical power generation, transmission, and distribution, it is the low-frequency spectrum that constitutes the area of application. Also, the use of higher voltages in the electrical power area necessarily requires detailed knowledge of how the electrical losses vary as a function of the electrical field. Since most electrical power apparatus operates at a fixed frequency of 50 or 60 Hz, the main variable apart from the temperature is the applied or operating voltage. At power frequencies the dipole losses are generally very small and invariant with voltage up to the saturation fields which exceed substantially the operating fields, being in the order of 107 kV cm–1 or more. However, both the space charge polarization and ionic losses are highly field-dependent. As the electrical field is increased, ions of opposite sign are increasingly segregated; this hinders their recombination and, in effect, enhances the ion charge carrier concentration. As the dissociation rate of the ionic impurities is further augmented by temperature increases, combined rises in temperature and field may lead to appreciable dielectric loss. Thus, for example, for a thin liquid film bounded by two solids, tand increases with voltage until at some upper voltage value the physical boundaries begin to finally limit the amplitude of the ion excursions, at which point tand commences a downward trend with voltage (Böning–Garton effect). The interfacial or space charge polarization losses may evince a rather intricate field dependence, depending upon the manner in which the discrete conductivities of the contiguous media change FIGURE 55.3 Frequency rangse of various dielectric test methods [Bartnikas, 1987]