Definition of Capacitance +Qo -Qo + - + - + - d A V Co = capacitance of a parallel plate capacitor in free space Qo = charge on the plates V = voltage o = absolute permittivity (8.854 pF/m or pC/(V. m) A = area of a plate d = distance of the space between the plates

Definition of Charge density +Qo -Qo V d A + -

(a) Parallel plate capacitor with free space between the plates. (b) As a slab of insulating material is inserted between the plates, there is an external current flow indicating that more charge is stored on the plates. (c) The capacitance has been increased due to the insertion of a medium between the plates.

Definition of Relative Permittivity or Dielectric constant r = relative permittivity or dielectric constant, Q = charge on the plates with a dielectric medium, Qo = charge on the plates with free space between the plates, C = capacitance with a dielectric medium, Co = capacitance of a parallel plate capacitor in free space

Dielectric constant is a material property that is frequency dependent Dielectric constant is a material property that is frequency dependent. For a dielectric, the voltage, at which an appreciable current flow (or breakdown) occurs, is called “dielectric strength”. Material r or k (measure at 1 kHz) Dielectric strength (kV/mm) Al2O3 (99.9%) 10.1 9.1 Al2O3 (99.5%) 9.8 9.5 BeO (99.5%) 6.5 10.2

Definition of Dipole Moment The definition of electric dipole moment. p = Qa p = electric dipole moment, Q = charge, a = vector from the negative to the positive charge

Three major polarization mechanisms Electronic polarization Ionic polarization Orientation (dipolar) polarization

The origin of electronic polarization.

Definition of Polarizability pinduced = E pinduced = induced dipole moment,  = polarizability, E = electric field Electronic Polarization pe = magnitude of the induced electronic dipole moment, Z = number of electrons orbiting the nucleus of the atom, x = distance between the nucleus and the center of negative charge,  = constant, E = electric field

Static Electronic Polarizability e = electronic polarizability Z = total number of electrons around the nucleus me = mass of the electron in free space (9.1094x10-31 kg) o = natural oscillation frequency (= 2fo) e = electron charge (1.60218x10-19 C)

Electronic polarizability and its resonance frequency versus the number of electrons in the atom (Z). The dashed line is the best-fit line.

(a) When a dilectric is placed in an electric field, bound polarization charges appear on the opposite surfaces. (b) The origin of these polarization charges is the polarization of the molecules of the medium. (c) We can represent the whole dielectric in terms of its surface polarization charges +QP and -QP.

Definition of Polarization Vector P = Polarization vector, p1, p2, ..., pN are the dipole moments induced at N molecules in the volume Definition of Polarization Vector P = Npav pav = the average dipole moment per molecule P = polarization vector, N = number of molecules per unit volume

Polarization and Bound Surface Charge Density P = p P = polarization, p = polarization charge density on the surface Definition of Electronic Susceptibility P = eoE P = polarization, e = electric susceptibility, o = permittivity of free space, E = electric field

Polarization charge density on the surface of a polarized medium is related to the normal component of the polarization vector.

Electric Susceptibility and Polarization e = electric susceptibility, o = permittivity of free space, N = number of molecules per unit volume, e = electronic polarizability Relative Permittivity and Electronic Susceptibility r = 1 + e r = relative permittivity, e = electric susceptibility

Relative Permittivity and Polarizability r = relative permittivity N = number of molecules per unit volume e = electronic polarizability o = permittivity of free space Assumption: Only electronic polarization is present

The electric field inside a polarized dielectric at the atomic scale is not uniform. The local field is the actual field that acts on a molecules. It can be calculated by removing that molecules and evaluating the field at that point from the charges on the plates and the dipoles surrounding the point.

Local Field in Dielectrics Eloc = local field, E = electric field, o = permittivity of free space, P = polarization Clausius-Mossotti Equation r = relative permittivity, N = number of molecules per unit volume, e = electronic polarizability, o = permittivity of free space

Example 1 The electronic polarization polarizability of the Ar atom is 1.7x10-40 Fm-2 . What is the static dielectric constant, r, of solid Ar (below 84 K) if its density is 1.8 g/cm3 and atomic mass = 39.95 g/mol, NA = 6.02x1023 atom/mol

(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 the covalent bonds are shifted very easily with respect to the positive ionic cores. The whole solid becomes polarized due to the collective shift in the negative charge distribution of the valence electrons. (Supplements)

Example 2 Consider a pure Si crystal that has r = 11.9. if its density is 2.33 g/cm3 and atomic mass = 28.09 g/mol, NA = 6.02x1023 atom/mol. What is the electronic polarization due to valence electrons per Si atom (if one could portion the observed crystal polarization to individual atom). If a Si crystal sample is electroded opposite faces and has voltage applied across it. By how much is the local field greater than the applied field? What is the resonant frequency fo corresponding to o.

