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• 1. MT-201A MATERIALS SCIENCE Electrical and Electronic Materials Module 7 Dielectric Materials Compiled by Dr. Vikram Dabhade Dept. of Metallurgical and Materials Engineering, Indian Institute of Technology Roorkee, Roorkee-247667, Uttrakhand.
• 2. INTRODUCTION&#x2022; Dielectric material: is one that is electrically insulating (non-metallic)and exhibits or may be made to exhibit an electric dipole structure; that is,there is a separation of positive and negative electrically charged entitieson a molecular or atomic level.&#x2022; While insulating materials are used to resist the flow of current, dielectricmaterials are used to store electrical energy.Capacitance&#x2022; When a voltage is applied across a capacitor, one plate becomes positivelycharged, the other negatively charged, with the corresponding electric fielddirected from the positive to the negative. The capacitance C is related to thequantity of charge stored on either plate Q by C=Q/Vwhere V is the voltage applied across the capacitor. The units of capacitanceare coulombs per volt, or farads (F).
• 3. &#x2022; Now, consider a parallel-plate capacitor with a vacuum in the region between the plates. The capacitance may be computed from the relationship C = &#x3B5;o A lwhere A represents the area of the plates and l is the distance between them.&#x2022; The parameter &#x3B5;o is called the permittivity of a vacuum, is a universalconstant having the value of 8.86 x 10-12 F/m.
• 4. If a dielectric material is inserted into the region within the plates then C=&#x3B5; A lwhere &#x3B5; is the permittivity of this dielectric medium, which will be greaterin magnitude than &#x3B5;o. The relative permittivity &#x3B5;r often called the dielectricconstant, is equal to the ratio &#x3B5;r = &#x3B5; &#x3B5;owhich is greater than unity and represents the increase in charge storingcapacity by insertion of the dielectric medium between the plates. Thedielectric constant is one material property that is of prime consideration forcapacitor design.
• 5. Dielectric Constant (Permittivity)As explained above, dielectric constant or permittivity of a material is definedas the &#x201C;ratio of capacitance of a capacitor with that material as dielectricbetween the conducting plates, to the capacitance of the same capacitor withvacuum as dielectric medium.&#x201D; &#x3B5;r = &#x3B5; / &#x3B5;o or &#x3B5;r = c / coThe relative permittivity of vacuum is 1.00 and that of air is 1.00058 whichis taken as unity. Gases have a relative permittivity slightly higher than unity,while polar liquids and ionic solids have high values of permittivity.
• 6. Dielectric Strength (breakdown voltage)&#x2022; Dielectric strength of an insulating material is the maximum electric fieldstrength that it can withstand intrinsically without breaking down, i.e., withoutexperiencing failure of its insulating properties or it is the minimum electricfield that produces breakdown in a given configuration of dielectric material.&#x2022; The dielectric strength is also know as the breakdown voltage i.e. the voltagebelow which the dielectric material remains stable but above which it results inthe destruction of insulating properties.&#x2022; The theoretical dielectric strength of a material is an intrinsic property of thebulk material and is dependent on the configuration of the material on whichthe field is applied.&#x2022; At breakdown, the electric field frees bound electrons. If the applied electricfield is sufficiently high, free electrons may become accelerated to velocitiesthat can liberate additional electrons during collisions with neutral atoms ormolecules in a process called avalanche breakdown.
• 7. &#x2022; Breakdown occurs quite abruptly (typically in nanoseconds)., resulting inthe formation of an electrically conductive path and a disruptive dischargethrough the material. For solid materials, a breakdown event severelydegrades, or even destroys, its insulating capability.&#x2022; Factors affecting dielectric strength1. It increases with the increase in thickness of the specimen. (Directly proportional)2. It decreases with the increase in operating temperature. (Inversely proportional)3. It decreases with the increase in frequency. (Inversely proportional)4. It decreases with the increase in humidity. (Inversely proportional)The field strength at which break down occurs in a given case is dependenton the respective geometries of the dielectric (insulator) and the electrodeswith which the electric field is applied, as well as the rate of increase at whichthe electric field is applied. Because dielectric materials usually containminute defects, the practical dielectric strength will be a fraction of theintrinsic dielectric strength seen for ideal, defect free, material.
• 8. Table: Dielectric strength (in MV/m) of various common materials: Substance Dielectric Strength (MV/m)Helium 0.15Air 3.0 (depends on pressure)Alumina 13.4Window glass 9.8 - 13.8Silicone oil, Mineral oil 10 - 15Benzene 16Polystyrene 19.7Polyethylene 18.9 - 21.7Neoprene rubber 15.7 - 27.6Ultra pure Water 30High Vacuum (field emission limited) ] 20 - 40 (depends on electrode shape)Fused silica 25 - 40Waxed paper 40 - 60PTFE (Teflon) 60Mica [11] 20 - 70Thin films of SiO2 in ICs &gt; 1000
• 9. Dielectric Loss&#x2022; The dielectric material separating two electrodes / conductors / plates isstressed when subjected to a potential. When the potential is reversed, thestress is also reversed.&#x2022; This change of stress involves molecular rearrangement within thedielectric. This involves energy loss with each reversal. This is because themolecules have to overcome a certain amount of internal friction in theprocess of alignment. The energy expended in the process is released as heatin the dielectric.&#x201C;The loss appearing in the form of heat due to reversal of electric stressescompelling molecular rearrangement is known as dielectric loss&#x201D;&#x2022; The dielectric loss is not appreciable at ordinary frequency of 50 Hz, but incommunication systems where frequencies of mega hertz are used, the heatreleased will be very high and can be observed by the increase in thetemperature of the dielectric material.
