8.1 Physical Quantities of Dielectrics
The physical quantities esntial to understanding the electric prop-erties of dielectric materials are briefly described in the following paragraphs.
8.1.1 Permittivity of Vacuum
The permittivity of a vacuum , sometimes called the permittivity of empty space by electrical engineers, is denoted ε0 and expresd in
F.m –1
. It is defined by Coulomb’s law in a vacuum. The modulus of the electrostatic force, F 12, expresd in newtons (N), between two point electric charges in a vacuum q 1 or q 2, expresd in coulombs (C), parated by a distance r 12 in meters (m), is given by the follow-ing equation:
F 12 = (1/4πε0)·q 1q 2/r 12·e r ,
with ε0 = 1/μ0.c 2 = 8.854 118 781 7 × 10–12 F.m –1
.
8.1.2 Permittivity of a Medium
The permittivity of a medium , denoted ε, expresd in farad per
meter (F.m –1
), is defined by Coulomb’s law applied to the medium. Actually, the modulus of the electrostatic force, F 12, in newtons (N) between two point electric charges in a medium q 1 or q 2, in cou-lombs (C), parated by a distance r 12 in meters (m), is given by the following equation:
F 12 = (1/4πε)·q 1q 2/r 12·e r
Insulators and Dielectrics
8.1.3 Relative Permittivity and Dielectric Constant
The relative permittivity , denoted εr , is a dimensionless physical quantity equal to the ratio
of the permittivity of the medium to the permittivity of a vacuum. It is also called the dielec-tric constant of the medium. Hence, it is defined by the equation ε = ε0·εr .
Moreover, the relative permittivity of an insulating material depends on the frequency, ν, in Hertz (Hz) of the applied electric field and can be described as a complex physical quan-tity, where the imaginary part is related to dielectric loss:
ε = εr '+ j·εr ".
Usually dielectric constants of materials listed in tables and databas are measured at a fre-quency of 1 MHz unless otherwi specified.
8.1.4 Capacitance
Two conducting bodies or electrodes parated by a dielectric constitute a capacitor (formerly a condenr ). If a positive charge is placed on one electrode, an equal negative charge is simul-taneously induced in the other electrode to maintain electrical neutrality. Therefore the prin-cipal characteristic of a capacitor is that it can store an electric charge Q , expresd in cou-lombs (C), that is directly proportional to the voltage applied, expresd in volts (V), according to the following equation:
Q = CV ,
where C is the capacitance expresd in farad (F). Hence the capacitance value is defined as 1 F when the electric potential difference (i.e., voltage) across the capacitor is 1 V, and a charging current of 1 A flows for 1 s. The required charging current, i , in amperes (A) is therefore defined as:
i = d Q /d t = C d V /dt.
The farad is a very large unit of measurement and is not encountered in practical applica-tions, so submultiples of the farad are commonly encountered; in decreasing order of u, they are the picofarad (pF), the nanofarad (nF), and the microfarad (μF). Another important point is that the dielectric properties of a medium relate to its ability to conduct dielectric lines. This must be clearly distinguished from its insulating properties, which relate to its ability not to conduct an electric current. For instance, an excellent electrical insulator can rupture dielectrically at low breakdown voltages.
8.1.5 Temperature Coefficient of Capacitance
The temperature coefficient of capacitance , denoted a or TCC and expresd in K –1
, is deter-mined accurately by measurement of the capacitance change at various temperatures from
a reference point usually t at room temperature (T 1) up to a required higher temperature (T 2) by means of an environmental chamber:
a = 1/T ·∂C /∂T .
In the electrical industry, the temperature coefficient of capacitance is usually expresd as the percent change in capacitance, or in parts per million per degree Celsius (ppm/°C). More-over, for industrial dielectrics, it is usually plotted in the temperature range –55°C to +125°C.
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Insulators and Dielectrics
8.1.6 Charging and Discharging a Capacitor
When charging a capacitor, with a capacitance C and an internal resistance R , by connecting it to a direct-current power supply of voltage E , at each instant the current that flows in the circuit is given by:
i (t ) = C d V /d t = (E – V )/R .
At the beginning, an important current flows but decreas exponentially until the voltage of the capacitor reaches that of the power supply, at which point the electrical charge is clo to CE .