A NaCl chain in the NaCl crystal without an applied field A NaCl chain in the NaCl crystal without an applied field. Average or net dipole moment per ion is zero. In the presence of an applied field the ions become slightly displaced which leads to a net average dipole moment per ion.

Orientation (dipolar) polarization (a) A HCl molecule possesses a permanent dipole moment p0. (b) In the absence of a field, thermal agitation of the molecules results in zero net average dipole moment per molecule. (c) A dipole such as HCl placed in a field experiences a torque that tries to rotate it to align p0 with the field E. (d) In the presence of an applied field, the dipoles try to rotate to align with the field against thermal agitation. There is now a net average dipole moment per molecule along the field.

Average Dipole Moment in Orientational Polarization pav = average dipole moment, po = permanent dipole moment, E = electric field, k = Boltzmann constant, T = temperature Dipolar Orientational Polarizability d = dipolar orientational polarizability, po = permanent dipole moment

Interfacial polarization (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.

Total Induced Dipole Moment pav = e Eloc + i Eloc + d Eloc pav = average dipole moment, Eloc = local electric field, e = electronic polarizability, i = ionic polarizability, d = dipolar (orientational) polarizability Clausius-Mossotti Equation r = dielectric constant, o = permittivity of free space, Ne = number of atoms or ions per unit volume, e = electronic polarizability, Ni = number of ion pairs per unit volume , i = ionic polarizability

Example 3 Consider the CsCl crystal which has one Cs+-Cl- pair per unit cell and a lattice parameter a of 0.412 nm. The electronic polarization of Cs+ and Cl- ions is 3.35x10-40 F m2, 3.40x10-40 respectively, and the mean ionic polarizibility per ion pair is 6x10-40 F m2. What is the dielectric constant at low frequencies and that at optical frequency?

Relaxation process The dc field is suddenly changed from Eo to E at time t = 0. The induced dipole moment p has to decrease from ad(0)Eo to a final value of ad(0)E. The decrease is achieved by random collisions of molecules in the gas.

Dipolar Relaxation Equation p = dipole moment, dp/dt = rate at which the induced dipole moment is changing, d = dipolar orientational polarizability, E = electric field,  = relaxation time Orientational Polarizability and Frequency (under ac field) d () = dipolar orientational polarizability as a function of ,  = angular frquency of the applied field,  = relaxation time, j is (-1).

(a) An ac field is applied to a dipolar medium (a) An ac field is applied to a dipolar medium. The polarization P(P = Np) is out of phase with the ac field. (b) The relative permittivity is a complex number with real (r') and imaginary (r'') parts that exhibit frequency dependence.

Complex Relative Permittivity r = dielectric constant r = real part of the complex dielectric constant r = imaginary part of the complex dielectric constant j = imaginary constant (-1)

The dielectric medium behaves like an ideal (lossless) capacitor of capacitance C which is in parallel with a conductance Gp.

Admittance of a Parallel Plate Capacitor Y = jC + GP Y = admittance,  = angular frequency of the applied field , C = capacitance, GP = conductance Loss Tangent tan = loss tangent or loss factor, r = real part of the complex dielectric constant, r = imaginary part of the complex dielectric constant

Dielectric Loss per Unit Volume Wvol = dielectric loss per unit volume,  = angular frquency of the applied field , E = electric field, o = permittivity of free space, r = real part of the complex dielectric constant, tan = loss tangent or loss factor

The frequency dependence of the real and imaginary parts of the dielectric constant in the presence of interfacial, orientational, ionic, and, electronic polarization mechanisms.

Real and imaginary part is of the dielectric constant, r' and r'' versus frequency for (a) a polymer, PET, at 115 C and (b) an ionic crystal, KCl, at room temperature. both exhibit relaxation peaks but for different reasons. SOURCE: Data for (a) from author’s own experiments using a dielectric analyzer (DEA), (b) from C. Smart, G.R. Wilkinson, A. M. Karo, and J.R. Hardy, International Conference on lattice Dynamics, Copenhagen, 1963, as quoted by D. G. Martin, “The Study of the Vibration of Crystal Lattices by Far Infra-Red Spectroscopy,” Advances in Physics, 14, no. 53-56, 1965, pp. 39-100.

Field in the cavity is higher than the field in the solid Field E2 in a small cavity (e.g. air) is higher than the field in the solid since er1 > er2

A thin slab of dielectric is placed in the middle of a parallel plate capacitor. The field inside the thin slab is E2

Free Charges and Field in a Dielectric En = electric field normal to a small surface area dA dA = small surface area Qfree = total (net) free charges enclosed inside the surface o = permittivity of free space r = dielectric constant

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.

