The electrical engineering handbook CH055

The electrical engineering handbook

Bartnikas, R “Dielectrics and Insulators”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

Dielectrics and Insulators

55.1 Introduction 55.2 Dielectric Losses 55.3 Dielectric Breakdown 55.4 Insulation Aging 55.5 Dielectric Materials Gases • Insulating Liquids • Solid Insulating Materials • Solid-Liquid Insulating Systems

55.1 Introduction

Dielectricsare materials that are used primarily to isolate components electrically from each other or ground

or to act as capacitive elements in devices, circuits, and systems Their insulating properties are directly attributable to their large energy gap between the highest filled valence band and the conduction band The number of electrons in the conduction band is extremely low, because the energy gap of a dielectric (5 to 7 eV)

is sufficiently large to maintain most of the electrons trapped in the lower band As a consequence, a dielectric, subjected to an electric field, will evince only an extremely small conduction or loss current; this current will

be caused by the finite number of free electrons available in addition to other free charge carriers (ions) associated usually with contamination by electrolytic impurities as well as dipole orientation losses arising with polar molecules under ac conditions Often the two latter effects will tend to obscure the miniscule contribution

of the relatively few free electrons available Unlike solids and liquids, vacuum and gases (in their nonionized state) approach the conditions of a perfect insulator—i.e., they exhibit virtually no detectable loss or leakage current

Two fundamental parameters that characterize a dielectric material are its conductivitys and the value of the real permittivity or dielectric constant «¢.By definition, s is equal to the ratio of the leakage current density

J l to the applied electric field E,

(55.1)

Since J l is in A cm–2 and E in V cm–1, the corresponding units of s are in S cm–1 or V–1 cm–1 Alternatively, when only mobile charge carriers of charge e and mobilitym, in cm2 V–1 s–1, with a concentration of n per cm3

are involved, the conductivity may be expressed as

The conductivity is usually determined in terms of the measured insulation resistance R in V; it is then given

by s = d/RA, where d is the insulation thickness in cm and A the surface area in cm2 Most practical insulating materials have conductivities ranging from 10–6 to 10–20 S cm–1 Often dielectrics may be classified in terms of their resistivity value r, which by definition is equal to the reciprocal of s

s = J

E

l

R Bartnikas

Institut de Recherche d’Hydro-Québec

The real value of the permittivity or dielectric constante¢ is determined from the ratio

(55.3)

where C represents the measured capacitance in F and C o is the equivalent capacitance in vacuo, which is

calculated for the same specimen geometry from C o= eo A/d; here eo denotes the permittivity in vacuo and is

equal to 8.854 3 10–14 F cm–1 (8.854 3 10–12 F m–1 in SI units) or more conveniently to unity in the Gaussian

CGS system In practice, the value of eo in free space is essentially the same as that for a gas (e.g., for air,eo =

1.000536) The majority of liquid and solid dielectric materials, presently in use, have dielectric constants

extending from approximately 2 to 10

55.2 Dielectric Losses

Under ac conditions dielectric lossesarise mainly from the movement of free charge carriers (electrons and

ions), space charge polarization, and dipole orientation [Bartnikas and Eichhorn, 1983] Ionic, space charge,

and dipole losses are temperature- and frequency-dependent, a dependency which is reflected in the measured

values of s and e¢ This necessitates the introduction of a complex permittivity e defined by

where e² is the imaginary value of the permittivity, which is equal to s/w Note that the conductivity s

determined under ac conditions may include the contributions of the dipole orientation, space charge, and

ionic polarization losses in addition to that of the drift of free charge carriers (ions and electrons) which

determine its dc value

The complex permittivity, e,is equal to the ratio of the dielectric displacement vector D to the electric field

vector E , i.e., e = D /E Since under ac conditions the appearance of a loss or leakage current is manifest as a

phase angle difference d between the D and E vectors, then in complex notation D and E may be expressed as