• 10. Dielectric Polarization&#x2022; A material is made up of atoms; each atom consists of a cloud of negativecharge (electrons) bound to and surrounding a positive point charge at itscenter. Because of the comparatively huge distance between them, none of theatoms in the dielectric material interact with one another.&#x2022; In the presence of an electric field the charge cloud is distorted, as shown inthe top right of the figure.&#x2022; This can be reduced to a simple dipole using the superposition principle. Adipole is characterized by its dipole moment, a vector quantity shown in thefigure as the blue arrow labeled M. It is the relationship between the electricfield and the dipole moment that gives rise to the behavior of the dielectric Figure: Electric field interaction with an atom under the classical dielectric model
• 11. Polar and Non-Polar DielectricsPolar Dielectrics&#x2022; Like water, alcohol, CO2, NH3, HCl etc. aremade of polar atoms/molecules.&#x2022; In polar molecules when no electric field isapplied centre of positive charges does notcoincide with the centre of negative charges.&#x2022; A polar molecule has permanent electric dipole moment in the absence ofelectric field also. But a polar dielectric has net dipole moment is zero in theabsence of electric field because polar moleculesare randomly oriented as shown in figure.&#x2022; In the presence of electric field polar molecules tends to line up in thedirection of electric field, and the substance has finite dipole moment.
• 12. Non - Polar Dielectrics&#x2022; Like N2, O2, Benzene, Methane etc. are made of non-polar atoms/molecules.In non-polar molecules, when no electric field is applied the centre of positivecharge coincides with the centre of negative charge in the molecule. Eachmolecule has zero dipole moment in its normal state.&#x2022; When electric field is applied, positive charge experiences a force in thedirection of electric field and negative charge experiences a force in thedirection opposite to the field i.e., molecules becomes induced electricdipole.
• 13. 7.1 Matter Polarization and Relative PermittivityRelative PermittivityConsider a parallel plate capacitor with vacuum as the dielectric mediumbetween the plates (Fig.(a)). The plates are connected to a constant voltagesupply V. Let Qo be the charge on the plates. The capacitance Co of theparallel plate capacitor in free space is defined by Co = Qo / V Co = capacitance of a parallel plate capacitor in free space Qo = charge on the plates V = voltage
• 14. When a dielectric slab (slab of non-conducting material) is inserted into thisparallel plate capacitor (Fig.b &amp; c) with V kept the same. Now due to theinsertion of the dielectric slab, there is an external current flow that indicatesthat there is additional charge being stored on the plates. The charge on theelectrodes increases from Qo to Q. Because now there is greater amount ofcharge stored on the plates, the capacitance of the system in Fig.(a) is largerthan that in Fig.(b) by the ratio Q to Qo.The relative permittivity (or the dielectric constant) &#x3B5;r is defined to reflect thisincrease in the capacitance or the charge storage capacity by virtue of having adielectric medium. If C is the capacitance with the dielectric medium (Fig.(c))then: &#x3B5;r = Q/Qo = C/CoThe increase in the stored charge is due to the polarization of the dielectric bythe applied field.
• 15. Dipole Moment and Electronic PolarizationAn electrical dipole moment is simply a separation between a negative andpositive charge of equal magnitude Q in a system of charges. In the simple caseof two point charges, one with charge + q and one with charge &#x2212; q, the electricdipole moment p is: p = Qawhere a is the displacement vector pointing from the negative charge to thepositive charge (a in the scalar form is the bond length in the molecule whichhas got polarized)
• 16. &#x2022; The net charge within a neutral atom is zero. In the absence of an electric fieldthe center of negative charge of the electrons coincides with the positivenuclear charge, means that the atom has no net dipole moment (Fig.7.3(a)).&#x2022; With an application of electric field induced dipole moment will take placecausing electrons being much lighter than the positive nucleus to get displacedby the field. This results in the separation of the negative charge center from thepositive charge center as shown in Fig.7.3(b).&#x2022; This separation of negative and positive charges and the resulting induceddipole moment are termed polarization. An atom is said to be polarized if itpossesses an effective dipole moment, that is, if there is a separation betweenthe centers of negative and positive charge distributions.&#x2022; The induced dipole moment depends on the electric field causing it. We definea quantity called the polarizability &#x3B1; to relate the induced dipole momentpinduced to the field E causing it, pinduced = &#x3B1;Ewhere &#x3B1; is a coefficient called the polarizability of the atom. Since thepolarization of a neutral atom involves the displacement of electrons &#x3B1; isgenerally called electronic polarization denoted as &#x3B1;e.
• 17. Polarization Vector P&#x2022; When a material is placed in an electric field, the atoms and molecules of thematerial become polarized, so we have a distribution of dipole moments in thematerial. We can visualize this effect with the insertion of a dielectric slab intothe parallel plate capacitor as shown in Fig.(a).&#x2022; The placement of the dielectric slab into an electric field polarizes themolecules in the material. The induced dipole moments all point in the directionof the field.
• 18. &#x2022; Consider a polarized medium alone, as shown in Fig.(b) in which everypositive charge has a negative charge next to it and vice versa. There istherefore no net charge within the bulk. But the positive charges of the dipolesappearing at the right hand face are not canceled by negative charges of anydipoles at this face. There is therefore a surface charge +Qp on the right handface that results from the polarization of the medium.&#x2022; Similarly, there is a negative charge -Qp with the same magnitude appearingon the left hand face due to the negative charges of the dipoles at this face.These charges are bound and are a direct result of the polarization of themolecules. They are termed surface polarization charges.&#x2022; Fig(c) emphasizes this aspect of dielectric behavior in an electric field byshowing the dielectric and its polarization charges only.&#x2022; We represent the polarization of a medium by a quantity called polarizationP, which is defined as the total dipole moment per unit volume, P = 1 [p1 + p2 + &#x2026;&#x2026;+ pN] VolumeWhere p1, p2,&#x2026;.pN are the dipole moments induced at N molecules in thevolume.