An exaggerated schematic illustration of a soft dielectric medium experiencing strong compressive forces to the applied voltage.

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. 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.

Some typical water trees found in field aged cables Some typical water trees found in field aged cables. (Left: Trees in a cable with tape and graphite insulation. Right: Trees in a cable with strippable insulation.) SOURCE: P. Werellius, P. Tharning, R. Eriksson,B . Holmgren. J. Gafvert, “Dielectric Spectroscopy for Diagnosis of Water Tree Deterioration in XLPE Cables” IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 8, February 2001, p 34, Figure 10 ( IEEE, 2001)

Coaxial cable connector with traces of corona discharge; electrical treeing. SOURCE: M. Mayer and G.H. Schröder , “Coaxial 30 kV Connectors for the RG220/U Cable: 20 Years of Operational Experience”IEEE Electrical Insulation Magazine , Vol. 16, March/April 2000, p 11, Figure 6. ( IEEE, 2000)

Tree and bush type electrical discharge structures (a) Voltage, V = 160 kV, gap spacing d = 0.06 m at various times. (b) Dense bush discharge structure, V = 300 kV, d = 0.06 m at various times. SOURCE: V. Lopatin, M.D. Noskov, R. Badent, K. Kist, A.J. Swab, “Positive Discharge Development in Insulating Oil: Optical Observation and Simulation” IEEE Trans. on Dielec and Elec. Insulation Vol. 5, No. 2, 1998, p. 251, Figure 2.( IEEE, 1998)

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 (based on L.A. Dissado and J.C. Fothergill, Electrical Degradation and Breakdown in Polymers, Peter Peregrinus Ltd. for IEE, UK, © 1992, p. 63)

Examples of dielectrics that can be used for various capacitance values.

Examples of dielectrics that can be used in various frequency ranges.

Sindle0 and multilayer dielectric capacitors.

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.

Aluminum electrolytic capacitor.

Solid electrolyte tantalum capacitor. A cross section without fine detail. An enlarged section through the Ta capacitor.

Comparison of dielectrics for capacitor applications Capacitor name Polypropylene Polyester Mica Aluminum, electrolytic Tantalum, electrolytic, solid High-K ceramic Dielectric Polymer film Anodized Al2O3 film Anodized Ta2O5 X7R BaTiO3 base er 2.2 – 2.3 3.2 – 3.3 6.9 8.5 27 2000 tand 4  10-4 4  10-3 2  10-4 0.05 - 0.1 0.01 Ebr (kV mm-1) DC 100 - 350 100 - 300 50 - 300 400 - 1000 300 - 600 10 d (typical minimum) 3 - 4 µm 1 µm 2 - 3 µm 0.1 µm 0.1 mm 10 µm Cvol (µF cm-3) 2 30 15 7,500a 24,000a 180 Rp = 1/Gp; C = 1 mF; 1000 Hz 400 kW 40 kW 800 kW 1.5 - 3 kW 16 kW Evol (mJ cm-3)b 8 1000 1200 100 Polarization Electronic Electronic and Dipolar Ionic Large ionic displacement NOTES: Typical values. h = 3 assumed. The table is for comparison purposes only. Breakdown fields are typical DC values, and can vary substantially, by at least an order of magnitude; Ebr depends on the thickness, material quality and the duration of the applied voltage. a Proper volumetric calculations must also consider the volumes of electrodes and the electrolyte necessary for these dielectrics to work; hence the number would have to be decreased. b Evol depends very sensitively on Ebr and the choice of h; hence it 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.

Capacitance per unit volume Cvol = capacitance per unit volume, o = permittivity of free space, r = dielectric constant, d = separation of the capacitor plates Maximum energy per unit volume Evol = maximum energy stored per unit volume, C = capacitance, Vm = maximum voltage, A = surface area of the capacitor plates, d = separation of the capacitor plates, o = permittivity of free space, r = real part of the complex dielectric constant,  = safety factor, Ebr = breakdown electric field

Dielectric loss per unit volume Wvol = dielectric loss per unit volume, Ebr = breakdown electric field,  = safety factor,o = permittivity of free space, r = real part of the complex dielectric constant, tan = loss tangent or loss factor, = angular frequency of the applied field

(a) A polymer dielectric that has dipolar side groups attached to the polymer chains. With no applied field, the dipoles are randomly oriented. (b) In the presence of an applied field, some very limited rotation enables dipolar polarization to take place. (c) Near the softening temperature of the polymer, the molecular motions are rapid and there is also sufficient volume between chains for the dipoles to align with the field. The dipolar contribution to r is substantial, even at high frequencies.

Real part of the dielectric constant, er', and loss tangent, tand, at 1 kHz vs. temperature from Dielectric Analysis, DEA [by Kasap and Maeda (1995)]