D o exp [j (vt – d)] and E o exp[jvt], respectively, where v is the radial frequency term, t the time, and D o and

E o the respective magnitudes of the two vectors From the relationship between D and E, it follows that

(55.5) and

(55.6)

It is customary under ac conditions to assess the magnitude of loss of a given material in terms of the value

of its dissipation factor , tand; it is apparent from Eqs (55.5) and (55.6), that

(55.7)

Examination of Eq (55.7) suggests that the behavior of a dielectric material may also be described by means

of an equivalent electrical circuit It is most commonplace and expedient to use a parallel circuit representation,

consisting of a capacitance C in parallel with a large resistance R as delineated in Fig 55.1 Here C represents

¢ =

Co

¢ =

E

o

o

cos

¢¢ =

E

o

o

sin

e

s we

= ¢¢

the capacitance and R the resistance of the dielectric For an applied voltage V across the dielectric, the leakage current is I l = V /R and the displacement current is I C = j vCV; since tand = I l /I C , then

(55.8)

It is to be emphasized that in Eq (55.8), the quantities R and C are functions of temperature, frequency, and voltage The equivalence between Eqs (55.7) and (55.8) becomes more palpable if I l and I C are expressed as

we²C o V and j we¢C o V, respectively.

Every loss mechanism will exhibit its own characteristic tand

loss peak, centered at a particular absorption frequency, vo for

a given test temperature The loss behavior will be contingent

upon the molecular structure of the material, its thickness, and

homogeneity, and the temperature, frequency, and electric field

range over which the measurements are performed [Bartnikas

and Eichhorn, 1983] For example, dipole orientation losses

will be manifested only if the material contains permanent

molecular or side-link dipoles; a considerable overlap may

occur between the permanent dipole and ionic relaxation

regions Ionic relaxation losses occur in dielectric structures

where ions are able to execute short-range jumps between two

or more equilibrium positions Interfacial or space charge

polarization will arise with insulations of multilayered

struc-tures where the conductivity and permittivity is different for

the individual strata or where one dielectric phase is

inter-spersed in the matrix of another dielectric Space charge traps also occur at crystalline-amorphous interfaces,

crystal defects, and oxidation and localized C-H dipole sites in polymers Alternatively, space charge losses will

occur with mobile charge carriers whose movement becomes limited at the electrodes This type of mechanism takes place often in thin-film dielectrics and exhibits a pronounced thickness effect If the various losses are considered schematically on a logarithmic frequency scale at a given temperature, then the tand and e¢ values will appear as functions of frequency as delineated schematically in Fig 55.2 For many materials the dipole and ionic relaxation losses tend to predominate over the frequency range extending from about 0.5 to 300 MHz, depending upon the molecular structure of the dielectric and temperature For example, the absorption peak of an oil may occur at 1 MHz, while that of a much lower viscosity fluid such as water may appear at approximately 100 MHz There is considerable overlap between the dipole and ionic relaxation loses, because the ionic jump distances are ordinarily of the same order of magnitude as the radii of the permanent dipoles Space charge polarization losses manifest themselves normally over the low-frequency region extending from

10–6 Hz to 1 MHz and are characterized by very broad and intense peaks; this behavior is apparent from Eq (55.7), which indicates that even small conductivities may lead to very large tand values at very low frequencies The nonrelaxation-type electronic conduction losses are readily perceptible over the low-frequency spectrum and decrease monotonically with frequency

The dielectric loss behavior may be phenomenologically described by the Pellat-Debye equations, relating the imaginary and real values of the permittivity to the relaxation time, t, of the loss process (i.e., the frequency

at which the e² peak appears: fo = 1/2 pt), the low-frequency or static value of the real permittivity, es, and the high- or optical-frequency value of the real permittivity, e` Thus, for a loss process characterized by a single relaxation time

(55.9)

tand

w

RC

FIGURE 55.1 (a) Parallel equivalent RC circuit and (b) corresponding phasor diagram.

