Transcript Definition of Capacitance

Definition of Capacitance
+Qo -Qo
+ + + d
Qo  o A
Co 

(unit  Farad)
V
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
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Definition of Charge density
+Qo -Qo
+ + + d
Qo  o A
Co 

V
d
Qo  o A

V
d
V
Qo  oV
Qo
 oV
2


 Do 
(unit  C/m )
A
d
A
d
D is charge density
A
2
(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.
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Definition of Relative Permittivity or
Dielectric constant
Q C
D
o A
r 


 C  r 
Qo Co Do
d
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
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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 Dielectric strength
Al2O3 (99.9%)
1 kHz)
10.1
(kV/mm)
9.1
Al2O3 (99.5%)
9.8
9.5
BeO (99.5%)
6.5
10.2
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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
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Three major polarization mechanisms
1. Electronic polarization
2. Ionic polarization
3. Orientation (dipolar) polarization
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The origin of electronic polarization.
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Definition of Polarizability
pinduced = E
pinduced = induced dipole moment,  = polarizability, E = electric field
Electronic Polarization
 Z 2e2 
E
pe  ( Ze) x  
 β 
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
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Static Electronic Polarizability
2
Ze
e 
2
me o
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)
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11
1/ 2
  

o  
 Zme 
2
Ze
e 
me o 2
Electronic polarizability and its resonance frequency versus the number of electrons in the
atom (Z). The dashed line is the best-fit line.
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(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.
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Definition of Polarization Vector
1
P =
[p1 + p2 +...+ pN ]
Volume
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
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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
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Polarization charge density on the surface of a polarized medium is related to the normal
component of the polarization vector.
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Electric Susceptibility and Polarization
e 
1
o
N e
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
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Relative Permittivity and Polarizability
 r  1
N e
o
r = relative permittivity
N = number of molecules per unit volume
e = electronic polarizability
o = permittivity of free space
Assumption: Only electronic polarization is present
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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.
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Local Field in Dielectrics
E loc
1
E 
P
3 o
Eloc = local field, E = electric field, o = permittivity of free space, P = polarization
Clausius-Mossotti Equation
 r  1 N e

 r  2 3o
r = relative permittivity, N = number of molecules per unit volume, e = electronic
polarizability, o = permittivity of free space
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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
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(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)
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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.
a. What is the electronic polarization due to valence
electrons per Si atom (if one could portion the
observed crystal polarization to individual atom).
b. 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?
c. What is the resonant frequency fo corresponding to o.
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(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.
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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.
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Average Dipole Moment in Orientational Polarization
2
1 po E
pav 
3 kT
pav = average dipole moment, po = permanent dipole moment, E = electric field, k =
Boltzmann constant, T = temperature
Dipolar Orientational Polarizability
2
1 po
d 
3 kT
d = dipolar orientational polarizability, po = permanent dipole moment
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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.
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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 1 1

( N e e  N i i )
 r  2 3 o
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
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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?
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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 d(0)Eo to a final value of ad(0)E. The decrease is
achieved by random collisions of molecules in the gas.
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Dipolar Relaxation Equation
dp
p   d (0)E

; E  Eo exp( jt )
dt

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)
p   d ( ) Eo exp( jt )
 d (0)
 d ( ) 
1  j
d () = dipolar orientational polarizability as a function of ,  = angular
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frquency of the applied field,  = relaxation time, j is (1).
(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.
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Complex Relative Permittivity
 r  r  jr
r = dielectric constant
r = real part of the complex dielectric constant
r = imaginary part of the complex dielectric constant
j = imaginary constant (1)
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The dielectric medium behaves like an ideal (lossless) capacitor of capacitance C
which is in parallel with a conductance Gp.
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Admittance of a Parallel Plate Capacitor
Y = jC + GP
Y = admittance,  = angular frequency of the applied field , C =
capacitance, GP = conductance
Loss Tangent
r
t an 
r
tan = loss tangent or loss factor, r = real part of the complex dielectric constant,
r = imaginary part of the complex dielectric constant
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Dielectric Loss per Unit Volume
Wvol = E  o r tan
2
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
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The frequency dependence of the real and imaginary parts of the dielectric constant in the
presence of interfacial, orientational, ionic, and, electronic polarization mechanisms.
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(a) 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.
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40
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 r1 > r2
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A thin slab of dielectric is placed in the middle of a parallel plate capacitor. The
field inside the thin slab is E2
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Free Charges and Field in a Dielectric

Surface
E n dA 
Qfree
 o r
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
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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.
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An exaggerated schematic illustration of a soft dielectric medium experiencing strong
compressive forces to the applied voltage.
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(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.
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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)
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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)
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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)
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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)
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Examples of dielectrics that can be used for various capacitance values.
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Examples of dielectrics that can be used in various frequency ranges.
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Sindle0 and multilayer dielectric capacitors.
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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.
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Aluminum electrolytic capacitor.
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Solid electrolyte tantalum capacitor.
(a) A cross section without fine detail.
(b) An enlarged section through the Ta capacitor.
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Comparison of dielectrics for capacitor applications
Capacitor name
Polypropylene
Polyester
Mica
Aluminum,
electrolytic
Tantalum,
electrolyt
ic, solid
High-K ceramic
Dielectric
Polymer film
Polymer film
Mica
Anodized Al2O3
film
Anodized
Ta2O5
film
X7R
BaTiO3 base
r
2.2 – 2.3
3.2 – 3.3
6.9
8.5
27
2000
tan
4  10-4
4  10-3
2  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 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
16 kW
Evol (mJ cm-3)b
10
15
8
1000
1200
100
Polarization
Electronic
Electronic and
Dipolar
Ionic
Ionic
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.
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Capacitance per unit volume
Cvol 
 o r
d
2
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
 o r
1
1
2
2
 CVm 

E
br
2
Ad 2 2
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
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Dielectric loss per unit volume
Wvol 
E br

2
2
o r tan
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
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(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.
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Real part of the dielectric constant, r', and loss tangent, tan, at 1 kHz vs.
temperature from Dielectric Analysis, DEA [by Kasap and Maeda (1995)]
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