When discharging a capacitor of capacitance C into a resistance R , the current intensity as a function of time is given by:
i (t ) = –C d V /d t = V /R .
Table 8.1. Charging and discharging a capacitor Quantity Charging
Discharging Voltage V (t) = E ·[1 – exp(–t /RC)] V (t) = E ·exp(–t /RC) Current I (t) = (E /R )·exp(–t /RC) I (t) = (E /R )·exp(–t /RC)
Charge Q(t) = CE ·[1 – exp(–t /RC)]Q(t) = CE ·exp(–t /RC)
8.1.7 Capacitance of a Parallel-Electrode Capacitor
The capacitance of a capacitor with parallel electrodes is directly proportional to the active electrode area and inverly proportional to the dielectric thickness as described by the following equation:
C = ε0·εr ·(A /d ).
8.1.8 Capacitance of Other Capacitor Geometries
For more complicated capacitor geometries the capacitance is given in Table 8.2.
Table 8.2. Capacitance of capacitors of different geometries Description Theoretical capacitance formula Parallel finite plates
C = ε0·εr ·(A /d )
Coaxial cylinders of infinite length of inner radius R 1 and outer radius R 2
C = 2π ε0·εr ·[1/ln(R 2/R 1)] Concentric spheres of inner radius R 1and outer radius R 2
C = 4π ε0·εr ·[R 2·R 1/(R 2 – R 1)] Two parallel wires of infinite length C = ε0·εr ·[1/ln(
D /r )]
8.1.9 Electrostatic Energy Stored in a Capacitor
The electrostatic energy , W, expresd in joules (J), stored in a capacitor is given by the fol-lowing equation:
W = 1/2 QV = 1/2 CV 2
= 1/2 Q 2
/C .
8.1.10 Electric Field Strength
The electric field strength , denoted by E , is a vector quantity directed from negative charge
regions to positive charge regions. Its module is expresd in volt per metre (V.m –1
). It is clearly defined by the vector equation as follows:
E = – ∇V .
For insulating materials, it is common to define the dielectric field strength , or sometimes improperly breakdown voltage , denoted by E d , which is the maximum electric field that the material can withst
and before the sparking begins (i.e., dielectric breakdown). The common non-SI units are the volt per micrometer (V/μm) or, in the US or UK systems, the volt per mil (V/mil). The dielectric field strength is a measure of the ability of a material to withstand a large electric field strength without the occurrence of an electrical breakdown.
8.1.11 Electric Flux Density
The electric flux density or electric displacement , denoted by D , is a vector quantity defined in a vacuum as the product of the electric field strength and the permittivity of vacuum. Its mod-ule is expresd in coulombs per square meter (C.m –2
). It is defined by the following equation:
D = ε0·
E .
In a medium, the electric flux density is defined as the product of the electric field strength by permittivity of the medium as follows:
D = ε0εr ·
E = ε·E .
8.1.12 Microscopic Electric Dipole Moment
Molecules having a dissymetric electron cloud distribution exhibit a permanent electric dipole moment. The electric dipole moment of a molecule, i , is a vector physical quantity, denoted by μi or p i , with a modulus expresd In some old textbooks, it was ex-presd in the obsolete unit called the debye (D):
1 D = 10–18
esu (E) = (10–19
/c) C.m (E) = 3.33564095198 × 10–30
<
The electric dipole moment between two identical electric charges, q , in coulombs (C) pa-rated by a distance d i , in meters (m), is given by the equation:
p i = q i d i .
NB: When an electric dipole is compod of two point charges +q and –q , parated by a distance r , the direction of the dipole moment vector is taken to be from the negative to the positive electric charge. However, the opposite convention was adopted in physical chemis-try but is to be discouraged. Moreover, the dipole moment of an ion depends on the choice of the origin.
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Insulators and Dielectrics
8.1.13 Polarizability
The polarizability of an atom or a molecule, which describes the respon of the electron cloud (i.e., Fermi gas) to an external electric field strength E , is the proportional factor exist-ing between the resulting dipole moment and the electric field strength; however, two equa-tions exist in the literature:
α=⋅ E p or [][]α
αα==ε=G G G
0E E D p . The first quantity, denoted α
, is expresd in C.m 2.V –1
, while the cond quantity [α ] is the absolute polarizability expresd in cubic meters (m 3
). The polarizability of a dielectric ma-terial, containing n atoms per unit volume (m –3
) and having a relative dielectric permittivity εr , is given by the Clausius–Mosotti equation:
[α ] = (3ε0/n )·[(εr – 1)/(εr + 2)].