-+

w t

s

(55.10)

In practice Eqs (55.9) and (55.10) are modified due to a distribution in the relaxation times either because several different loss processes are involved or as a result of interaction or cooperative movement between the discrete dipoles or the trapped and detrapped charge carriers in their own particular environment Since the relaxation processes are thermally activated, an increase in temperature will cause a displacement of the loss peak to higher frequencies In the case of ionic and dipole relaxation, the relaxation time may be described by the relation

(55.11)

where h is the Planck constant (6.624 3 10–34 J s–1), k the Boltzmann constant (1.38 3 10–23 J K–1), DH the activation energy of the relaxation process, R the universal gas constant (8.314 3 103 J K–1 kmol–1), and DS the

entropy of activation For the ionic relaxation process, t may alternatively be taken as equal to 1/2G, where G denotes the ion jump probability between two equilibrium positions Also for dipole orientation in liquids, t may be approximately equated to the Debye term h/4pr3T, where h represents the macroscopic viscosity of

the liquid and r is the dipole radius [Bartnikas, 1994] With interfacial or space charge polarization, which may

arise due to a pile-up of charges at the interface of two contiguous dielectrics of different conductivity and permittivity, Eq (55.10) must be rewritten as [von Hippel, 1956]

(55.12)

FIGURE 55.2 Schematic representation of different absorption regions [Bartnikas, 1987].

-+

¥

w t

ë

ê ê

ù û

ú

-é ë

ê ê

ù û

ú ú

h kT

H RT

S R

+

æ è

ö ø

¥

wt t

wt

w t

1 2

2 2

1

K

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¢1d21e¢2d 1)/(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¢1and e¢2are 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

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]

(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, d o 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

t t

1 2

i t kT

vt n E

( )= ( )

do

-¢ +

e e 1 2

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].

with applied voltage and temperature [as is apparent from the nature of Eqs (55.12) and (55.13)] The exact frequency value at which the space charge loss exhibits its maximum is contingent upon the value of the relaxation time t Figure 55.4 depicts typical tand versus applied voltage characteristics for an oil-impregnated paper-insulated cable model at two different temperatures, in which the loss behavior is primarily governed by ionic conduction and space charge effects The monotonically rising dissipation factor with increasing applied voltage at room temperature is indicative of the predominating ionic loss mechanism, while at 85°C, an incipient decrease in tand is suggestive of space charge effects

55.3 Dielectric Breakdown

As the voltage is increased across a dielectric material, a point is ultimately reached beyond which the insulation will no longer be capable of sustaining any further rise in voltage and breakdown will ensue, causing a short

to develop between the electrodes If the dielectric consists of a gas or liquid medium, the breakdown will be self-healing in the sense that the gas or liquid will support anew a reapplication of voltage until another breakdown recurs In a solid dielectric, however, the initial breakdown will result in a formation of a permanent conductive channel, which cannot support a reapplication of voltage The dielectric breakdown processes are distinctly different for the three states of matter

In the case of solid dielectrics the breakdown is dependent not only upon the molecular structure and morphology of the solid but also upon extraneous variables such as the geometry of the material, the temper-ature, and the ambient environment Since breakdown often occurs along some fault of the material, the breakdown voltage displays a readily perceptible decrease with area and thickness of the specimen due to increased incidence of faults over larger volumes This is indeed part of the reason why thin-film inorganic dielectrics, which are normally evaluated using small-diameter dot counter electrodes, exhibit exceptionally high dielectric strengths With large organic dielectric specimens, recessed electrodes are used to minimize

electrode edge effects, leading to greatly elevated breakdown strengths in the order of 106 to 107 kV cm–1, a range of values which is considered to represent the ultimate breakdown strength of the material or its intrinsic breakdown strength; as the intrinsic breakdown occurs in approximately 10–8 to 10–6 s, an electronic mechanism

is implicated

The breakdown strength under dc and impulse conditions tends to exceed that at ac fields, thereby suggesting the ac breakdown process may be partially of a thermal nature An additional factor, which may lower the ac breakdown strength, is that associated with the occurrence of partial discharges either in void inclusions or at the electrode edges; this leads to breakdown values very much less than the intrinsic value In practice, the breakdowns are generally of an extrinsic nature, and the intrinsic values are useful conceptually insofar as they provide an idea of an upper value that can be attained only under ideal conditions The intrinsic breakdown theories were essentially developed for crystalline dielectrics, for which it was assumed that a very small number