• 19. &#x2022; If pav is the average dipole moment per molecule, then an equivalentdefinition of P is P = Npav&#x2022; To calculate the polarization P for the polarized dielectric we need to sum allthe dipoles in the medium and divide by the volume Ad as in eqn.1. Howeverthe polarized medium can be simply represented as in Fig.(c) in terms ofsurface charge +QP and -QP, which are separated by the thickness distance d.&#x2022; We can view this arrangement as one big dipole moment per unit volume, themagnitude of P is P = ptotal / volume = Qpd / Ad = Qp / ABut Qp / A is the surface polarization charge density &#x3C3;p, so P = &#x3C3;p&#x2022; Polarization is a vector and the above equation only gives its magnitude. Forthe rectangular slab in Fig.7.5., the direction of P is normal to the surface. For+&#x3C3;p (right face), it comes out from the surface and for -&#x3C3;p (left face), it isdirected into the surface. If Pnormal is the component of P normal to the surfacewhere the polarization charge density is &#x3C3;p, as shown in Fig.7.6, then, Pnormal = &#x3C3;p
• 20. Local Field Eloc&#x2022; The electronic polarizability &#x3B1;e is related to relative permittivity &#x3B5;r by therelation &#x3B5;r = 1 + N&#x3B1;e / &#x3B5;o. Relative permittivity &#x3B5;r is a macroscopic propertywhile electronic polarizability &#x3B1;e is related to microscopic polarizationmechanisms. This equation assumes that the field acting on an individualatom or molecule is the field E, which is assumed to be uniform within thedielectric.&#x2022; However the induced polarization depends on the actual field experiencedby the molecule. But there are polarized molecules within the dielectic withtheir negative and positive charges separated so that the field is not constanton the atomic scale as we move through the dielectric. This is depicted inFig.7.7.&#x2022; The field experienced by an individual molecule is actually different than E,which represents the average field in the dielectric. As soon as the dielectricbecomes polarized, the field at some arbitrary point depends not only on thecharges on the plates (Q) but also on the orientations of all the other dipolesaround this point in the dielectric. When averaged over some distance, say athousand molecules, this field becomes E, as shown in Fig.7.7.
• 21. &#x2022; The actual field experienced by a molecule in a dielectric is defined as thelocal field and denoted by Eloc. It depends not only on the free charges on theplates but also on the arrangement of all the polarized molecules around thispoint. In evaluating Eloc we simply remove the molecule from this point andcalculate the field at this point coming from all sources, including neighbouringpolarized molecules as shown in Fig.7.7.
• 22. 7.2 Electronic Polarization: Covalent Solids&#x2022; When a field is applied to a solid substance, the constituent atoms ormolecules become polarized as shown in Fig.7.8. The electron clouds withineach atom becomes shifted by the field, and this gives rise to electronicpolarization.&#x2022; This type of electronic polarization within an atom, however, is quite smallcompared with the polarization due to the valence electrons in the covalentbonds within the solid.&#x2022; For example, in crystalline silicon, there are electrons shared withneighboring Si atoms in covalent bonds as shown in Fig.7.8. These valenceelectrons form bonds (i.e. become shared) between the Si atoms because theyare already loosely bound to their parent atoms. Thus, they readily respond toan applied field and become displaced.&#x2022; This type of electronic polarization, due to the displacement of electrons incovalent bonds is responsible for the large dielectric constants of covalentcrystals.
• 23. (a) Valence electrons in covalent bonds in the absence of an applied field.(b) When an electric field is applied to a covalent solid, the valence electrons in thecovalent bonds are shifted very easily with respect to the positive ionic cores. Thewhole solid becomes polarized due to the collective shift in the negative chargedistribution of the valence electrons.
• 24. 7.3 Polarization MechanismsIn addition to electronic polarization, there are a number of other polarizationmechanisms such as:1. Ionic polarization2. Orientational (Dipolar) Polarization3. Interfacial Polarization and4. Total Polarization (which is the sum of electronic, ionic and dipolar)
• 25. Ionic Polarization&#x2022; This type of polarization occurs in ionic crystals such as NaCl, KCl and LiBr.Ionic crystals have distinctly identifiable ions, ex, Na+ and Cl-, located at welldefined lattice sites, so each pair of oppositely charged neighboring ions has adiple moment.&#x2022; As an example, we consider the one-dimensional NaCl crystal depicted as achain of alternating Na+ and Cl- ions as shown in Fig.7.9a. In the absence ofand applied field, the solid has no net polarization because the dipole momentsof equal magnitude are lined up head to head and tail to tail so that the netdipole moment is zero. The dipole moment p+ in the positive direction has thesame magnitude as p- in the negative x direction, so the net dipole moment pnetis zero.&#x2022; In the presence of a field E along the x direction, however, the Cl- ions arepushed in the &#x2013;x direction and the Na+ ions in the +x direction about theirequilibrium positions. Consequently, the dipole moment p+ in the +x directionincreases to p+ and the dipole moment p- decreases to p- as shown in Fig.7.9b.The net dipole moment, or the average dipole moment, per ion pair is now (p+ -p-), which depends on the electric field E.
• 26. (a) A NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion is zero.(b) In the presence of an applied field the ions become slightly displaced which leads to a net average dipole moment per ion.
• 27. Orientational (Dipolar) Polarization&#x2022; Certain molecules exhibit permanent dipole moments as discussed earlier. Forexample HCl molecule shown in Fig.7.10a has a permanent dipole moment pofrom the Cl- ion to the H+ ion.&#x2022; In the liquid or gas phases, these molecules, in the absence of an electric field,are randomly oriented as a result of thermal agitation as shown in Fig.7.10b.&#x2022; When a electric field E is applied E tries to align the dipoles parallel to itself,as depicted in Fig.7.10c. The Cl- and H+ charges experience forces in oppositedirections. But the nearly rigid bond between Cl- and H+ holds them together,which means that the molecule experiences a troque &#x3C4; about its center of mass.&#x2022; This torque acts to rotate the molecule to align po with E. If all the moleculeswere to simply rotate and align with the field, the polarization of the solidwould be P = NpoWhere N is the number of molecules per unit volume.&#x2022; However, due to their thermal energy, the molecules move around randomlyand collide with each other and with the walls of the container. These collisionsdestroy the dipole alignments. Thus the thermal energy tries to randomize theorientations of the dipole moments.
• 28. &#x2022; A snapshot of the dipoles in the material in the presence of a field can bepictured in Fig.7.10d in which the dipoles have different orientations. There is,never less, a net average dipole moment per molecule Pav that is finite anddirected along the field. Thus the material exhibits net polarization, which leadsto a dielectric constant that is determined by this orientational polarization.