8.1.14 Macroscopic Electric Dipole Moment
The macroscopic electrical dipole moment, μ, expresd in C.m, is the summation of all the contributions of individual microscopic electric dipole moments:
μ = ∑μi .
8.1.15 Polarization
The dielectric polarization , or simply polarization , within dielectric materials is a vector
physical quantity, denoted by P , and its module is expresd in C.m –2
. Electric polarization aris due to the existence of atomic and molecular forces and appears whenever electric charges in a material are displaced with respect to one another under the influence of an applied external electric field strength, E . On the other hand, the electric polarization repre-nts the total electric dipole moment contained per unit volume of the material averaged
over the volume of a crystal cell lattice, V, expresd in cubic meters (m 3
):
P = N μ/V = n ·μ.
The negative charges within the dielectric polarization are displaced toward the positive region, while the positive charges shift in the opposite direction. Becau electric charges are not free to move in an insulator owing to atomic forces involving them, restoring forces are activated that either do work or cau work to be done on the circuit, i.e., energy is trans-ferred. On charging a dielectric material, the polarization effect opposing the applied field draws charges onto the electrodes, storing energy. By contrast, on discharge, this energy is relead. As a result of the above microscopic interaction,
materials that posss easily po-larizable charges will greatly influence the degree of charge that can be stored in a material. The proportional increa in storage ability of a dielectric material with respect to a vacuum is defined as the relative dielectric permittivity (sometimes called dielectric constant) of the material. The degree of polarization P is related to the relative permittivity and to the electric field strength E as follows:
P = D – ε0E = (εr – 1)ε0E .
8.1.16 Electric Susceptibility
The electric susceptibility of a material, denoted χe , is a dimensionless physical quantity
defined as the ratio of the electric polarization over the electric flux density, according to the following equation:
P = ε0χe E .
Therefore, the relation existing between electric susceptibility and relative permittivity is given by:
χe = (εr – 1).
In fact, when the electric field strength is greater than the value of the interatomic electric field (i.e., 100 to 10,000 MV/m), the relationship between the polarization and the electric field strength becomes nonlinear and the relation can be expanded in a Taylor ries:
P = ε0 [χe (1)E + (1/2)χe (2)E 2 + (1/2)χe (3)E 3
+ ... ],
where χe (1)
is the linear dielectric susceptibility and χe (2)
and χe (3)
are, respectively, the first and
cond hypersusceptibilities (sometimes called optical susceptibilities) expresd in C 3.m 3J –2
and C 4.m 4.J –3
. In a medium that is anisotropic, the electric susceptibilities are tensor quan-tities of rank 2, 3, and
4, while for an isotropic medium (e.g., gas, liquids, amorphous sol-ids, and cubic crystals) or for a crystal with a centrosymmetric unit cell, the cond hyper-susceptibility is zero by symmetry.
8.1.17 Dielectric Breakdown Voltage
The dielectric breakdown voltage of a dielectric material, which is sometimes shortened to breakdown voltage, is the maximum value of the potential difference that the material can sustain without losing its insulating properties and before sparking appears (i.e., dielectric failure). It is commonly symbolized by V b and expresd in volts (V). However, the break-down voltage is a quantity that depends on the thickness of the insulator and hence is not an intrinsic property that describes the material. Therefore, it is preferable to u the dielectric field strength.
8.1.18 Dielectric Absorption
Dielectric absorption is the measurement of a residual electric charge on a capacitor after it is discharged and is expresd as the percent ratio of the residual voltage to the initial charge voltage. The residual voltage, or charge, is attributed to the relaxation phenomena of polari-zation. Actually, the polarization mechanisms can relax the applied electric field. The inver situation, whereby there is a relaxation on depolarization, or discharge, also applies. A small fraction of the polarization may i
n fact persist after discharge for long time periods and can be measured in the device with a high impedance voltmeter. Dielectrics with higher dielec-tric permittivity, and therefore more polarizing mechanisms, typically display more dielec-tric absorption than lower dielectric permittivity materials.