FIGURE 55.4 Loss characteristics of mineral oil-impregnated paper.

of thermally activated electrons can be thermally excited to move from the valence to the conduction band and that under the influence of an external field they will be impelled to move in the direction of the field, colliding with the lattice of the crystalline dielectric and dissipating their energy by phonon interactions [Bartnikas and Eichhorn, 1983] Accordingly, breakdown is said to occur when the average rate of energy gain by the electrons,

A(E, T, T e, j), exceeds that lost in collisions with the lattice, B(T, T e, j) Hence, the breakdown criterion can

be stated as

where E is the applied field, T the lattice temperature, T e the electron temperature, and j an energy distribution constant Thus in qualitative terms as the temperature is increased gradually, the breakdown voltage rises because the interaction between the electrons and the lattice is enhanced as a result of the increased thermal vibrations of the lattice Ultimately, a critical temperature is attained where the electron–electron interactions surpass in importance those between the electrons and the lattice, and the breakdown strength commences a monotonic decline with temperature; this behavior is borne out in NaCl crystals, as is apparent from Fig 55.5 [von Hippel and Lee, 1941] However, with amorphous or partially crystalline polymers, as for example with polyethylene, the maximum in breakdown strength is seen to be absent and only a decrease is observed [Oakes, 1949]; as the crystalline content is increased in amorphous-crystalline solids, the breakdown strength is reduced The electron avalanche concept has also been

applied to explain breakdown in solids, in particular

to account for the observed decrease in breakdown

strength with insulation thickness Since breakdown

due to electron avalanches involves the formation of

space charge, space charges will tend to modify the

conditions for breakdown Any destabilization of the

trapping and detrapping process, such as may be

caused by a perturbation of the electrical field, will

initiate the breakdown event [LeGressus and Blaise,

1992] The detrapping of mobile charge carriers will

be accompanied by photon emission and formation of

the plasma breakdown channel, resulting in the

dissi-pation of polarization energy If dipole interaction is

neglected, the polarization energy due to a trapped

charge is of the order of 5c eV, where c is the dielectric

susceptibility The release of the polarization energy

will be accompanied by electrical tree growth in and

melting of the polymer

The breakdown process in gases is relatively well understood and is explained in terms of the avalanche theory A free electron, occurring in a gas due to cosmic radiation, will be accelerated in a field and upon collision with neutral molecules in its trajectory will eject, if its energy is sufficient, other electrons that will in turn undergo additional collisions resulting in a production of more free electrons If the electric field is sufficiently high, the number of free electrons will increase exponentially along the collision route until ulti-mately an electron avalanche will form As the fast-moving electrons in the gap disappear into the anode, they leave behind the slower-moving ions, which gradually drift to the cathode where they liberate further electrons with a probability g When the height of the positive ion avalanche becomes sufficiently large to lead to a regeneration of a starting electron, the discharge mechanism becomes self-sustaining and a spark bridges the two electrodes The condition for the Townsend breakdown in a short gap is given by

g[exp(ad) 21] = 1 (55.17) where d is the distance between the electrodes and a represents the number of ionizing impacts per electron

per unit distance The value of g is also enhanced by photoemission at the cathode and photon radiation in

FIGURE 55.5 Dielectric breakdown characteristics of

sodium chloride [von Hippel and Lee, 1941] and polyethylene [Oakes, 1949].

the gas volume by the metastable and excited gas atoms or molecules In fact, in large gaps the breakdown is governed by steamer formation in which photon emission from the avalanches plays a dominant role Break-down characteristics of gases are represented graphically in terms of the Paschen curves, which are plots of the

breakdown voltage as a function of the product of gas pressure p and the electrode separation d Each gas is characterized by a well-defined minimum breakdown voltage at one particular value of the pd product.