• 29. &#x2022; The term interfacial polarization arises because the positive chargesaccumulating at the interface and the remainder of negative charges in the bulktogether constitute dipole moments that appear in the polarization vector P.&#x2022; Grain boundaries frequently lead to interfacial polarization as they can trapcharges migrating under the influence of an applied field, as indicated inFig.7.11c. Dipoles between the trapped charges increase the polarization vector.(a) A crystal with equal number of mobile positive ions and fixed negative ions. In the absence of a field, there is no net separation between all the positive charges and all the negative charges.(b) In the presence of an applied field, the mobile positive ions migrate toward the negative charges and positive charges in the dielectric. The dielectric therefore exhibits interfacial polarization.(c) Grain boundaries and interfaces between different materials frequently give rise to interfacial polarization.
• 30. Total Polarization&#x2022; In the presence of electronic, ionic, and dipolar polarization mechanisms,the average induced dipole moment per molecule will be the sum of all thecontributions in terms of the local field, Pav = &#x3B1;eEloc + &#x3B1;iEloc + &#x3B1;dEloc&#x2022; Each effect adds linearly to the net dipole moment per molecule. Interfacialpolarization cannot be simply added to the above equation as it occurs atinterfaces and cannot be put into an average polarization per molecule in thebulk.
• 31. 7.4 Frequency Dependence: Dielectric Constant and Dielectric Loss&#x2022; The static dielectric constant is an effect of polarization under dc conditions.When the applied field, or the voltage across a parallel plate capacitor, is asinusoidal signal, then the polarization of the medium under these ac conditionsleads to an ac dielectric constant that is generally different than the static case.&#x2022; Lets consider the orientation polarization involving dipolar molecules. Thesinusoidal varying field changes magnitude and direction continuously, and ittries to line up the dipoles one way and then the other way and so on.There are two factors opposing the immediate alignment of the dipoles with thefield:(i) First is that thermal agitation tries to randomize the dipole orientations. Collisions, for example, aid the randomization of the dipole orientations.(ii) Second, the molecules rotate in a viscous medium by virtue of their interactions with neighbors, which is particularly strong in the liquid and solid states and means that the dipoles cannot respond instantaneously to the changes in the applied field.
• 32. Induced dipole moment dc fieldThe dc field is suddenly changed from Eo to E at time t = 0. The induced dipolemoment p has to decrease from ad(0)Eo to a final value of ad(0)E. The decrease isachieved by random collisions of molecules in the gas.
• 33. &#x2022; If the field changes too rapidly, then the dipoles cannot follow the field and asa consequence, remain randomly oriented. At high frequencies, therefore, &#x3B1;dwill be zero as the field cannot induce a dipole moment. At low frequencies, ofcourse, the dipoles can respond rapidly to follow the field and &#x3B1;d has itsmaximum value.&#x2022; Suppose that after a prolonged application, corresponding to dc conditions,the applied field across the dipolar gaseous medium is suddenly decreased fromEo to E at a time we define as zero, as shown in Fig.7.12. The field E is smallerthan Eo, so the induced dc dipole moment per molecule should be smaller givenby &#x3B1;d(0)E where &#x3B1;d(0) is &#x3B1;d at &#x3C9; = 0, dc conditions. Therefore the induceddipole moment per molecule has to decrease, or relax from &#x3B1;d(0)Eo to &#x3B1;d(0)E.&#x2022; In a gas medium the molecules would be moving around randomly and theircollisions with each other and the walls of the container randomize the induceddipole per molecule. Thus the decrease or the relaxation process, in the induceddipole moment is achieved by random collisions. Assuming that &#x3C4; is the averagetime, called the relaxation time, between molecular collisions, then this is themean time it takes per molecule to randomize the induced dipole moment.
• 34. &#x2022; At low frequencies the rate of relaxation (1/&#x3C4;) is much faster than thefrequency of the field while at high frequencies the rate of relaxation (1/&#x3C4; ) ismuch slower than the frequency of the field.&#x2022; The dielectric constant is given by the relation: &#x3B5;r = &#x3B5;r - j&#x3B5;rwhere &#x3B5;r is the real part and &#x3B5;r is the imaginary part, both being frequencydependent as shown in Fig.7.13b. The real part &#x3B5;r decreases from its maximumvalue to 1 at high frequencies while the imaginary part &#x3B5;r is zero at low andhigh frequencies. The real part &#x3B5;r represents the relative permittivity i.edelectric constant that we would use in calculating the capacitance, theimaginary part &#x3B5;r represents the energy lost in the dielectric medium as thedipoles are oriented against random collisions one way and then the other wayand so on by the field.
• 35. (a) An ac field is applied to a dipolar medium. The polarization P(P = Np) is out of phase withthe ac field.(b) The relative permittivity is a complex number with real (&#xF065;r) and imaginary (&#xF065;r)parts that exhibit frequency dependence.
• 36. &#x2022; Although we considered only orientational polarization, in general a dielectricmedium will also exhibit other polarization mechanisms and certainlyelectronic polarization since there will always be electron clouds aroundindividual atoms, or electrons in covalent bonds. We can represent the generalfeatures of the frequency dependence of the real and imaginary parts of thedielectric constant as in Fig.7.15. The frequency dependence of the real and imaginary parts of the dielectric constant in the presence of interfacial, orientational, ionic, and, electronic polarization mechanisms.
• 37. Debye Equations and Cole-Cole PlotsDebye Equations:The Debye equations reflect the behaviour of &#x3B5;r and &#x3B5;r as a function offrequency &#x3B5;r = &#x3B5;r&#x221E; + &#x3B5;rdc - &#x3B5;r&#x221E; 1 + (&#x3C9;&#x3C4;)2 &#x3B5;r = (&#x3B5;rdc - &#x3B5;r&#x221E;) (&#x3C9;&#x3C4;) 1 + (&#x3C9;&#x3C4;)2&#x3B5;r&#x221E; is the relative permittivity (dielectric constant) at high frequencies.&#x3B5;rdc is the static relative permittivity (dielectric constant)&#x3C9; is the frequency&#x3C4; is the relaxation timeThe above equations reflect the behaviour of &#x3B5;r and &#x3B5;r as a function offrequency as shown in Figure below and discussed earlier in Fig.7.13.