The breakdown process in liquids is perhaps the least understood due to a lack of a satisfactory theory on the liquid state The avalanche theory has been applied with limited success to explain the breakdown in liquids,

by assuming that electrons injected from an electrode surface exchange energy with the atoms or molecules of the liquid, ultimately causing the atoms and molecules to ionize and thus precipitating breakdown Recent investigations, utilizing electro-optical techniques, have demonstrated that breakdown involves steamers with tree- or bushlike structures that propagate from the electrodes [Bartnikas, 1994] The negative streamers emerging from the cathode form due to electron emission, while positive steamers originating at the anode are due to free electrons in the liquid itself The breakdown of liquids is noticeably affected by electrolytic impurities

as well as water and oxygen content; also, macroscopic particles may form bridges between the electrodes along which electrons may hop with relative ease, resulting in a lower breakdown As in solids, there is a volume effect and breakdown strength decreases with thickness; a slight increase in breakdown voltage is also observed with viscosity

In both solid and liquid dielectrics, the breakdown strength under dc and impulse fields is markedly greater than that obtained under ac fields, thus suggesting that under ac conditions the breakdown may be partially thermal in nature Thermal breakdown occurs at localized hot spots where the rate of heat generated exceeds that dissipated by the surrounding medium The temperature at such hot spots continues to rise until it becomes sufficiently high to induce fusion and vaporization, causing eventually the development of a channel along which breakdown ensues between the opposite electrodes Since a finite amount of time is required for the heat buildup to occur to lead to the thermal instability, thermally induced breakdown is contingent upon the time

of the alternating voltage application and is thus implicated as the leading cause of breakdown in many dielectrics under long-term operating conditions However, under some circumstances thermal instability may develop over a very short time; for example, some materials have been found to undergo thermal breakdown when subjected to very short repetitive voltage pulses In low-loss dielectrics, such as polyethylene, the occur-rence of thermal breakdown is highly improbable under low operating temperatures, while glasses with signif-icant ionic content are more likely to fail thermally, particularly at higher frequencies

The condition for thermal breakdown may be stated as

where the left-hand side represents the heat transfer in J s–1 along a length l (cm) of sectional area A (cm2) of the dielectric surface in the direction of the temperature gradient due to the temperature difference DT, in °C, such that the units of the thermal conductivity constant K are in J °C –1 cm–1 s–1 The right-hand side of

Eq (55.18) is equal to the dielectric loss dissipated in the dielectric in J s–1, where E is the external field, e¢ the

real value of the permittivity, and tand the dissipation factor at the radial frequency v

Other causes of extrinsic breakdown are associated with particular defects in the dielectric or with the environmental conditions under which the dielectric material is employed For example, some dielectrics may contain gas-filled cavities that are inherent with the porous structure of the dielectric or that may be inadvert-ently introduced either during the manufacturing process or created under load cycling If the operating electrical field is sufficiently elevated to cause the gas within the cavities to undergo discharge, the dielectric will be subjected to both physical and chemical degradation by the partial discharges; should the discharge process be sustained over a sufficiently long period, breakdown will eventually ensue

With overhead line insulators or bushings of electrical equipment, breakdown may occur along the surface rather than in the bulk of the material Insulator surfaces consisting of porcelain, glass, or polymeric materials (usually elastomers), may become contaminated by either industrial pollutants or salt spray near coastal areas, leading to surface tracking and breakdown below the normal flashover voltage Surface tracking is enhanced

in the presence of moisture, which increases the surface conductivity, particularly in the presence of ionic contaminants [Bartnikas, 1987] The latter is measured in S or V–1 and must be distinguished from the volume