• 38. The above equations reflect the behaviour of &#x3B5;r and &#x3B5;r as a function offrequency as shown in Figure below and discussed earlier in Fig.7.13.
• 39. Cole-Cole Plots:&#x2022; In dielectric studies of materials it is quite common to find a plot of theimaginary part (&#x3B5;r) versus the real part (&#x3B5;r) as a function of frequency &#x3C9;. Suchplots are called Cole-Cole plots after their originators. The Debye equationsprovide the necessary values for &#x3B5;r and &#x3B5;r to be plotted for the present simpledipolar relaxation mechanism that has only a single relaxation time &#x3C4;. Bysimply putting in &#x3C4; = 1 second, we can calculate and plot &#x3B5;r versus &#x3B5;r for &#x3C9; = 0(dc) to &#x3C9; &#x2192; &#x221E; as shown in Fig. below. The result is a semicircle. For certainsubstances like gases and certain liquids, the Cole-Cole plots generate asemicircle, for many dielectrics, the circle is typically flattened and asymmetric,and not a semicircle.&#x2022; The Cole-Cole plot is useful in several ways. Suppose for example that only apart of the, dispersion is covered by the frequency range available, as is oftenthe case, then by drawing the best fitting arc of the Cole-CoIe plot one is able todeduce, with an accuracy depending on the proportion of the total arcdetermined, both the zero-frequency dielectric constant (&#x3B5;r) and the limitinghigh-frequency value (&#x3B5;r), assuming that only one absorption occurs in the totalfrequency range.
• 40. &#x3B5;r = &#x3B5;r&#x221E; + &#x3B5;rdc - &#x3B5;r&#x221E; 1 + (&#x3C9;&#x3C4;)2 &#x3B5;r = (&#x3B5;rdc - &#x3B5;r&#x221E;) (&#x3C9;&#x3C4;) 1 + (&#x3C9;&#x3C4;)2Cole-Cole plot is a plot of &#xF065;&#xF0B2;r vs. &#xF065;&#xF0A2;r as a function of frequency, &#xF077; . As the frequency ischanged from low to high frequencies, the plot traces out a circle if Debye equationsare obeyed.
• 41. 7.6 Dielectric Strength and Insulation BreakdownDielectric Strength&#x2022; The voltage across a dielectric material and hence the field within it cannot,however be increased without limit. Eventually a voltage is reached that causesa substantial current to flow between the electrodes, which appears as a shortbetween the electrodes and leads to what is called dielectric breakdown.&#x2022; In gaseous and many liquid dielectrics, the breakdown does not generallypermanently damage the material. This means that if the voltage causingbreakdown is removed, then the dielectric can again sustain voltages until thevoltage is sufficiently high to cause breakdown again. In solid dielectrics thebreakdown process is invariably leads to the formation of a permanentconducting channel and hence to permanent damage.&#x2022; The dielectric strength Ebr is the maximum field that can be applied to aninsulating medium without causing dielectric breakdown. Beyond Ebr dielectricbreakdown takes place.
• 42. &#x2022; The dielectric strength of solids depends on a number of factors besides simply the molecular structure, such as the:1. impurities in the material,2. microstructural defects (e.g. microvoids, cracks),3. sample geometry,4. nature of the electrodes,5. temperature,6. ambient conditions (e.g. humidity),7. duration and frequency of the applied field8. thickness&#x2022; Dielectric strength is different under dc and ac conditions. There are alsoaging effects that slowly degrade the properties of the insulator and reduce thedielectric strength.
• 43. Dielectric Breakdown and Partial Discharges: Gases&#x2022; Consider a gas between two charged plates connected to a field. The gas willalways have a few free electrons due to cosmic radiation. If the field issufficiently large, then one of these electrons can be accelerated to sufficientlylarge kinetic energies to impact ionize a neutral gas molecule and produce anadditional free electron and a positively charged gas ion.&#x2022; Both the first and the liberated electrons are now available to accelerate inthe field again and further impact ionize more neutral gas molecules, and so on.Thus, an avalanche of impact ionization processes creates many free electronsand positive gas ions in the gas, which give rise to a discharge current betweenthe electrodes.&#x2022; The breakdown in gases depends on the pressure. The concentration of gasmolecules is greater at higher pressures. This means the mean separationbetween molecules and hence the mean free path of a free electron is shorter.Shorter mean free paths inhibit the free electrons from accelerating to reachimpact ionization energies unless the field is increased. Thus, generally Ebrincreases with the gas pressure.
• 44. Corona and Partial Discharges:(a) The field is greatest on the surface of the cylindrical conductor facing ground. If the voltage is sufficiently large this field gives rise to a corona discharge.(b) The field in a void within a solid can easily cause partial discharge.(c) The field in the crack at the solid-metal interface can also lead to a partial discharge.
• 45. &#x2022; A partial discharge occurs when only a local region of the dielectric isexhibiting discharge, so the discharge does not directly connect the twoelectrodes. For ex, the cylindrical conductor carrying a high voltage above agrounded plate (Fig.7.25a) the electric field is greatest on the surface of theconductor facing the ground. This field initiates discharge locally in this regionbecause the field is sufficiently high to give rise to an electron avalanche effect.Away from the conductor, however the field is not sufficiently strong tocontinue the electron avalanche discharge. This type of local discharge in highfield regions is termed corona discharge.&#x2022; Voids and cracks occurring within solid dielectrics and discontinuities at thedielectric-electrode interface can also lead to partial discharges as the field inthese voids is higher than the average field in the dielectric and further thedielectric strength in the gas in the void is less than that of the continuous solidinsulation. Fig.7.25b and c depict two examples of partial discharges occurringin voids, one inside the solid (an air bubble introduced during the processing ofthe dielectric) and the other (in the form of a crack) at the solid-electrodeinterface.
• 46. Dielectric Breakdown: Liquids&#x2022;The process of breakdown in liquids is not as clearly understood as in gases.Every liquid has some impurities with small conductive particles insuspension, it is believed that these impurities coalesce end to end to form aconducting bridge between the electrodes and thereby give rise to discharge.&#x2022; In some liquids, the discharge initiates as partial discharges in gas bubblesentrapped in the liquid. These partial charges can locally raise the temperatureand vaporize more of the liquid and hence increase the size of the bubble.Moisture (H2O which is polar) absorption and absorption of gases from theambient (polar gas molecules) also play a role in the breakdown.
• 47. Dielectric Breakdown: Solids&#x2022; There are various mechanisms that can lead to dielectric breakdown in solids.Most of them depend on the dielectric materials condition and sometimes onextrinsic factors such as the ambient conditions, moisture absorption being atypical example.&#x2022; The various dielectric breakdown mechanisms in solids are listed below: 1. Intrinsic Breakdown or Electronic Breakdown 2. Thermal Breakdown 3. Electromechanical Breakdown and Electrofracture 4. Internal Discharges 5. Insulation Ageing 6. External Discharges
• 48. 1. Intrinsic Breakdown or Electronic Breakdown:&#x2022; This is the most common type of dielectric breakdown. A free electron in theconduction band (CB) of a dielectric in the presence of a large field can beaccelerated to sufficiently large energies to collide with and ionize a host atomof the solid. The electron gains an energy eEbr&#x2113; when it moves a distance &#x2113;under an applied field Ebr. If this energy is greater than the bandgap energy Eg,then the electron as a result of a collision with the lattice vibrations, can excitean electron from the valence band to the conduction band, that is, break a bond.&#x2022; Both the primary and the released electron can further impact ionize otherhost atoms and thereby generate an electron avalanche effect that leads to asubstantial current. The initial conduction electrons for the avalanche are eitherpresent in the CB or are injected from the metal into the CB as a result of field-assisted thermal emission from the Fermi energy in the metal to the CB in thedielectric.&#x2022; If dielectric breakdown does not occur by an electron avalanche effect (due toshort mean free paths in the insulator), then another insulation breakdownmechanism is the enormous increase in the injection of electrons from the metalelectrode into the insulator at very high fields as a result of field-assistedemission.
• 49. 2. Thermal Breakdown&#x2022; Conduction and dielectric losses generate heat within a dielectric. If this heatcannot be removed from the solid quickly by thermal conduction or othermeans, then the temperature of the dielectric will increase. The increase in thetemperature invariably increases the conductivity of an insulator.&#x2022; The increase in the conductivity then leads to more Joule heating and hencefurther rises in the temperature and so on. If the heat cannot be conducted awayto limit the temperature, then the result is a thermal runaway condition in whichthe temperature and the current increase until a discharge occurs throughvarious sections of the solid.&#x2022; As a consequence of sample inhomogeneities, frequent thermal runaway issevere in certain parts of the solid that become hot spots and suffer localmelting and physical and chemical erosion. Local breakdown at various hotspots eventually leads to a conducting channel connecting the oppositeelectrodes and hence to a dielectric breakdown.
• 50. 3. Electromechanical Breakdown and Electrofracture&#x2022; A dielectric medium between oppositely charged electrodes experiencescompressional forces because the opposite charges +Q and &#x2013;Q on the platesattract each other, as shown in Fig.7.26. As the voltage increases, so does thecompressive load, and the dielectric becomes squeezed, of the thickness d getssmaller.An exaggerated schematic illustration of a soft dielectric medium experiencing strongcompressive forces to the applied voltage.
• 51. &#x2022; At each stage, the increase in the compressive load is normally balanced bythe elastic deformation of the insulation to a new smaller thickness. However,if the elastic modulus is sufficiently small, then compressive loads cannot besimply balanced by the elastic modulus of the solid.&#x2022; Hence there is a mechanical runaway due to the following reasons. Thedecrease in d, due to the compressive load, leads to higher field (E = V/d) andalso to more charges on the electrodes. This in turn leads to a greatercompressive load, which further decreases d, and so on, until the shear stresseswithin the insulation cause the insulation to flow plastically. Eventually theinsulation breaks down.&#x2022; Another possibility is the initiation and growth of internal cracks by internalstresses around inhomogeneous regions inside the dielectric. Combined effectsof both large shear stresses and large electric field eventually lead to crackpropagation and hence dielectric failure. This type of process is sometimescalled electrofracture.
• 52. 4. Internal Discharges&#x2022; These are partial discharges that take place in microstructural voids, cracks,or pores within the dielectric where the gas atmosphere (usually air) has lowerdielectric strength. As explained earlier in dielectric breakdown in gases(Fig.7.25) the discharge current in a void, such as those in Fig.7.25b and c, canbe easily sustained under ac conditions. Initially the pores size (or number ofpores) may be small and the partial discharge insignificant, but with time thepartial discharge erodes the internal surfaces of the void.&#x2022; Partial discharges can locally melt the insulator and can easily cause chemicaltransformations. Eventually, an electrical tree type of discharge develops froma partial discharge that has been eroding the dielectric as shown in Fig.7.27a fora high voltage cable in which there is a tiny void at the interface between thedielectric and the inner conductor.&#x2022; The erosion of the dielectric by the partial discharge propagates like abranching tree. The &#x201C;tree branches&#x201D; are erosion channels (hollow filaments ofvarious sizes) in which gaseous discharge takes place and forms a conductingchannel during operation.
• 53. (a) A schematic illustration of electrical treeing breakdown in a high voltage coaxial cable which was initiated by a partial discharge in the void at the inner conductor - dielectric interface.(b) A schematic diagram of a typical high voltage coaxial cable with semiconducting polymer layers around the inner conductor and around the outer surface of the dielectric.
• 54. 5. Insulation Aging&#x2022; Aging is a term used to describe the deterioration in the properties of theinsulation. Aging therefore determines the useful life of the insulation. Thereare many factors that either directly or indirectly affect the properties andperformance of an insulator in service:1. Insulation will experience physical and chemical aging whereby its physicaland chemical properties change considerably, even in the absence of andelectric field.2. An insulation that is subjected to temperature and mechanical stressvariations can develop structural defects, such as microcracks.3. Irradiation by ionizing radiation such as X-rays, exposure to severe ambientconditions such as excessive humidity, ozone, etc.4. Oxidation of a polymeric insulation with time is another form of chemicalaging.&#x2022; Chemical aging processes are generally accelerated with temperature.Electrical trees develop as a result of electrical aging because, the ac field givesrise to continual partial discharges in an internal or surface microcavity, whichthen erodes the region around it and slowly grows like a branching tree asshown in figure below.
• 55. Some typical water trees found in field aged cables. (Left: Trees in a cable withtape and graphite insulation. Right: Trees in a cable with strippable insulation.)
• 56. 6. External Discharges&#x2022; There are many examples where the surface of the insulation becomescontaminated by ambient conditions such as:(a) Excessive moisture(b) Deposition of pollutants(c) Dirt and dust(d) Salt spraying&#x2022; Eventually the contaminated surface develops sufficient conductance to allowdischarge between the electrodes at a field below the normal breakdownstrength of the insulator. This type of dielectric breakdown over the surface ofthe insulation is termed surface tracking.
• 57. Relationship between the breakdown field and the time to breakdown.&#x2022; There are a number of dielectric breakdown mechanisms and the one thatcauses eventual breakdown depends not only on the properties and quality ofthe material but also on the operating conditions, environmental factors beingno less important. Time to breakdown and the field at breakdown, Ebr, are interrelated and depend on the mechanism that causes the insulation breakdown. External discharges have been excluded
• 58. 7.7 Capacitor Dielectric Materials&#xF0D8; Selection criteria of dielectric materials for capacitors:&#x2022; Capacitance value&#x2022; Frequency of application&#x2022; Maximum tolerable loss&#x2022; Maximum working voltage&#x2022; Size and cost C = &#x3B5;o&#x3B5;r A l Large capacitances can be achieved by using high &#x3B5;r dielectrics, thin dielectrics, and large areas.
• 59. Examples of dielectrics that can be used for various capacitance values.Examples of dielectrics that can be used in various frequency ranges.
• 60. (a) Single and multilayer ceramic capacitors:Fig (a). shows a typical single layerceramic capacitor. The thin ceramicdisk or plate has suitable metal electrodes,and the whole structure is encapsulatedin an epoxy by dipping it in a thermosetting resin. The epoxy coating prevents moisturefrom degrading the dielectric properties ofthe ceramic. One way to increase the capacitance is to connect N number of these in parallel, and this is done in a efficient way by using a multilayer ceramic structure as shown in Fig (b).
• 61. (b) Polymeric film capacitors:Fig.7.33 shows one arrangement by which a polymeric film capacitor can beconstructed. Two polymeric tapes having metallized electrodes (vacuumdeposited / coated Al) on one surface leaving a margin on one side. The twotapes together are rolled up (like a Swiss-roll cake) and the opposite sides areelectroded using suitable conducting glue. Concept is similar to multilayerceramic capacitor except that the layers are rolled up to form a circular crosssection. Two polymer tapes in (a), each with a metallized film electrode on the surface (offset from other), can be rolled together (like a Swiss roll) to obtain a polymer film capacitor as in (b). As the two separate metal films are lined at opposite edges, electroding is done over the whole side surface.
• 62. (c) Electrolytic CapacitorsElectrolytic capacitors provide large values of capacitance while maintaining atolerable size. In aluminium electrolytic capacitors, the metal electrodes are twoAl foils, typically 50-100 &#xB5;m thick, that are separated by a porous papermedium soaked with a liquid electrolyte. The two foils together are wound intoa cylindrical form and held within a cylindrical case as shown in Fig.7.34. Thedielectric medium is the thin alumina Al2O3 layer grown on the roughenedsurface of one of the foils as shown in Fig. 7.34b. This foil is then called theanode. The capactive behavior is due to the Al/(Al2O3)/electrolyte structure.
• 63. Comparison of dielectrics for capacitor applications Capacitor name Polypropylene Polyester Mica Aluminum, Tantalum, High-K ceramic electrolytic electrolyt ic, solid Dielectric Polymer film Polymer film Mica Anodized Al2O3 Anodized X7R film Ta2O5 BaTiO3 base film &#xF065;r&#xF0A2; 2.2 &#x2013; 2.3 3.2 &#x2013; 3.3 6.9 8.5 27 2000 tand 4 &#xF0B4; 10-4 4 &#xF0B4; 10-3 2 &#xF0B4; 10-4 0.05 - 0.1 0.01 0.01 Ebr (kV mm-1) DC 100 - 350 100 - 300 50 - 300 400 - 1000 300 - 600 10 d (typical minimum) 3 - 4 &#xB5;m 1 &#xB5;m 2 - 3 &#xB5;m 0.1 &#xB5;m 0.1 mm 10 &#xB5;m Cvol (&#xB5;F cm-3) 2 30 15 7,500a 24,000a 180 Rp = 1/Gp; C = 1 mF; 400 kW 40 kW 800 kW 1.5 - 3 kW 16 kW 16 kW 1000 Hz Evol (mJ cm-3)b 10 15 8 1000 1200 100 Polarization Electronic Electronic and Ionic Ionic Ionic Large ionic Dipolar displacementNOTES: Typical values. h = 3 assumed. The table is for comparison purposes only. Breakdown fields are typical DC values, and canvary substantially, by at least an order of magnitude; Ebr depends on the thickness, material quality and the duration of the appliedvoltage. a Proper volumetric calculations must also consider the volumes of electrodes and the electrolyte necessary for thesedielectrics to work; hence the number would have to be decreased. b Evol depends very sensitively on Ebr and the choice of h; henceit can vary substantially. Polyester is PET, or polyehthylene terephthalate. Mica is potassium aluminosilicate, a muscovite crystal.X7R is the name of a particular BaTiO3-based ceramic solid solution.
• 64. 7.8 Piezoelectricity, Ferroelectricity, and Pyroelectricity Piezoelectricity&#x2022; Certain crystals like quartz (crystalline SiO2) and BaTiO3, become polarizedwhen they are mechanically stressed. Charges appear on the surfaces of thecrystal as depicted in Fig.7.38a and b. Appearance of surface charges leads to avoltage difference between the two surfaces of the crystal.&#x2022; The same crystals also exhibit mechanical strain or distortion when theyexperience an electric field, as shown in Fig.7.38c and d. The direction ofmechanical deformation (expansion or compression) depends on the directionof the applied field. The two effects are complementary and is know aspiezoelectricity. Only certain crystals exhibit piezoelectricity because thephenomenon requires a crystal structure that has &#x201C;no center of symmetry&#x201D;
• 65. Crystals exhibiting center of symmetry&#x2022; Consider a NaCl type cubic unit cell (Fig.7.39a), this unit cell has a center ofsymmetry at O because if we draw a vector from O to any charge and then drawthe reverse vector, we will find the same type of charge. When unstressed, thecenter of mass of the negative charges at the corners of the unit cell coincideswith the positive charge at the center, as shown in Fig.7.39a. There is thereforeno net polarization in the unit cell and P = 0.&#x2022; Under stress the unit cell becomes strained as shown in Fig.7.39b, but thecenter of mass of the negative charges still coincides with the positive chargeand the net polarization is still zero. Thus, the strained crystal still has P = 0.&#x2022; This result is generally true for all crystals that have a center of symmetry.The centers of mass of negative and positive charges in the unit cell remaincoincident when the crystal is strained.
• 66. Crystals exhibiting center of symmetryA NaCl-type cubic unit cell has a center of symmetry.(a) In the absence of an applied force, the centers of mass for positive and negativeions coincide.(b) This situation does not change when the crystal is strained by an applied force.
• 67. Crystals exhibiting no center of symmetry&#x2022; Piezoelectric crystals have no center of symmetry. For ex., the hexagonal unitcell shown in Fig.7.40a exhibits no center of symmetry. If we draw a vectorfrom point O to any charge and then reverse the vector, we will find an oppositecharge. The unit cell is said to be noncentrosymmetric. When unstressed, asshown in Fig.7.40a, the center of mass of the negative charges coincides withthe center of mass of the positive charges, both at O.&#x2022; However, when the unit cell is stressed as shown in Fig.7.40b, the positivecharge at A and the negative charge at B both become shifted and there is nowa net polarization P. Thus, an applied stress produces a net polarization P inthe unit cell, and in this case P appears to be in the same direction as the appliedstress, along y.&#x2022; Suppose Tj is the applied mechanical stress along some j direction and Pj isthe induced polarization along some i direction; then the two are linearly relatedby Pj = dij Tj Where dij is the piezoelectric coefficient.
• 68. A hexagonal unit cell has no center of symmetry. (a) In the absence of an applied force the centers ofmass for positive and negative ions coincide. (b) Under an applied force along y the centers of mass forpositive and negative ions are shifted which results in a net dipole moment P along y. (c) When the forceis along a different direction, along x, there may not be a resulting net dipole moment in that directionthough there may be a net P along a different direction (y).
• 69. &#x2022; Piezoelectric crystal are essentially electromechanical transducersbecause they convert an electric field / signal to a mechanical strain, andvice versa. They are used in many engineering applications like ultrasonictransducers, microphones, accelerometers etc. Piezoelectric transducers are widely used to generate ultrasonic waves in solids and also to detect such mechanical waves. The transducer on the left is excited from an ac source and vibrates mechanically. These vibrations are coupled to the solid and generate elastic waves. When the waves reach the other end they mechanically vibrate the transducer on the right which converts the vibrations to an electrical signal.
• 70. &#x2022; It is clear that an important engineering factor in the use of piezoelectrictransducers is the electromechanical coupling between electrical andmechanical energies. The electromechanical coupling factor k is defined interms of k2 by: k2 = Electrical energy converted to mechanical energy Input of electrical energy or equivalently by k2 = Mechanical energy converted to electrical energy Input of mechanical energy
• 71. Ferroelectricity&#x2022; Certain crystals are permanently polarized even in the absence of anapplied field. The crystals already possesses a finite polarization vector dueto the separation of positive and negative charges in the crystal. These crystalsare called ferroelectric.&#x2022; Barium titanate (BaTiO3) is probably the best example exhibitingferroelectricity. Above approximately 130oC, the crystal structure of BaTiO3 iscubic as shown in Fig.7.44a. There is therefore no net polarization and P = 0.Above 130oC, therefore barium titanate crystal exhibits no permanentpolarization and is not ferroelectric. However below 130oC, the structure ofbarium titanate is tetragonal as shown in Fig.7.44c. The crystal is thereforepolarized by the separation of the centers of mass of the negative and positivecharges. The crystal possesses a finite polarization vector P and isferroelectric. The critical temperature at which ferroelectric property is lost, inthis case 130oC, is called the Curie temperature (TC)&#x2022; The nonlinear nature of ferroelectric materials can be used to makecapacitors with tunable capacitance.
• 72. BaTiO3 has different crystal structures above and below 130 &#xF0B0;C that lead to different dielectricproperties.
• 73. &#xF0D8; All ferroelectric crystals are also piezoelectric, but the reverse is not true: not all piezoelectric crystals are ferroelectric Piezoelectric properties of BaTiO3 below its Curie temperature.
• 74. Pyroelectricity&#x2022; Pyroelectricity is the ability of certain materials to generate a temporaryvoltage when they are heated or cooled. The change in temperature slightlymodifies the positions of the atoms within the crystal structure, such that thepolarization of the material changes. This polarization change gives rise to avoltage across the crystal.&#x2022; Pyroelectric crystals are widely used as infrared detectors The heat absorbed by the crystal increases the temperature by dT which induces a change dP in the polarization. This is the pyroelectric effect. The change dP gives rise to a change dV in the voltage which can be measured.